THE UNIVERSITY OF NEW SOUTH WALES
SCHOOL OF MECHANICAL AND MANUFACTURING ENGINEERING
Active Vibration Control of Finite Thin-Walled
Cylindrical Shells
Timothy McGann
3158829
Bachelor of Engineering (Mechanical)
October 2008
Supervisor: Dr N. J. Kessissoglou
i
Abstract
Cylindrical shell-like structures exist in pipelines, pressure vessels, aircraft fuselages,
ship hulls and submarine hulls. Improved understanding of the dynamic behaviour and
control of vibration in these applications can reduce the associated problems of
unwanted fatigue stresses, component misalignment, increased wear, energy loss, sonar
detectable acoustic signatures of submarines and passenger discomfort due to both noise
and vibrations in aircraft.
Active control is a technique that involves using a feedforward control loop to inject
energy of the right magnitude and phase so as to cancel out any unwanted oscillatory
energy in the system. Very few experiments have been documented regarding the use of
this technique to control vibrations within thin-walled cylindrical shells. The purpose of
this thesis is to begin filling this existing gap in research.
An experimental investigation was conducted into the use of active control to attenuate
vibrations generated by single frequency excitations. A thin-walled mild steel
cylindrical tube with two thick circular plates fixed at each end was used as the test
structure in this thesis. The modal characteristics of the system were experimentally
determined and validated by comparison with literature. The cylinder was then excited
in the axial direction by an inertial shaker driven at one of three selected system natural
frequencies. Active control was applied by using a secondary inertial shaker mounted at
the opposing cylinder end for each of the chosen frequencies. The use of multiple error
sensor control has also been investigated.
ii
The application of active vibration control effectively produced global attenuation of
cylinder vibrations for two out of the three selected natural frequencies. Further
validation and improvements in resolution of the system natural frequencies are
expected to yield higher performance control. The use of multiple error sensors with a
single control actuator was found to deteriorate active control performance in
comparison to single sensor/single actuator control.
iii
Statement of Originality
The thesis presented herein contains no material or subject that has been accepted
previously for the award of any other degree or diploma in any education institution. To
the best of my knowledge and belief, all the material presented in this thesis, except
where stated and otherwise referenced, is my own original work. I consent that this
thesis be made available for loan and photocopying.
Tim McGann
October 2008
iv
Acknowledgements
I would like to extend a special thankyou to my thesis supervisor Dr Nicole
Kessissoglou for her encouragement and belief in my abilities at all points of this
challenging project. Despite her family commitments with her newborn son, I am very
grateful that she could still to find the time and energy to share as much relevant
guidance and experience as possible. Without such wisdom this project would not have
been possible.
I would also like to thank Russell Overhall whose extensive experience in the use of
acoustics and vibrations technologies was invaluable. Many issues were encountered
during experimentation, and Russell’s passion and willingness to assist aside from his
busy schedule was always greatly appreciated.
Finally I wish to thank all of my family and friends for their support throughout the
duration of this thesis and for the continued encouragement I have received during my
years of university study.
Thank you.
