models of a standing human body in structural vibration
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Models of a Standing Human Body
in Structural Vibration
A thesis submitted to The University of Manchester for the degree of
Doctor of Philosophy in the Faculty of Engineering and Physical Sciences
2013
Qingwen Zhang
School of Mechanical, Aerospace and Civil Engineering
List of Contents List of Figures ..................................................................................................................................... 5 List of Tables ....................................................................................................................................... 8 Abstract ............................................................................................................................................. 10 Declaration .........................................................................................................................................11 Copyright Statement .........................................................................................................................11 Publications ....................................................................................................................................... 12 Dedication ......................................................................................................................................... 13 Acknowledgements ........................................................................................................................... 14 Notation ............................................................................................................................................. 15 1 Introduction ................................................................................................................................... 17
1.1 Introduction ........................................................................................................................ 17 1.2 Aim, Objectives and Research Strategy ........................................................................... 19 1.3 Outline of the Thesis ........................................................................................................... 20
2 Literature Review .......................................................................................................................... 24 2.1 Introduction ........................................................................................................................ 24 2.2 Human Body Models .......................................................................................................... 24
2.2.1 Biomechanics models .............................................................................................. 25 2.2.2 Conventional models in structural vibration ........................................................ 28 2.2.3 Human-structure interaction models .................................................................... 29 2.2.4 Continuous body models ......................................................................................... 34 2.2.5 Higher Degree of Freedom Models ........................................................................ 36
2.3 Structure response due to human action .......................................................................... 38 2.4 Human response to structure vibration ........................................................................... 43 2.5 Tuned-mass-dampers ......................................................................................................... 46 2.6 Summary ............................................................................................................................. 48
3 Experiment Test Set-up ................................................................................................................ 49 3.1 Introduction ........................................................................................................................ 49 3.2 Test rig ................................................................................................................................. 49
3.2.1 Test Rig ..................................................................................................................... 49 3.2.2 Improvements to the Test Rig ................................................................................. 50
3.3 The Data Collection System............................................................................................... 51 3.4 The Vibration Control System .......................................................................................... 52 3.5 Test Procedure .................................................................................................................... 54
3.5.1 Free vibration .......................................................................................................... 54 3.5.2 Forced vibration ...................................................................................................... 57
3.6 Summary ............................................................................................................................. 59
4 A Continuous Model of a Standing Human Body in Vertical Vibration ................................. 60 4.1 Introduction ........................................................................................................................ 60 4.2 A Continuous Standing Body Model ................................................................................. 61
4.2.1 Assumptions ............................................................................................................. 61 4.2.2 Mass and stiffness distributions of the model ....................................................... 63
4.3 Identification of the Stiffness ............................................................................................. 66 4.3.1 Method of identification .......................................................................................... 66
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4.3.2 Comparison of the models ...................................................................................... 71 4.4 Vertical Dynamic Characteristics of a Standing Body .................................................... 72
4.4.1 Mode shapes ............................................................................................................. 72 4.4.2 Modal properties ..................................................................................................... 73
4.5 Improved Continuous Model ............................................................................................ 74 4.6 Numerical verification ....................................................................................................... 76 4.7 Conclusions ......................................................................................................................... 78
5 Parameter Identification of the Interaction Model Using Available Measurements .............. 82 5.1 Introduction ........................................................................................................................ 82 5.2 Parameter Identification Method ..................................................................................... 82
5.2.1 Extraction of experimental data ............................................................................ 82 5.2.2 Verification of the method ...................................................................................... 83
5.3 Parameter identification for the interaction models ....................................................... 88 5.3.1 Model 1c ................................................................................................................... 88 5.3.2 Model 2e ................................................................................................................... 94
5.4 Comparison of the Human-Structure Models ................................................................. 97 5.4.1 Comparison between the same Human-Structure Models using different parameters ........................................................................................................................ 99 5.4.2 Comparison between different Human-Structure Models ................................ 100
5.5 Conclusions ....................................................................................................................... 103
6 Frequency Characteristics of Human-Structure Models in Forced Vibration ...................... 105 6.1 Introduction ...................................................................................................................... 105 6.2 Basic Equations and Models ............................................................................................ 105 6.3 Parametric Study .............................................................................................................. 109
6.3.1 Effect of the mass ratio ..........................................................................................110 6.3.2 Effect of the frequency ratio ..................................................................................112 6.3.3 Effect of the body damping ratio ..........................................................................113
6.4 Critical Positions ...............................................................................................................114 6.5 Experimental verification .................................................................................................118
6.5.1 Experiment cases ....................................................................................................118
6.5.2 FRFs of two H-S Models ........................................................................................119 6.5.3 The effect of 2γ α ............................................................................................... 121 6.5.4 The effect of Hξ ................................................................................................... 123 6.5.5 Validation of the H-S Models ................................................................................ 124 6.5.6 Comparison between experimental and theoretical results ............................... 126
6.7 Conclusions ....................................................................................................................... 129
7 Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration131 7.1 Introduction ...................................................................................................................... 131 7.2 Test procedure and result ................................................................................................ 131
7.2.1 Test procedure ........................................................................................................ 131 7.2.2 Data processing ...................................................................................................... 133 7.2.3 Experimental Results ............................................................................................ 137
7.3 Simulation of free vibration of 2DOF interaction model .............................................. 138 7.4 Experimental identification of the natural frequency and damping ratio of a human body ......................................................................................................................................... 141
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7.5 Sensitivity study ................................................................................................................ 146
7.5.1 Hf and Hξ ......................................................................................................... 146 7.5.2 HSf and HSξ ....................................................................................................... 147 7.5.3 SM ........................................................................................................................ 148
7.6 Conclusion ......................................................................................................................... 149
8 Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio ................................................................................................................................ 151
8.1 Introduction ...................................................................................................................... 151 8.2 The Method ....................................................................................................................... 151 8.3 Equivalent damping ratio ................................................................................................ 156 8.4 Application ........................................................................................................................ 162 8.5 Conclusions ....................................................................................................................... 165
9 Conclusions and Further Work ................................................................................................. 167 9.1 Conclusions ....................................................................................................................... 167 9.2 Further Work .................................................................................................................... 170
References ....................................................................................................................................... 172
Word count in the thesis: 33518
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List of Figures Figure 1.1: The relationships between the main chapters in the thesis .............................................. 22 Figure 1.2: The relationships between the human body models and between human-structure systems
presented in this study........................................................................................................... 23 Figure 2.1: Discrete biomechanical models (Matsumoto and Griffin, 2003) ..................................... 25 Figure 2.2: an undamped two degrees-of-freedom model .................................................................. 27 Figure 2.3: Conventional models in structural vibration .................................................................... 28 Figure 2.4: An individual standing on a SDOF structure system (the human object comes from
(Hemera Technology, 2001) .................................................................................................. 29 Figure 2.5: Human-structure interaction model ................................................................................. 33 Figure 2.6: A two-part continuous model ........................................................................................... 35 Figure 2.7: 15 DOF spring mass system (Nigam and Malik, 1987) ................................................... 36 Figure 2.8: 7 DOF spring mass system (Tregoubov, 2000) ................................................................ 37 Figure 3.1: Test rig ............................................................................................................................. 49 Figure 3.2: Cross-section of the test rig (Yao et al., 2004) ................................................................. 50 Figure 3.3: 3D view of the prop ......................................................................................................... 51 Figure 3.4: Accelerometer .................................................................................................................. 52 Figure 3.5: CED Power 1401 ............................................................................................................. 52 Figure 3.6: Spike 2 Version 6 ............................................................................................................. 52 Figure 3.7: Shaker APS 113 ............................................................................................................... 53 Figure 3.8: Power Amplifier APS 125 ............................................................................................... 53 Figure 3.9: Vibration Control Unit VCU13.2S .................................................................................. 53 Figure 3.10: VCS201Vibration Control Version 1.2.1.0 .................................................................... 54 Figure 3.11: Free vibration tests ......................................................................................................... 55 Figure 3.12: Acceleration-time history and frequency spectrum ....................................................... 55 Figure 3.13: Free vibration tests ......................................................................................................... 58 Figure 4.1: Distribution of body mass ................................................................................................ 64 Figure 4.2: Continuous standing body models ................................................................................... 66 Figure 4.3: Relationships between 2 1/k k and 2 1/f f for the four models ................................... 70 Figure 4.4: The Mode shapes of Model 3 .......................................................................................... 73 Figure 4.5: Continuous standing body models ................................................................................... 75 Figure 4.6: A continuous body model on a SDOF structure system forming a human-structure system
.............................................................................................................................................. 77 Figure 4.7: Human-structure models .................................................................................................. 77 Figure 5.1: Comparison between experimental (——) and identified (------) results (Matsumoto and
Griffin, 2003) ........................................................................................................................ 83 Figure 5.2: Reproduced curves from original measurements (Matsumoto and Griffin, 2003) using
123 pairs of data .................................................................................................................... 83 Figure5.3: Biomechanics models (Matsumoto and Griffin, 2003) ..................................................... 84 Figure 5.4: Mean normalised apparent masses and mean phase ........................................................ 86 Figure 5.5: Model 1c .......................................................................................................................... 88 Figure 5.6: Comparison of the fitting of the Normalised apparent masses and phase between Models
1b and 1c ............................................................................................................................... 92
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Figure 5.7: Model 2e .......................................................................................................................... 94 Figure 5.8: Comparison of the fitting of the normalised apparent masses and phase between Model
2d and 2e ............................................................................................................................... 96 Figure 5.9: Human-Structure models with different body models ..................................................... 98 Figure 6.1: An interaction model on a SDOF structural system ....................................................... 106 Figure 6.2: Acceleration FRFs (Equation 6.14) of the structure and human models ( 0.3Hξ = ,
0.5β = ) ...............................................................................................................................110 Figure 6.3: Acceleration FRFs (Equation 6.14) of the structure and human models ( 0.3Hξ = ,
1.0β = ) ............................................................................................................................... 111 Figure 6.4: Acceleration FRFs (Equation 6.14) of the structure and human models ( 0.3Hξ = ,
2.0β = ) .............................................................................................................................. 111 Figure 6.5: Acceleration FRFs (Equation 6.14) of the structure and human models ( 0.3Hξ = ,α =0.3)
.............................................................................................................................................112 Figure 6.6: Acceleration FRFs (Equation 6.14) of the structure and human models ( 0.3Hξ = ,α =1)
.............................................................................................................................................112 Figure 6.7: Acceleration FRFs (Equation 6.14) of the structure and human models ........................113 ( 0.3Hξ = ,α = 3) .............................................................................................................................113 Figure 6.8: Acceleration FRFs (Equation 6.14) of a human-structure Model ( 1.0α = , 1β = ) ....113 Figure 6.10: Acceleration FRFs (Equation 6.11) of a human-structure system ................................116 Figure 6.11: the curve of Equation 6.18 ............................................................................................116 Figure 6.12: Human-Structure systems with different body models ................................................ 120 Figure 6.14: Comparison between experimental and theoretical FRFs with two values of 2γ α
(Case 2.3) ............................................................................................................................ 122 Figure 6.15: Comparison between experimental and theoretical FRFs with three damping ratios
(Case 1.2) ............................................................................................................................ 123 Figure 6.16: Comparison between experimental and theoretical FRFs with three damping ratios
(Case 2.3) ............................................................................................................................ 124 Figure 6.17: FRFs for the test rig and two human body models (Case 1.2) ..................................... 125 Figure 6.18: FRFs for the test rig and two human body models (Case 2.3) ..................................... 125 Figure 6.19: Case 1.0-the bare rig 1 ................................................................................................. 127 Figure 6.20: Case 1.1-one person standing on Rig 1 ........................................................................ 127 Figure 6.21: Case 1.2-two people standing on Rig 1 ....................................................................... 128 Figure 6.22: Case 1.3-three people standing on Rig 1 ..................................................................... 128 Figure 6.23: Case 1.4-four people standing on Rig 1 ....................................................................... 128 Figure 6.24: Case 2.0-the bare rig 2 ................................................................................................. 128 Figure 6.25: Case 2.1- one person standing on Rig 2 ....................................................................... 128 Figure 6.26: Case 2.2 - two people standing on Rig 2 ..................................................................... 128 Figure 6.27: Case 2.3 - three people standing on Rig 2 ................................................................... 128 Figure 6.28: Case 2.4 –f our people standing on Rig 2 .................................................................... 128 Figure 7.1: Case 2.0-the bare rig ...................................................................................................... 134 Figure 7.2: Case 2.1-the rig with subject P1 .................................................................................... 134 Figure 7.3: Case 2.2-the rig with subject P2 .................................................................................... 134 Figure 7.4: Case 2.3-the rig with subject P3 .................................................................................... 134 Figure 7.5: Case 2.4-the rig with subject P4 .................................................................................... 135
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Figure 7.6: Case 5.0-the bare rig ...................................................................................................... 135 Figure 7.7: Case 5.1-the rig with subject P1 .................................................................................... 135 Figure 7.8: Case 5.2-the rig with subject P2 .................................................................................... 135 Figure 7.9: Case 5.3-the rig with subject P3 .................................................................................... 136 Figure 7.10: Case 5.4-the rig with subject P4 .................................................................................. 136 Figure 7.11: Case 9.0-the bare rig .................................................................................................... 136 Figure 7.12: Case 9.1-the rig with subject P1 .................................................................................. 136 Figure 7.13: Case 9.2-the rig with subject P2 .................................................................................. 137 Figure 7.14: Case 9.3-the rig with subject P3 .................................................................................. 137 Figure 7.15: Case 9.4-the rig with subject P4 .................................................................................. 137 Figure 7.16: Human-structure system ........................................................................................... 139 Figure 7.17: the acceleration-time history and the acceleration spectrum of the rig with a standing
person .................................................................................................................................. 140 Figure 7.18: The damped natural frequency of human-structure system based on different natural
frequencies and damping ratio of human body ................................................................... 143 Figure 7.19: The damping ratio of human-structure system based on different natural frequencies and
damping ratio of human body ............................................................................................. 143 Figure 7.20: The intersection of the plane HSf and the curved surface ( , )H Hf f ξ ................... 144 Figure 7.21: The intersection of the plane HSξ and the curved surface ( , )H Hfξ ξ .................... 144 Figure 7.22: The two curves obtained from Figures 7.20 and 7.21 .................................................. 145 Figure 8.1: A damped 2DOF systems ............................................................................................... 153 Figure 8.2: FRFs of SDOF structure model and 2DOF TMD-structure model ................................ 153 Figure 8.3: Curved surface of eξ
∆ (The mass ratio α =0.05) ....................................................... 158 Figure 8.4: Contours of eξ
∆ (The mass ratio α =0.05) ................................................................. 158
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List of Tables Table 2.1: Optimised model parameters of six biomechanical models (Matsumoto and Griffin, 2003)
....................................................................................................................................... 26 Table 2.2: Frequencies observed at Twickenham (Ellis and Ji, 1997). ............................................... 40 Table 2.3: Frequencies observed on the beam (Ellis and Ji, 1997). ................................................... 40 Table 2.4: Natural frequencies of a retractable grandstand (Littler, 2000). ........................................ 42 Table 2.5: Vibration criteria over times (Murray, 1999) ..................................................................... 44 Table 2.6: VDVs at which various degrees of adverse comment may be expected (BS 6472) .......... 45 Table 2.7: Possible VDV ranges for grandstands (Ellis and Littler, 1994) ......................................... 45 Table 2.8: Suggested acceptable vibration levels and their extrapolation to VDVs (Ellis and Littler,
1994) ..................................................................................................................................... 46 Table 3.1: Mass and height of the participants ................................................................................... 56 Table 3.2: Experiment cases ............................................................................................................... 56 Table 3.3: Weight and Height of the four people ............................................................................... 58 Table 3.4: Experiment cases ............................................................................................................... 58 Table 4.1: Mass distribution of 15 ellipsoidal segments (Bartz and Gianotti ,1975; Nigam and Malik,
1987) ..................................................................................................................................... 65 Table 4.2: The length and the number of the element ........................................................................ 67 Table 4.3: Stiffnesses and natural frequencies of a standing body ..................................................... 71 Table 4.4: Modal mass and stiffness of a standing body .................................................................... 74 Table 4.5: Stiffnesses and natural frequencies of a standing body ..................................................... 75 Table 4.6: Modal mass and stiffness of a standing body based on Model 5 ....................................... 76 Table 4.7: Comparison of the first three natural frequencies of a human-structure system using
different body models (Hz) ................................................................................................... 81 Table 5.1: Comparison of optimised model parameters ..................................................................... 87 Table 5.2: Identified parameters of Model 1c .................................................................................... 90 Table 5.3: Identified parameters of Model 1c .................................................................................... 92 Table 5.4: Identified parameters of Model 1b and 1c ......................................................................... 93 Table 5.5: Identified parameters of Model 2e .................................................................................... 95 Table 5.6: Identified parameters of Model 2d and 2e ......................................................................... 97 Table 5.7: Comparison of the first three natural frequencies of H-S Model 4 and 5 with different
parameters (Hz) .................................................................................................................. 101 Table 5.8: Comparison of the first three natural frequencies of a human-structure model using
different body models (Hz) ................................................................................................. 102 Table 6.1: Mass and height of the participants ..................................................................................118 Table 6.2: Experiment cases ..............................................................................................................119 Table 6.3: Measured and predicted resonance frequencies (Hz) ...................................................... 128 Table 7.1: Mass and height of the participants ................................................................................. 132 Table 7.2: Experiment cases ............................................................................................................. 133 Table 7.3: Measured result summary ............................................................................................... 138 Table 7.4: Summary of the result ..................................................................................................... 145 Table 7.5 Sensitivity study of Hf and Hξ ................................................................................... 147 Table 7.6 Sensitivity study of HSf and HSξ ................................................................................. 148
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Table 7.7 Sensitivity study of SM ................................................................................................. 149 Table 8.1a: Optimum design parameters ( β and Hξ ) of a TMD: 1%Sξ = .............................. 159 Table 8.1b: Optimum design parameters ( β and Hξ ) of a TMD: 1.5%Sξ = ............................ 160 Table 8.1c: Optimum design parameters ( β and Hξ ) of a TMD: 2%Sξ = ................................ 161 Table 8.1d: Optimum design parameters ( β and Hξ ) of a TMD: 5%Sξ = ............................... 162
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Abstract It has been widely accepted that a stationary human body, such as a person when sitting or standing, acts as a single degree of freedom (SDOF) system in structural vibration. However, it is not clear what form the SDOF model should take and what are the appropriate parameters for the model. The significance of considering human body models in structural vibration comes from the fact that human involvement affects the dynamic behaviour of the structure when a crowd is present and that human body response is different from structural vibration. This forms the basis of this study.
This thesis presents both experimental and theoretical studies to develop human body models. It examines the characteristics of two interaction human body models, determines the parameters of the two body models in structural vibration and explores their applications.
A continuous model of a standing human body in vertical vibrations is first developed using an anthropomorphic model and two available natural frequencies obtained from shaking table tests. A standing human body is represented as a bar with seven mass segments using the anthropomorphic model and two stiffnesses of the model are identified using the two natural frequencies. The relationships between the continuous model and discrete body models are provided.
The masses, damping ratios and stiffnesses of two interaction body models are identified by curve fitting of the measured apparent mass curves from shaking table tests in published biomechanics studies. In this identification process it was identified that one or two conditions have to be applied which can be derived from the outcome of the continuous body model.
The characteristics of human-structure interaction models are investigated using both theoretical and experimental Fourier Response Functions. The comparative studies based on 10 tests help to show that the interaction body model is more appropriate than the conventional body model used in structural vibration, and identify the appropriate parameters for the interaction model. The theoretical study shows that the response of stationary people is always larger than structural vibration when human loads are applied, such as walking, jumping and bouncing. The conditions for observing two resonance frequencies are provided graphically for a human-structure system where the interaction body model is used.
A method is proposed to identify the parameters of the interaction model through 45 free vibration tests of a standing person on a test rig. The identified values of the natural frequency and damping ratio of a standing body are not close to those from the biomechanics tests. Sensitivity studies show that the two parameters are sensitive to the input data, the damped natural frequency and damping ratio of the human-structure system, which are obtained from free vibration tests.
As an extension of the application of FRF and the human-structure model, the optimum parameters of a tuned-mass-damper are obtained based on the concept of equivalent damping ratio of a SDOF structure system. The results are tabulated for practical use. An example of floor vibration induced by rhythmic crowd loads is provided to demonstrate the use of the optimum TMDs and shows the effect of vibration reduction.
This thesis entitled “Models of a Standing Human Body in Structural Vibration” is submitted to the University of Manchester by Qingwen Zhang for the degree of Doctor of Philosophy in 2013.
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Declaration
No portion of the work referred to in the thesis has been submitted in support of an
application for another degree or qualification of this or any other university or other
institute of learning;
Copyright Statement i. The author of this thesis (including any appendices and/or schedules to this thesis)
owns any copyright in it (the “Copyright”) and s/he has given The University of
Manchester the right to use such Copyright for any administrative, promotional,
educational and/or teaching purposes.
ii. Copies of this thesis, either in full or in extracts, may be made only in accordance
with the regulations of the John Rylands University Library of Manchester. Details
of these regulations may be obtained from the Librarian. This page must form part
of any such copies made.
iii. The ownership of any patents, designs, trade marks and any and all other intellectual
property rights except for the Copyright (the “Intellectual Property Rights”) and any
reproductions of copyright works, for example graphs and tables (“Reproductions”),
which may be described in this thesis, may not be owned by the author and may be
owned by third parties. Such Intellectual Property Rights and Reproductions cannot
and must not be made available for use without the prior written permission of the
owner(s) of the relevant Intellectual Property Rights and/or Reproductions.
iv. Further information on the conditions under which disclosure, publication and
exploitation of this thesis, the Copyright and any Intellectual Property Rights and/or
Reproductions described in it may take place is available from the Head of School
of Mechanical, Aerospace and Civil Engineering and for its candidates.
11
Publications
The following papers have been published or to be published during the study:
• Zhang, Q. and Ji, T., (2012), "Optimum design parameters for a tuned-mass-damper
to maximise the equivalent damping ratio", Advances in Vibration Engineering.
Vol.11(4), pp.349-360.
• Ji, T., Zhou, D., and Zhang, Q., In Press. "Models of a standing human body in
vertical vibration", Structures and Buildings. DOI: 10.1680/stbu.12.00010, Abstract
• Wang, D., Ji, T., Zhang, Q., and Duarte, E., In Press. " Resonance frequencies of a
highly damped two degree-of-freedom system", Journal of Engineering Mechanics.
ASCE. DOI: org/10.1061/(ASCE)EM.1943-7889.0000668
• Zhang, Q. and Ji, T., (2011), "Stiffness and mass distributions of continuous models
of a standing human body subject to vertical vibrations", The Thirteenth
International Conference on Civil, Structural and Environmental Engineering
Computing, pp 16. 6-9 September.
• Zhang, Q. and Ji, T., (2010), "Representation of a standing human body in vertical
vibration", The 45th UK Conference on Human response to Vibration, pp 12. 6-8
September.
The following paper is to be submitted for journal publication:
• Zhang, Q. and Ji, T., (2012) "Frequency characteristics of a human-structure
system".
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Dedication
To my lovely wife Yu Zhang
To the dear parents
To all my relatives and friends
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Acknowledgements I would like to express my sincere gratitude and appreciation to my supervisor Dr
Tianjian Ji for his supervision, encouragement and help, which he provided throughout
the course of this work. PhD student usually faces economic difficulties; Dr Ji not only
generously gave me a lot of opportunities to work as a Demonstrator in the University
of Manchester, but also provided me an excellent opportunity to work in a consultancy
project and earn extra money for my family, helping me go through my PhD study.
I would like to express my special thanks to Dr Jyoti Sinha, who is the examiner of the
first and second year report of my PhD research, for his invaluable help and discussions
throughout the work. His friendship will always be remembered.
The scholarship provided by the University of Manchester (tuition fee) and Chinese
Scholar Council (living expenses) is essential for my PhD research and living in the UK.
The generous giving and help from the University of Manchester is highly appreciated.
Finally, I am most grateful to my wife Yu Zhang for their love, patience and support
throughout all these years of my education.
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Notation
( )c x the distributed axial damping of a human body
1 2, H HC C the modal damping coefficients of discrete human-body models
SC the damping coefficient of a SDOF structure system
Sξ the damping ratios of a SDOF structure system
1 2, H Hξ ξ the damping ratios of discrete human-body models
HSξ the damping ratios of human-structure system
Sf the natural frequency of a SDOF structure system
1Hf , 2Hf the first and second natural frequencies of a standing human body
HSf the damped natural frequency of human-structure system
Rf the resonance frequency of human-structure system
( )k x the distributed axial stiffness of a human body
1k , 2k the axial stiffnesses of the lower and upper parts of a bar
1 2, H HK K the modal stiffnesses of discrete human-body models
SK the stiffness of a SDOF structure system
L the height of a standing human body
( )m x the distributed mass of a human body
0HM the whole-body mass.
1HM , 2HM the participating masses of the first and second modes of a human body
respectively
11HM , 22HM the modal masses of the first and second modes of a human body respectively
SM the mass of a SDOF structure system
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( , )Hu x t the absolute movement of a human body
1( )Hu t , 2 ( )Hu t the absolute movements of the first and second modes of a human body
( , )HRu x t the movement of a standing human body relative to a supporting structure
1( )HRu t , 2 ( )HRu t the movements of the first and second modes of a human body relative to a
supporting structure
( )Su t the movement of a SDOF structure system
1( )xf , 2 ( )xf the shapes of the first and second modes of vertical vibration throughout the
height of a standing human body relative to a supporting structure
11 /H SM Mα = the ratio of the modal mass of the first mode of the body to the modal mass of
the structure
1 /H Sβ ω ω= the ratio of the natural frequency of the first mode of the body to the natural
frequency of the SDOF structure system
0 /H SM Mη = the ratio of the whole body mass to the modal mass of the structure
1 /H SM Mγ = the ratio of the participating mass of the first mode of the body to the modal
mass of the structure
Sω the natural Angular frequency of a SDOF structure system
1Hω , 2Hω the first and second natural Angular frequencies of a standing human body
1ω , 2ω the first and second natural Angular frequencies of a 2DOF body-structure
system
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Chapter 1: Introduction
1 Introduction
1.1 Introduction
In recent years, there have been an increasing number of problems related to
human-induced vibrations of floors, bridges, assembly structures and stairs due, in part,
to a gradual increase of span lengths in new constructions. Also higher-strength and
lighter-weight construction materials have been used in many new structures. As a result,
the natural frequencies of the structures reduce to a point where the resonant or
near-resonant vibration may be induced by human actions and this, in turn, can lead to
unacceptable levels of vibration. At the same time the users’ expectation of these
structures as a working environment increases. The situation has now been reached
where the dynamic behaviour of some structures is the critical factor in their design. It is
expected that spans of floors will be even longer and the human expectation of the
quality of the working environment will become even greater in the future.
Consequently, human-structure interaction needs to be considered when designing new
structures excited and/or occupied by people.
Human–structure interaction is a complex and increasingly important issue that is not
yet well understood. There are two key questions relating to human–structure
interaction:
• How does structural vibration affect human response/comfort?
• How does crowd involvement influence the dynamic behaviour and response of a
structure?
When considering these two questions, it is necessary to consider both the structure and
the human body. And for detailed investigation it is important that an appropriate model
of the human body is used in the study.
