models of a standing human body in structural vibration

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

http://www.manchester.ac.uk/mace

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".

12

http://dx.doi.org/10.1680/stbu.12.00010

http://www.mace.manchester.ac.uk/people/staff/academic-staff/profile/journalpaperabstract/index.htm?id=2948&staffId=152&title=Ji+T%2C+Zhou+D%2C+Zhang+Q.+In+Press.+%22Models+of+a+Standing+Human+Body+in+Vertical+Vibration%22.+Proceedings+of+ICE%2C+Structures+and+Buildings.+

http://dx.doi.org/org/10.1061/(ASCE)EM.1943-7889.0000668

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

15

( , )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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http://en.wikipedia.org/wiki/Angular_frequency



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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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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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


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 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 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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


(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


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


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


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


(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


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









56


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


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


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


58


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


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


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


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


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


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


(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


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


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


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


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


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


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


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


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


Modal mass 22HM (kg) 36.67 49.0%


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


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


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


Modal mass 11HM (kg) 24.48 32.66%


Second Mode


Modal mass 22HM (kg) 41.19 54.95%


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


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


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


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


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


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


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


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


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


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


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


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


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)


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)


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


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


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


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


• 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


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


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


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


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


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


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


• 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


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


Third natural frequency


101


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


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


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


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


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


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


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


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


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


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


a) the structure model b) the body model


( 0.3Hξ = , 1.0β = )



( 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 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


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.



( 0.3Hξ = ,α =0.3)



( 0.3Hξ = ,α =1)

112




( 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.


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.


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


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


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


(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


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


• 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.


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



Bare rig Rig+S1 Rig+S1+

S2

Rig+S1+

S2+S3

Rig+S1+

S2+S3+S4



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


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


• 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


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


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


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


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


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





127


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


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


• 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



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



Bare rig Rig+P1 Rig+P2 Rig +P3 Rig+P4 Sf (Hz)










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


(a)The acceleration time-history and spectrum (b) The damping ratio curve fitting

Figure 7.1: Case 2.0-the bare rig


Figure 7.2: Case 2.1-the rig with subject P1





134










135










136








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


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


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


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.


Figure 7.17: the acceleration-time history and the acceleration spectrum of the rig with a

standing person

140


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


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


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


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


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


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.

146


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


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.

148


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

149


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


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


(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


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


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


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


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


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


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

159


Table 8.1b: Optimum design parameters ( β and Hξ ) of a TMD: 1.5%Sξ =


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

160


Table 8.1c: Optimum design parameters ( β and Hξ ) of a TMD: 2%Sξ =


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

161


Table 8.1d: Optimum design parameters ( β and Hξ ) of a TMD: 5%Sξ =


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


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


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


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


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


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


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


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


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

References

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