v
Table of Contents
Abstract.........................................................................................................................i
Statement of Originality ............................................................................................iii
Acknowledgements.....................................................................................................iv
Table of Contents ........................................................................................................v
List of Figures...........................................................................................................viii
List of Tables ............................................................................................................xiii
List of Symbols ..........................................................................................................xv
Chapter 1 Introduction and Literature Review .....................................................1
1.1 Introduction .....................................................................................................1
1.2 Thesis objectives .............................................................................................3
1.3 Literature review .............................................................................................4
1.3.1 Dynamic response of cylindrical shells............................................4
1.3.2 Active control .................................................................................11
1.4 Thesis Layout ................................................................................................16
Chapter 2 Vibrations Theory ...............................................................................17
2.1 General structural dynamics..........................................................................17
2.1.1 Vibration fundamentals ..................................................................17
2.1.2 Frequency response function .........................................................20
2.1.3 Coherence ......................................................................................21
2.1.4 Modal analysis ...............................................................................22
2.2 Theory of cylinder vibration .........................................................................24
vi
Chapter 3 Active Control Theory ........................................................................30
3.1 Basic principles .............................................................................................30
3.2 Adaptive feedforward control .......................................................................32
3.3 Active control system design ........................................................................33
Chapter 4 Experimental Testing and Results.....................................................36
4.1 Introduction ...................................................................................................36
4.2 Experimental arrangement ............................................................................36
4.2.1 Experimental rig ............................................................................36
4.2.3 Description of equipment ...............................................................40
4.3 Determination of natural frequencies............................................................42
4.3.1 Free response experimental procedure..........................................42
4.3.2 Free response experimental results ...............................................44
4.3.3 Forced response experimental procedure......................................47
4.3.4 Forced response experimental results ...........................................47
4.3.5 Natural frequencies........................................................................50
4.3 Determination of mode shapes......................................................................51
4.3.1 Mode shape mapping procedure ....................................................51
4.3.2 Mode shape results.........................................................................54
Chapter 5 Active Control .....................................................................................62
5.1 SISO control method.....................................................................................62
5.1.1 SISO control mode 1 results...........................................................64
5.1.2 SISO control mode 2 results...........................................................66
5.1.3 SISO control mode 3 results...........................................................68
5.2 Dual error sensor control method..................................................................69
vii
5.2.1 Dual error sensor control results...................................................70
Chapter 6 Discussion.............................................................................................72
6.1 Mode 227Hz..................................................................................................72
6.1.1 Mode shape ....................................................................................72
6.1.2 Single error sensor control ............................................................73
6.1.3 Dual error sensor control ..............................................................74
6.2 Mode 478Hz..................................................................................................77
6.2.1 Mode shape ....................................................................................77
6.2.2 Single error sensor control ............................................................79
6.2.3 Comparison to theory.....................................................................79
6.3 Mode 546Hz..................................................................................................81
6.3.1 Mode shape ....................................................................................81
6.3.2 Failed control.................................................................................81
Chapter 7 Conclusions and Future Work...........................................................83
7.1 Conclusions ...................................................................................................83
7.2 Future Work ..................................................................................................85
References ..................................................................................................................87
Appendix A Tabulated Experimental Data......................................................91
Appendix B Engineering Drawings ................................................................102