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Chapter 1: Introduction
There is little research on the assessment of human response to floor vibration although
floors are primarily for human use and floor vibrations are sometimes assessed. For
practical significance, human body response to structural vibration may be predicted if a
human body can be modelled correctly. These may provide the tools that allow a
designer to assess the responses of the human body and structures where stationary
people are involved. Although this is a very challenging task, this research aims to
contribute new knowledge to this area, in particular on the models and dynamic
properties of a standing human body in structural vibration. However, this is very
difficult. To find the most appropriate human body model is the key task in this
research.
The human body can be modelled in various ways with five representations being
considered here:
1) Biomechanics models that were developed based on the results of shaking table
tests;
2) Conventional models that were developed based on a fixed base and often used
in structural vibration;
3) Interaction models that were developed based on a vibrating structure;
4) Continuous models that describe a standing person using continuous stiffness
and mass functions;
and
5) Higher degree of freedom models that require a finite element solution.
These models were developed independently but they are related. This thesis considers
these human body models, in particular the interaction models. It includes both
experimental and theoretical methods, and thereby contributes new understanding and
knowledge to modelling the human body and to the study of human-structure
interaction.
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Chapter 1: Introduction
1.2 Aim, Objectives and Research Strategy
The aim of the this research is to develop appropriate human body models, by
identifying more accurate parameters for modelling people by studying the
characteristics of human-structure systems. The human body models require both the
correct method to model the body and the parameters to use in these models.
The objectives of this investigation are to:
1. Develop a continuous model of a standing human body from available natural
frequency measurements of a standing human body and using an anthropomorphic
model.
2. Determine the parameters of the interaction human body models using available
biomechanics measurements;
3. Identify the parameters of the interaction body model through forced vibration tests
on human-structure systems;
4. Identify the parameters of the interaction body model through free vibration tests of
human-structure systems;
5. Provide an optimum design of a tuned-mass-damper using the Fourier Response
Functions developed from the study of human-structure interaction.
Analytical methods often provide an understanding of a solution but they are only
applicable to solve relatively simple problems. In contrast, numerical methods can be
applied to a broad range of problems for providing a complete solution, but they are
unlikely to give a general solution. Both analytical and numerical solutions are
developed based on a number of assumptions. Experimental methods do not require
such assumptions and provide true solutions, but measurements can only be taken on
selected situations. However, the three methods are complementary. The experimental
studies are particularly important in this study for identifying the parameters for the
human body models.
19
Chapter 1: Introduction
The research methods used in this study are as follows:
Experiments provide the key information for identifying the parameters of a
human body model and validating appropriate body models in structural
vibration. Therefore, in this study the available measurements from body
biomechanics studies are used and forced vibration and free vibration tests on
human-structure systems are conducted. Thus the parameters of the interaction
body models can be assessed in several different ways.
Analytical expressions of Fourier Response Functions (FRFs) for acceleration of
human-structure systems are derived, which simulate the forced vibration tests.
As the measurements can only be taken on the test rig, and not on the body of a
subject, the simulation effectively facilitates the study and contributes
significantly to achieving a good understanding of human body models
determining the characteristics of human-structure systems. A further application
of the FRF is to determine the optimum design parameters of
tuned-mass-dampers for reducing floor vibration induced by rhythmic crowd
loads.
The finite element method is also used to develop a continuous human body
model and used for verification when analytical solutions cannot be applied.
1.3 Outline of the Thesis
The thesis is organised into nine chapters of which the first is this introduction. Chapter
2 provides a literature review of human body models, human response to the structural
vibration and structure response induced by human movements. The following five
chapters provide the main contents of the thesis:
Chapter 3 introduces the experimental set-up, including the test rig and the equipment
used in the tests. The free vibration and forced vibration test procedures are described
and the various tests explained.
20
Chapter 1: Introduction
Chapter 4 develops a continuous human body model including seven segments of a
standing human body based on an anthropomorphic model. The two stiffnesses of the
upper and lower body are identified using two available measured natural frequencies of
a standing body. The modal properties of the standing body are also determined and
linked to those of discrete body models.
Chapter 5 identifies the parameters of the interaction models based on published
apparent mass curves obtained from shaking-table tests. Curve fitting is used to abstract
the parameters. The quality of these models is assessed through eigenvalue analysis of
different human-structure models.
The experimental and theoretical studies of human-structure interaction systems are
presented in Chapter 6. Forced vibration tests are conducted on a test rig with different
numbers of standing people and the results are presented using Fourier Response
Functions (FRFs) for acceleration. In parallel to the experimental study, theoretical
expressions of two degree-of-freedom human-structure systems are derived. The
comparison between the measurements and predictions helps to identify the parameters
of the interaction body models and shows that the interaction model is more appropriate
for studying human-structure interaction than the conventional body model. Also, the
necessary conditions for the presence of two resonance frequencies in some
human-structure experiments, an observation which has been previously noted but not
understood, are given graphically.
Chapter 7 identifies the natural frequency and damping ratio of the interaction body
model based on free vibration tests. This study explores the possibility if the two
parameters can be determined based on free vibration tests of a simple human-structure
system. In the free vibration tests only one damped natural frequency and one damping
ratio of the human-structure system can be obtained from the measurements. Four
subjects participate in the free vibration test. The experiment results based on four
different persons also show the variability of human dynamic properties.
21
Chapter 1: Introduction
Chapter 8 extends the use of the FRFs for acceleration and for displacement to obtain
the optimum design parameters of a tuned-mass-damper based on the concept of
equivalent damping ratio. The outcome is then used in vibration reduction of a dance
floor.
Finally, Chapter 9 summarises the main conclusions obtained from this study and
presents ideas for further work.
The relationships between the key chapters of the thesis are given in the chart in Figure
1.1. The relationships between the human body models and between human-structure
systems presented in this study are illustrated in Figure 1.2.
Experimental Study Theoretical Study
Figure 1.1: The relationships between the main chapters in the thesis
Chapter 5: Parameter Identification of the Interaction Model Using Available
Measurements
Chapter 7: Experimental Identification of Human Body
Model in Free Vibration
Chapter 3: Experiment Test Set-up
Chapter 4: A Continuous Model of a Standing Human Body in
Vertical Vibration
Chapter 6: Frequency Characteristics of Human Body
Models in Forced Vibration
Chapter 8: Optimum Design Parameters of a
Tuned-Mass-Damper
22
Chapter 1: Introduction
Figure 1.2: The relationships between the human body models and between human-structure systems presented in this study
23
Chapter 2: Literature Review
2 Literature Review
2.1 Introduction
There are two different aspects to consider when studying human–structure interaction:
structure response due to human action and human response to the structural vibration.
The human body model is the fundamental item in this study. In recognition of this,
human body models are reviewed in section 2.2. Structure response due to human action
and human response to structural vibration are then reviewed in sections 2.3 and 2.4
respectively. In section 2.5 tuned-mass-dampers are reviewed, which uses similar
models to those for human-structure interaction. This review provides an understanding
of human-structure interaction and suggests where further investigation of
human-structure models would be beneficial.
2.2 Human Body Models
A standing human body is a continuum in which mass and stiffness are distributed
unevenly throughout the height of the body. Many models have been developed to
represent the standing body and in this study five typical types are considered. These
were developed in different ways and have respective advantages and limitations. The
five body models are reviewed in sections 2.2.1~2.2.5.
24
Chapter 2: Literature Review
2.2.1 Biomechanics models
Discrete biomechanical models of individual human bodies have been developed
(Griffin, 1990), including single degree-of-freedom (SDOF) models, two SDOF models
and two degree-of-freedom (2DOF) models. These models were developed from the
study of body biomechanics of seated and standing subjects using a shaking table
(Matsumoto and Griffin, 1998). The format of these models was intuitively provided
while the parameters of the models, such as damping coefficients, stiffnesses and
masses, were identified based on the best fit between measured and predicted apparent
masses (Wei and Griffin, 1998; Matsumoto and Griffin, 2003). These models captured
the biomechanical or dynamic characteristics of a whole-body. Six discrete
biomechanical models are shown in Figure 2.1 and the parameters for these models are
given in Table 2.1.
Figure 2.1: Discrete biomechanical models (Matsumoto and Griffin, 2003)
25
Chapter 2: Literature Review
Table 2.1: Optimised model parameters of six biomechanical models (Matsumoto and
Griffin, 2003)
The SDOF models are shown in Figure 2.1a (Model 1a) and Figure 2.1b (Model 1b).
The difference between the two models is that Model 1a had a massless support at its
base, whereas the bottom structure in Model 1b had a mass m0 (Matsumoto and Griffin,
2003). The two SDOF models are the simplest biodynamic models of the human body.
Another four models have been developed from the SDOF model. The four models are
shown in Figure 2.1c to 2.1f (Model 2a–2d). There are two kinds of connections
between these 2DOF models. One kind is when the second DOF is attached to the first
DOF as shown in Figures 2.1c and 2.1d; another is when the second DOF is completely
independent of the first DOF as shown in Figures 2.1e and 2.1f. The support structure in
Model 2a and c had no mass, whereas the support structure in Model 2b and 2d had a
mass m0. Wei and Griffin (1998) suggest that the reason why the non-vibration mass m0
contributes only mass is that it represents the effect of other modes that are above the
frequency range of interest.
Biomechanics researchers have usually obtained dynamic characteristics of the human
body experimentally by placing a person on a shaking table in laboratory conditions.
The experimental data were then used to calculate apparent mass ( )M f (Griffin,
1990). By curve-fitting to the apparent mass, the dynamic properties of the biodynamic
human models were identified (Matsumoto and Griffin, 1998).
26
Chapter 2: Literature Review
Matsumoto and Griffin (1998) studied the apparent mass of standing human bodies on a
shaking table that was subjected to vertical vibration from a 1-m stroke
electro-hydraulic vibrator. 12 male subjects were subjected to random vertical vibration
in the frequency range between 0.5 and 30 Hz at vibration magnitudes between 0.125
and 2.0 2ms− r.m.s. It was found that the resonance frequency of the apparent mass in
a normal posture decreased from 6.75 Hz to 5.25 Hz when the vibration magnitude
increased from 0.125 to 2.0 2ms− r.m.s. Their further work (Matsumoto and Griffin,
2003) provided discrete models to represent a standing person, including two SDOF
models, two 2DOF models and two other models, each consisting of two SDOF systems.
The parameters for the models were determined by comparing the measured and
calculated apparent masses.
When examining the apparent mass of a stationary body on a vibrating structure, these
models are ideal for either theoretical or experimental investigation.
Randall et al. (1997) developed the undamped 2DOF human–structure system shown in
Figure 2.2. They determined the natural frequencies of 113 standing individuals in the
range 9 to 16 Hz based on a structure with a fundamental frequency of about 40 Hz.
Interestingly, the identified natural frequencies were significantly higher than those
reported in damped discrete models.
Figure 2.2: an undamped two degrees-of-freedom model 27
Chapter 2: Literature Review
2.2.2 Conventional models in structural vibration
For determining structural response, it was suggested that a standing body can be
represented as a SDOF system and a structure as another SDOF system. When
combined the two SDOF systems form the two-degree-of-freedom (2DOF)
human-structure system (Ellis and Ji, 1997; Zhou and Ji, 2006) shown in Figure 2.3.
Figure 2.3: Conventional models in structural vibration
The equation of motion of this human-structure system is:
1 1 1 1
11 1 1 1 1 1
00
S S S H H S S H H S
H H H H H H H H
M u C C C u K K K uM u C C u K K u
+ − + − + + = − −
(2.1)
Equation 2.1 has been used in most studies of human-structure interaction in structural
vibration (Ellis and Ji, 1997; Sachse et al., 2004; Alexander 2006, Sim et al., 2006,
Zhou and Ji, 2006). This model was useful in the early study of human-structure
interaction for interpreting observations from experiments. However, for this modelling
procedure, a human body was first considered on a fixed base, i.e. a non-vibrating
environment, and then placed on a structure, a vibrating environment. This implies that
the human body model on a fixed base is the same as that on a vibrating structure and
this assumption should be examined (Ji et. al, 2012).
28
Chapter 2: Literature Review
2.2.3 Human-structure interaction models
These models were developed when a standing body was placed on a SDOF structure
(Ji et al., 2012). Because the interaction models are going to be used in the following
chapters, the derivation of this model is represented in detail in this section.
A standing body is considered on a SDOF structure system, i.e. in a vibrating
environment, as shown in Figure 2.4. Due to the movement of the SDOF structure
system ( )Su t , the vertical movement of the body, ( , )Hu x t , when the first two modes
of vibration of the body are considered, can be described as:
1 1 2 2( , ) ( ) ( , ) ( ) ( ) ( ) ( ) ( )H S HR S HR HRu x t u t u x t u t u t x u t xf f= + = + + (2.2)
Equation 2.2 includes the vibration of the SDOF structure, ( )Su t .
Figure 2.4: An individual standing on a SDOF structure system (the human object comes from
(Hemera Technology, 2001))
To establish the equation of motion of the human-structure model (Figure 2.4) subjected
to small amplitude vibration, the Lagrange equation can be used.
The total elastic energy of the combined structure and body system is:
29
Chapter 2: Literature Review
2 2
0
1 1 ( , )( )( )2 2
L HRS H S S
u x tU U U K u k x dxx
∂= + = +
∂∫ (2.3)
The total kinetic energy of the system is:
2 2
0
1 1 ( )[ ( ) ( , )]2 2
L
S H S S S HRT T T M u m x u t u x t dx= + = + +∫
(2.4)
The energy dissipation for the system is:
2 2
0
1 1 ( )[ ]2 2
L
S H S S HRR R R C u c x u dx= + = + ∫
(2.5)
For free vibration the Lagrange equations are (Thomson, 1966):
( )S S S
d T U Rdt u u u
∂ ∂ ∂+ = −
∂ ∂ ∂ (2.6a)
1 1 1
( )HR HR HR
d T U Rdt u u u
∂ ∂ ∂+ = −
∂ ∂ ∂ (2.6b)
2 2 2
( )HR HR HR
d T U Rdt u u u
∂ ∂ ∂+ = −
∂ ∂ ∂ (2.6c)
Substituting equations (2.3-2.5) into equations (2.6) gives the following governing
differential equations:
1 1 2 20 0 0( ) ( ) ( ) ( ) ( ) 0
L L L
s S S HR HR S S S SM u m x dx u m x x dx u m x x dx u C u K uf f+ ⋅ + ⋅ + ⋅ + + =∫ ∫ ∫
(2.7a)
21 1 1 1 2 20 0 0
2 21 2 11 1 2 10 0 0
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )( ) 0
L L L
S HR HR
L L L
HR HR HR
m x x dx u m x x dx u m x x x dx u
d d dc x x dx u k x dx u k x dx udx dx dx
f f f f
f f ff
⋅ + ⋅ + ⋅
+ ⋅ + ⋅ + ⋅ =
∫ ∫ ∫
∫ ∫ ∫
(2.7b) 2
2 1 2 1 2 20 0 0
2 21 2 22 2 1 20 0 0
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )( ) 0
L L L
S HR HR
L L L
HR HR HR
m x x dx u m x x x dx u m x x dx u
d d dc x x dx u k x dx u k x dx udx dx dx
f f f f
f f ff
⋅ + ⋅ + ⋅
+ ⋅ + ⋅ + ⋅ =
∫ ∫ ∫
∫ ∫ ∫
(2.7c)
Equation 2.7 can be written in matrix form as follows:
30
Chapter 2: Literature Review
0 1 2
1 11 1 1 1 1 1
2 22 2 2 2 2 2
00 0
0 0
S H H H S S S S S
H H HR H HR H HR
H H HR H HR H HR
M M M M u C u K uM M u C u K uM M u C u K u
+ + + =
(2.8)
where
0 0( )
L
HM m x dx= ∫ (2.9a)
1 10( ) ( )
L
HM m x x dxf= ∫ (2.9b)
2 20( ) ( )
L
HM m x x dxf= ∫ (2.9c) 2
11 10( ) ( )
L
HM m x x dxf= ∫ (2.9d)
12 1 20( ) ( ) ( ) 0
L
HM m x x x dxf f= =∫ (2.9e) 2
22 20( ) ( )
L
HM m x x dxf= ∫ (2.9f)
211 0
( )( )( )L
HxK k x dx
xf∂
=∂∫ (2.9g)
222 0
( )( )( )L
HxK k x dx
xf∂
=∂∫ (2.9h)
1 2
12 0
( ) ( )( ) 0L
Hd x d xK k x dx
dx dxf f
= =∫ (2.9i)
1 1 1 112H H H HC Mξ ω= (2.9j)
2 2 2 222H H H HC Mξ ω= (2.9k)
21 1 11/H H HK Mω = ;
22 2 22/H H HK Mω = (2.9l)
In equation 2.9 11HM , 22HM , 1 2, H HC C and 1 2, H HK K are the modal masses,
modal damping coefficients and modal stiffnesses of the first two modes of the human
body system in which the continuous body model is considered; 0HM is the
whole-body mass. The damping and stiffness matrices are diagonal as the damping
forces and spring forces are proportional to the relative velocities and displacements,
31
Chapter 2: Literature Review
which are used in Equation 2.8. The mass matrix is not diagonal and the coupling
between the structure and the body can be seen in the expression of the kinetic energy.
The relationship between the absolute and relative displacements of the body-structure
system can be obtained from equation 5 and written in matrix form:
{ }1 1
2 2
1 0 01 1 01 0 1
S S
HR H
HR H
u uu u T uu u
= − = − (2.10)
Substituting Equation 2.10 into Equation 2.8 and pre-multiplying the transformation
matrix, TT to both sides of Equation 2.8, gives
0 111 11 2 22
22 1 2
1 11 11 1
2 22 22 2
1 2 1 2 1 2 1 2
1 1 1 1
2 2 2
2 2
S H HH H H H
H H H S
H H H H
H H H H
S H H H H S S H H H H
H H H H
H H H
M M MM M M M
M M M uM M M uM M M u
C C C C C u K K K K KC C u K KC C u
+ + + − − + − − − +
−
+ + − − + + − − − + − −
1 1
2 2 2
000
S
H H
H H H
uu
K K u
= −
(2.11)
This is the equation of motion of the human-structure system using absolute coordinates
when the first two modes of vibration of a standing human-body on a SDOF structure
system are considered. If only the fundamental mode of the standing body on the SDOF
structure system is considered, all the terms relating to 2Hu , 2Hu , 2Hu and 2f in
equation 2.11 should be removed. Therefore equation 2.11 reduces to a 2DOF
human-structure system:
0 11 1 1 11 1 1
1 11 11 1 1 1 1
1 1
1 1 1
2
0
0
S H H H H H S S H H S
H H H H H H H
S H H S
H H H
M M M M M M u C C C uM M M u C C u
K K K uK K u
+ + − − + − + + − −
+ − + = −
(2.12)
32
Chapter 2: Literature Review
Thus equations 2.11 and 2.12 can be represented by a 3DOF model and a 2DOF model
respectively shown in Figure 2.5. In other words, equations 2.11 and 2.12 can be
derived based on the 3DOF and the 2DOF models shown in Figure 2.5.
a) A 3DOF human-structure system
b) A 2DOF human-structure system
Figure 2.5: Human-structure interaction model
The parameters of the model were defined by exact mathematical expressions, giving a
qualitatively correct model. However, the parameters of the model shown in Figure 2.5
cannot be derived without assumptions of the distributions of mass and stiffness
throughout the height of a standing body and the mode shapes. One of the characteristics
of the model (Figure 2.5) is that a mass device is presented which has not been seen
anywhere else. The mass device can be an analogy with a spring device and a damper
device which have been widely used already. As an analogy to a damper or a spring, a
mass device is introduced. The mass device has a coefficient, m, the force produced by
the mass device is proportional to acceleration, x , and is expressed as mx− . It is the
33
Chapter 2: Literature Review
mass device that couples the mass matrix and is the main differences between
conventional models and interaction models. An illustration of the mass device is given
in Figure 2.6 in conjunction with the conventional spring and damping devices. It
should be noted that the mass device is not a degree of freedom and that the mass device
is not physical but is a mathematical analogy to the spring and damping devices.
Figure 2.6: The spring, damping and mass devices
2.2.4 Continuous body models
It is reasonable to represent a standing body as a continuous model because the body is
a continuum. The difficulties for the continuous model are the provision of the mass and
stiffness distributions over the height of the body. Ji (1995) employed a continuous bar
with two segments of different masses and stiffnesses (Figure 2.6) to simulate the
vertical vibration of the body. The heights of the upper and lower parts of people were
the same and the mass of the upper part was twice that of the lower part. The mass
distribution was simplified based on the data provided by Bartz and Gianotti (1975) and
Nigam and Malik (1987). The unknown parameters in the model were the stiffnesses of
the lower and upper parts of the body. An assumption about the ratio of the two
stiffnesses of the body was made and parametric studies were conducted. It was
assumed that the stiffness 2k equals 0.5 1k , 1k or 2 1k . Based on this assumption, it
was concluded that when the assumed ratio of the axial stiffness of the upper and lower
part of the human body varies up to 300%, the corresponding modal mass fluctuates
about 33% and the human body frequency changes about 10%. The continuous model
allowed examination of the vertical vibration over the height of a standing body but it
was a purely theoretical study. The model would be more useful if experimental results
could be used to determine the unknowns in the model.
34
Chapter 2: Literature Review
Figure 2.7: A two-part continuous model
Ji (1995) then used the theoretically derived lumped mass to estimate the stiffness of an
undamped SDOF human model. Thereby, he simplified the human body to be
undamped SDOF systems. Hence, the human–structure system is an undamped 2DOF
system in which the human DOF is connected to the structural SDOF. This led to a
method to estimate the body natural frequency using vibration tests of a single structure
and the human-structure model:
(1) Measure the natural frequency of the empty structure ( Sf );
(2) Measure one of the two natural frequencies of the undamped 2DOF human-structure
system ( 1f or 2f );
(3) Calculate the lumped masses of the structure ( Sm ) and the human occupant ( Hm ).
Knowing these parameters, the natural frequency of the DOF representing the human
occupant Hf could be estimated as follows:
2 22
2
22
1
SH
SH
S
f fffm
m f
−=
+ − (2.13)
35
Chapter 2: Literature Review
By applying this equation, Ji (1995) concluded that the natural frequency of a standing
person ranges from 9.96 to 11.87 Hz, which is larger than that of the biomechanics
models.
2.2.5 Higher Degree of Freedom Models
When a human body is represented by a three or more degree of freedom model, the
difficulties in determining the parameters of the models are unavoidable. If more
degrees of freedom are considered, the representation of the body appears more
reasonable, but it becomes even harder to determine correct parameters for the body.
Nigam and Malik (1987) provided a 15 DOF spring mass system (Figure 2.7). This
model was based on an anthropomorphic model of the average male body in a standing
posture with the body modelled using ellipsoidal segments. The provision of the mass
distribution might be reasonable but it would be extremely difficult to define the
stiffnesses of the fourteen springs linking the 15 masses accurately.
Figure 2.7: 15 DOF spring mass system (Nigam and Malik, 1987)
36
Chapter 2: Literature Review
Tregoubov (2000) also developed mechanical models with multi-joint links, as shown in
Figure 2.8. It was established that the mechanical model parameters can be determined
uniquely if both a transfer function and an input mechanical impedance of a human
body are known. However, no parameters were given for these models.
Figure 2.8: Mechanical models with multi-joint links (Tregoubov, 2000)
The human body is a complex system. Even considering whole-body vibrations, there
are large variations in experimentally determined dynamic properties of a human body.
In a sitting position, the natural frequency of the 12 human bodies had an average value
of 6 Hz with a standard deviation of 0.47 (Griffin, 1990). When sitting in a chair with a
back support, the natural frequency varied about 1 Hz. In a standing position, the natural
frequency of the whole-body is about 6 Hz while one lifted one leg with the other leg
standed, the natural frequencies reduced to about 3.5 Hz (Subashi et al, 2005).
There have been a number of publications relating to human-structure interaction [BRE,
Sheffield, Oxford, Manchester]. However, there are no recognized human body models
that can be used to engineering designs reflecting practical situations.
37
Chapter 2: Literature Review
2.3 Structure response due to human action
It has been demonstrated through site measurements and laboratory tests (Ellis and Ji,
1997; Ji, 2003; Duarte and Ji, 2009) that in structural vibration a human body acts as a
dynamic load only when the body and feet move, such as walking or jumping; or as a
SDOF system and a dynamic load when the body moves while the feet remain in
contact with a supporting surface, as occurs when bouncing.
Ji and Ellis (1994) provided a method for determining the response of a floor to human
loading produced by dancing and aerobics. They observed that the mean value of the
time history of a vertical load corresponding to bouncing to music on toes or to
rhythmic jumping was always equal to the weight of the performer. And they also
suggest that more research needed to be done on dance type loads on other structure,
such as grandstands.
Yao et al. (2003) studied the effect of human jumping on a flexible rig. The subjects
were asked to jump for 20 seconds on a rig that was set to different natural frequencies
ranging from 2.0 to 16.0Hz. They concluded that it was impossible for the subject to
jump at exactly the natural frequency of the structure. The maximum force induced by
the human jumping varied between 1.8 and 4.0 times the subject’s weight.
Parkhouse et al. (2004) provided a method of analysing the jumping pattern of one
subject. Each participant was asked to jump at 1.50, 2.00, 2.67, and 3.50 Hz. The
authors found that the dynamic factor loads of the first harmonic ranged between 1.30
and 1.70 and that 2.0 Hz was the easiest frequency for the subject to follow, and 3.50
Hz was the hardest.
Sim (2006) studies human-structure interaction in cantilever grandstands, with emphasis
on modelling the passive and jumping crowds, and found that the passive crowd adds
38
Chapter 2: Literature Review
significant mass and damping to the structure and these effects vary with the natural
frequency of the structure. Experimental individual jumping and bobbing tests were
conducted at six beat frequencies to look at the variations of the impulse shape and
degree of synchronisation with the beat frequency.
Comer et al. (2010) describes the design, construction and use of a new laboratory rig
which could be used for the study of dynamic crowd-structure interaction in cantilever
grandstands.
People present on civil engineering structures not only excite the structure, but also
change the modal properties of the structure (Ellis et al., 1994; Ellis and Littler, 2004a;
Ellis and Littler, 2004b; Ellis and Ji, 2004). Therefore, modal properties of the
human–structure system need to be considered during the design process.
Ellis and Ji (1997) made dynamic response measurements on Twickenham stadium, and
indicated that humans were acting as a dynamic mass–spring–damper system rather
than as a simple additional mass. If occupied by spectators, the tested assembly
structure clearly showed an additional peak (Figure 2.9). They hypothesized that this
additional mode was caused by human occupants adding a SDOF to the structure.
(a) The empty stadium (b) occupied by a crowd
Figure 2.9: Dynamic responses of the Twickenham stadium (Ellis and Ji, 1997)
39
Chapter 2: Literature Review
Ellis and Ji (1997) estimated natural frequencies of the empty and the crowd-occupied
structure by curve-fitting single degree-of-freedom (SDOF) and 2DOF models,
respectively. The fundamental frequencies of the three sections of the empty structure
were between 7.24 Hz and 8.55 Hz (Table 2.2). Under human occupation, the
fundamental natural frequencies were 5.13 to 5.44 Hz, and second natural frequencies
from 7.89 to 8.72 Hz were identified (Table 2.2).
Table 2.2: Frequencies observed at Twickenham (Ellis and Ji, 1997).
Truss Empty structure Human-occupied structure
5 8.55Hz 5.44 Hz and 8.72 Hz
9 8.32Hz 5.41 Hz and 7.91 Hz
11 7.24Hz 5.13 Hz and 7.89 Hz
Moreover, another laboratory tests was conducted by Ellis and Ji (1997). Several tests
were conducted using a simple beam. The results of frequency measurements made on
the beam are given in Table 2.3. They concluded that when the person was stationary on
the beam, he acted as a spring–mass–damper on the structure and when the person was
moving on the beam, he acted solely as a load and the structural characteristics were
those of the empty beam. They also noted that when a person stood on a beam,
thereby producing a 2DOF system, only one mode could be identified from the
experimental measurements and this needed explaining.
Table 2.3: Frequencies observed on the beam (Ellis and Ji, 1997).
Description of experiments Measured frequency (Hz)
Bare beam 18.68
Beam plus dead mass of 100 lb 15.75
Beam plus dead mass of 200 lb 13.92
Beam with T. Ji standing 20.02
Beam with T. Ji sitting on a high stool 19.04
Beam with T. Ji jumping on spot 18.68
Beam with T. Ji walking on spot 18.68
40
Chapter 2: Literature Review
A general literature review of the studies on vibration serviceability of stadia structures
was published in 2011 (Jones et al., 2011). This paper gave a critical review of
information pertinent to the behaviour of stadia structures subjected to dynamic crowd
loading. It introduced and explained key concepts in the fields and summarized the
development of current guidance and methods.
Reynolds et al. (2004) conducted the modal tests in the Bradford Stadium during 20
football matches and 9 rugby matches. The structure had natural frequencies between
3.28 Hz and 5.75 Hz. Test results showed that the natural frequencies reduced when the
stand was occupied by seated and standing spectators. It was observed that the natural
frequencies reduced greater when the spectators were standing than when seated. And
an increase in the damping ratio was also reported.
Littler measured natural frequencies of the vertical and two horizontal modes of a
retractable grandstand when it was empty and when all 99 places were occupied by
sitting or standing people (Littler, 1998). The data in Table 2.4 (Littler, 2000) show that
the modes of the structure were affected by standing or sitting human occupants.
Moreover, Littler (1998) claimed that damping of the empty and the two types of
occupation varied. Hence, a dynamic human model has probably different for standing
and sitting crowds. Importantly, the crowd not only changed the vertical natural
frequencies, but also of the horizontal modes, as shown in Table 2.4. Therefore,
dynamic human models have to be considered in the design of civil engineering
structures against human-induced vertical and horizontal forces.
Human occupation of structures can increase damping, change the fundamental natural
frequency and also add an additional natural frequency (Littler, 2000). Therefore,
appropriate dynamic models of human occupants have to be used. This is important in
the design of slender assembly structures because they can be subjected to high levels of
human-induced forces and their dynamic properties can be changed significantly (Table
2.4).
41
Chapter 2: Literature Review
Table 2.4: Natural frequencies of a retractable grandstand (Littler, 2000).
Configuration Front-to-back Mode
(Hz)
Sway Mode
(Hz)
Vertical Mode
(Hz)
Empty stand 3.05 3.66 13.6
Standing
occupants 3.30 3.54 9.16
Sitting occupants 1.71 1.83 9.03
Reynolds and Pavic (2002) modelled grandstands with an emphasis on their dynamic
behaviour. Discrepancies between the calculated and measured modal properties were
found for a grandstand in a football stadium due to additional stiffness provided by the
joints between the main structural members. Another example was the City of
Manchester Stadium (Reynolds et al. 2005) in which the perimeter concrete block wall
was found to have a significant influence on the natural frequencies of the structure.
Dynamic forces induced by crowds are an issue of great concern (Kasperski, 2001).
Although it has been found that dynamic loads induced by groups of people are higher
than those induced by individuals, the human-induced forces do not increase linearly
with the number of people. People are synchronized by a prompt (Kasperski and
Niemann, 1993) which can be provided by music, movements of other people, or
perceptible movements of the occupied structure.
Guidance from different countries varies. The USA and Canadian guidance (Murray et
al., 2003) include human mass in calculating responses of a floor subjected to crowd
rhythmic loading, while the UK guidance and code (BS6399, 2002; Ellis and Ji, 2004;
Wilford and Young, 2006; Smith and Hicks, 2007; Institution of Structural Engineers,
2008) do not include human mass in such calculations. When part of a group of people
are jumping and the other part are standing or sitting, the stationary human bodies will
contribute significant damping which may attenuate the response produced by the
people jumping. This type of scenario has been considered in the guidance for
permanent grandstands (Institution of Structural Engineers, 2008).
42
Chapter 2: Literature Review
2.4 Human response to structure vibration
Human acceptance of the levels of vibration depends on the environment and activity in
which they are involved. Some tolerance criteria have been proposed for design in the
last few years. These criteria are based on the limitation of static stiffness (Murray, 1999)
and acceleration. People are sensitive to vertical floor motion. A half sinusoidal
amplitude of 1.0mm or an acceleration of 0.5%g will annoy people in quiet
environments, such as residences or offices (Murray, 1999). The tolerance level will
increase, if the environment is more noisy, such as in shopping mall or outdoor. When
the people are attending active events, such as lively concerts or dancing clubs, the
tolerance level will increase.
Researchers have been attempting to quantify the response of the human body to the
floor vibration for many years. Murray (1999) gave two types of design criteria: (1)
criteria for human response to known or measured vibration, and (2) design criteria
related to human response that include an estimation of dynamic floor response. Table
2.5 shows the chronological list of human acceptance criteria for floor vibrations.
The serviceability evaluation for human response relates more to tolerance than
perception of vibration. However, there is no actual guidance in the British Standards
for this situation, although it is suggested that VDVs (vibration dose values) provide a
sensible means for comparing different events.
The VDV which provides a method of assessing the cumulative effect of vibration, is
defined in BS 6472, and is used as an indicator of when various degrees of adverse
comment can be expected in buildings. This is recommended as being a sensible means
for comparing various events. But it is only necessary to sample the waveform when it
is at high magnitude; periods at low magnitude may be omitted without effecting on the
resulting value.
43
Chapter 2: Literature Review
41
0
4 ])([∫=
==
Tt
tdttaVDV (2.14)
Where
VDV is the vibration does value (in m/s1.75);
( )a t is the frequency-weighted acceleration;
T is the total period of the day (in s) during which vibration may occur.
Table 2.5: Vibration criteria over times (Murray, 1999)
Date Reference Loading Application Comments 1931 Reiher and Meister Steady
State
General Human response criteria
1966 Lenzen Heel-drop Office Design criterion using Modified Reiher
and Meister scale
1970 HUD Heel-drop Office Design criterion for manufactured
housing
1974 International Standards
Organization
Various Various Human response criteria
1974 Wiss and Parmelee Footstep Office Human response criteria
1974 McCormick Heel-drop Office Design criterion using Modified Reiher
and Meister scale
1975 Murray Heel-drop Office Design criterion using Modified Reiher
and Meister scale
1976 Allen and Rainer Heel-drop Office Design criterion using modified ISO
scale
1981 Murray Heel-drop Office Design criterion based on experience
1984 Ellingwood and Tallin Walking Commercial Design criterion
1985 Allen, Rainer and Pernica Crowds Auditorium Design criterion related to ISO scale
1986 Ellingwood et al Walking Commercial Design criterion
1988 Ohlsson Walking Residential/Office Lightweight Floors
1989 International Standard
ISO 2231-2
Various Buildings Human response criteria
1989 Clifton Heel-drop Office Design criterion
1989 Wyatt Walking Office/Residential Design criterion based on ISO 2631-2
1990 Allen Rhythmic Gymnasium Design criterion for aerobics
1993 Allen and Murray Walking Office/Commercial Design criterion using ISO 2631-2
44
Chapter 2: Literature Review
Within the standards there are several guidance rules and Griffin (1990) gives a logical
extrapolation based on information given in BS 6472 to provide VDVs for different
situations. This guidance is shown in Table 2.6.
Table 2.6: VDVs at which various degrees of adverse comment may be expected (BS
6472)
Place Low probability of
adverse comment
Adverse comment
possible
Adverse comment
probable
Critical areas 0.1 0.2 0.4
Residential
buildings 0.2-0.4 0.4-0.8 0.8-1.6
Office 0.4 0.8 1.6
Workshops 0.8 1.6 3.2
The lower values for the critical areas relate to the threshold of human perception,
whereas the workshop values are probably linked more closely to vibrations that can be
tolerated. It is therefore reasonable to see whether the VDVs suggested for workshops
are appropriate for use with grandstands. If the serviceability limits for workshops are
expanded slightly, to assign ranges for VDVs, the values in Table 2.7 are obtained.
Table 2.7: Possible VDV ranges for grandstands (Ellis and Littler, 1994)
VDV range Reaction
<0.6 OK but may be perceptible
0.6~1.2 Low probability of adverse comment
1.2~2.4 Adverse comment possible
2.4~4.8 Adverse comment probable
>4.8 Unacceptable
For guidance on acceptable vibration levels in grandstands it seems appropriate to
examine measurements made on grandstands. A number of experiments with groups of
45
Chapter 2: Literature Review
people jumping on cantilever grandstands have been undertaken by Ellis, B. R. and J. D.
Littler and recommendations for low-frequency vibration have been made. These are
given in the first and second columns of Table 2.8.
Table 2.8: Suggested acceptable vibration levels and their extrapolation to VDVs (Ellis
and Littler, 1994)
Vibration level: %g Reaction Event VDV: m/s1.75
<5 Reasonable for passive persons <0.66
5~18 Disturbing 0.66.2.38
18~35 Unacceptable 2.38.4.64
>35 Probably causing panic >4.64
2.5 Tuned-mass-dampers
There are many publications on tuned-mass-dampers (TMDs) which can be used to
reduce structural vibration (Setreh and Hanson, 1992) and on their optimum design
(Setreh and Hanson, 1992; Satareh, 2002; Den Hartog, 1956; Warburton, 1982; Leung
and Zhang, 2009; Lee, et al., 2004; Fujino and Abe, 1993). The optimization of a
passive TMD for a single degree-of-freedom (SDOF) structure under a harmonic
excitation were well explored with widely known publications, such as Den Hartog
(Den Hartog, 1956), Warburton (Warburton, 1982). The studies on other inputs using
different optimization objectives and different optimization methods can be found in
many recent publications, such as (Leung and Zhang, 2009; Lee, et al., 2004, Li and Qu,
2006; Fujino and Abe, 1993; Hoang and Warnitchai, 2008; Marano, et al., 2008).
Leung and Zhang studied the design parameters for TMDs using particle swarm
optimization and considered different types of excitation (Leung and Zhang, 2009) and
provided a list of references. The objective function defined in their study was the mean
square displacement response. Detailed optimum design parameters of the TMDs were
given in a tabular form for different excitations and the damping ratio of the structure 46
Chapter 2: Literature Review
SDOF system. The results showed that, when the damping ratio of the structure SDOF
system is given, the objective function decreases monotonically as the mass ratio of the
TMD to the structure increases. This implies that the mass ratio may not be a variable
for optimization.
Lee, et al (2004), estimated the equivalent damping ratio of a structure with various
added damping devices. Then this ratio was added to the damping ratio of the structure
in the conventional equation of motion for a SDOF system based on the mode
superposition method. Assumptions had to be made to obtain the equivalent damping
ratio. Fujino (1993) and Hoang (2008) provided formulae for optimum design of TMDs
for seismic applications. Marano et al (2008) studied the optimum design of TMD
considering random vibration mitigation. TMDS are normally used to reduce vibration
in a particular mode vibration, Warnitchai and Hoang (2006) investigated the optimum
placement of multiple TMDs to suppress vibrations in several modes.
There is an increasing use of TMDs on floors that are subjected to rhythmic crowd loads
(Setreh and Hanson, 1992; Satareh, 2002). Rhythmic crowd loads are normally
generated with music beats at pop concerts or dances and the loads consist of several
harmonics of the beat frequency. In other words, the vibration of dance floors includes
several modes and is induced by several harmonics. Experience of assessing floor
vibration shows that the fundamental mode often dominates the response of a floor at
resonance. Therefore, TMDs are suitable for use on dance floors to reduce the vibration
induced by rhythmic crowd loads. This background has led to a further study of the
optimum parameters for a TMD subject to a harmonic load. The optimum parameters
for a TMD depend on input, such as a load on a structure or ground motion, harmonic or
white noise, measurement variable, (displacement, velocity or acceleration), and
objective function, such as response with and without statistical consideration.
47
Chapter 2: Literature Review
2.6 Summary
The literature review shows that the human-structure interaction is a relatively new
topic that needs to be further investigated. Conclusions and further works in this topic
are identified as follows:
• It is reasonable to represent a standing body as a continuous model. However, the
mass and stiffness distributions of the existing continuous body model are assumed.
A further study of a continuous model should remove some of the assumptions and
improve the model of the actual standing body.
• The parameters of the interaction model were defined by exact mathematical
expressions, giving a qualitatively correct model. Further work should be to
quantify the values of the parameters.
• The frequency characteristics of the interaction model should be examined to gain a
better understanding of the interaction effects.
• The natural frequency and damping ratio of a human body were usually identified
based on shaking table experiments. However, the vibration level of the shaking
table is much higher than structural vibration. It may be possible to use structural
dynamics methods to identify these parameters.
• Human bodies contribute significant damping to structures. This effect has not been
considered in current designs.
The following chapters focus on the study of the first four items and contribute to an
improved understanding of human body models and human-structure interaction.
48
Chapter 3: Experiment Test Set-up
3 Experiment Test Set-up
3.1 Introduction
This chapter describes the equipment and set-up used in the experiments. The set-up
includes a test rig, a vibration control system and a data collection system. These are
described in sections 3.2, 3.3 and 3.4. Section 3.5 describes the procedure for free
forced vibration tests. Section 3.6 provides a summary.
3.2 Test rig
3.2.1 Test Rig
The test rig shown in Figure 3.1 (Yao et al., 2004) is designed to behave like a single
degree-of-freedom system. It consists of a steel cantilever with an adjustable span, a platform
that is support at the end of the cantilever and a support frame.
Figure 3.1: Test rig
A cross-section of the platform is shown in Figure 3.2. The frame supports vertical rails,
and the platform slides up and down the rails on linear bearings with low friction. The
49
Chapter 3: Experiment Test Set-up
stiffness required to provide a restoring force to the platform is provided by the
cantilever. The adjustable sliding prop support is used to vary the span of the cantilever
in order to obtain the required natural frequency for each test. For safety, the platform
has a guard frame on each side.
Figure 3.2: Cross-section of the test rig (Yao et al., 2004)
3.2.2 Improvements to the Test Rig
During free vibration tests, it was found that the web of the adjustable sliding prop
became unstable and swayed. This could have posed a safety hazard when a person
stood, bounced or jumped on the rig. Therefore the web of the prop was stiffened. The
original design of the adjustable sliding prop is shown in Figure 3.3a.
The force applied on the top plate is transmitted through the web to the lower plate that
is supported by two small vertical plates. The web has a height of 236mm. It appears
that the web is the weakest part of the prop. Therefore two side plates were placed
perpendicular to the web and welded to the web and the upper and lower plates. An
additional small vertical plate was placed directly underneath the two side plates,
parallel to the two existing small vertical plates, to reduce the bending of the lower plate.
The improved design is shown in Figure 3.3b. The improved design prevented excessive
sway of the web. 50
Chapter 3: Experiment Test Set-up
(a) Original design (b) Improved design
Figure 3.3: 3D view of the prop
3.3 The Data Collection System
A A223 servo-accelerometer (range +/− 2g standard) was used to monitor vertical
motion of the platform. The accelerometer was fixed to a metal cube standing on three
legs standing (Figure 3.4). The signal from the accelerometer was digitised using a a
CED Power 1401 16 bit high performance interface system (Figure 3.5) with a PGF 8
channel gain and tracking filter and recorded on a laptop computer. The laptop, installed
with the CED software Spike 2 version 6 (Figure 3.6), was used on site to monitor,
record and process the acceleration records. A sampling rate of 200Hz is taken. The
sample duration is set 5 seconds for the free vibration and 170 seconds for forced
vibration. Each test is repeated at least once to ensure that the test results are repeatable.
The recorded accelerations were analysed using the Spike 2 package and the data have
been averaging to improve the pattern of the spectra. By averaging the data, the spectra
value at zero frequency would decrease to zero, which would help to identify the peak
of spectra. The natural frequency was identified in the spectra using Matlab programme.
As the pattern and peak of the spectra are clear enough and without any disturbance, no
Side plates
Vertical plate
Web
51
Chapter 3: Experiment Test Set-up
window and filtering are used to process the data.
Figure 3.4: Accelerometer
Figure 3.5: CED Power 1401
Figure 3.6: Spike 2 Version 6
3.4 The Vibration Control System
The vibration control system consists of a APS113 Shaker, Power Amplifier and
Vibration Control Unit. The shaker shown in Figure 3.7 is a force generator.
52
Chapter 3: Experiment Test Set-up
Figure 3.7: Shaker APS 113
The APS 125 Power Amplifier, shown in Figure 3.8, is designed to provide power for
electrodynamics shakers. The amplifier can be operated in either a voltage or current
amplifier mode, selectable from the front panel.
Figure 3.8: Power Amplifier APS 125
In addition to shaker and amplifiers, Vibration Control Unit VCU13.2S (Figure 3.9) and
VCS201Vibration Control software (Figure 3.10) were used to control the shaker.
Figure 3.9: Vibration Control Unit VCU13.2S
53
Chapter 3: Experiment Test Set-up
Figure 3.10: VCS201Vibration Control Version 1.2.1.0
3.5 Test Procedure
3.5.1 Free vibration
Free vibration tests can be conducted in two simple ways:
1. Free vibration induced by an initial velocity, i.e. an impact test. An impact is applied
to the test rig to generate vibration. A person stands on a ladder next to the rig and uses
one foot to give a vertical knock to the platform to generate vertical vibration (Figure
3.11a).
2. Free vibration induced by an initial displacement. An initial displacement of the test
rig is given and then released suddenly to generate vibration. A person stands on the
platform using one of his legs to generate a vertical displacement of the rig. He then
moves from the test rig onto the ladder, as shown in Figure 3.11b, thereby inducing
vertical free vibration of the rig.
54
Chapter 3: Experiment Test Set-up
(a) Free vibration induced (b) Free vibration induced
by an initial velocity by an initial displacement
Figure 3.11: Free vibration tests
Figure 3.12 shows and compares 5 seconds acceleration records and corresponding
frequency spectra for the two test methods.
(a) Free vibration induced (b) Free vibration induced
by an initial velocity by an initial displacement
Figure 3.12: Acceleration-time history and frequency spectrum
It can be seen that the initial velocity and initial displacement methods give the almost
identical values of natural frequency (6.03 and 6.04 Hz). As the initial velocity method
is more convenient to use it is adopted in later studies (Chapter 7).
55
Chapter 3: Experiment Test Set-up
Four people took part in the free vibration tests to provide data for studying the
interaction between a standing body and the rig. The details of the four people are
summarised in Table 3.1.
Table 3.1: Mass and height of the participants
P1(M) P2(F) P3(M) P4(F)
Weight(kg) 94.4 62.4 75.6 58.6
Height(cm) 176 158 181 162
Age 30 27 26 29
The stiffness of the test rig was adjusted to create different experimental conditions
relating to the fundament natural frequency of the rig (last column in Table 3.2). The
test cases for each natural frequency are summarised in Table 3.2. The first digit in the
experimental cases in the table indicates the setting of the test rig and the second shows
the participant’s number. Each test was repeated at least once to ensure that the test
results were repeatable.
Table 3.2: Experiment cases
Bare rig Rig+P1 Rig+P2 Rig +P3 Rig+P4 Sf (Hz)
Rig 1 Case 1.0 Case 1.1 Case 1.2 Case 1.3 Case 1.4 6.55
Rig 2 Case 2.0 Case 2.1 Case 2.2 Case 2.3 Case 2.4 7.19
Rig 3 Case 3.0 Case 3.1 Case 3.2 Case 3.3 Case 3.4 8.02
Rig 4 Case 4.0 Case 4.1 Case 4.2 Case 4.3 Case 4.4 8.91
Rig 5 Case 5.0 Case 5.1 Case 5.2 Case 5.3 Case 5.4 9.76
Rig 6 Case 6.0 Case 6.1 Case 6.2 Case 6.3 Case 6.4 11.85
Rig 7 Case 7.0 Case 7.1 Case 7.2 Case 7.3 Case 7.4 13.57
Rig 8 Case 7.0 Case 8.1 Case 8.2 Case 8.3 Case 8.4 15.36
Rig 9 Case 9.0 Case 9.1 Case 9.2 Case 9.3 Case 9.4 15.63
56
Chapter 3: Experiment Test Set-up
3.5.2 Forced vibration
In order to identify all possible resonance frequencies of the human-rig system, it was
subjected to a forced vibration test considering the frequency range between 1 Hz and
15 Hz with the sweeping rate of 5 Hz/min. The force input was controlled by the
acceleration at the head of the shaker at the magnitude of 5 m/s2. Four people took part
in the forced vibration test. The mass ratio of the human body to the test rig was
adjusted by using different numbers of subjects. The details of the four people are
summarised in Table 3.3. The subjects were asked to stand still on the platform as
shown in Figure 3.13. The test rig is set at two natural frequencies, 7.05 Hz and 5.66Hz,
forming Case 1.0 and Case 2.0. Eight human-structure interaction forced vibration tests
were conducted. The test cases and the mass ratios are summarised in Table 3.4 where
the first digit indicates the setting of the test rig and the second shows the number of
subjects involved. Each test was repeated at least once to ensure that the test results
were repeatable.
All the tests were conducted in the structures laboratory at the University of Manchester.
The test rig was checked every year, potential minor risks were assessed and prevention
measures were provided. For the experiments conducted in Chapter 6 and 7, a total of 4
subjects were involved. They needed to use a ladder to reach the test rig at a height of
1.2 meters and might experience rig vibration in the vertical direction for up to 170
seconds. This vibration level on the test rig was comparable to that passengers
experience in a bus or a tram in Manchester. The study was not required an approval by
the Ethics Committee of the University, but the consents were obtained from all 4
participants before the tests. In addition, the related safety issues were explained to the
participants.
57
Chapter 3: Experiment Test Set-up
Figure 3.13: Free vibration tests
Table 3.3: Weight and Height of the four people
S1(F) S2(M) S3(M) S4(M)
Weight(kg) 60.2 86.6 75.8 80.8
Height(cm) 162 178 180 181
Age 32 29 26 29
Table 3.4: Experiment cases
Bare rig Rig+S1 Rig+S1+S2 Rig+S1+S2+S3 Rig+S1+S2+S3+S4
Rig 1 Case 1.0 Case 1.1 Case 1.2 Case 1.3 Case 1.4
Rig 2 Case 2.0 Case 2.1 Case 2.2 Case 2.3 Case 2.4
58
Chapter 3: Experiment Test Set-up
3.6 Summary
This chapter provides details of the test system and test equipment. The test rig was
improved to enable the tests to be conducted safely. The data collection system and
vibration system are introduced. Both free and forced vibration tests are also described.
The experimental results will be presented in the following chapters.
59
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
4 A Continuous Model of a Standing Human Body in
Vertical Vibration
4.1 Introduction
Human body response to structural vibration may be predicted if the human body can be
modelled correctly. However, it is very difficult to model a human body accurately.
Standing is perhaps the simplest and most common posture among all possible postures.
Thus modelling a standing human body is a reasonable start before modelling other
postures. A standing human body is a continuum in which the mass and stiffness are
distributed unevenly throughout the height of the body. Many models have been
developed to represent the whole body of a standing person and these may be classified
into five types:
1) simple single or two degree-of-freedom models in which the parameters were
determined based on shaking table tests in the study of body biomechanics;
2) Conventional models that were developed based on a fixed base and often used
in structural vibration;
3) human-structural interaction models;
4) continuous body models;
and
5) high degree-of-freedom models in which an anthropomorphic model of the
average male body in a standing posture was used.
Their respective advantages and limitations were introduced in Chapter 2 (Section 2.2).
It can be observed that there are no links between the five types of model although they all
have been used to represent a standing body and that these models complement each
other. 60
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
In this chapter a new model is developed that can link the four other models. A
continuous standing human body model in vertical vibrations is developed using the
anthropomorphic model and the two available natural frequency measurements in the
biomechanics models. A standing human body is considered as a bar with particular
mass and stiffness distributions in the vertical direction in section 4.2. The mass
distribution of a standing body is formed using the mass distribution of fifteen body
segments in the anthropomorphic model (Bartz and Gianotti. 1975, Nigam and Malik,
1987). The axial stiffness of the model is determined based on the best match between
the first two natural frequencies of the proposed models and the two available natural
frequencies (Matsumoto and Griffin, 2003) in section 4.3. Four similar models are
assessed using finite element parametric analysis. The best of the four models has seven
uniform mass segments and two uniform stiffness segments. The continuous model is
able to show the shapes of vibration modes throughout the height of the standing body.
In addition, relationships between the continuous and simple discrete models can be
derived quantitatively. The modal mass and modal stiffness of the continuous model are
evaluated, which are related to the human-structure interaction models in section 4.4.
Two continuous models are improved in section 4.5. Finally A numerical verification of
the discrete human body models is provided in Section 4.6 in which eigenvalue analysis
of three human-structure systems is conducted using the two interaction models and the
newly proposed continuous body model. Conclusions from this study are summarised in
Section 4.7.
4.2 A Continuous Standing Body Model
4.2.1 Assumptions
To establish a human body model, it is important to present and clarify the basic
assumptions involved in its development. The assumptions used in establishing the model
and justification of the assumptions are given as follows:
61
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
1. Local vibrations in a standing body are neglected.
This model considers only global behaviour which may be used to investigate the
responses of a standing individual or a crowd and of the structure that supports the people.
Thus the local vibrations of the human body, such as arm vibration or eye-ball vibration,
are insignificant for the purpose of the study and can be neglected.
2. Each of the fifteen segments of a body has a uniform density.
The standing body consists of 15 mass segments in this study, which was defined in the
paper by Bartz and Gianotti (1975) and Nigam and Malik (1987). The densities of the
segments are different. However, it is assumed that each segment has a uniformly
distributed density.
3. The first two natural frequencies of a typical standing body (Table 2.1) adopted in
this study are considered to be correct.
The first two natural frequencies of the human body were obtained from shaking table
tests (Matsumoto and Griffin, 2003). They are considered to be correct and reliable. Thus
they are directly used as a basis for identifying the two stiffness values in the continuous
models of a standing body.
4. The axial stiffness of a standing body is represented by constant upper and lower
body stiffnesses.
This assumption is likely to affect the accuracy of the model. However, only two known
natural frequencies of a standing body are available, which restrict the model from
having more unknown parameters.
Based on the above assumptions, the vertical vibration of the human body can be
studied as the axial vibration of a column assembled from several uniform bars having
different properties.
62
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
4.2.2 Mass and stiffness distributions of the model
Table 4.1 summarises the mass distribution of the fifteen segments based on the papers
of Bartz and Gianotti (1975) and Nigam and Malik, (1987). The second column in Table
4.1 describes the segments and the third and fourth columns show the mass and length
of each segment. The left half of Figure 4.1a shows a standing body where the values of
segmental masses are indicated together with the lengths of the segments. Based on the
second assumption, the mass density along the height is the ratio of the mass to the
corresponding length of the segment. The right half of Figure 4.1a shows the
distribution of mass density along the height of the body and the combination of some
segments. Based on the second assumption, a further treatment of body mass
distributions are considered for the purpose of the modeling of vertical vibration. This
includes: a) the two arms and the upper torso are grouped as a whole and named as the
upper torso; b). the two thighs are merged into one, similarly the two legs and the two
feet, in the model. Therefore, the body mass distribution is represented by seven
distinctive components along the height of the body based on the anthropomorphic
model (Nigam and Malik, 1987). Figure 4.1b shows the mass distribution of the
continuous standing body model with the given height for each component (Nigam and
Malik, 1987). The stiffness distribution will be discussed in each model.