viii
List of Figures
Figure 1.1 Stringer stiffened cylinder.........................................................................8
Figure 1.2 Ring stiffened cylinder..............................................................................8
Figure 1.3 Mode shapes of ring stiffened circular cylindrical shell support by shear
diaphragm [4] ..........................................................................................10
Figure 2.1 a) SDOF system b) and free body diagram.............................................17
Figure 2.2 Frequency response of a forced SDOF system. ......................................20
Figure 2.3 Example of a time and frequency domain transformation for a vibrating
beam. .......................................................................................................21
Figure 2.4 First mode of vibration in a tensioned string. .........................................23
Figure 2.5 Mode separation of frequency response function [28]............................24
Figure 2.6 Cylindrical shell co-ordinate system [1]. ................................................24
Figure 2.7 First three longitudinal mode shapes of a cylinder [4]............................27
Figure 2.8 First 4 circumferential mode shapes of a cylinder [4].............................27
Figure 2.9 Combined longitudinal and circumferential modes [1] ..........................28
Figure 2.10 Natural Frequencies of unstiffened cylinder, a = 2m, L = 6m, h = 0.02m
.................................................................................................................29
Figure 2.11 Natural Frequencies of unstiffened cylinder, a = 2m, L = 3m, h = 0.02m
.................................................................................................................29
Figure 3.1 Basic principle of superposition..............................................................30
Figure 3.2 First patented active noise control concept .............................................31
Figure 3.3 Adaptive feedforward vibration control system......................................32
Figure 3.4 Adaptive feedforward control system functional diagram [30]. .............35
Figure 4.1 Cylinder end-plate and shaker assembly sketch......................................37
ix
Figure 4.2 A-frame assembly ...................................................................................38
Figure 4.3 (a) Original surface of inertial shaker. (b) Inertial shaker with adapter-
plate attachment ......................................................................................39
Figure 4.4 Modified assembly with force transducer ...............................................39
Figure 4.5 Equipment Configuration for free response testing. ...............................43
Figure 4.6 Hammer impact locations and accelerometer location. ..........................44
Figure 4.7 Frequency response functions of the cylinder from an impulse excitation
at different locations................................................................................45
Figure 4.8 Coherence functions of the cylinder from an impulse excitation at
different locations. ..................................................................................45
Figure 4.9 Frequency response function of the cylinder from an impulse excitation
at point 13................................................................................................46
Figure 4.11 Equipment configuration for forced response testing. ............................47
Figure 4.12 Frequency response functions of the cylinder at multiple accelerometer
locations using a forced broadband excitation. .......................................48
Figure 4.13 Coherence functions of the cylinder at multiple accelerometer locations
using a forced broadband excitation. ......................................................48
Figure 4.14 Frequency response function of the cylinder at point 19 using a
broadband excitation. ..............................................................................49
Figure 4.15 Coherence function of the cylinder at point 19 using a broadband
excitation. ................................................................................................49
Figure 4.16 Experimental mesh definition for 11 x 16 point mesh............................52
Figure 4.17 3-D cylinder mesh plot of uncontrolled 227Hz mode shape...................55
Figure 4.18 Longitudinal mode shape at 227Hz measured along b = 4 using 33 point
mesh. .......................................................................................................56
x
Figure 4.19 Circumferential mode shape at 227Hz measured about x = 16 using 32
point mesh. ..............................................................................................56
Figure 4.20 3-D cylinder mesh plot of uncontrolled 478Hz mode shape...................57
Figure 4.21 Longitudinal mode shape at 478Hz measured along b = 8 using 33 point
mesh. .......................................................................................................58
Figure 4.22 (a) Circumferential mode shape at 478Hz measured about x = 8 using 32
point mesh. (b) Circumferential mode shape at 478Hz measured about x
= 24 using 32 point mesh. .......................................................................58
Figure 4.23 3-D cylinder mesh plot of uncontrolled 546Hz mode shape...................59
Figure 4.24 Longitudinal mode shape at 546Hz measured along b = 16.5 using 33
point mesh. ..............................................................................................60
Figure 4.25 Circumferential mode shape at 546Hz measured about x = 16 using 32
point mesh. ..............................................................................................60
Figure 4.26 (a) Circumferential mode shape at 546Hz measured about x = 8 using 32
point mesh (b) Circumferential mode shape at 546Hz measured about x =
24 using 32 point mesh. ..........................................................................61
Figure 5.1 SISO Active control hardware configuration..........................................62
Figure 5.2 (a) 3-D cylinder mesh plot of uncontrolled magnitudes at 227 Hz. (b) 3-
D cylinder mesh plot of controlled magnitudes at 227Hz. .....................64
Figure 5.3 Controlled and uncontrolled magnitudes at 227Hz measured along the
cylinder length through b = 4 using a 33 point mesh..............................65
Figure 5.4 Controlled and uncontrolled response at 227Hz measured around the
circumference through x = 16 using a 32 point mesh. ............................65
Figure 5.5 (a) 3-D cylinder mesh plot of uncontrolled magnitudes at 478 Hz. (b) 3-
D cylinder mesh plot of controlled magnitudes at 478Hz. .....................66