In incorporating the stiffness into the continuous model, there are four possible models
that are summarised in the last four columns in Table 4.1 and described as follows:
Model 1: The upper nine segments are grouped into the upper part of the body while the
lower six segments are classified as the lower part. Each of the parts has a uniform mass
distribution. Two different stiffnesses are assigned to the upper and lower parts
respectively, as shown in Figure 4.2a. It can be noted from the fifth column in Table 4.1
that the upper and lower parts have almost the same height. This model was originally
developed by Ji (1995) and is relatively simple. The reason for lumping the fifteen
segments into upper and lower parts is that only two stiffness values can be assigned
63
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
based on the fourth assumption and each part is given the same mass and stiffness.
Model 2: This model takes the mass distribution defined in Figure 4.1b, i.e. the model
has seven different mass densities distributed over the height of the body. The two axial
stiffnesses are assigned to the same heights as that in Model 1. The model appears more
accurate than Model 1 as the mass distribution is more reasonable than that of Model 1.
(a) Anthropomorphic model and densities of the main parts of a body (kg/m)
(b) Continuous body model with known masses and
unknown stiffness
Figure 4.1: Distribution of body mass
Model 3: This model is almost the same as Model 2 except for the assignment of the
stiffness. It can be seen from the seventh column in Table 4.1 and Figure 4.2c that
stiffness 1k is assigned to the four lower parts in this model while to the three lower
parts in Model 2. As shown in Figure 4.2c, the ratio of the length assigned 1k to the
height of the human body is 0.613 (approximately equal to the golden ratio 0.618).
Model 4: This model is similar to Model 1 as the masses are distributed into only upper
and lower parts. The difference is that the lower part includes the lower torso in this
model while the upper part contains the lower torso in Model 1. Consequently, the
stiffness distribution is altered too as shown in Figures 4.2a and 4.2d.
64
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
Table 4.1: Mass distribution of 15 ellipsoidal segments (Bartz and Gianotti ,1975;
Nigam and Malik, 1987)
No. Segment
designation
Anthropomorphic Model
Model 1 M (kg)
L (m)
Model 2 M (kg)
L (m)
Model 3 M (kg)
L (m)
Model 4 M (kg)
L (m) M (kg) L (m)
1 Head pivot 3.044 0.154
11M =
49.96
11L =
0.842
2k
21M =3.251
21L =0.173
31M =3.251
31L =0.173
41M =
37.37
41L =
0.653
2k
2 Neck pivot 0.207 0.019
3 Right upper arm 2.322 0.291 22M =17.57
22L =0.146
32M =17.57
32L =0.146
4 Left upper arm 2.322 0.291
5 Right lower arm 1.910 0.378
6 Left lower arm 1.910 0.378
7 Upper torso 9.105 0.146
8 Centre torso 16.55 0.334
23M =16.55
23L =0.334
33M =16.55
33L =0.334
9 Lower torso 12.59 0.189
24M =12.59
24L =0.18,
34M =12.59
34L =0.189
42M =
37.53
42L =
1.034
1k
10 Right upper leg 7.827 0.432
12M =
24.94
12L =
0.845
1k
25M =15.65
25L =0.432
35M =15.65
35L =0.432 11 Left upper leg 7.827 0.432
12 Right Lower leg 3.445 0.359 26M =6.890
26L =0.359
36M =6.890
36L =0.359
13 Left Lower leg 3.445 0.359
14 Right foot 1.198 0.054 27M =2.396
27L =0.054
37M =2.396
37L =0.054
15 Left foot 1.198 0.054
Sum 74.9 1.687 M =74.9
L =1.687 M =74.9
L =1.687 M =74.9
L =1.687 M =74.9
L =1.687
65
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
(a) Model 1 (b) Model 2 (c) Model 3 (d) Model 4
Figure 4.2: Continuous standing body models
For the four models defined above, the mass distributions are known and the pattern of
stiffness distribution is given while the values of the two stiffness 1k and 2k are
unknowns and are to be determined.
4.3 Identification of the Stiffness
4.3.1 Method of identification
Parametric free vibration analysis is conducted to identify the two stiffnesses for the
four models using the finite element method.
The height of the human anthropomorphic model has a height of 1.687m. For modeling,
338 bar elements (LINK 8) with an equal length of 0.005m in ANSYS, making a total
body height of 1.690m. The relatively fine mesh should give a good presentation of
higher mode shapes. The procedure to implement the FE model and analysis is
straightforward:
1. Define and create the 338bar elements in series with the length 0.005m.
2. Assign the mass density and stiffness for each element based on Table 4.2 and
Figure 4.2
66
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
3. Provide constraints in the horizontal direction for all elements. Otherwise, the model
becomes a mechanism.
4. Assign the boundary condition at the bottom of the model to prevent free body
movement in the vertical direction.
5. Conducted the free eigenvalue analysis and obtained the natural frequencies and
mode shapes for the first 20 modes
The finite element model is shown in Figure 4.3. The length and the number of each
element is shown in the Table 4.2.
Figure 4.3: Numerical model in ANSYS
Table 4.2: The length and the number of the element
No. Segment
designation Li
(m) Length of the element(m)
Number of the element
Li
(m)
1,2 Head and Neck
pivot 0.173 0.005 35 0.175
3,4,5,6,7 Upper torso and
two arm 0.146 0.005 29 0.145
8 Centre torso 0.335 0.005 67 0.335 9 Lower torso 0.189 0.005 38 0.190
10,11 Upper leg 0.432 0.005 86 0.430 12,13 Lower leg 0.359 0.005 72 0.360 14,15 foot 0.054 0.005 11 0.055
Sum 1.687 338 1.69
In free vibration analysis, the stiffness and mass of the models is normally the input and
the natural frequency is the output. However, it is unlikely that the stiffness of the
67
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
models can be determined directly from the two given natural frequencies. Therefore is
helpful to understand the dynamic characteristics of the models before any parametric
finite element analysis and identification are conducted.
The first two natural frequencies of the body models can be qualitatively expressed as:
2
0
2
0
( )( )( )1 1, 22 ( ) ( )
Li
i L
i
xk x dxxf i
m x x dx
f
π f
∂∂= =
∫∫
(4.1)
Where ( )k x and ( )m x are the stiffness and mass distributions over the height of the
body models; ( )i xf is the shape function of the i th mode. If the correct values of
( )k x , ( )m x and ( )i xf can be provided, it will lead to an exact solution for the natural
frequency. Due to the complexity of a human body, it is highly unlikely that these
functions can be determined.
The ratio of the second natural frequency to the first natural frequency is
22 2
10 02
2211 200
( )( )( ) ( ) ( )( ) ( ) ( )( )( )
L L
LL
xk x dx m x x dxf xxf m x x dxk x dx
x
ff
f f
∂∂∂= ×
∂ ∂∂
∫ ∫∫∫
(4.2)
For the studied cases, the stiffness can be expressed as
11 12
21
1 0< (lower part)
( ) ( ) < (upper part)
k x Lk x k k S xk
k L x Lk
ββ
≤ = = = ≤
(4.3)
68
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
where 2
1
1 0<( )
<
x LS x k L x L
k
β
β
≤= ≤
(4.4)
and β is the ratio of the length of lower part to the height of the model. Substituting
Equation 4.3 into Equation 4.2 gives
22 21 10 02
2211 2 200
( )( )( ) ( ) ( )( ) ( ) ( )( )( )
L L
LL
xS x dx m x x dxf xxf m x x dxS x dx
x
ff
f f
∂∂= ×
∂∂
∫ ∫∫∫
(4.5)
It can be observed from Equations 4.1, 4.4 and 4.5 that:
1. Equation 4.1 indicates that if ( )k x is scaled to 2 ( )c k x , the natural frequency if
becomes icf where c is a constant.
2. Equations 4.4 and 4.5 indicate that the frequency ratio 2 1f f only relate to the
stiffness ratio 2 1k k rather than the absolute values of 1k and 2k . In other words,
if 1k and 2k become 1ck and 2ck respectively, the frequency ratio 2 1f f ,
remains unchanged.
A qualitative understanding of the relationships between the natural frequency and
stiffness can effectively simplify the identification process. Instead of identifying the
two unknown stiffnesses simultaneously, the ratio of the two stiffnesses is identified to
match the target frequency ratio (Table 4.1) as closely as possible; then the two stiffness
values are multiplied by the same scalar to match the measured fundamental natural
frequency in Table 4.1.
This identification strategy can be easily realised in the finite element analysis which
requires an input of the two stiffnesses. A value of 1k , saying 100kN, is given and fixed,
and different values of 2k are provided based on the ratio of 2 1k k varying from 0.01
69
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
to 2.0 with an increment of 0.001. A do-loop is used to generate the values of 2k . Thus
a series of values of 1f and 2f are calculated for the four models. The relationships
between 2 1f f and 2 1k k can be presented graphically as shown in Figure 4.3 where
the target frequency ratio of 2.29 and the minimum frequency ratio are also indicated.
The stiffness ratio corresponding to the minimum natural frequency ratio (Figure 4.3) is
selected as the solution for each model because the minimum natural frequency ratio is
the closest of all values to the target ratio. If the ratio of the measured natural frequency
(5.88 Hz) to the calculated fundamental natural frequency corresponding to the
minimum frequency ratio is c, the first two calculated natural frequencies are multiplied
by c and the two stiffness values corresponding to the minimum frequency ratio are
multiplied by 2c . Table 4.3 summarises the two identified stiffness values and the
corresponding first two natural frequencies of the four models together with a
comparison between the minimum and target frequency ratios.
1.5
2
2.5
3
3.5
4
0 0.3 0.6 0.9 1.2 1.5
k2/k1
f 2/f 1
1.5
2
2.5
3
3.5
4
0 0.3 0.6 0.9 1.2 1.5
k2/k1
f 2/f 1
(a) Model 1 (b) Model 2
1.5
2
2.5
3
3.5
4
0 0.3 0.6 0.9 1.2 1.5
k2/k1
f 2/f 1
1.5
2
2.5
3
3.5
4
0 0.3 0.6 0.9 1.2 1.5
k2/k1
f 2/f 1
(c) Model 3 (d) Model 4
Figure 4.3: Relationships between 2 1/k k and 2 1/f f for the four models
70
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
Table 4.3: Stiffnesses and natural frequencies of a standing body
1k (kN) 2k (kN) 1Hf (Hz) 2Hf (Hz) 2
1
Hi
H
ff
δ =
Ratio
0iδ δ
( 0δ =2.29)
Experiment results 5.88 13.5 2.29 Model 1 160.6 35.33 5.88 16.81 2.86 125% Model 2 142.9 37.15 5.88 16.79 2.85 124% Model 3 134.9 24.01 5.88 14.89 2.53 110% Model 4 130.5 23.36 5.88 14.97 2.55 111%
4.3.2 Comparison of the models
The results given in Table 4.3 provide an improved understanding of the four
continuous body models. It can be observed that:
• The difference between Model 1 and Model 2 is the mass distribution. The
results of the two models indicate that the effect of mass distribution on the ratio
of the two natural frequencies is not significant (2.86 for Model 1 and 2.85 for
Model 2). This observation is confirmed by comparing the same information for
Models 3 and 4 (2.53 for Model 3 and 2.55 for Model 4)
• The differences between Model 1 and Model 4 are the position of the lower
torso to the lower or the upper part of the body and the stiffness distribution. As
the effect of mass distribution is insignificant, the change of the frequency ratio
from 2.86 to 2.53 is mainly due to the change of the distribution of the stiffness
and Model 3 appears better than Model 2.
• The frequency ratio for Model 3 is the smallest among the four models and is
closest to the ratio of the measured natural frequencies. Model 3 has the same
fundamental natural frequency as the measured one while its second natural 71
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
frequency is 10% larger than the measurement.
• As only two values of stiffness can be used in the model this limits the accuracy
of the model. Models 3 and 4 are better than Models 1 and 2 to represent a
standing body as a continuous model. Model 3 is slightly better than Model 4 in
terms of accuracy but Model 4 has much a simpler mass distribution than Model
3.
4.4 Vertical Dynamic Characteristics of a Standing Body
As Model 3 appears the best of the four, it is used to examine the vertical dynamic
characteristics of the standing body.
4.4.1 Mode shapes
An eigenvalue analysis of Model 3 was conducted. The shapes of the first two modes
are given in Figure 4.5 together with the corresponding natural frequencies. It can be
observed from Figure 4.5 that:
• The fundamental mode of vibration of the standing body is dominated by the
upper part (head neck, upper torso and centre torso) of the body. This mode shows
that all parts of the human body vibrate in the same direction and the head has the
maximum movement while the feet has the least.
• The second mode shows that the upper torso and the head move in the opposite
direction to the other segments of the body. The lower torso has the largest
movement while the bottom of the upper torso has little movement.
72
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
y = -0.0898x2 + 0.7018x
y = -5.0281x2 + 9.7206x - 3.6941
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1
y = -3.2436x3 + 0.7776x2 + 2.3633x
y = -184.63x4 + 657.49x3 - 853.76x2 + 474.77x - 94.646
The First Mode (5.88 Hz) The Second Mode (14.89 Hz)
Figure 4.4: The Mode shapes of Model 3
4.4.2 Modal properties
The modal properties of Model 3 can be calculated from the finite element analysis. The
total mass can be determined using the following formula:
01
n
H jj
M M=
=∑ (4.6)
Where iM is the i th element and n is the total number of elements of the model. The
participating factor and modal mass of the model are according to the definitions:
,1
n
Hi j i jj
M M f=
=∑ 2,
n
Hii j i ji j
M M f=
=∑ i = 1, 2 (4.7)
where jM is the mass of the j th element and ,i jf is the movement at the j th node in
the i th mode. The modal stiffness of the model can be determined using the following
formula:
2 21 1, 1 , , 1 ,
1 1
( )n ni j i j i j i j
Hi i ij j
K k x kx x
f f f f+ ++ +
= =
− − = ∆ = ∆ ∆ ∑ ∑ i = 1, 2 (4.8)
where x∆ is the length of the element. The modal mass and stiffness for the first two
modes of Model 3 are given in Table 4.4. The last column in the table shows the ratio of
the mass properties to the total mass. 73
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
Table 4.4: Modal mass and stiffness of a standing body
Model number
Model 3
Results Ratio to the total mass
Total mass 0HM (kg) 74.9
First Mode
Mode factor 1HM (kg) 40.00 53.4%
Modal mass 11HM (kg) 29.34 39.2%
Modal stiffness 1HK (N/m) 39844
Second Mode
Mode factor 2HM (kg) 22.10 29.5%
Modal mass 22HM (kg) 36.67 49.0%
Modal stiffness 2HK (N/m) 320064
4.5 Improved Continuous Model
Table 4.3 also shows that the frequency ratio becomes closer to the target ratio when the
lower part assigned as 1k becomes longer. This observation suggests that further
increase of the lower part may lead to meeting the target ratio. Thus Model 5 is
developed, which is almost the same as Model 3 except for the distribution of stiffness.
It can be seen in Figure 4.5 that stiffness 1k is assigned to the four lower parts and the
lower part of the centre torso in this model while to the four lower parts in Model 3. The
length of the lower part of the centre torso is identified in an iterative manner. The
centre torso was divided into 67 elements. The bottom element is assigned by stiffness
1k , an eigenvalue analysis is conducted and the frequency ratio is indentified using the
method in Section 4.2. A do-loop is used to identify how many elements need to be
74
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
assigned the stiffness 1k . As a result, when 30 parts were assigned the stiffness 1k ,
instead of 2k in Model 3, the frequency ratio matches the target frequency ratio. Thus
the length of all the parts assigned by stiffness 1k is 1.185m, leaving the upper part a
length of 0.505 m where 2k is assigned.
(a) Model 3 (b) Model 4 (c) Model 5 (d) Model 6
Figure 4.5: Continuous standing body models
Model 6 is similar to Model 4 in which two mass densities are needed. The difference
between the two models is that the lower part of the centre torso is assigned by 1k in
Model 6. Table 4.5 shows the stiffness and natural frequencies of Models 5 and 6,
together with those of Model 3 and 4 for comparison.
Table 4.5: Stiffnesses and natural frequencies of a standing body
1k (kN) 2k (kN) 1Hf (Hz) 2Hf (Hz) 2
1
Hi
H
ff
δ =
Ratio
0iδ δ
( 0δ =2.29)
Experiment results 5.88 13.47 2.29 Model 3 134.9 24.01 5.88 14.89 2.53 110% Model 4 130.5 23.36 5.88 14.97 2.55 111% Model 5 135.3 13.53 5.88 13.46 2.29 100% Model 6 126.4 14.7 5.88 13.56 2.31 101%
The modal mass and modal stiffness for the first two modes of Model 5 are given in 75
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
Table 4.6.
Table 4.6: Modal mass and stiffness of a standing body based on Model 5
Model number Model 5
Results Ratio to the total mass
Total mass 0HM (kg) 74.9
First Mode
Mode factor 1HM (kg) 35.66 47.58%
Modal mass 11HM (kg) 24.48 32.66%
Modal stiffness 1HK (N/m) 33222
Second Mode
Mode factor 2HM (kg) 25.26 33.70%
Modal mass 22HM (kg) 41.19 54.95%
Modal stiffness 2HK (N/m) 293251
4.6 Numerical verification
The human body models can be used to study human-structure interaction, including the
responses of both structure and human systems. However, it is necessary to assess
whether the human body models correctly represent a continuous body model in
structural vibration. For verification, four human-structure models are considered for
eigenvalue analysis so that the natural frequencies of the four models can be compared.
Model 3 is chosen for this step. Figure 4.6 shows the continuous body model 3 on a
SDOF structure system.
ANSYS is used for the eigenvalue analysis of the model. The first three natural
frequencies are determined from this model for comparison with those of the other 76
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
human-structure systems where the discrete human body models (Figure 4.7) are used.
Figure 4.6: A continuous body model on a SDOF structure system forming a human-structure
system
(a) 2DOF model (b) 2DOF interaction model (c) 3DOF interaction model
Figure 4.7: Human-structure models
The natural frequencies of the TDOF human-structure system (Figure 4.7a) can be
calculated by solving the eigenvalue equation when the damping terms are removed.
The two-mass SDOF body model and a SDOF structure system form the 2DOF
human-structure system that is shown in Figure 4.7b. The parameters for the body
models were determined based on the mode shapes determined from a FE eigenvalue
analysis of the continuous human body model 3. The numerical values for the discrete
body model are provided in Table 4.4 when the first mode of vibration is considered.
The three-mass two-SDOF body model and a SDOF structure system form the
77
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
human-structure system shown in Figure 4.7c. The mass and stiffness parameters for the
body model are given in Table 4.4. The natural frequencies of the systems can be
calculated by solving the eigenvalue equation when the damping terms are removed.
For the analysis nine cases are considered. The parameters of the SDOF structure can be
altered to achieve a particular ratio of the total body mass to the modal mass of the
SDOF structure η , and a particular ratio of the fundamental natural frequency of the
human-body to that of the SDOF structure β . The selected ratios are η = 0.01, 0.1 and
1.0 and β = 0.5, 1.0 and 2.0.
The natural frequencies of the continuous body model 3 and SDOF structure system
(Figure 4.6) and the three discrete human-structure systems (Figures 4.7) for the nine
cases are calculated and listed in Table 4.7. The comparison of the results in Table 4.7
shows that:
• The two proposed human-body models in structural vibration correspond
reasonably well with the continuous body model as the natural frequencies of the
models on a SDOF structure model are close. In particular, the natural frequencies
based on the three-mass, two-SDOF model are identical to those based on the
continuous model. This shows the correctness of the derivation, pattern and
definitions of the interaction body models presented in the last section.
• When the mass ratio is very small (η =0.01), the natural frequencies of all discrete
body models are close to those of the base model. The natural frequencies of the
human-structure system with the SDOF body model have larger differences than
those with the two-mass SDOF body model and the largest difference is
approximately 20% at η =1 and β = 0.5 for the fundamental frequency.
4.7 Conclusions
Continuous standing body models in vertical vibrations are developed and assessed
78
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
using two available natural frequencies of a biomechanics model and mass distributions
of anthropomorphic model. Six similar models are considered and analysed in this paper.
The most appropriate vertical stiffnesses of the models are identified based on the best
match with the measured natural frequencies. The study shows that the effect of
stiffness distribution is more significant than that of mass distribution throughout the
height of the body on the accuracy of the models. Models 3 and 5 are two possible
selections. Model 3 appears reasonable as the stiffness of centre torso is different from
that of the lower parts of the body. However, Model 5 becomes the best if only the
match to the two available natural frequencies of a standing human body is considered.
The continuous model is able to show the shapes of vibration modes throughout the
height of the standing body. The fundamental mode shows that the upper part of the body
has more significant movement than the lower part of the body while the second mode
indicates that the low torso has the largest movement but the upper torso moves
insignificantly.
The modal properties for the first two modes are also provided based on the continuous
model (Table 4.4). These parameters can be used for the human-structure interaction
model. However, the parameters of the interaction model can be determined more
directly through curve fitting to actual measurements, which would give a more
accurate representation.
Numerical verification of discrete human body models in structural vibration is
conducted by comparing the natural frequencies of four human-structure systems in
which both continuous and discrete human-body models are placed on the same SDOF
structure system. This shows that the derivation, pattern and definition of the discrete
human-body models in structural vibration are valid. However, this does not mean that
the values of the mass and stiffness of the models are accurate. This is because the
parameters of the continuous model of a standing body are approximate.
79
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
The suggested continuous models for a standing body in this study appears an
improvement of the previous continuous model (Ji, 1995) due to an improved body
mass distribution and curve-fitting based on two available measured natural frequencies
of a standing body (Matsumoto and Griffin, 2003). The limitations of the suggested
body models come from the assumptions relating to the distributions of body stiffness
and body mass used in this study. The body stiffness along its height is complicated and
the use of two uniform stiffnesses for the upper and lower bodies would lead to obvious
differences. The mass distribution shown in Figure 4.2 clearly presents the differences
between the model and reality. These modeling errors are partly compensated by the
curve fitting with the two true natural frequency measurements. The actual errors cannot
be quantified at present as there are no more accurate continuous models available.
The discrete body models developed from the study of human-structure interaction (Ji,
et ac, 2012) are qualitatively correct as the distributions of body stiffness and body mass
are described in general terms, k(x) and m(x). However, the parameters of the discrete
models presented in Table 4.6 are derived based on assumed stiffness distributions, i.e.
two constant stiffnesses in the upper and lower bodies (Figure 4.2) and the assumed
mass distributions (Figure 4.2). Thus the parameters are quantitatively approximate.
Further work is required to determine the parameters of the body models through
experimental methods, such as curve fitting of the apparent mass measured in
shaking-table tests. Thus how the stationary human body contribute damping to
structural vibrations can be better understood and used in the design of structures when
a crowd of people are involved.
80
Chapter 4: A Continuous Model of a Standing Human Body in Vertical Vibration
Table 4.7: Comparison of the first three natural frequencies of a human-structure system using different body models (Hz)
Body Model η = 0.01 η = 0.1 η = 1
0.5β = 1.0β = 2.0β = 0.5β = 1.0β = 2.0β = 0.5β = 1.0β = 2.0β =
Fundamental natural frequency Continuous body model 3 (Figure 4.6) 5.87 5.63 2.92 5.81 5.10 2.77 5.27 3.72 2.02
SDOF model (Figure 4.7a) 5.88 5.70 2.93 5.84 5.33 2.87 5.54 4.32 2.42 2SDOF body Model (Figure 4.7b) 5.87 5.63 2.92 5.81 5.11 2.78 5.27 3.73 2.03 3SDOF body Model (Figure 4.7c) 5.87 5.63 2.92 5.81 5.11 2.78 5.27 3.72 2.03
Second natural frequency Continuous body model 3 (Figure 4.6) 11.7 6.13 5.91 11.6 6.66 6.14 10.8 8.07 7.47
SDOF model (Figure 4.7a) 11.8 6.07 5.90 11.8 6.49 6.03 12.5 8.00 7.13 2DOF body Model (Figure 4.7b) 11.8 6.13 5.91 11.7 6.67 6.14 11.6 8.21 7.54 3SDOF body Model (Figure 4.7c) 11.7 6.13 5.91 11.6 6.67 6.14 10.8 8.07 7.47
Third natural frequency Continuous body model 3 (Figure 4.6) 14.9 14.9 14.9 15.2 15.1 15.0 17.2 16.4 16.2
3SDOF body Model (Figure 4.7c) 14.9 14.9 14.9 15.2 15.1 15.0 17.2 16.4 16.2
81
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
5 Parameter Identification of the Interaction Model
Using Available Measurements
5.1 Introduction
Chapter 4 provides parameters for a discrete interaction model of a standing human
body in vertical vibrations. The parameters are determined from the continuous model
based on available mass distribution information (Bartz, 1975; Nigam, 1987) and the
two available natural frequency measurements (Matsumoto and Griffin, 2003). In this
chapter, the parameters of the body model are determined using available published
experimental measurements. Section 5.2 provides the method and criteria for the
parameter identification. Section 5.3 identifies the parameters of the interaction model.
A comparison of human-structure models is given in Section 5.4. Section 5.5 gives
the concluding remarks and summarises the findings from this study.
5.2 Parameter Identification Method
5.2.1 Extraction of experimental data
In order to identify the parameters for the interaction model using available published
experimental measurements, two steps have been taken. The data from the apparent
mass curves (Figure 5.1) (Matsumoto and Griffin, 2003) are first extracted and
digitised. Then the extracted data are used to determine the parameters of the
interaction model by curve fitting.
To extract the coordinates of the published curves (Matsumoto and Griffin, 2003)
accurately, the curves are first converted from a PDF file to a JPG file (Figure 5.1).
Then a program, called “GetData.Graph.Digitizer.v2.23”, is used to obtain 123 pairs of
82
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
data for the mean normalised apparent mass and mean phase respectively in Figure 5.2.
It can be observed in Figure 5.1 and 5.2 that the abstraction process accurately
re-generates the original measurements.
a) Mean normalised apparent mass b) Mean phase
Figure 5.1: Comparison between experimental (——) and identified (------) results (Matsumoto
and Griffin, 2003)
a) Apparent mass against frequency b) Phase against frequency
Figure 5.2: Reproduced curves from original measurements (Matsumoto and Griffin, 2003)
using 123 pairs of data
5.2.2 Verification of the method
The parametric identification method used in (Matsumoto and Griffin, 2003) is tested
here to determine the basic parameters of Model 1a, 1b, 2c and 2d (Figure 5.3).
Optimised parameters are obtained by a non-linear parameter search method, based on
the Nelder–Mead simplex method, which is provided within MATLAB (MathWorks
Inc.). The Nelder–Mead method is a commonly used nonlinear optimization technique,
which is a well-defined numerical method for twice differentiable problems. The
method uses the concept of a simplex, which is a special polytope of N + 1 vertices in N
dimensions (McKinnon, 1999). The parameter identification method requires initial
values for each model parameter. The initial values of the natural frequencies were 83
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
selected as 3, 4, or 5 Hz for one of the mass-spring systems, and 10 or 15 Hz for the
second mass-spring system. The selection of different initial parameters in the
parameter search resulted in the same sets of optimum parameters for all the models. By
undertaking the test, an understanding of the method can be acquired before it is used
for identifying the parameters of the interaction models.