xi
Figure 5.6 Controlled and uncontrolled magnitudes at 478Hz measured along
cylinder length through b = 8 using a 33 point mesh..............................67
Figure 5.7 (a) Controlled and uncontrolled response at 478Hz measured around the
circumference through x = 8 using a 32 point mesh. (b) Controlled and
uncontrolled response around measured around the circumference
through x = 24 using a 32 point mesh. ....................................................67
Figure 5.8 Location of error sensor in 546Hz active control attempt.......................68
Figure 5.9 Dual sensor active control configuration. ...............................................69
Figure 5.10 Controlled and uncontrolled responses at 227Hz using two error sensors
at (16, 4) and (16, 12) and measured along cylinder length through b = 4
using a 33 point mesh..............................................................................70
Figure 5.11 Controlled and uncontrolled response at 227Hz using two error sensors
at (16, 4) and (16, 12) and measured around the circumference through x
= 16 using a 32 point mesh. ....................................................................71
Figure 6.1 The (m, n) = (1, 2) mode shape found by magnitude measurements
compared with the measurement of imaginary components in Goodwin
[1]. ...........................................................................................................72
Figure 6.2 Comparison of lengthwise control results between the use of 2 error
sensors and a single error sensor.............................................................75
Figure 6.3 Control results about the circumference comparing the use of 2 error
sensors and a single error sensor.............................................................76
Figure 6.4 Cylinder beginning, middle and end circumferential mode shapes at
478Hz ......................................................................................................78
Figure 6.5 n =2 circumferential mode shapes measured at ¼ and ¾ along the
cylinder length at 478Hz .........................................................................78
xii
Figure 6.6 Expected control results for the m = 2 lengthwise mode shape [22] ......80
Figure 6.7 Active control results comparison to expected theoretical control at
478Hz ......................................................................................................81
xiii
List of Tables
Table 4.1: List of components used during experimentation ..................................40
Table 4.2: Natural frequency comparison from impact testing................................50
Table 4.3: Natural frequency comparison from forced testing ................................50
Table A1: Global uncontrolled mode shape data at primary frequency of 227Hz ..91
Table A2: Global controlled mode shape data at primary frequency of 227Hz ......91
Table A3: Lengthwise mode shape data for both controlled and uncontrolled
responses at primary frequency of 227Hz during active control ............92
Table A4: Circumferential mode shape data for both controlled and uncontrolled
responses at primary frequency of 227Hz during active control ............93
Table A5: Global uncontrolled mode shape data at primary frequency of 478Hz ..94
Table A6: Global controlled mode shape data at primary frequency of 478Hz ......94
Table A7: Lengthwise mode shape data for both controlled and uncontrolled
responses at primary frequency of 478Hz during active control ............95
Table A8: Circumferential mode shape data for both controlled and uncontrolled
responses at primary frequency of 478Hz during active control with error
sensor at point (8,8) and accelerometer about circumference x = 8 .......96
Table A9: Circumferential mode shape data for both controlled and uncontrolled
responses at primary frequency of 478Hz during active control with error
sensor at point (8,8) and accelerometer about circumference x = 24 .....97
Table A10: Global uncontrolled mode shape data at primary frequency of 546Hz ..98
Table A11: Lengthwise mode shape data for uncontrolled response at 546Hz.........98
Table A12: Circumferential mode shape data for uncontrolled response at 546Hz ..99
xiv
Table A13: Lengthwise mode shape data for both controlled and uncontrolled
responses at primary frequency of 227Hz under active control using two
error sensors ..........................................................................................100
Table A14: Circumferential mode shape data for both controlled and uncontrolled
responses at a primary frequency of 227Hz under active control using
two error sensors ...................................................................................101
xv
List of Symbols
M Elemental mass of a single degree of freedom model [kg]
C Damping coefficient [kg/s]
K Spring constant [N/m]
ω Frequency of input forcing [rad/s]
ωn Undamped natural frequency [rad/s]
ωd Damped natural frequency [rad/s]
ζ Damping ratio
φ Phase angle of forced response [rad]
F0 Magnitude of general harmonic forcing [N]
γxy(ω) Coherence function
H(ω) Frequency response function
X(ω) Input autospectrum of frequency response function
Y(ω) Output autospectrum of frequency response function
Gxx Input auto-spectral density
Gyy Output auto-spectral density
Gxy Cross spectral density
)t(x Time domain input function
)t(y Time domain output response
u Cylinder axial displacement [m]
v Cylinder circumferential displacement [m]
w Cylinder radial displacement [m]
[L] Donnell Mushtari differential matrix operator
{ui} Displacement vector
xvi
x Axial coordinate reference
θ Tangential coordinate reference
z Radial coordinate reference
a Cylinder radius [m]
ρ Material density [kg/m3]
E Young’s modulus [N/m2]
v Poisson’s ratio
s Non-dimensional cylinder length
h Shell thickness [m]
β2 Non-dimensional thickness parameter
n Circumferential mode number
m Longitudinal mode number
b Circumferential mesh point coordinate
1
Chapter 1 Introduction and Literature Review
1.1 Introduction
Vibrations are inherently present in all aspects of everyday life. Examples of industries
where knowledge in the area of vibrations is deemed important include the transport,
construction, aerospace, naval, manufacturing, military and music industries to name a
few. These applications all contain mechanical systems, which can be viewed upon as
comprising of distributed elements with characteristics of mass, stiffness and damping.
A vibrating response in these systems occurs when an external or internal force excites
the system. Such a force is generally either periodic or random in nature. Periodic
loadings are most often a result of mass imbalances in machinery such as motors and
propellers or cyclic impacts from reciprocating compressors and punching machines.