Figure5.3: Biomechanics models (Matsumoto and Griffin, 2003)
In matrix form the equation of motion of Model 1b in Figure 5.3(b) is:
0 1 1 1 1
1 1 1 1 1 1 1 1
( )0
S S S SM x C C x K K x P tM x C C x K K x
− − + + = − −
(5.1)
Alternatively the equation of motion of a two DOF model (Equation 5.1) can be
written as:
0 1 1 1 1( ) ( ) ( )S S S SM x C x x K x x P t− − − − = (5.2)
1 1 1 1 1 1( ) ( ) 0S SM x C x x K x x+ − + − = (5.3)
Adding Equation 5.3 to Equation 5.2, and dividing both sides of the equation by Sx
gives:
11 0
( )S
S S
P t xM Mx x
= +
(5.4)
84
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
Where ( )S SP t x is the apparent mass.
Equation 5.3 can be rewritten as follow:
1 1 1 1 1 1 1 1S SM x C x K x C x K x+ + = + 5.5)
The responses of the model subjected to a harmonic motion can be expressed as
follows:
1i tx Ae ω= i t
Sx Be ω= (5.6)
Substituting Equation 5.6 into Equation 5.5 gives:
21 1 1 1 1
i t i t i t i t i tM Ae iC Ae K Ae iC Be K Beω ω ω ω ωω ω ω− + + = + (45.7)
Rearranging Equation 5.7 gives:
1 1 12
1 1 1
i t
i tS
x iC KAex Be M iC K
ω
ω
ωω ω
+= =
− + +
(5.8)
Substituting Equation 5.8 into Equation 5.4, the apparent mass is a function of iω :
1 1 11 02
1 1 1
( )( )( )b
M iC KM i MM iC K
ωωω ω
+= +
− + + (5.9)
Following the same method, the apparent mass for the three other models are:
Model 1a: 1 1 11 2
1 1 1
( )( )( )a
M iC KM iM iC K
ωωω ω
+=
− + + (5.10)
Model 2c: 1 1 1 2 2 22 2 2
1 1 1 2 2 2
( ) ( )( )( ) ( )c
M iC K M iC KM iM iC K M iC K
ω ωωω ω ω ω
+ += +
− + + − + + (5.11)
Model 2d: 2 2 0( ) ( )d cM i M i Mω ω= + (5.12)
The derived Equations 5.9~5.12 are identical to those in (Matsumoto and Griffin,
2003). For curve fitting, the apparent masses in Equation 5.9~5.12 are normalized to
the body weight. Figure 5.4 provides a comparison between the measured apparent
mass and phase (Figure 5.2) in solid lines and the predicted values (Equations,5.9~
5.12) in dashed lines for the four models. Table 5.1 compares the identified parameters
of the four models between Matsumoto’s results (Matsumoto and Griffin, 2003) and
85
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
the current results. The identification process shows that 0M tends to be zero for
Models 1b and 2d, when no constraint conditions are given. Thus 0M is assumed to be
10% of the upper mass which is the same as Matsumoto and Griffin (2003) used. For
Model 1b 0 110%M M= and for Model 2d, 0 1 210%( )M M M= + . To indicate the
differences between the results from (Matsumoto and Griffin, 2003) and the current
analysis, the ratios of the two are given in Table 5.1.
a) Model 1a
b) Model 1b
c) Model 2c
d) Model 2d
Figure 5.4: Mean normalised apparent masses and mean phase
The apparent masses calculated using Eq. 5.9~5.12 for each of the four models were
compared with the measured apparent mass curve of standing subjects (Matsumoto and
Griffin, 2003). The model parameters, natural frequencies and damping ratios, were
optimised through minimising the following error function:
2
1
1 ( ) ( )n
m ci
err M i f M i fn =
= ∆ − ∆∑ (5.13)
where mM is the measured apparent mass and cM is the calculated apparent mass. ,
f∆ is the frequency increment and 0.1 Hz is taken in the curve fitting process. 86
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
Table 5.1: Comparison of optimised model parameters
1K
1 1( )Nm kg− − 2K
1 1( )Nm kg− − 1C
1 1( )Nsm kg− −
2C
1 1( )Nsm kg− − 0M 1M 2M
Model 1a
(Matsumoto and
Griffin, 2003) 1340 - 51.6 - - 1.03 -
Equation 5.10 1338 - 51.7 - - 1.03 - Ratio (%) 99.9 - 100.2 - - 100.0 -
Model 1b
(Matsumoto and
Griffin, 2003) 1300 - 43.1 0.0955 0.955
Equation 5.9 1341 - 51.7 - 9E-12 1.03 - Ratio (%) 103.2 - 119.9 - 0.00 107.9 -
Model 1b 0
110%M
M=
(Matsumoto and
Griffin, 2003) 1300 - 43.1 - 0.0955 0.955 -
Equation 5.10 1292 - 43.2 - 0.0957 0.9566 - Ratio (%) 99.4 - 100.2 - 100.2 100.2 -
Model 2c
(Matsumoto and
Griffin, 2003) 2370 849 24.8 16.5 - 0.345 0.633
Equation 5.11 2363 845 25 16.3 - 0.348 0.629 Ratio (%) 99.7 99.5 100.8 98.8 - 100.9 99.4
Model 2d
(Matsumoto and
Griffin, 2003) 1820 893 14.2 17.6 0.0909 0.254 0.655
Equation 5.12 2363 845 25 16.3 3E-12 0.348 0.629 Ratio (%) 129.8 94.6 176.1 92.6 0.00 137.0 96.0
Model 2d
0
1 210%(M
M M=
+
(Matsumoto and
Griffin, 2003) 1820 893 14.2 17.6 0.0909 0.254 0.655
Equation 5.12 1799 892 14.1 17.6 0.0908 0.254 0.654
Ratio (%) 98.9 99.9 99.3 100.0 99.9 100.0 99.9
The results in Table 5.1 demonstrate that the method presented is valid which will be
used for identifying the parameters of the interaction models.
87
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
5.3 Parameter identification for the interaction models
5.3.1 Model 1c
The parameters of the interaction model are identified in this section using the method
given in Section 5.2.
The differences between Model 1b and Model 1c (Figure 5.5) are:
• A mass device is present in Model 1c with a value of 1 11H HM M− , which is
defined in Section 2.2.
• The sum of the top and bottom masses equals the total body mass 0HM in
Model 1c, while the total mass in Model 1b is 0.0955+0.955=1.05 times the
body mass.
The apparent mass of Model 1c, the interaction model, can be given theoretically using
complex functions in a similar form to those for Model 1a, 1b, 2c and 2d.
Figure 5.5: Model 1c
The basic equation of motion of Model 1c:
0 11 1 1 11
1 11 11 1 1
1
2
( )
0
H H H H H S H H S
H H H H H
H H S S
H H
M M M M M x C C xM M M x C C x
K K x P tK K x
+ − − − + + − −
− + = −
(5.13)
88
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
Equation 5.13 can be written as:
0 11 1 1 11 1 1 1( 2 ) ( ) ( ) ( ) ( )H H H S H H H S H S SM M M x M M x C x x K x x P t+ − + − − − − − = (5.14)
1 11 11 1 1 1( ) ( ) ( ) 0H H S H H S H SM M x M x C x x K x x− + + − + − = (5.15)
Adding Equation 5.15 to Equation 5.14, and dividing both sides of the equation by Sx ,
gives:
11 0 1
( )SH H H
S S
P t xM M Mx x
= + −
(5.16)
Equation 5.15 can be rewritten as follows:
11 1 1 1 1 11( )H H H H H S H S H SM x C x K x M M x C x K x+ + = − − + + (5.17)
The absolute motion is:
1i tx Ae ω= i t
Sx Be ω= (5.18)
Substituting Equation 5.18 into Equation 5.17 gives:
2 211 1 11( )i t i t i t i t i t i t
H H H H H H HM Ae iC Ae K Ae M M Be iC Be K Beω ω ω ω ω ωω ω ω− + + = − + +
(5.19)
Equation 5.19 can be re-arranged as:
21 1 11
211
( )i tH H H H
i tS H H H
x M M iC KAex Be M iC K
ω
ω
ωω ω
− + += =
− + +
(5.20)
Substituting Equation 5.20 into Equation 5.16 gives:
21 1 11
1 0 1211
(( ) )( )( )
H H H H Hc H H
H H H
M M M iC KM i M MM iC K
ω ωωω ω
− + += + −
− + + (5.21)
211 1(2 )H HK M fπ= (5.22)
1 11 12 (2 )H HC M fξ π= (5.23)
Substituting Equation 5.22 and Equation 5.23 into Equation 5.21 gives:
2 21 1 11 1 11 1 11 1
1 0 12 211 1 11 1 11 1
(( )(2 ) 2 (2 )(2 ) (2 ) )( )( (2 ) 2 (2 )(2 ) (2 ) )
H H H H H H Hc H H
H H H H H
M M M f i M f f M fM i M MM f i M f f M f
π ξ π π πωπ ξ π π π
− + += + −
− + +
(5.24) 89
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
As 1 ( )cM iω is the normalised mass, 0HM =1. There are four unknown parameters
0 1 11 1 1and H H H H HM M M fξ, , , in Equation 5.24. Case 1 is considered following the
same treatment in (Matsumoto and Griffin, 2003).
Case 1: 0 1 110%H H HM M M− =
If the bottom mass 0 1H HM M− in Figure 5.5 is assumed to be 10% of the upper
mass 1HM , which follows the treatment in (Matsumoto and Griffin, 2003), the
parameters are as shown in Table 5.2.
Using these parameters, the first diagonal element in the mass matrix in Equation 5.13
becomes 0 11 12H H HM M M+ − 1 0.7933 2 0.9090= + − × = 0.0247 0− < . Physically the
diagonal element in the mass matrix should be positive. Hence this is not a valid case.
Table 5.2: Identified parameters of Model 1c
HK HC 1HM 11HM 1Hf 1Hξ
0 1 1=10%H H HM M M− 1052 41.5 0.9090 0.7933 5.80 0.7176
Due to the invalid results following the treatment in (Matsumoto and Griffin, 2003), a
new treatment is suggested here. Both the numerator and denominator of Equation 5.24
are divided by 11HM , Equation 5.24 becomes:
2 211 1 1 1
111 12 2
1 1 1
(( 1)(2 ) 2 (2 )(2 ) (2 ) )( ) 1
( (2 ) 2 (2 )(2 ) (2 ) )
HH H H H
Hc H
H H H
MM f i f f fMM i M
f i f f f
π ξ π π πω
π ξ π π π
− + += + −
− + + (5.25)
There are four unknown parameters 1 1 11 1/H H H HM M M ξ, , and 1 Hf in Equation
5.25. It is noted in the identification, that the results are dependent on the initial values
of 1HM and 11HM . If 1Hξ and 1 Hf are given before the identification, the number
of unknown parameters in Equation 5.25 reduces from four to two, but the identified
90
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
results are still dependent on the initial values of 1HM and 11HM . When the ratio
1 11/H HM M is given, the results become stable and do not change with the initial
values of 1HM and 11HM .
The apparent mass of model 1b (Equation 5.9) can be rewritten as follows:
21 1 1 1
1 02 21 1 1
( 2 (2 )(2 ) (2 ) )( )( (2 ) 2 (2 )(2 ) (2 ) )
H H Hb
H H H
M i f f fM i Mf i f f f
ξ π π πωπ ξ π π π
+= +
− + + (5.26)
Comparing Equation 5.25 and 5.26, it can be noted that an additional item,
21 11( / 1)(2 )H HM M fπ− , is present in the numerator in Equation 5.25. When the ratio
1 11/H HM M is given, the format of the two equations becomes the same. This may
explain the reason why the identified results are not stable unless the ratio 1 11/H HM M
is given.
Case 2: The ratio of 1 11/H HM M is given in the range of 1.1 and 1.8.
Based on the definition of 1HM and 11HM (Equation 3.7), 1HM should always be
larger than 11HM . Several trial ratios 1 11/H HM M between 1.1~1.8 are given for the
identification process. The ratio of 1 11/H HM M =1.36 is also included, which is based
on Model 3 (Table 4.4) in the Chapter 4. The identified parameters of the interaction
model (Figure 5.5) are provided in Table 5.3 for the given ratios of 1 11/H HM M .
91
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
Table 5.3: Identified parameters of Model 1c
1 11/H HM M HK HC 1HM 11HM 1Hf 1Hξ 21 11/H HM M
1.8 426 16.8 0.579 0.321 5.80 0.717 1.04
1.7 478 18.8 0.613 0.360 5.80 0.717 1.04
1.6 539 21.3 0.651 0.409 5.80 0.717 1.04
1.5 614 24.2 0.694 0.463 5.80 0.717 1.04
1.4 705 27.8 0.744 0.531 5.80 0.717 1.04
1.36 747 29.4 0.766 0.563 5.80 0.717 1.04
1.3 817 32.2 0.801 0.616 5.80 0.717 1.04
1.2 959 37.8 0.868 0.723 5.80 0.717 1.04
1.1 1141 45.0 0.947 0.861 5.80 0.717 1.04
Figure 5.6 compares the measured and identified normalised apparent mass and phase
against frequency of Model 1b and 1c, where the solid lines indicate the measurements
and the dashed lines the theoretical predictions based on the identified parameters
when 1 11/H HM M =1.36.
(a) Model 1b (b) Model 1c 1 11/H HM M =1.36
Figure 5.6: Comparison of the fitting of the Normalised apparent masses and phase between
Models 1b and 1c
The results in Table 5.3 and Figure 5.6 show that:
92
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
• The first natural frequency 1f and the damping ratio 1ξ of the interaction
model do not change with the ratio 1 11/H HM M .
• The parameter 21 11/H HM M is a constant (1.04), when 1 11/H HM M is changed
during the identification.
Table 5.4 summaries the identified results of Model 1b and Model 1c separately.
Table 5.4: Identified parameters of Model 1b and 1c
1Hf (Hz) 1Hξ error
Model 1b 5.87 0.612 1.2771 Model 1c 5.80 0.717 1.0164
The results in Table 5.4 show that:
• 1Hf of Model 1c is smaller than 1Hf of Model 1b while 1Hξ of Model 1c is
greater than 1Hξ of Model 1b.
• The dynamic properties of Models 1b obtained from equations 5.9 are the same
as that quoted from ref (Matsumoto and Griffin, 2003), which verifies that the
curving fitting procedure and equations used in this study are correct.
• The interaction models (Models 1c) provide smaller fitting errors than Models
1b (Matsumoto and Griffin, 2003), which may indicate that the interaction
models are more appropriate representations of a standing human body than
Model1b.
• As Model 1b and 1c are physically different (Figure 5.3 and 5.5), it is expected
that the basic parameters of the two models would have some differences
although they are determined from the same sets of measurements.
• The mass device in Model 1c decreases the natural frequency, and increases the
damping ratio of the human body model, because this is the only difference
between Model 1b and 1c.
93
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
Using the masses determined with 1 11/H HM M =1.36, the first diagonal mass in
Equation 5.13 is 0 11 12H H HM M M+ − =1.0+0.563-2 × 0.766=3.1%. This value
corresponds to the mass of the feet (1.198×2/74.9=3.2%) (Table 4.1) in Chapter 4.
5.3.2 Model 2e
Model 2e has been described in Section 2.3 and re-presented in Figure 5.7.
Figure 5.7: Model 2e
The differences between Model 2e and Model 2d are that:
• Two mass devices are presented in Model 2e with a magnitude of 1 11H HM M−
and 2 22H HM M− respectively.
• The sum of the top and bottom masses is 0HM =1.0, in Model 2e, while the
total mass in model mass in Model 2d is 0.0909+0.655+0.254=0.9999.
Equation 2.11 gives the basic equation of motion of Model 2e. Following the same
method presented in Section 5.3.1, the apparent mass for Model 2e is:
2 211 1 1 1
112 2 2
1 1 1
2 222 2 2 2
221 22 2
2 2 2
(( 1)(2 ) 2 (2 )(2 ) (2 ) )( )
( (2 ) 2 (2 )(2 ) (2 ) )
(( 1)(2 ) 2 (2 )(2 ) (2 ) )1
( (2 ) 2 (2 )(2 ) (2 ) )
HH H H H
He
H H H
HH H H H
HH H
H H H
MM f i f f fMM i
f i f f fMM f i f f fM M M
f i f f f
π ξ π π πω
π ξ π π π
π ξ π π π
π ξ π π π
− + += +
− + +
− + ++ + − −
− + +
(5.27)
Following the same the identification process as Model 1c, 3.2% of the total mass is
assigned to the bottom mass in the parameter identification, i.e.
94
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
0 1 2H H HM M M− − =3.2% 0HM for model 1c or 0 11 22 1 22 2H H H H HM M M M M+ + − −
=3.2% 0HM for Model 2e, it is found that the identified results are not in a reasonable
range.
The apparent mass of model 2d (Equation 5.12) can be rewritten as follows:
21 1 1 1
2 2 21 1 1
22 2 2 2
02 22 2 2
( 2 (2 )(2 ) (2 ) )( )( (2 ) 2 (2 )(2 ) (2 ) )
( 2 (2 )(2 ) (2 ) )( (2 ) 2 (2 )(2 ) (2 ) )
H H Hd
H H
H H H
H H H
M i f f fM if i f f f
M i f f f Mf i f f f
ξ π π πωπ ξ π π π
ξ π π ππ ξ π π π
+= +
− + +
++ +
− + +
(5.28)
Comparing Equation 5.27 and 5.28, it can be noted that two additional items,
21 11( / 1)(2 )H HM M fπ− , 2
2 22( / 1)(2 )H HM M fπ− , are present in the numerator in
Equation 5.27. When the ratios 1 11/H HM M and 2 22/H HM M are given, the format
of the two equations becomes the same.
Case 1: Let 1 11/H HM M =1.36, 2 22/H HM M =0.61 based on the continuous body
Model 3 (calculated from Table 4.4), the identified results are shown in Table 5.5.
Table 5.5: Identified parameters of Model 2e
1HM 11HM 1Hf 1Hξ 2HM 22HM 2Hf 2Hξ
Case 1 0.484 0.356 5.78 0.369 0.562 0.921 13.2 0.445
Case 2 0.533 0.431 5.78 0.369 0.296 0.256 13.2 0.445
Case 3 0.507 0.391 5.78 0.369 0.409 0.487 13.2 0.445
The bottom mass becomes 0 1 2H H HM M M− − = 1 − 0.484 − 0.562 = − 0.046<0.
Physically the value of the bottom mass should be positive, hence this is not a valid
case. But the same phenomena can be observed as for Model 1c. The parameters
21 11/H HM M and 2
2 22/H HM M are constants (0.66 and 0.34), when 1 11/H HM M and
2 22/H HM M are changed during the identification process.
95
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
Therefore, the two parameters 21 11/H HM M and 2
2 22/H HM M are identified. So the
parameterc can be calculated by giving either, 1HM and 2HM , or, 11HM and
22HM .
Case 2: If let 1HM =0.533, 2HM =0.296 based on the continuous body model 3
(calculated from Table 4.4), the identified results are shown in Table 5.5.
For Case 2, 2 22(0.296) (0.256)H HM M> which is different from the characteristics of
the continuous model, i.e. 2HM should be smaller than 22HM . Hence it is not a valid
case.
Case 3: If let 11HM =0.391, 22HM =0.487 based on the continuous body model 3
(calculated from Table 4.4), the identified results are shown in Table 5.5.
For case 3, Figure 5.8 compares the measured and identified normalised apparent mass
and phase against frequency of Model 2d and 2e, where the solid lines indicate the
measurements and the dashed lines the theoretical predictions.
(a) Model 2d (b) Model 2e
Figure 5.8: Comparison of the fitting of the normalised apparent masses and phase between
Model 2d and 2e
Table 5.7 compares the identified natural frequencies and damping ratios of Model 2d
and Model 2e separately.
96
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
Table 5.6: Identified parameters of Model 2d and 2e
1Hf (Hz) 1Hξ 2Hf (Hz) 2Hξ error
Model 2d 5.88 0.364 13.5 0.330 0.1573 Model 2e 5.78 0.369 13.2 0.445 0.0595
The results in Table 5.6 and Figure 5.8 show that:
• 1Hf and 2Hf of Model 2e are slightly smaller than those of Model 2d, while
2Hξ of Model 2e is greater than that of Model 2d.
• The dynamic properties of Models 2d obtained from equations 5.12 are the
same as that quoted from ref (Matsumoto and Griffin, 2003), which verifies
that the curving fitting procedure and equations used in this study are correct.
• The interaction models (Models 2e) provide smaller fitting errors than Model2d
(Matsumoto and Griffin, 2003), which may indicate that the interaction models
are more appropriate representations of a standing human body than Model 2d.
• As Model 2d and 2e are physically different (Figures 2.1d and 5.7), it is
expected that the basic parameters of the two models would have some
differences although they are determined from the same sets of measurements.
5.4 Comparison of the Human-Structure Models
It is necessary to assess whether parameters of the human body models, Model 1c and
Model 2e in Figure 5.5 and 5.7 are correct. For verification, the continuous body model
(Model 3 in Chapter 3), two biomechanics models (Model 1b and Model 2d) and the
two newly derived models (Model 1c and Model 2e) are placed on a SDOF structure
model to form human-structure models as shown in Figure 5.9. The eigenvalue
analysis of the human-structure models are calculated for comparison.
97
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
a) H-S model 1 b) H-S model 2 c) H-S model 3
d) H-S model 4 e) H-S model 5
Figure 5.9: Human-Structure models with different body models
The human-structure models in Figure 5.11 are explained as follows with each model
being placed on the same SDOF structural model:
H-S Model 1: The continuous human body (Model 3 in Chapter 4) has seven different
mass densities distributed over the height of the body (Table 4.1). The two axial
stiffnesses are assigned to the Model 3 as shown in Figure 5.9a. The stiffness
1=134.9kNk is assigned to the four lower parts while 2 =24.01kNk is assigned to the
three upper parts.
H-S Model 2: Model 1b using the parameters in Table 5.1.
H-S Model 3: Model 2d using the parameters in Table 5.1.
H-S Model 4: Model 1c using the parameter identified in Table 5.3.
H-S Model 5: Model 2e using the parameters in Table 5.5. 98
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
The four body models in the last four H-S models are all abstracted from the same
measured apparent mass. In the analysis, the parameters of the SDOF structure model
are altered to obtain particular values of the ratio of the total body mass to the modal
mass of the SDOF structure, 0= /H SM Mη , as are the values of the ratio of the
fundamental natural frequency of the human-body to that of the SDOF structure
1= /H Sf fβ . Choosing η = 0.01, 0.1 and 1.0 and β = 0.5, 1.0 and 2.0 gives nine
combinations.
The eigenvalue analysis of H-S Model 1 (Model 3 on a SDOF structure) is conducted
using ANSYS, while the natural frequencies of the other H-S Models are solved based
on the exact mathematical expressions.
5.4.1 Comparison between the same Human-Structure Models using different
parameters
As mentioned in Section 5.3.1 and 5.3.2, in order to know the effect of the different
parameters, the three natural frequencies of H-S Models 4 and 5 with different
parameters are listed in Table 5.8. The parameters of two cases in Tables 5.2 and 5.3,
and three cases in Table 5.5 are used to assess the effect on the natural frequency of
H-S Models 4 and 5 (Figure 5.9).
The comparison of the results in Table 5.7 shows that:
• When η is very small (η = 0.01), there is little difference between the natural
frequencies of H-S Models 4 and 5.
• When η is small (η = 0.1), the corresponding natural frequencies of H-S
Models 4 and 5 are similar.
• When η is large (η = 1.0), the difference between of H-S Model 4 and 5
become slightly larger when the order of the natural frequency increases. 99
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
• The three natural frequencies of H-S models 4 and 5 are unchanged with the
different parameters based on the identified results.
For H-S Model 4, the solution of Equation 2.12 without considering the damping terms
leads to the expressions of the natural frequencies:
2 2 2 2 2 2 2 22
1 2
(1 ) [ (1 ) ] 4(1 / )2(1 / )
S H S H S Hf f f f f ff
η η η γ αη γ α
+ + − + + − + −=
+ − (5.29a)
2 2 2 2 2 2 2 22
2 2
(1 ) [ (1 ) ] 4(1 / )2(1 / )
S H S H S Hf f f f f ff
η η η γ αη γ α
+ + + + + − + −=
+ − (5.29b)
where 0H
S
MM
η = ; 11H
S
MM
α = ; 1H
S
MM
γ = ; (5.30)
There are four parameters η , Sf , Hf , 2 /γ α in Equation 5.29. The values of η
and Sf are given for each case. The value of Hf and 2 /γ α are identified in Tables
5.2~5.4. Interestingly, Hf and 2 /γ α are the same value for both case 1 and case 2
of Model 1c. This would explain why the natural frequencies are unchanged with the
different parameters for H-S Model 4. The reason for H-S Model 5 should be the same,
although the expressions of the natural frequencies cannot be written directly.
5.4.2 Comparison between different Human-Structure Models
The natural frequencies of the H-S models 2, 3 , 4 and 5 and the first three natural
frequencies of the H-S model 1 with different mass ratios 0= /H SM Mη and frequency
ratios = /H Sf fβ are listed in Table 5.8.
100
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
Table 5.7: Comparison of the first three natural frequencies of H-S Model 4 and 5 with different parameters (Hz)
Human-Structure Models η = 0.01 η = 0.1 η = 1
3Sf Hz= 6Sf Hz= 12Sf Hz= 3Sf Hz= 6Sf Hz= 12Sf Hz= 3Sf Hz= 6Sf Hz= 12Sf Hz=
Fundamental natural frequency
H-S Model 4 Case 1 2.98 5.60 5.79 2.82 5.04 5.71 2.05 3.66 5.10 Case 2 2.98 5.60 5.79 2.82 5.04 5.71 2.05 3.66 5.10
H-S Model 5 Case 1 2.98 5.63 5.77 2.83 5.15 5.72 2.07 3.78 5.26 Case 2 2.98 5.63 5.77 2.83 5.15 5.72 2.07 3.78 5.26 Case 3 2.98 5.63 5.77 2.83 5.15 5.72 2.07 3.78 5.26
Second natural frequency
H-S Model 4 Case 1 5.84 6.22 12.0 6.18 6.92 12.2 8.69 9.72 13.9 Case 2 5.84 6.22 12.0 6.18 6.92 12.2 8.69 9.72 13.9
H-S Model 5 Case 1 5.81 6.14 11.9 6.01 6.59 11.4 7.09 7.62 9.94 Case 2 5.81 6.14 11.9 6.02 6.60 11.4 7.09 7.62 9.95 Case 3 5.81 6.14 11.9 6.01 6.59 11.4 7.09 7.62 9.94
Third natural frequency
H-S Model 5 Case 1 13.2 13.2 13.3 13.4 13.5 14.1 15.6 15.9 17.5 Case 2 13.2 13.2 13.3 13.4 13.5 14.1 15.6 15.9 17.5 Case 3 13.2 13.2 13.3 13.4 13.5 14.1 15.6 15.9 17.5
101
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
Table 5.8: Comparison of the first three natural frequencies of a human-structure model using different body models (Hz)
Human-Structure model η = 0.01 η = 0.1 η = 1
3Sf Hz= 6Sf Hz= 12Sf Hz= 3Sf Hz= 6Sf Hz= 12Sf Hz= 3Sf Hz= 6Sf Hz= 12Sf Hz=
Fundamental natural frequency Continuous Model H-S 1: Model 3 2.98 5.68 5.87 2.83 5.17 5.81 2.07 3.77 5.29
Biomechanics Models (Matsumoto and Griffin, 2003)
H-S 2: Model 1b 2.98 5.65 5.86 2.82 5.09 5.78 2.03 3.67 5.17
H-S 3: Model 2d 2.98 5.70 5.87 2.83 5.19 5.82 2.07 3.80 5.33
Interaction Model H-S 4: Model 1c 2.98 5.60 5.79 2.82 5.04 5.71 2.05 3.66 5.10
H-S 5: Model 2e 2.98 5.63 5.77 2.83 5.15 5.72 2.07 3.78 5.26 Second natural frequency
Continuous Model H-S 1: Model 3 5.91 6.20 12.0 6.14 6.72 11.8 7.48 8.11 11.0
Biomechanics Models (Matsumoto and Griffin, 2003)
H-S 2: Model 1b 5.91 6.23 12.0 6.22 6.89 12.1 8.28 9.16 13.0
H-S 3: Model 2d 5.91 6.18 11.9 6.12 6.66 11.5 7.23 7.77 10.3
Interaction Model H-S 4: Model 1c 5.84 6.22 12.0 6.18 6.92 12.2 8.69 9.72 13.9 H-S 5: Model 2e 5.81 6.14 11.9 6.01 6.59 11.4 7.09 7.62 9.94
Third natural frequency
Continuous Model H-S 1: Model 3 14.9 14.9 14.9 15.0 15.0 15.2 16.2 16.4 17.3
Biomechanics Models (Matsumoto and Griffin, 2003)
H-S 3: Model 2d 13.5 13.5 13.6 13.7 13.7 14.1 15.2 15.4 16.7
Interaction Model H-S 5: Model 2e 13.2 13.2 13.3 13.4 13.5 14.1 15.6 15.9 17.5
102
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
The comparison of the results in Table 5.8 shows that:
• When η is very small (η = 0.01), there is little difference between the natural
frequencies of the last four H-S models.
• When η is small (η = 0.1), the corresponding natural frequencies from the last
four models are similar.
• When η is large (η = 1.0), the difference between Matsumoto’s models and
the interaction models become slightly larger when the order of the natural
frequency increases.
• For the two DOF models (H-S Models 2 and 4), the following condition hold:
1 2( , )S Hf f f f≤ ≤ .
• There are no obvious difference between the biomechanics and interaction
models when 0.1η ≤ .
• The interaction models show better agreements with the continuous model than
the biomechanics models. Especially, H-S Models 5 shows the best agreement
with the H-S Model 1.
5.5 Conclusions
This chapter determines the parameters of the proposed interaction body models
(Figures 5.5 and 5.7) using the available measurements (Matsumoto and Griffin, 2003).
The expressions of the apparent mass of the two models are derived for curve fitting to
the measurements. Then the abstracted parameters are taken to assess the effects of
different parameters on the natural frequencies of (H-S Models 4 and 5 and on the
natural frequencies of different H-S Models (Figure 5.9). The conclusions drawn from
this study are:
• Similar to the parametric identification in (Matsumoto and Griffin, 2003), the
parameters identified are not unique as one additional condition has to be given
for Models 1b and 2d. For the interaction models (Figures 5.5 and 5.7), one 103
Chapter 5: Parameter Identification of the Interaction Model Using Available Measurements
additional condition is required for Model 1c and two conditions for Model 2e.
This could lead to several sets of parameters, but with the results from the
continuous model (Table 4.4 in Chapter 4), reasonable parameters of the two
interaction models are identified.
• The quality of the curve fitting for the interaction model is as good as (Model 1c)
and is slightly better (Model 2e) than the published results (Models 1b and 2d).
• Based on Model 2e, 1f is identified as 5.78Hz, and 1ξ of the interaction
model is 0.369. 2f is identified as 13.2Hz, and 2ξ of the interaction model is
0.445. These parameters can be used in further calculations.
• There are no obvious differences between the biomechanics and interaction
models when 0.1η ≤ .
104
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
6 Frequency Characteristics of Human-Structure
Models in Forced Vibration
6.1 Introduction
This chapter examines the frequency characteristics of the human-structure interaction
model when a force is applied to the structural model and the frequency response
functions are examined. The basic equations are shown in Section 6.2. This simulates
actual dynamic experiments, allows an investigation of the effects of the key parameters
of the system and identifies the frequency characteristics of the human-structure model.
The key parameters of the model include the mass ratio, the frequency ratio and the
human body damping ratio. A Matlab programme is used to present graphically the
effects of the body damping ratio, mass ratio and frequency ratio on the normalized
acceleration FRFs in Section 6.3. Section 6.4 studies the conditions when the body
response is larger than the structural response and shows that the body response is
always larger than the structural response when the structure is subjected to normal
human loading due to walking and jumping. Human-structure interaction tests are
conducted in Section 6.5 and the interaction human body model is validated. The
conclusions are summarised in Section 6.6.
6.2 Basic Equations and Models
This chapter conducts theoretical and experimental studies to assess the frequency
characteristics of the SDOF biomechanics and interaction models on a SDOF structure
model. Effects of the body damping ratio, mass ratio and frequency ratio on the
normalized acceleration FRFs are to be discussed. There are two reasons to select the
two SDOF body models: 1) The vibration measurements from conventional dynamic
tests cannot reveal sufficient evidence to abstract the dynamic parameters of a human 105
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
body with two modes of vibration considered. 2) The SDOF body models capture most
physics of whole-body vibration and are easier to be investigated as a start. The SDOF
human interaction model of a standing body is now considered in a vibrating
environment, i.e. the body model is placed on a SDOF structural system as shown in
Figure 6.1. A harmonic force ( )SP t is applied to the structural model.
Figure 6.1: An interaction model on a SDOF structural system
The equation of motion of the TDOF human-structure system is (Figure 6.1):
0 11 1 1 11 1 1
1 11 11 1 1 1 1
1 1
1 1 1
2
( )
0
S H H H H H S S H H S
H H H H H H H
S H H S S
H H H
M M M M M M u C C C uM M M u C C u
K K K u P tK K u
+ + − − + − + + − −
+ − + = −
(6.1)
For a frequency response analysis, the form of the Fourier transformation of the
Equation 6.1 is:
0 11 1 1 11 1 12
1 11 11 1 1
1 1
1 1 1
2(
( ) ( ) )
( ) 0
S H H H H H S H H
H H H H H
S H H S S
H H H
M M M M M M C C Ci
M M M C C
K K K u PK K u
ω ω
ω ωω
+ + − − + − − + − −
+ − + = −
(6.2)
or
[ ] 11 12
1 21 22
( ) ( ) ( )( ) ( )( )
( ) ( ) ( )0 0S S S
H
u H HP PH
u H Hω ω ωω ω
ωω ω ω
= =
(6.3)
Where ( )H ω is the frequency response function (FRF) and is expressed as
106
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
2 211 1 1 1 11 1 1
2 21 11 1 1 0 11 1 1 1
1( )( )
( )( ) ( 2 ) ( ) ( )
H H H H H H H
H H H H S H H H H S H S
HZ
M i C K M M i C KM M i C K M M M M i C C K K
ωω
ω ω ω ωω ω ω ω
= ×
− + + − + +
− + + − + + − + + + +
(6.4) 2 2
11 1 1 0 11 1 1
2 21 1 11 1 1
( ) [ ][ ( 2 ) ( )
( )] ( ( ) )H H H S H H H H S
H S H H H H
Z M i C K M M M M i C C
K K M M i C K
ω ω ω ω ω
ω ω
= − + + − + + − + +
+ + − − + +(6.5)
Expanding Equation 6.3, the FRF for acceleration of the structural model and human
model can be expressed as follows:
211 11 11 11
( ) ( ) ( )( ) ( ) Re[ ( )] Im[ ( )]( ) ( ) ( )
S S
S
u R iEH A A AP X iY
ω ω ωω ω ω ω ωω ω ω
+= − = − = = +
+
(6.6a)
2121 21 21 21
( ) ( ) ( )( ) ( ) Re[ ( )] Im[ ( )]( ) ( ) ( )
H H
S
u R iEH A A AP X iY
ω ω ωω ω ω ω ωω ω ω
+= − = − = = +
+
(6.6b)
where
( )2 2 211( )S H HR Mω ω ω ω= − (6.7a)
2 2 2111
11
( ) [ ( 1) ]HH H H
H
MR MM
ω ω ω ω= + − (6.7b)
311( ) 2 H H HE Mω ξ ω ω= (6.7c)
( )2
4 2 2 20111 0
11
(1 ) 1 4 1HHS H H H S H S H H
H S
MMX M M MM M
ω ω ω ξ ξ ω ω ω ω = + − − + + + +
(6.7c)
( ) ( ) 20112 1 H
S H H H S H S H HS
MY M MM
ω ω ω ξ ξ ξ ξ ω ω = + − + +
(6.7d)
where Sω and Hω are the natural frequencies of the SDOF structure and SDOF body
models respectively and ω is the unknown frequency of the TDOF system. The
following parameters are defined:
0H
S
MM
η = ; 11H
S
MM
α = ; 1H
S
MM
γ = ; (6.8a)
S
ωλω
= ; H
S
ωβω
= ; 2 SS
S
KM
ω = ; 21
11
HH
H
KM
ω = (6.8b)
107
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
where η is the ratio of the total body mass to the structural mass; α is the ratio of
the modal body mass to the structural mass; γ is the ratio of the mass factor to the
structural mass; β is the frequency ratio and λ is the normalised frequency variable.
Substituting Equation 6.8 into Equation 6.7 leads to:
4 2 2 2 4( ) ( ) ( )S S S S S SR M M Rλ α ω λ β λ α ω λ= − = (6.9a)
4 2 2 2 4( ) [ ( 1) ] ( )H S S S S HR M M Rγλ α ω λ β λ α ω λα
= + − = (6.9b)
4 3 4( ) 2 ( )S S H S SE M M Eλ α ω βλ ξ α ω λ= = (6.9c)
22 4 4 2 2 2 2 4( ) { [1 ] [(1 ) 4 1] } ( )S S S H S SX M M Xγλ α ω λ η β η β ξ ξ β λ α ω λ
α= + − + − + + + = (6.9d)
2 4 2 2 4( ) 2 { ( ) [ (1 ) ] } ( )S S S H S H S SY M M Yλ λα ω β βξ ξ ξ η βξ λ α ω λ= + − + + = (6.9e)
Where,
2 2 2( ) ( )SR λ λ β λ= − (6.10a)
2 2 2( ) [ ( 1) ]HR γλ β λ λα
= + − (6.10b)
3( ) 2 HE λ βλ ξ= (6.10c)
24 2 2 2( ) [1 ] [(1 ) 4 1]S HX γλ λ η β η β ξ ξ β λ
α= + − + − + + + (6.10d)
2( ) 2 { ( ) [ (1 ) ] }S H S HY λ λ β βξ ξ ξ η βξ λ= + − + + (6.10e)
Therefore, the FRFs for acceleration of the structure and the body models in Equation
6.6a and 6.6b can be express as:
( ) ( ) ( )( ) ( )
2 2
11 2 21 S
S
R EA
M X Y
λ λλ
λ λ
+=
+ (6.11a)
( ) ( ) ( )( ) ( )
2 2
21 2 21 H
S
R EA
M X Y
λ λλ
λ λ
+=
+ (6.11b)
SM is a constant that affects the magnitude of the function rather than the shape of the
108
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
function. By multiplying both sides of the Equation 6.11 by SM , the dimensionless
acceleration of the structure and the human body models become:
( ) ( )( ) ( )
2 2
2 2S
S
R Ea
X Yλ λ
λ λ
+=
+ (6.12a)
( ) ( )( ) ( )
2 2
2 2H
H
R Ea
X Yλ λ
λ λ
+=
+ (6.12b)
There are six parameters, η , γ , α , β , Hξ , and Sξ in Equation 6.12. The
relationships between η , γ and α have been identified in Chapter 4 (Table 4.4). If
α is assumed, η and γ can be calculated. A nominal damping ratio of the base
system, Sξ , is taken as 0.01 as this value is within a reasonable range and a
variation of this value does not have significant effect on the resonance frequencies of
the system. Therefore the remaining three parameters, α , β and Hξ need to be
considered in this evaluation.
6.3 Parametric Study
According to the study of the biomechanics of human whole body vibration, the
damping ratio of a standing person may range between 30% and 50% (Matsumoto, Y. et
al.,1997). Three Hξ values of 0.3, 0.4 and 0.5 were selected in this study. The mass
ratio of a crowd to a structure is considered to vary between 0.01 and 3.0 because this
covers most possible practical cases. For example, when a crowd of people stands on a
temporary grandstand, the mass of human bodies may be as large as 300% of the mass
of the temporary grandstand. If a few people stand on a concrete floor, the body mass
may be as small as 1% of the mass of the floor. Therefore six mass ratios are considered
to cover this range, i.e. 0.01,0.03,0.1,0.3,1.0,3.0α = . The frequency ratio of the crowd
to the structure is taken to vary between 0.25 and 2.0 with increments of 0.25 being 109
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
evaluated. If the natural frequency of a human body is considered to be 6 Hz, the
structures in the evaluation will have a frequency 3 Hz or over, which covers almost all
engineering structures for human use which are considered to be safe. Eight frequency
ratios 0.25,0.5,0.75,1.0,1.25,1.5,1.75,2.0β = were used in the analysis. This gives a
total of 144 cases.
The effects of the mass ratio, the frequency ratio and the body damping ratio of the
human-structure system on the resonance frequencies and peak accelerations were
investigated.
6.3.1 Effect of the mass ratio
Figure 6.2~6.4 shows six FRFs of the structure mass and human models for the six
mass ratios with Hξ =0.3 and β =0.5, 1.0 and 2.0 respectively.
a) the structural model b) the body model
Figure 6.2: Acceleration FRFs (Equation 6.14) of the structure and human models
( 0.3Hξ = , 0.5β = )
110
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
a) the structure model b) the body model
Figure 6.3: Acceleration FRFs (Equation 6.14) of the structure and human models
( 0.3Hξ = , 1.0β = )
a) the structure model b) the body model
Figure 6.4: Acceleration FRFs (Equation 6.14) of the structure and human models
( 0.3Hξ = , 2.0β = )
The acceleration FRFs of the structure model show that:
• The two resonance frequencies cannot always be observed simultaneously.
• The first resonance frequency decreases with increases of the mass ratio α .
The acceleration FRFs of the human-body model show that:
• The two resonance frequencies cannot always be observed simultaneously.
• The resonance frequencies decreases with the increase of the mass ratio α .
• The value of the highest peak decreases with increases of the mass ratio α .
111
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
6.3.2 Effect of the frequency ratio
Figure 6.5~6.7 shows six FRFs for the six frequency ratios with Hξ =0.3 and α =0.3,
1.0 and 3.0.
a) the structure model b) the body model
Figure 6.5: Acceleration FRFs (Equation 6.14) of the structure and human models
( 0.3Hξ = ,α =0.3)
a) the structure model b) the body model
Figure 6.6: Acceleration FRFs (Equation 6.14) of the structure and human models
( 0.3Hξ = ,α =1)
112
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
a) the structure model b) the body model
Figure 6.7: Acceleration FRFs (Equation 6.14) of the structure and human models
( 0.3Hξ = ,α = 3)
It can be seen from Figures 6.5~6.7 that for both structure and human models:
• The peak value increases as the frequency ratio β increases.
• The two resonance frequencies cannot always be observed simultaneously.
6.3.3 Effect of the body damping ratio
Figure 6.8 shows three FRFs for the six damping ratios with 0.3α = and β =1.0.
a) the structure model b) the body model
Figure 6.8: Acceleration FRFs (Equation 6.14) of a human-structure Model
( 1.0α = , 1β = )
Acceleration FRFs for the three damping ratios are plotted in Figure 6.8. The
acceleration FRFs of the structure and body models show that the value of the highest 113
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
peak of the acceleration FRF amplitude decreases with increases of the damping ratio
of the human body Hξ .
6.4 Critical Positions
It is useful to know the conditions when the body model vibrates more than the
structural model and vice versa. The critical position being when the two models
vibrate with the same amplitude.
The ratio of the acceleration amplitudes of the body model to the structure model
(Equation 6.10) is
2 2
2 2 2 2 3 22 2 2 2
2 2 2 2 2 2 3 22 2
2 2
( ) ( )( ( ( 1) )) (2 )( ) ( ) ( ) ( )
( ) ( ) ( ( )) (2 )( ) ( )( ) ( )
HH
H H
S S HS
R EX Ya R E
a R ER EX Y
λ λ γλ β λ βλ ξλ λ λ λ αµλ λ λ β λ βλ ξλ λ
λ λ
++ − ++ +
= = = =+ − ++
+
(6.13)
When 1µ > , the body model vibrates more significantly than the structure model,
When 1µ = , the body model and the structure model have the same vibration
amplitude, meaning that human moves as a rigid body with the structure. When 1µ < ,
the body model vibrates less than the structure model.
In current practice, only the structural vibration is assessed and this index is used to
assess indirectly the human comfort without considering whether the human body
vibrates more or less than the structural vibration.
By letting 1µ = , Equation 6.13 becomes
2 2 2[( 1) 1] 2 ( ) 0γ γλ βα α− − + = (6.14)
Rearrange Equation 6.14 gives:
114
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
2
2( )
1 ( 1)
γαλ βγα
=− −
(6.15)
Using body model 3 in Chapter 3, i.e. 1
11
1.36H
H
MM
γα= = . The relationship between λ
and β can be obtained from Equation 6.15 and is expressed as:
( ) 1.768λ β β= (6.16)
Equation 6.16 indicates that λ is related to β , and independent of α and Hξ .
Consider two cases:
(a) 0.01Sξ = , 0.3Hξ = , 0.03α = , 0.75β =
(b) 0.01Sξ = , 0.5Hξ = , 0.3α = , 0.75β =
Where α and Hξ are significantly different, but β is the same. λ is calculated
using Equation 6.16 and is 1.33. The acceleration FRF (Equation 6.14) of the two cases
for both the body model and structure model are plotted in Figure 6.10. It can be
observed from Figure 6.10 that
• When λ <1.33, the body model vibrates more significantly than the structure
model.
• When λ >1.33, the structure model vibrates more significantly than the body
model.
115
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
(a) Case 1 (b) Case 2
Figure 6.10: Acceleration FRFs (Equation 6.11) of a human-structure system
Although the statements are correct, they are not straightforward for applications as
Sλ ω ω= has to be determined. For practical application, Equation 6.13 is further
investigated. The parameters in Equation 6.13 are
11H
S
MM
α = ; 1H
S
MM
γ = ; S
ωλω
= ; H
S
ωβω
= ; 2 fω π= ; 2H Hfω π= ; 2S Sfω π= ; (6.17)
Substituting Equation 6.17 into Equation 6.13 leads to:
2 8 2 2 2 6 4 4
8 2 2 6 4 4
( 1) (2 ( 1) 4 )
(4 2)
H H H H
H H H
f f f f f f
f f f f f
γ γ ξα αµ
ξ
− + − + +=
+ − + (6.18)
Where f is the load frequency.
Consider a human model to have parameters of Hf = 6.0Hz and γ α =1.36 as an
example. Equation 6.18 is plotted in Figure 6.11 with two damping ratios.
Figure 6.11: the curve of Equation 6.18
A horizontal line 1µ = is also given in Figure 6.11 which divides the figure into two 116
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
areas, one above the line, showing larger vibration of the body model, and one below
the line, showing large vibration of the structure model. The intersection point of the
Equation 6.18 and 1µ = is f =10.61Hz, for both damping ratios. Based on the
assuming Hf = 6.0Hz, the conclusions drawn from Figure 6.11 are that:
• When the frequency of a harmonic is less than 10.61Hz, the body model will
vibrate more significantly than the structural model. In practical situations, the
frequency of the human load is smaller or far smaller than 10.61. Thus, the
human body response would be larger than the structural response of a
human-structure system.
• When Hξ =0.3 and f =5.77Hz, the maximum acceleration of the body model
can be 2.50 times of that of the structural model; when Hξ =0.5 and f =5.47Hz,
the ratio is 1.71.
In order to study the variation of human body, difference γ α and Hf are used to
calculate the frequency of critical position. 20% variation of γ α , and Hf are used to
examine the frequency of critical position. Table 6.1 provides the results. The cell in
second row and second column are based on γ α and Hf without variations; others
cells give the calculated f based on γ α and Hf with 20% variations and the
relative errors are also given in cells.
Table 6.1: The frequency of critical position
γ α
Hf (Hz)
4.8 (80%) 6.0(100%) 7.2 (120%)
1.09(80%) 7.12 (67%) 8.90 (84%) 10.67 (101%) 1.36(100%) 8.49 (80%) 10.61 (100%) 12.73 (120%) 1.63(120%) 11.16 (105%) 13.95 (131%) 16.74 (158%)
It can be observed from Table 6.1 that:
• 20% variation of γ α and Hf could lead to 58% error of f .
117
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
• The resulting error on f is linear with variation of Hf .
6.5 Experimental verification
The experimental study in this section aims to
• Verify the correctness of the human body models
• Verify the condition of the presence of the two resonance frequencies
• Examine the effect of the human damping ratio
6.5.1 Experiment cases
The details of the experimental set- up are given in Chapter 3. The test rig is set at two
natural frequencies, 7.05 Hz and 5.66 Hz, forming Case 1.0 and Case 2.0. Four people
took part in these tests. The mass ratio of the human body to the test rig is adjusted by
using different numbers of participants. The mass and height of the participants are
shown in Table 6.1. Experiment cases are summarised as follows. The first digit in
experimental case indicates the setting of the test rig and the second shows the number
of participants involved in Table 6.2.
Table 6.1: Mass and height of the participants
S1(F) S2(M) S3(M) S4(M)
Weight(kg) 60.2 86.6 75.8 80.8
Height(cm) 162 178 180 181
Age 32 29 26 29
118
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
Table 6.2: Experiment cases
Bare rig Rig+S1 Rig+S1+
S2
Rig+S1+
S2+S3
Rig+S1+
S2+S3+S4
Rig 1 Case 1.0 Case 1.1 Case 1.2 Case 1.3 Case 1.4
Rig 2 Case 2.0 Case 2.1 Case 2.2 Case 2.3 Case 2.4
0H SM Mη = _ 0.33 0.82 1.24 1.69
11H SM Mα = _ 0.13 0.32 0.48 0.66
6.5.2 FRFs of two H-S Models
In order to compare the experimental and theoretical results, the equation for
acceleration FRFs on the structure model is:
( ) ( )( ) ( )
2 2
2 2S
S
R Ea
X Yλ λ
λ λ
+=
+ (6.19)
For Conventional model (Figure 6.12a), the parameters in Eq. 6.19 are
( )2 2 2( )SR λ λ β λ= − (6.20.a)
3( ) 2 HE λ βξ λ= (6.20.b)
( )4 2 2 2( ) 1 4 1S HX λ λ α β βξ ξ λ β = − + + + + (6.20.c)
( ) ( ){ }2( ) 2 1S H S HY λ λ β βξ ξ ξ α βξ λ = + − + + (6.20.d)
For Interaction model (Figure 5.12b), the parameters in Eq. 6.19 are
2 2 2( ) ( )SR λ λ β λ= − (6.21.a)
3( ) 2 HE λ βλ ξ= (6.21.b)
24 2 2 2( ) [1 ] [(1 ) 4 1]S HX γλ η λ η β ξ ξ β λ β
α= + − − + + + + (6.21.c)
2( ) 2 { ( ) [ (1 ) ] }S H S HY λ λ β βξ ξ ξ η βξ λ= + − + + (6.21.d)
Where
119
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
11H
S
MM
α = , 1H
S
MM
γ = , 0H
S
MM
η = (6.22)
a) Conventional model b) Interaction model
Figure 6.12: Human-Structure systems with different body models
To compare the predictions (Equation 6.20 and 6.21) with the measurements, the
parameters in the two equations should be given. In addition, the measurements may
help to define more appropriate values for the parameters. The parameters are:
• 2γ α , two values are considered as follows
2γ α =0.73, based on continuous model 3, calculated based on the data in
Table 4.4.
2γ α =1.04, based on Model 1c, calculated based on the data in Table 5.3.
• Hξ , the damping ratio of a standing body was reported between 30% and 50%
(Matsumoto, et al., 1997), therefore three values, 0.3, 0.4 and 0.5 are considered
in the analysis.
• Hf =5.88Hz (Matsumoto and Griffin, 2003). Although there is significant
person-to-person variability, for the sake of simplicity, only a single, mid-range
natural frequency of the whole body is selected for this study, which effectively
reduces the amount of calculations.
120
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
• β : for Case 1 5.88 / 7.05 0.83β = = and for Case 2 5.88 / 5.66 1.04β = =
• η are given in Table 5.3
Comparing Eq.6.20 and Eq. 6.21, it is noted that ( )X λ and ( )Y λ in Equation 6.27
and 6.28 are different.
6.5.3 The effect of 2γ α
In order to study the effect of 2γ α , Hξ =0.3 is chosen. Cases 1.2 and 2.3 are used to
show the effect. FRFs with 2γ α =0.73, 1.04 are shown in Figures 6.13 and 6.14.
a) Experiment
b) Eq. 6.19 ( 2γ α =0.73) c) Eq. 6.19( 2γ α =1.04)
Figure 6.13: Comparison between experimental and theoretical FRFs with two values of
2γ α (Case 1.2)
121
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
a) Experiment
b) Eq. 6.19 ( 2γ α =0.73) c) Eq. 6.19( 2γ α =1.04)
Figure 6.14: Comparison between experimental and theoretical FRFs with two values of 2γ α
(Case 2.3)
As shown in Figure 6.13 and 6.14, the second resonance frequency increases, when the
parameter 2γ α increases. Figure 6.13b and 6.14b ( 2γ α =0.73) shows a better
agreement with the experimental result (Figure 6.13a and 6.14a). Thus 2γ α =0.73 will
be used in the following study.
122
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
6.5.4 The effect of Hξ
Take cases 1.2 and 2.3 as examples, FRFs with Hξ =0.3, 0.4 and 0.5 for 2γ α =0.73
together with measurements are shown in Figure 6.15 and 6.16.
a) Experiment b) Eq. 6.19( Hξ =0.3)
c) Eq. 6.19( Hξ =0.4) d) Eq. 6.19 ( Hξ = 0.5)
Figure 6.15: Comparison between experimental and theoretical FRFs with three damping ratios
(Case 1.2)
123
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
a) Experiment b) Eq. 6.19 ( Hξ =0.3)
c) Eq. 6.19( Hξ =0.4) d) Eq. 6.19 ( Hξ = 0.5)
Figure 6.16: Comparison between experimental and theoretical FRFs with three damping ratios
(Case 2.3)
As shown in Figure 6.15 and 6.16, the second resonance frequency increases with
increases in the damping ratio of the human body. Figure 6.15b and 6.16b ( Hξ =0.3)
shows a better agreement with the experiment result.
6.5.5 Validation of the H-S Models
Based on the comparison in the last two sections, Hξ =0.3 and 2γ α =0.73 are the most
appropriate values. Considering the effect of different human body models, the FRFs for
a conventional model (Figure 6.12a) and the interaction model (Figure 6.12b) are shown
in Figure 6.17 and 6.18.
124
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
a) Experiment
b) Interaction model c) Conventional model
Figure 6.17: FRFs for the test rig and two human body models (Case 1.2)
a) Experiment
b) Interaction model c) Conventional model
Figure 6.18: FRFs for the test rig and two human body models (Case 2.3) 125
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
As shown in Figure 6.17 and 6.18, the patterns of the FRFs of the human-structure
systems using the conventional model are significantly different from those using the
interaction model. The predicted FRFs of the H-S system using the interaction model
are closer to the measurements.
6.5.6 Comparison between experimental and theoretical results
By comparing the experimental and theoretical results, 2γ α = 0.73 and Hξ =0.3 are
the best parameters and are chosen to draw the FRFs of the test rig in Figure 6.20~6.28.
All eight cases are simulated using the interaction model. In the predictions, 2γ α =
0.73 and Hξ =0.3 selected from the studies in Section 6.5.3 and 6.5.4, are used as input
for theoretical predictions. The FRFs for acceleration based on measurements and
predictions are compared in Figure 6.20~6.28 (the blue line is the experiment result,
while the red dashed line is the simulation result).
Figure 6.20: Case 1.1 Figure 6.21: Case 1.2
126
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
Figure 6.22: Case 1.3 Figure 6.23: Case 1.4
Figure 6.25: Case 2.1 Figure 6.26: Case 2.2
Figure 6.27: Case 2.3 Figure 6.28: Case 2.4
127
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
Table 6.3: Measured and predicted resonance frequencies (Hz)
Measurements Predictions
1Rf 2Rf 1Rf 2Rf
Case 1.1 _ 8.79 _ 8.81 Case 1.2 5.27 9.38 4.44 11.2 Case 1.3 4.88 11.5 4.22 10.6 Case 1.4 4.69 13.6 4.01 10.3 Case 2.1 5.08 _ 5.11 _ Case 2.2 4.88 9.33 4.24 10.5 Case 2.3 4.30 9.57 4.18 10.4 Case 2.4 4.10 9.38 3.96 10.2
The resonance frequencies identified from both measurements and predictions in
Figures 6.19~6.28 are given in Table 6.3.
It can be observed from Figures 6.19~6.24 and Table 6.3 that:
• The patterns of FRFs for acceleration from experiment and from prediction
(Equation 6.19) match reasonably well. The difference in magnitudes of the curves
is because Equation 6.19 has been normalised to the modal mass.
• The two resonance frequencies are not always seen. Only one resonance frequency
is identified in Cases 1.1 and 2.1 and two resonance frequencies are observed in
the remaining cases.
• The first resonance frequency decreases with increases of the mass ratio α (Table
6.3), and the measurements and predictions have a good agreement. There are
large differences between the measured and calculated second natural frequency.
This is because some of the second natural frequencies cannot be clearly identified
in the measured FRFs.
The difference between Case 1.1 and 1.2 is that one more person is involved in the
interaction test, which changes the mass ratio from 0.13 (=0.39×60.2/180) to 0.32
(=0.39× (60.2+86.6)/180). However, the acceleration FRFs for both experiment and
prediction show one peak for Case 1.1 but two peaks in Case 1.2, indicating that the
mass ratio affects the visible presence of the resonance frequencies.
128
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
6.6 Conclusions
This chapter investigates theoretically and experimentally the frequency characteristics
and acceleration FRF of a TDOF human-structure system in the frequency domain. The
conclusions obtained can be summarised as follows:
1. The theoretical study of the frequency and response characteristics of the interaction
H-S model show that:
• The value of the highest peak of the acceleration FRF amplitude decreases with
increases of the damping ratio of the human body Hξ and/or the mass ratio α .
• When the frequency of a harmonic is less than 10.61Hz, the body model will
vibrate more significantly than the structural model. In practical situation, the
frequency of human loading is smaller or far smaller than 10.61. Thus, the
human body response would be larger than the structural response of a
human-structure system.
• When Hξ =0.3 and f =5.77Hz, the maximum acceleration of the body model
can be 2.50 times that of the structural model; when Hξ =0.5 and f =5.47Hz,
the ratio is 1.71.
2. The experimental and theoretical studies of human-structure systems have led to the
following findings
• The patterns of the FRFs of the human-structure systems using the conventional
model (Figure 6.12a) are significantly different from those using the interaction
model (Figure 6.12b). The predicted FRFs of the H-S system using the
interaction model are closed to the measurements.
129
Chapter 6: Frequency Characteristics of Human-Structure Models in Forced Vibration
• Comparison of 12 case studies shows a good agreement between the measured
and calculated FRF based on the interaction body model.
• The forced vibration tests help to identify reasonable parameters of the
interaction body model, Hξ =0.3 and 2γ α =0.73.
130
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
7 Experimental Identification of Hf and Hξ for the
Interaction Model in Free Vibration
7.1 Introduction
In practice, only one damped natural frequency of a human-structure system is often
measured in either free vibration or forced vibration tests. This chapter explores
whether the natural frequency and damping ratio of a standing body can be identified
through free vibration tests in which only one damped natural frequency and one
damping ratio of the human-structure system are measured. Section 7.2 shows the test
procedure, data processing and experimental results. Numerical simulation and solution
of the 2DOF equation of free vibration are provided in Section 7.3. A Matlab program
was used to facilitate the numerical solution and present normalize acceleration spectra.
A method of identification is suggested in Section 7.4, and used to determine the natural
frequency and the damping ratio of the human model from experimental results. Section
7.5 discusses the limitations of the method. Section 7.6 summarises the findings from
this study.
7.2 Test procedure and result
7.2.1 Test procedure
The experiment set-up is described in Chapter 3. A total of 4 individuals (2 women and
2 men) took part in the tests. Before the experiments, the weight of each individual was
recorded. Then the individuals stood on the test rig in their own footwear. The standing
person and the rig form a simple human-structure system. The details of the four people
are summary in Table 7.1.
131
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
Table 7.1: Mass and height of the participants
P1(M) P2(F) P3(M) P4(F)
Weight(kg) 94.4 62.4 75.6 58.6
Height(cm) 176 158 181 162
BMI(kg/m2) 30.5 25.0 23.1 22.3
Age 30 27 26 29
The location of the support prop can be adjusted to select the vertical natural
frequencies of the rig. Thus, nine test structure settings and a further 36 human-structure
cases were created. The test cases together with the natural frequencies of the structure
are summarised in Table 7.2. The first digit in experimental cases in Table 7.2 indicates
the setting of the test rig and the second shows the particular subject. Each test was
repeated at least once to assure that the test results were repeatable.
The test procedure is summarised as follows:
1. The free vibration of the bare test rig is conducted using the initial velocity method
(Section 3.5.1). The acceleration time-history of the test rig is recorded.
2. One subject stands on the rig. An impact is generated by the subject standing on his
toes and dropping onto his heels (the heel-drop test). The acceleration of the rig is
recorded.
3. Repeat procedure 2 for each individual.
4. The frequency of the test rig is modified by changing the location of the adjustable
sliding prop.
5. Repeat procedure 1~3 for a total of nine structural cases with different natural
frequencies of the rig.
132
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
Table 7.2: Experiment cases
Bare rig Rig+P1 Rig+P2 Rig +P3 Rig+P4 Sf (Hz)
Rig 1 Case 1.0 Case 1.1 Case 1.2 Case 1.3 Case 1.4 6.55
Rig 2 Case 2.0 Case 2.1 Case 2.2 Case 2.3 Case 2.4 7.19
Rig 3 Case 3.0 Case 3.1 Case 3.2 Case 3.3 Case 3.4 8.02
Rig 4 Case 4.0 Case 4.1 Case 4.2 Case 4.3 Case 4.4 8.91
Rig 5 Case 5.0 Case 5.1 Case 5.2 Case 5.3 Case 5.4 9.76
Rig 6 Case 6.0 Case 6.1 Case 6.2 Case 6.3 Case 6.4 11.85
Rig 7 Case 7.0 Case 7.1 Case 7.2 Case 7.3 Case 7.4 13.57
Rig 8 Case 7.0 Case 8.1 Case 8.2 Case 8.3 Case 8.4 15.36
Rig 9 Case 9.0 Case 9.1 Case 9.2 Case 9.3 Case 9.4 15.63
7.2.2 Data processing
The procedure for data processing is summarised as follow:
1. Draw the acceleration time history graph.
2. Use the Matlab toolbox “periodogram” to obtain the acceleration spectrum based on
the acceleration time history data. The damped natural frequency of the bare rig Sf
or human-rig system HSf can be identified which corresponds to the peak in the
acceleration spectrum.
3. Use the Matlab toolbox “lsqcurvefit” for curve fitting of the acceleration-time
history and for extracting the damping ratio of the bare rig Sξ or the damping ratio
of the human-rig system HSξ .
Taking cases 2, 5, 9 as examples. The acceleration-time histories and the acceleration
spectra of the bare rig for various settings and the acceleration-time histories and the
acceleration spectra of the rig with a standing person (P1, P2, P3, P4) are shown in
Figure 7.1~7.15 separately.
133
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.1: Case 2.0-the bare rig
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.2: Case 2.1-the rig with subject P1
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.3: Case 2.2-the rig with subject P2
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.4: Case 2.3-the rig with subject P3
134
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.5: Case 2.4-the rig with subject P4
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.6: Case 5.0-the bare rig
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.7: Case 5.1-the rig with subject P1
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.8: Case 5.2-the rig with subject P2
135
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.9: Case 5.3-the rig with subject P3
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.10: Case 5.4-the rig with subject P4
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.11: Case 9.0-the bare rig
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.12: Case 9.1-the rig with subject P1
136
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.13: Case 9.2-the rig with subject P2
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.14: Case 9.3-the rig with subject P3
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.15: Case 9.4-the rig with subject P4
7.2.3 Experimental Results
For most test cases, one damped natural frequency of the human-rig system is obtained,
as shown in Cases 2 and 5. In a few cases two damped natural frequencies can be
obtained, such as Case 9. The aim of this Chapter is to use one damped natural
frequency and the corresponding damping ratio of the human-rig system to identify the
natural frequency and damping ratio of the human body. The natural frequencies,
corresponding to the highest peaks in the spectra, and damping ratios of all cases are 137
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
summarised in Table 7.3.
Table 7.3: Measured result summary
Bare rig P1 P2 P3 P4
Case Sf Sξ HSf HSξ HSf HSξ HSf HSξ HSf HSξ
Rig 1 6.55 0.057 5.35 0.089 5.38 0.169 5.26 0.084 5.53 0.106 Rig 2 7.19 0.044 6.34 0.168 5.84 0.129 5.68 0.117 5.73 0.144 Rig 3 8.02 0.049 7.04 0.153 6.90 0.148 7.35 0.133 7.17 0.166 Rig 4 8.91 0.029 8.02 0.101 8.03 0.103 8.17 0.099 8.47 0.141 Rig 5 9.76 0.024 9.18 0.041 9.10 0.092 9.12 0.079 9.37 0.081 Rig 6 11.85 0.032 11.26 0.068 11.79 0.127 11.32 0.065 10.90 0.115 Rig 7 13.57 0.030 12.28 0.073 13.25 0.055 13.47 0.065 13.75 0.054 Rig 8 15.36 0.020 14.34 0.054 14.60 0.049 15.31 0.164 15.09 0.086 Rig 9 15.63 0.015 15.28 0.061 15.46 0.024 15.42 0.022 15.42 0.050
The information in Table 7.3 can be summarised as follows:
• The damped natural frequencies of the human-rig systems are always smaller
than those of the bare rig. This is different from the conclusions of natural
frequencies made from both conventional models and interaction models. The
relationships between the natural frequencies of both conventional models and
interaction models are 1 2( , )S Hf f f f< < (Ellis and Ji, 1997). As shown in Case
9, the second damped natural frequency is smaller than the damped natural
frequency of the bare rig.
• The damped natural frequency of the human-rig system increases with increases
of the damped natural frequency of the bare rig.
• The damping ratios of the human-rig systems are always larger than the
damping ratios of the bare rigs.
7.3 Simulation of free vibration of 2DOF interaction model
Consider the interaction model shown in Figure 7.16. The equation of motion in free
vibration is defined in Equation 2.12.
138
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
Figure 7.16: Human-structure system
4S S S SC M fπ ξ= ; 114H H H HC M fπ ξ= (7.1)
2 24S S SK M fπ= ; 2 2114H H HK M fπ= (7.2)
0H
S
MM
η = ; 11H
S
MM
α = ; 1H
S
MM
γ = ; (7.3)
Substituting Equations 7.1~7.3 into Equation 2.12 gives the following governing
differential equations:
1 1
2 2 2 2 2 2
2 2 2
4 4 41 24 4
4 4 4 +
4 4
S S S H H H H S
H H H H H H
S H H
H
u f f f uu f f u
f f ff
π ξ πη ξ πη ξη α γ γ απη ξ πη ξγ α α
π π η π ηπ η π
+ −+ + − − + −−
+ −−
21
00
S
HH
uufη
=
(7.4)
There are seven parameters, η , α , γ , Hf , Sf , Hξ and Sξ in Equation 7.4. In the
equation, two parameters α , γ have already been identified in Table 4.4. η can be
identified by measuring the weight of the human body and the test rig. The natural
frequency Sf and damping ratio Sξ of the test rig can be identified directly from the
experiment. Therefore, the remaining two parameters, Hf and Hξ , are to be assumed
in the free vibration simulation.
The simulation procedure is:
1. Input all the parameters of Equation 7.4.
139
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
2. Use the Matlab toolbox “ODE 45” to solve Equation 7.4. Obtain the
acceleration data from the simulation result.
3. Follow the same procedure as used to examine the experimental data to obtain
one damped natural frequency and one damping ratio of the 2DOF model.
When estimating the acceleration responses using MatLab “ODE 45” code, the initial
values of acceleration were set at 0.05g, 0.1g and 0.2g respectively. The selected initial
values resulted in the same predicted natural frequencies and damping ratios for all
cases. This indicates that the three initial acceleration values do not affect the required
outcomes. Therefore, the initial value of acceleration of 0.1g was used for further
analysis. For example, let 180SM kg= , 75HM kg= , 0.01Sξ = , 5Hf Hz= , 0.3Hξ =
and 8Sf Hz= , the acceleration-time history and the acceleration spectrum of the 2DOF
model is shown in Figure 7.17, and the human-rig systems has a damped natural
frequency of 6.38 Hz and a damping ratio of 0.083.
(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting
Figure 7.17: the acceleration-time history and the acceleration spectrum of the rig with a
standing person
140
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
7.4 Experimental identification of the natural frequency and damping
ratio of a human body
The simulation of free vibration in the last section shows that a given input pair Hf and
Hξ lead to a unique pair of HSf and HSξ , i.e.
( , )HS H Hf f f ξ= (7.5)
( , )HS H Hf fξ ξ= (7.6)
Although the analytical expressions of equations 7.5 and 7.6 are not available, they can
be presented graphically after a sufficient amount of data is calculated in which Hf
and Hξ data are varied within reasonable ranges. Substituting a real value of HSf
from the measurement in Table 7.3 to equation7.5, gives a plane parallel to the Hf and
Hξ plane. The intersection of the plane HSf and the curved surface ( , )H Hf f ξ is a
curve. All points on the curve have the same value of HSf or the same height of HSf .
Similarly, another curve at a height of HSξ can be obtained from equation 7.6. Then the
two curves are placed in the Hf ~ Hξ plane. The intersection point of the two curves
provides the solution and the corresponding values of Hf and Hξ are thereby
identified.
The procedure for realising the method can be described as follows:
1. Input the experiment results ( SM , 0HM , Sf and Sξ ) into the simulation
programme.
2. Assume an Hf value within the interval [3.00, 9.00] with steps of 0.01 and Hξ
within the interval [0.010, 0.600] with steps of 0.001 and follow the simulation
procedure described in Section 7.3 to obtain HSf and HSξ . 141
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
3. Based on the simulation result, the surfaces of the damped natural frequency
HSf (Figure 7.18) and the damping ratio HSξ (Figure 7.19) of the
human-structure system with respect to the natural frequencie Hf and damping
ratio Hξ of the human body can be plotted.
4. Use the experiment result HSf and HSξ in Table 7.3 to identify the two curves
for HSf and HSξ , as shown in Figures 7.20 and 7.21.
5. The intersection point of the two curves identifies the results, Hf and Hξ , as
shown in Figure 7.22.
6. The identified values of Hf and Hξ are then substitute to the Equation 7.4 to
check the simulation leads to the same values of HSf and HSξ .
Take case 1.3 ( 6.55Sf Hz= , 0.057Sξ = , 5.26HSf = and 0.084HSξ = ) in Table 7.3 as an
example. Figures 7.18 and 7.19 show the damped natural frequency and the damping
ratio of the human-structure system with respect to the damped natural frequencies of
the human body and damping ratios of the human body. Two relationships can be
identified between Hf and Hξ in Figure 7.20~7.21. So Hf and Hξ can be solved by
this numerical method, which is shown in Figure 7.22.
142
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
Figure 7.18: The damped natural frequency of human-structure system based on different natural
frequencies and damping ratio of human body
Figure 7.19: The damping ratio of human-structure system based on different natural frequencies
and damping ratio of human body
143
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
Figure 7.20: The intersection of the plane HSf and the curved surface ( , )H Hf f ξ
Figure 7.21: The intersection of the plane HSξ and the curved surface ( , )H Hfξ ξ
144
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
Figure 7.22: The two curves obtained from Figures 7.20 and 7.21
The identified result is 5.01Hf Hz= and 0.332Hξ = . The identified results for all the
experimental cases are listed in Table 7.4.
Table 7.4: Summary of the result
Bare rig P1 P2 P3 P4
Case Sf Sξ Hf Hξ Hf Hξ Hf Hξ Hf Hξ
Rig 1 6.55 0.057 4.34 0.118 3.73 0.277 5.01 0.332 4.02 0.418 Rig 2 7.19 0.044 3.33 0.138 4.16 0.294 4.34 0.328 4.19 0.244 Rig 3 8.02 0.049 3.26 0.098 3.75 0.334 4.24 0.388 3.37 0.261 Rig 4 8.91 0.029 3.17 0.044 4.36 0.067 4.64 0.085 3.62 0.257 Rig 5 9.76 0.024 4.06 0.063 3.87 0.289 4.01 0.064 4.25 0.051 Rig 6 11.85 0.032 3.78 0.084 4.50 0.200 3.64 0.256 4.22 0.377 Rig 7 13.57 0.030 5.02 0.064 3.66 0.121 4.21 0.092 5.18 0.112 Rig 8 15.36 0.020 5.81 0.052 5.63 0.067 6.45 0.189 5.31 0.185 Rig 9 15.63 0.015 4.67 0.112 4.61 0.035 4.57 0.025 4.78 0.123 Mean - - 4.16 0.086 4.25 0.187 4.57 0.195 4.33 0.225
It can be noted from Table 7.4 that: The identified natural frequency of the human body
varies from 3.17 to 5.81 Hz. The mean natural frequencies of the four individuals are
4.16, 4.25, 4.57 and 4.33 Hz respectively. The identified damping ratio of the human
body varies from 0.025 to 0.388. The mean damping ratios of the four individuals are 145
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
0.086, 0.187, 0.195 and 0.225 respectively. The results show the variations of the
identified body dynamic properties.
There are several possible reasons that may affect the results: a) the input excitations in
the free vibration experiments are not the constant. The heel drop forces induced by
different subjects and by the same subject in different tests varied. Different input
forces generate free vibration with different amplitudes, which may affect the
experimental results. The damping ratio of the human body may be affected by the
Body Mass Index. It shows that the body damping ratio decreases with the Body Mass
Index (BMI values of each person are calculated in Table 7.1). Subject P1 has the
largest BMI (30.5) and the smallest damping ratio (0.086), while Subject P4 has the
smallest BMI (22.3) and the largest damping ratio (0.225). c) The identification method
may be sensitive to errors in the experimental results, which will be discussed in next
section.
7.5 Sensitivity study
Sensitivity studies are conducted in this section to examine the quality of the
identification method.
7.5.1 Hf and Hξ
The basic data used for the study are Sf =8Hz, Sξ =0.01, SM =180kg, and 0HM =75kg.
The base values Hf =5.00Hz and Hξ =0.30 are given. A 5% error of Hf , and/or Hξ
are used to examine the outcome. Table 7.5 provides the results. The first two columns
are the input Hf and Hξ with and without 5% variations; the third and fourth
columns give the calculated HSf and HSξ and the relative errors are given in the last
two columns.
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Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
Table 7.5 Sensitivity study of Hf and Hξ
Input Output Ratio
Hf (Hz) Hξ HSf (Hz) HSξ Rf (%) HSξ (%)
5.00 0.300 6.38 0.083 5.25(+5%) 0.300 6.39 0.073 0.20 -12.1 4.75(-5%) 0.300 6.37 0.095 -0.16 14.5
5.00 0.315(+5%) 6.40 0.081 0.30 -2.41 5.00 0.285(-5%) 6.35 0.085 -0.47 2.40
5.25(+5%) 0.315(+5%) 6.41 0.072 0.50 -13.3 4.75(-5%) 0.315(+5%) 6.40 0.092 0.30 10.8 5.25(+5%) 0.285(-5%) 6.37 0.074 -0.16 -10.8 4.75(-5%) 0.285(-5%) 6.33 0.097 -0.78 16.9
The information in Table 7.5 can summarised as follows:
• The damped natural frequency of the human-structure system is not sensitive to
the natural frequency and damping ratio of the human body. An error of less
than 1% is induced by the 5% input error.
• The damping ratio of the system is sensitive to the natural frequency and
damping ratio of the human body. An error of up to 17% is produced by the 5%
input error.
7.5.2 HSf and HSξ
The same base values as those in Section 7.5.1 are given. The base data HSf =6.38Hz
and HSξ =0.083 are given in the third row of the first two columns of table 7.6. The
identified results, Hf and Hξ , are identical to the input data in Section 7.5.1. A 5%
error of HSf and HSξ are used to examine their effects on Hf and Hξ . The outcome
is given in the third and fourth columns of the table. The last two columns show the
errors of the identified results relative to the base results in the third row.
147
Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
Table 7.6 Sensitivity study of HSf and HSξ
Input Identified result Error ratio
HSf (Hz) HSξ Hf (Hz) Hξ Hf (%) Hξ (%)
6.38 0.083 5.00 0.300 6.70(+5%) 0.083 3.82 0.528 -23.6 76.0 6.06(-5%) 0.083 4.90 0.162 -2.00 -46.0
6.38 0.087(+5%) 4.87 0.304 -2.60 1.30 6.38 0.079(-5%) 5.04 0.305 0.80 1.70
6.70(+5%) 0.087(+5%) 3.81 0.495 -23.8 65.0 6.06(-5%) 0.087(+5%) 4.88 0.166 -2.40 -44.7 6.70(+5%) 0.079(-5%) 3.80 0.555 -24.0 85.0 6.06(-5%) 0.079(-5%) 4.92 0.156 -1.60 -48.0
It can be observed from Table 7.6 that:
• The effect of errors in HSf on Hf and Hξ are more significant than those of
HSξ .
• The resulting error on Hf is less than that on Hξ . The 5% input error could
lead to 24% error on Hf and 85% on Hξ .
7.5.3 SM
The same base values as those in Section 7.5.2 are given. The base data SM =180kg is
given in the third row of the first column of table 7.7. Errors of 5% or 10% in SM
are used to examine the effects on Hf and Hξ . The inputs are shown in the first three
columns. The results are given in the fourth and fifth columns in Table 7.7 and the last
two columns show the errors of the identified results relative to the base results in the
third row.
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Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
Table 7.7 Sensitivity study of SM
Input Identified result Error
SM (kg) HSf (Hz) HSξ Hf (Hz) Hξ Hf (%) Hξ (%)
180 6.38 0.083 5.00 0.300 171(-5%) 6.38 0.083 4.94 0.340 -1.20 13.3 189(+5%) 6.38 0.083 5.04 0.271 0.80 -9.67 162(-10%) 6.38 0.083 4.82 0.389 -3.60 29.7 198(+10%) 6.38 0.083 5.06 0.245 1.20 -18.3
It can be observed from Table 7.7 that the resulting error on Hf is less than that on
Hξ . A 10% input error could lead to a 3.60% error in Hf and a 29.7% error in Hξ .
Sensitivity studies explain that the identified results are sensitive to the input
parameters HSf and HSξ , and require quality measurements of HSf and HSξ . In order
to identify the exact frequency and damping ratio of the human body, the errors in the
measurements must be as small as possible. This may prove difficult with this particular
test method.
7.6 Conclusion
This chapter identifies the natural frequency Hf and damping ratio Hξ of a human
body using combined experimental and theoretical methods in free vibration
experiments. The conclusions obtained can be summarised as follows:
• The proposed method is to abstract HSf and HSξ through free vibration
experiments of a human-structure system and then to derive Hf and Hξ of
the human body model based on the measured values HSf and HSξ . Using the
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Chapter 7: Experimental Identification of Hf and Hξ for the Interaction Model in Free Vibration
method, Hf and Hξ can be identified through free vibration tests of the bare
rig and the human-rig system.
• The proposed identification method is sensitive to the input HSf and HSξ . This
would lead to relatively large errors in the identification.
• Derivation of Hf and Hξ based on HSf and HSξ is more sensitive to errors
than derivation of HSf and HSξ based on Hf and Hξ .
Further work should be conducted to assure more accurate results from experiments,
including SM , Sf , Sξ , HSf and HSξ .
150
Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
8 Optimum Design Parameters for a
Tuned-Mass-Damper to Maximise the Equivalent
Damping Ratio
8.1 Introduction
The 2DOF human-structure model is similar to a 2DOF tuned-mass-damper (TMD)
system (Setreh and Hanson, 1992). However, the objectives of studying the two systems
are different. The former focuses on how people affect the structure and how structural
vibration affects people, as studied in previous chapters. The later aims to determine the
optimum parameters of the tuned-mass-damper system in order to minimise the
dynamic response of a structure. There are many publications in this area (Setreh and
Hanson, 1992; Satareh, 2002; Den Hartog, 1956; Warburton, 1982; Leung and Zhang,
2009; Lee, et al., 2004; Fujino and Abe, 1993).
In this chapter, the FRF studied in the previous chapters, is used in conjunction with the
concept of an equivalent damping ratio, to derive the best design parameters of a TMD.
Section 8.2 provides the concept of the proposed method. Section 8.3 derives the peak
responses at resonance of a SDOF structure system and a 2DOF structure-TMD system
using Fourier Response Functions. Expressions for the equivalent damping ratio for
displacement and acceleration are given in Section 8.4. Tables of the optimum design
parameters of a TMD and the corresponding equivalent damping ratios are provided
when a structural SDOF system has typical damping ratios. Section 8.5 provides a case
study of floor vibration induced by rhythmic crowd loads to demonstrate the use of the
optimum parameters of TMDs and illustrates the advantage of taking the equivalent
damping ratio in design for attenuating vibration.
151
Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
8.2 The Method
Figure 8.1 shows a SDOF structure system, a TMD and a 2DOF TMD-structure system.
When the design parameters of the structure and the TMD systems are given, the
Fourier Response Functions (FRF) for the structure and TMD-structure systems can be
obtained. For example, when the damping ratios of the structural model and the TMD
are 1.5%Sξ = and 6%Hξ = respectively; the mass ratio and natural frequency ratio
of the TMD to the structure system are / 0.1H SM Mα = = and / 0.99H Sf fβ = = .
The structure and the TMD-structure systems are subjected to the same harmonic
load 0( ) sinS pP t P tω= . Figure 8.2 shows the FRFs for the normalised displacements of
the structure system (the dashed line) and of the structural mass of the TMD-structure
system (the solid line with two peaks). When the damping ratio of the structure system
is adjusted to make the maximum response equal to that of the structure mass of the
2DOF system, as shown in Figure 8.2, the corresponding damping ratio is defined as the
equivalent damping ratio. This ratio has a clear physical meaning and indicates how
much vibration reduction is achieved. For the studied case, the equivalent damping ratio
is 4.94%. The damping ratio of the structure divided by the equivalent damping ratio is
0.015/0.049=0.304, which shows that the maximum response of the structure with the
TMD is about 30% of that of the structure without the TMD. Thus the equivalent
damping ratio indicates the effectiveness of the TMD for reducing the maximum
vibration of the structural system. The smaller the ratio of the two damping ratios of the
structure without and with a TMD, the more effective the TMD and the greater the
vibration reduction. The following studies demonstrate how the parameters ,Hξ α and
β of a TMD can be determined to obtain the maximum equivalent damping ratio.
152
Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
(a) SDOF structure system (b) A TMD (c) A Structure-TMD system
Figure 8.1: A damped 2DOF systems
Figure 8.2: FRFs of SDOF structure model and 2DOF TMD-structure model
( 0.1α = , 0.99β = , 6%Hξ = , 1.5%Sξ = )
The resonance dynamic displacement and acceleration of the SDOF structural system
subjected to a harmonic load of 0( ) sinS pP t P tω= are:
0 12SS
S S
PK
δξ
= (8.1)
0 12SS
S S
PaM ξ
= (8.2)
Where SM and SK are the mass and stiffness of the structure system. Equations 8.1
and 8.2 are obtained from the FRF of the system when the load frequency is the same as
153
Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
the natural frequency. If the damping ratio of the structure is less than 10%, there is little
difference between the resonance responses and the responses given in equations 8.1
and 8.2.
The TMD-structure system is considered (Figure 8.1c) where a TMD has mass HM ,
stiffness HK and damping coefficient HC . The equation of motion of the system can be
expressed in a matrix form:
( )0
S S S H H S S H H S S
H H H H H H H H
M x C C C x K K K x P tM x C C x K K x
+ − + − + + = − −
(8.3)
The form of the Fourier transformation of equation 8.3 is
( )( )
2 ( )0
SS S H H S H H S
HH H H H H
xM C C C K K K Pi
xM C C K Kω ω
ω ωω
− + − + − + + = − − −
or
( )( ) ( ) ( ) ( )
( ) ( )11 12
21 22
( ) ( )0 0
S S S
H
x H HP PH
x H Hω ω ωω ω
ωω ω ω
= = (8.4)
where
( )( ) ( ) ( )
2
21 H H H H H
H H S S H S H
M iC K iC KH
iC K M i C C K KZω ω ω
ωω ω ωω
− + + + = + − + + + +
(8.5)
Equation 8.5 is the Frequency Response Function (FRF) of the 2DOF system where
( ) ( ) ( ) ( )22 2S S H S H H H H H HZ M K K i C C M K iC K iCω ω ω ω ω ω = − + + + + − + + − +
(8.6)
The FRFs for displacement and acceleration of the structure mass are:
211 11
( ) ( )( ) ( )( ) ( )
S S
S S
x xH HP P
ω ωω ω ωω ω
= = −
(8.6)
154
Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
where Sω and Hω are the natural frequencies of the structure and TMD models
respectively and ω is the unknown frequency of the 2DOF system. To simplify the
equations and reduce the number of unknowns, the following parameters are introduced:
; ; H H
S S S
MM
ω ωα β λω ω
= = = ; S HS H
S H
K KM M
ω ω= = (8.7)
where α is the mass ratio; β is the frequency ratio and λ is the normalised
frequency variable. The maximum dynamic displacement and acceleration of the
structural mass in the 2DOF system can be presented as
( ) ( )( ) ( )
2 2
00 11 0 11 2 2max{ ( )} ( ) S
SS
R EPP H P HK X Y
λ λδ λ λ
λ λ
+= = =
+ (8.8)
( ) ( )( ) ( )
2 2
00 11 0 11 2 2max{ ( )} ( ) S
SS
R EPa P A P AM X Y
λ λλ λ
λ λ
+= = =
+ (8.9)
Where,
( )2 2 2( )SR λ λ β λ= − (8.10.a)
3( ) 2 HE λ βξ λ= (8.10.b)
( )4 2 2 2( ) 1 1 4 S HX λ λ α β βξ ξ λ β = − + + + + (8.10.c)
( ) ( ){ }2( ) 2 1S H S HY λ λ ξ α βξ λ β βξ ξ = − + + + + (8.10.d)
Where λ in Equations 8.8 and 8.9 is the normalized resonance frequency
corresponding to the largest peak value in the FRF, as illustrated in Figure 8.2. It can be
seen from equations 8.8-8.10 that the peak responses and the normalised resonance
frequency of the structure are functions of four parameters, Sξ ,α , β and Hξ . When a
set of the four parameters is defined, λ can be determined by identifying the
155
Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
maximum value of the FRF for the 2DOF system shown in Figure 8.2. This can be
easily achieved using a MATLAB routine.
8.3 Equivalent damping ratio
Equating the maximum responses of the structure SDOF model (equations 8.1 and 8.2)
to those of the structure mass in the 2DOF model (equations 8.8 and 8.9) leads to the
expressions for the equivalent damping ratio, i.e. Sδ = SSδ and Sa = SSa :
( ) ( )( ) ( )
2 2
0 02 2
12S e S
R EP PK K X Y
λ λξ λ λ
∆
+=
+ (8.11)
( ) ( )( ) ( )
2 2
0 02 2
12 a
S e S
R EP PM M X Y
λ λξ λ λ
+=
+ (8.12)
Substituting equation 8.10 into equation 8.11 and 8.12 gives equivalent damping ratios
aeξ for acceleration and eξ
∆ for displacement respectively:
( )( ) ( ) ( ){ }( )( ) ( )
224 2 2 2 2
2 24 2 2
1 1 4 2 112 [ 2 ]
S H S H S Hae
H
λ α β βξ ξ λ β λ ξ α βξ λ β βξ ξξ
λ β λ βξ λ
− + + + + + − + + + + =− +
(8.13)
( )( ) ( ) ( ){ }( )( ) ( )
224 2 2 2 2
22 22 2
1 1 4 2 112 2
S H S H S H ae e
H
λ α β βξ ξ λ β λ ξ α βξ λ β βξ ξξ λ ξ
β λ βξ λ∆
− + + + + + − + + + + = =− +
(8.14)
It can be observed from equations 8.13 and 8.14 that:
The equivalent damping ratio is a function of α , β , Sξ and Hξ , and is not
explicitly dependent on the masses and stiffnesses of the structure and TMD systems,
and is independent of the magnitude of the load.
156
Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
The analytical expressions of the equivalent damping ratios contain the normalised
resonance frequency λ that is also a function of the four basic parameters.
Although λ is a function ofα , it varies in a limited range and is not sensitive toα .
eξ∆ and a
eξ monotonically increase as α increases. Therefore, α should not be
considered as a variable for maximising the equivalent damping ratio.
When α and Sξ are given, eξ∆ (or a
eξ ) is a function of β and Hξ only. Thus,
eξ∆ can be represented using a curved surface or a contour plot; hence the maximum
equivalent damping ratio and the corresponding optimum frequency ratio and
damping ratio of the TMD can be visualised and identified without using particular
optimization methods.
For illustration, a mass ratioα =0.05 and a damping ratio Sξ =0.015 of the structural
system are taken. The frequency ratio β is considered within the range 0.8 and 1.1
with an increment of 0.001 and the damping ratio of the TMD Hξ between 1% and
25% with an increment of 0.1%. This leads to 300 x 250 = 75000 points in a contour
plot or a curved surface. For each point, the FRF is first produced; a search is
consequently conducted to identify the normalised resonance frequency λ that
corresponds to the largest response peak of the 2DOF system (Figure 8.2). Then plots
are generated using equation 8.14. The maximum equivalent damping ratio eξ∆ and the
corresponding optimum frequency ratio β and the damping ratio of the TMD Hξ can
be determined from the plots. Figures 8.3 and 8.4, which are based on 75000 points,
show the equivalent damping ratio for displacement on a curved surface and on a
contour plot as a function of β and Hξ . It can be seen that the maximum equivalent
damping ratio is 9.06%eξ∆ = and the corresponding optimum frequency ratio of
0.948β = and damping ratio of the TMD of 13.2%Hξ = .
157
Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
Figure 8.3: Curved surface of eξ∆ (The mass ratio α =0.05)
Figure 8.4: Contours of eξ∆ (The mass ratio α =0.05)
Table 8.1 provides the optimum design parameters and the corresponding maximum
equivalent damping ratios for four typical structural damping ratios, 0.01, 0.015, 0.02
and 0.05 and mass ratios from 0.01 to 0.12 with an increment of 0.05. The equivalence
is considered for both dynamic displacement and acceleration.
158
Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
Table 8.1a: Optimum design parameters ( β and Hξ ) of a TMD: 1%Sξ =
Equivalence for displacement Equivalence for acceleration
α β Hξ (%) eξ∆ (%) β Hξ (%) a
eξ (%)
0.005 0.994 4.40 3.37 0.998 4.60 3.39 0.010 0.989 6.20 4.38 0.996 6.10 4.42 0.015 0.983 7.80 5.16 0.993 8.00 5.22 0.020 0.978 8.80 5.82 0.991 8.90 5.91 0.025 0.973 9.90 6.40 0.989 9.70 6.51 0.030 0.968 10.9 6.91 0.986 11.0 7.06 0.035 0.964 11.1 7.39 0.984 11.6 7.57 0.040 0.959 11.9 7.84 0.982 12.2 8.04 0.045 0.954 12.9 8.26 0.979 13.3 8.49 0.050 0.949 13.9 8.63 0.977 13.8 8.92 0.055 0.945 13.8 9.00 0.975 14.4 9.32 0.060 0.940 14.9 9.36 0.972 15.3 9.72 0.065 0.936 14.9 9.68 0.970 15.8 10.1 0.070 0.931 15.9 10.0 0.968 16.2 10.5 0.075 0.927 15.9 10.3 0.966 16.7 10.8 0.080 0.922 17.1 10.6 0.963 17.5 11.2 0.085 0.918 17.2 10.9 0.960 18.3 11.5 0.090 0.914 17.3 11.2 0.959 18.4 11.8 0.095 0.909 18.6 11.5 0.957 18.8 12.1 0.100 0.905 18.8 11.7 0.955 19.2 12.4 0.105 0.901 19.0 12.0 0.952 19.9 12.8 0.110 0.897 19.2 12.2 0.950 20.3 13.1 0.115 0.893 19.5 12.4 0.948 20.7 13.3 0.120 0.889 19.9 12.7 0.945 21.4 13.6
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Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
Table 8.1b: Optimum design parameters ( β and Hξ ) of a TMD: 1.5%Sξ =
Equivalence for displacement Equivalence for acceleration
α β Hξ (%) eξ∆ (%) β Hξ (%) a
eξ (%)
0.005 0.993 4.60 3.81 0.998 4.80 3.82 0.010 0.988 6.20 4.83 0.996 6.50 4.87 0.015 0.982 7.90 5.60 0.994 7.70 5.67 0.020 0.977 8.90 6.26 0.992 8.70 6.35 0.025 0.972 9.90 6.84 0.989 10.1 6.95 0.030 0.967 10.9 7.36 0.987 10.9 7.50 0.035 0.962 11.7 7.83 0.985 11.5 8.01 0.040 0.957 12.4 8.26 0.982 12.7 8.48 0.045 0.952 13.2 8.66 0.980 13.3 8.93 0.050 0.948 13.2 9.06 0.978 13.8 9.36 0.055 0.943 14.3 9.43 0.975 14.8 9.76 0.060 0.938 15.2 9.77 0.973 15.3 10.2 0.065 0.934 15.1 10.1 0.971 15.8 10.5 0.070 0.929 16.3 10.4 0.969 16.3 10.9 0.075 0.925 16.2 10.7 0.966 17.1 11.3 0.080 0.920 17.4 11.0 0.964 17.6 11.6 0.085 0.916 17.4 11.3 0.962 18.0 11.9 0.090 0.912 17.4 11.6 0.959 18.8 12.3 0.095 0.907 18.7 11.9 0.957 19.2 12.6 0.100 0.903 18.8 12.1 0.955 19.6 12.9 0.105 0.899 19.0 12.4 0.954 19.7 13.2 0.110 0.895 19.2 12.6 0.951 20.4 13.5 0.115 0.890 20.5 12.8 0.949 20.8 13.8 0.120 0.886 20.8 13.1 0.946 21.5 14.1
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Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
Table 8.1c: Optimum design parameters ( β and Hξ ) of a TMD: 2%Sξ =
Equivalence for displacement Equivalence for acceleration
α β Hξ (%) eξ∆ (%) β Hξ (%) a
eξ (%)
0.005 0.993 4.50 4.26 0.999 4.50 4.28 0.010 0.987 6.50 5.28 0.997 6.30 5.31 0.015 0.981 7.80 6.04 0.994 8.10 6.11 0.020 0.976 9.00 6.71 0.992 9.10 6.79 0.025 0.971 9.70 7.28 0.990 10.0 7.40 0.030 0.966 10.4 7.79 0.988 10.8 7.94 0.035 0.961 11.4 8.26 0.985 12.0 8.45 0.040 0.956 12.0 8.69 0.983 12.7 8.92 0.045 0.951 12.9 9.11 0.981 13.3 9.37 0.050 0.946 13.8 9.49 0.978 14.3 9.80 0.055 0.941 14.6 9.85 0.976 14.8 10.20 0.060 0.937 14.4 10.2 0.974 15.4 10.59 0.065 0.932 15.5 10.5 0.972 15.9 10.97 0.070 0.927 16.4 10.8 0.970 16.4 11.33 0.075 0.923 16.4 11.2 0.967 17.2 11.69 0.080 0.918 17.5 11.4 0.965 17.7 12.03 0.085 0.914 17.4 11.7 0.963 18.1 12.37 0.090 0.910 17.5 12.00 0.960 18.9 12.69 0.095 0.905 18.7 12.3 0.958 19.3 13.01 0.100 0.901 18.8 12.5 0.957 19.4 13.32 0.105 0.897 18.9 12.8 0.954 20.1 13.63 0.110 0.892 20.2 13.0 0.952 20.5 13.92 0.115 0.888 20.4 13.3 0.950 20.9 14.22 0.120 0.884 20.6 13.5 0.947 21.6 14.51
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Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
Table 8.1d: Optimum design parameters ( β and Hξ ) of a TMD: 5%Sξ =
Equivalence for displacement Equivalence for acceleration
α β Hξ (%) eξ∆ (%) β Hξ (%) a
eξ (%)
0.005 0.988 4.90 7.02 1.002 5.30 7.05 0.010 0.981 6.80 7.99 1.001 6.80 8.06 0.015 0.974 8.20 8.73 0.999 8.30 8.84 0.020 0.968 9.40 9.36 0.997 9.50 9.50 0.025 0.962 10.4 9.90 0.995 10.6 10.1 0.030 0.957 11.0 10.4 0.993 11.5 10.6 0.035 0.951 12.2 10.9 0.991 12.3 11.1 0.040 0.946 12.6 11.3 0.989 13.1 11.6 0.045 0.941 13.1 11.7 0.987 13.8 12.1 0.050 0.935 14.4 12.0 0.985 14.5 12.5 0.055 0.930 15.1 12.4 0.982 15.5 12.9 0.060 0.925 15.7 12.7 0.980 16.1 13.3 0.065 0.920 16.3 13.1 0.979 16.3 13.6 0.070 0.915 16.9 13.4 0.976 17.2 14.0 0.075 0.911 16.4 13.7 0.973 18.1 14.4 0.080 0.906 17.2 13.9 0.971 18.6 14.7 0.085 0.901 18.2 14.2 0.969 19.1 15.0 0.090 0.897 17.9 14.5 0.968 19.2 15.4 0.095 0.892 18.8 14.7 0.965 20.0 15.7 0.100 0.888 18.6 15.0 0.963 20.5 16.0 0.105 0.883 19.8 15.2 0.961 20.9 16.3 0.110 0.879 19.7 15.5 0.959 21.3 16.6 0.115 0.874 20.9 15.7 0.957 21.8 16.9 0.120 0.870 20.9 15.9 0.955 22.2 17.2
8.4 Application
This section is based on a real engineering case. A PhD student Tianxin Zheng set up
the FE model of the dance floor, and calculated the response of the floor by software
“ANSYS”. The optimum parameters of TMD were taken from my work. Based on the
optimum parameters, the response of floor has been recalculated by Tianxin based on
his FE model. And then I did the comparison between the floor with and without TMD.
Finally I summarised this case study in this section.
A dance floor in a nightclub was reported to experience significant vibrations when pop 162
Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
concerts were held. The vibration was believed to lead to fatigue problems of some
supporting beams. Thus remedial measures were considered to remove the vibration
problem. The dynamic measurements showed that the floor had a fundamental natural
frequency of 3.55 Hz and damping ratio of Sξ =1.5%. Analysis of the floor subjected to
rhythmic crowd movements where jumping was involved showed that several modes of
vibration of the floor contributed to the vibration but the floor response was dominated
by the second harmonic load component when the load had a frequency of 1.875 Hz.
The maximum dynamic displacement of the floor was calculated at Sδ =11mm and the
modal mass of the floor for the fundamental mode was 163654SM kg= . TMDs were
considered for suppressing the vibration.
The target displacement of the floor was STδ =3.5mm. The required equivalent damping
ratio can be estimated using the concept that the resonance response is inversely
proportional to the damping ratio, i.e.
11.00 0.015 0.04713.5
Se S
ST
δξ ξδ
∆ = = × = (8.15)
From Table 8.1b, 4.83%eξ∆ = satisfies the above requirement with the design
parameters of 0.01α = , 0.988β = and 6.2%Hξ = . It should be noted that the
optimum parameters are determined based on a 2DOF model subjected to a single
harmonic load but the studied case included the vibration from several modes and three
harmonic loads (Ellis and Ji, 2004). Therefore, the vibration reduction on the dance
floor is unlikely to be as efficient as an idealised model and a higher value of the
equivalent damping ratio is considered. 0.02α = was taken in the design and the
other parameters in Table 8.1b are 0.977β = , 8.90%Hξ = and 6.26%eξ∆ = . The
mass and the natural frequency of the TMDs are
0.02 0.02 163654 3273kgH SM M= = × = and 0.977 3.55 3.47HzH Sf fβ= = × =
respectively. As the maximum response and the damping ratio of the floor are known 163
Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
and the equivalent damping ratio of the floor with the TMDs is given, the maximum
response of the floor with the TMDs can be estimated as follows:
0.015 11.0 0.240 11.0 2.64mm0.0626
SSTMD S
e
ξδ δξ ∆= = × = × = (8.16)
Figure 8.5 shows the FRFs with the largest dynamic displacement at the critical point of
the floor with and without the TMDs. The FRFs are obtained based on a unit uniformly
distributed harmonic load on the dance area of the floor. The ratio of the two peaks with
and without the TMDs is 0.67/2.6 = 0.258 (Figure 8.5). This ratio is close to the ratio of
the two damping ratios, 0.240, in equation 8.16. It is expected that there are some
differences between the simple prediction and the FE analysis as the FE analysis
considers the whole floor where several modes contribute to the vibration. The results
indicate that the effect of vibration in other modes is not significant.
Figure 8.6 shows the response spectra at the critical point on the floor with and without
the TMDs. The spectra envelopes the possible maximum responses induced by the
rhythmic crowd loads containing three harmonics in the range of load frequency
between 1.5 Hz and 2.8 Hz (Ellis and Ji, 2004). It can be noted from Figure 8.6 that:
The ratio of the two peak responses of 4.19/11.0=0.381, which is larger than the ratio
of the peak responses of 0.258 in the FRFs, reflecting the effect of several harmonics.
There is no need to use larger TMDs for this case, as the response is then dominated
by other modes when 0.02α = is considered.
The TMDs with the optimum design parameters are still effective at reducing the
vibration induced by three harmonic loads although the responses of the floor are
larger than the target value.
164
Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
Figure 8.5: FRFs at the critical point (TP5) of the floor with and without the TMDs
Figure 8.6: Response spectra at the critical point (TP5) of the floor with and without
the TMDs
8.5 Conclusions
The chapter introduces the equivalent damping ratio as the objective function to find the
optimum design parameters of TMDs. The study is based on the FRFs of a 2DOF
system. The equivalent damping ratio has a clear physical meaning that is inversely
proportional to the resonance response. The procedure for obtaining the maximum
equivalent damping ratio does not require the use of any particular optimization method
as the objective function and the optimum results can be visualized. The conclusions
obtained from this study are that:
165
Chapter 8: Optimum Design Parameters for a Tuned-Mass-Damper to Maximise the Equivalent Damping Ratio
The mass ratio of the TMD to the structure is not a variable for maximising the
equivalent damping ratio. In general, the larger the mass ratio, the bigger the
vibration reduction.
The maximum equivalent damping ratio and optimum parameters of TMDs provided
in Tables 8.1 are convenient for design of vibration reduction.
A case study shows that the optimum TMDs can be used to reduce the floor vibration
induced by rhythmic crowd loads that include three harmonic terms.
166
Chapter 9: Conclusions and Further Work
9 Conclusions and Further Work
9.1 Conclusions
This thesis develops human body models, examines the characteristics of human body
models, determines the parameters of an interaction human body model in structural
vibration and explores their applications experimentally and theoretically. The research
provides a basis for further experimentation and investigation. The main items
undertaken in this study, the principal results and conclusions are summarised as
follows:
1. Development of a continuous human body model:
• Continuous standing body models in vertical vibrations are developed and
assessed using two available natural frequencies of a biomechanics model and
the mass distributions of an anthropomorphic model.
• The continuous model is able to show the shapes of vibration modes over the
height of the standing body. The fundamental mode shows that the upper part of
the body has much more significant movement than the lower part of the body
while the second mode indicates that the lower torso has the largest movement
while the upper torso moves insignificantly.
• The modal properties for the first two modes are estimated based on the
continuous model (Table 4.4). These parameters can be used for further
investigation of the human-structure interaction models.
• Numerical verification of discrete human body models in structural vibration is
conducted by comparing the natural frequencies of four human-structure
systems in which both continuous and discrete human-body models are placed
on the same SDOF structural system. This shows that the derivation, pattern and
definition of the discrete human-body models in structural vibration are valid. 167
Chapter 9: Conclusions and Further Work
2. Determination of the parameters of the interaction human body models using
available measurements:
• The modal parameters of the two interaction models are estimated with one/two
additional conditions from the outcome of the continuous model.
• The quality of curve fitting of the interaction model is as good as (Model 1c) and
is slightly better (Model 2e) than the published results (Models 1b and 2d)
(Matsumoto and Griffin, 2003). This may indicate the reasonability of the
interaction models.
• Based on Model 2e, 1f is identified at 5.78Hz, and 1ξ of the interaction
model is 0.369. 2f is identified at 13.2Hz, and 2ξ of the interaction model is
0.445. These findings are close to those from biomechanics studies. The
identified parameters are based on the apparent mass experiment of 12 subjects.
Considering the variability of human bodies, more experimental data are
needed.
3. Examination of the characteristics of the interaction human body models:
• Considering a human model to have parameters of Hf = 6.0Hz and γ α =1.36
as an example, When the frequency of a harmonic is less than 10.61Hz, the
body model will vibrate more significantly than the structural model. In
practical situation, the frequency of human load is smaller or far smaller than
10.61. These may give an idea to the designer that the human body response
would be larger than the structural response of a human-structure system
induced by human movements
• The predicted FRFs of the H-S system using the interaction model are close to
that of the measurements. A comparison of 12 case studies shows a good
agreement between the measured and calculated FRFs based on the interaction
body model. The patterns of the FRFs of the human-structure system using the
168
Chapter 9: Conclusions and Further Work
conventional model (Figure 6.12a) are significantly different from that using the
interaction model (Figure 6.12b).
4. Provision of a method to identify the parameters of the interaction model through 45
free vibration tests of standing individuals on a test rig:
• The proposed method is to abstract HSf and HSξ through free vibration
experiments of a human-structure system and then to derive Hf and Hξ of a
human body model based on the measured HSf and HSξ . Using this method,
Hf and Hξ can be identified through free vibration tests of the bare rig and
human occupied rig system.
• The identified natural frequency of the human body varies from 3.17 to 5.81 Hz.
The mean natural frequencies of the four individuals are 4.16, 4.25, 4.57 and
4.33 Hz respectively. The identified damping ratio of the human body varies
from 0.025 to 0.388. The mean damping ratios of the four individuals are
0.086, 0.187, 0.195 and 0.225 respectively. The results show the variation of
human dynamic properties.
• The proposed identification method is sensitive to the input HSf and HSξ . This
would lead to relatively large errors in the identification process.
5. Determination of the optimum design parameters of TMDs for maximising the
equivalent damping ratio:
The equivalent damping ratio monotonically increases as the mass ratio of the TMD
to the structure increases. The mass ratio should not be considered as a variable for
maximising the equivalent damping ratio.
The maximum equivalent damping ratio and optimum parameters of TMDs provided
in Tables 8.1 are convenient for engineers to use for vibration reduction.
The optimum TMDs have been used in the remedial scheme of a dance floor.
169
Chapter 9: Conclusions and Further Work
9.2 Further Work
The research work presented in this thesis will lead further studies as follows:
9.2.1 Examination of displacement and velocity response of a human-structure
system in addition to acceleration response
In practice and research, accelerations are normally used to measure the response of
human-structure system. It requires strong conditions on the mass ratio and frequency
ratio of the human body to the structure to observe two resonance frequencies of a
human-structure system. The presence of two resonance frequencies of the occupied
cantilever tier of the Twickenham Stadium was based on velocity measurements (Ellis
and Ji, 1997). This might indicate that taking velocity measurements may be more
appropriate to reveal the two resonance frequencies of the human-structure system than
taking acceleration measurements. Thus the further work on this topic can be specified
as follows:
a) Revising the equations 6.6 and 6.7 to the displacement and velocity responses
respectively and studying the frequency characteristics as that has been done for
accelerations. This will examine if the FRFs for displacement and velocity can
be better to reveal the two resonance frequencies of a simple human-structure
system.
b) Conducting experimental studies on the human-rig system in which
displacement, velocity and acceleration measurements are to be taken in forced
vibration tests.
9.2.2 Determination of the dynamic parameters of a standing human body using
forced vibration tests
Although the interaction body models are qualitatively correct, the dynamic properties
of the models are qualitatively inaccurate. It requires further studies to determine these
parameters more accurately. One of feasible ways is to explore the use of forced
vibration tests and curve fitting between theoretical and experimental FRF spectra. The
study in 9.2.1 may provide appropriate spectra for curve fitting. 170
Chapter 9: Conclusions and Further Work
9.2.3Determination of the dynamic parameters of a sitting human body using
forced vibration tests
The sitting human body has the same interaction model as the standing human body, but
they have different dynamic properties. The method developed in 9.2.2 will be
applicable to the sitting human body.
171
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