The system responses from such harmonic forcings are generally steady state motion
whilst the response from a single random excitation is expected to be a decaying
oscillation. In all cases where the structure is surrounded by a fluid, it is possible for
noise generation to occur due to the fluctuating pressure disturbance that arises from
vibrating motion.
The specific area of vibrations in thin cylindrical shells is applicable to understanding
and controlling the dynamic behaviour of aircraft fuselages, submarine hulls, ship hulls,
satellite launches, pipelines and pressure vessels where vibrations and the associated
noise are considered an issue. Excitations caused by the operation of propellers, motors
and other machinery in these applications can generate potentially damaging fatigue
stresses, component misalignment, increased wear, energy loss, passenger stress and
discomfort from both noise and vibration and finally sonar detectable acoustic
2
signatures in submarines. In order to reduce these undesired effects it is necessary to
have a knowledge base of the dynamic behaviour of cylindrical systems and of
strategies that can be employed to attenuate the vibration and noise levels.
Each cylindrical system, like all other mechanical systems, has a series of natural
vibration frequencies and mode shapes determined by the system geometry, size,
material properties and boundary conditions. It is important to note that structural
discontinuities such as shell stiffeners, bulkheads, junctions, changes in diameter and
end closures and other complicating factors such as fluid loading and fluid dynamic
effects should be considered if a more realistic cylindrical shell vibration analysis is
desired. Studies have shown that these factors can play a significant part in determining
the free response of the system.
Once the free response characteristics such as resonant frequencies of a system have
been understood, active and passive control methods can be implemented to reduce the
undesired effects of vibration. Passive control involves modifying the mass, stiffness
and damper properties to more effectively absorb radiated energy resulting from system
disturbances. Active control involves the use of feedback and feedforward control loops
to detect the unwanted disturbance and apply a secondary force to minimise the
resulting structural response.
3
1.2 Thesis objectives
This thesis is an extension of the experimental work conducted by Goodwin [1] on the
active control of low frequency vibrations in a thin cylindrical shell under an applied
harmonic axial excitation. An area of significant research in which this knowledge base
can be applied is the global control of low frequency vibration modes in submarines
resulting from fluctuating disturbances transmitted through the propeller shafting
system. Attenuation of such responses is highly important in military applications where
stealth is of the essence and the radiated acoustic signature due to shell vibrations is
undesired. The experiment in Goodwin [1] investigated the control of cylindrical
vibrations for a single mode shape and concluded that global attenuation of this mode
was effective by use of a single error sensor and single actuator. Goodwin [1] based this
conclusion on measurements taken along a single axial and a single circumferential anti-
nodal line with the assumption that attenuation also occurred in the unmeasured regions
of the cylinder.
The objectives of this thesis are to:
• Verify and improve the results obtained by Wayne Goodwin on the existing
experimental rig by taking measurements over the entire cylinder to confirm
global control.
• Investigate the effectiveness of the active control of higher order mode shapes
within the frequency limitations of the available EZ-ANCII controller.
• Compare the performance of multiple error sensor control against single error
sensor control as applied to cylinder vibrations.
4
1.3 Literature review
1.3.1 Dynamic response of cylindrical shells
The dynamic behaviour of vibrating cylindrical shells has been an area of research
interest since the 1960s and 70s [2]. Knowledge in this field is useful in many
engineering applications including aircraft fuselages, submarines, ship hulls, pipelines
and pressure vessels. To effectively attenuate undesired vibrations in these applications
it is highly necessary to determine the natural frequencies and mode shapes of the
cylindrical shell structures involved. Acoustic radiation and structural deflection is
strongest at these resonant frequencies and hence they are more appropriate to control
[3].
Shell structures are complex forms of plate structures, having all the same
characteristics as plates but with the addition of curvature. Unlike beams and plates, the
equations of motion and the effects of boundary conditions in thin cylindrical shells are
much more complex. This is due to strong interrelation between angular, axial and
circumferential displacements within cylindrical shapes. A coupling effect takes place,
whereby an axial force or excitation can generate a displacement in radial and tangential
directions as well as the expected axial direction. Common agreement has been met
over the classical bending theory in plates, which utilize fourth order equations,
however literature still remains divided over the most appropriate theory for cylindrical
shells, which generally use eighth order equations. Many theories have been developed
over the years with differences usually attributed to the point during their derivation at
which simplifying assumptions are made, and or the choice of assumptions themselves
[4]. These simplifications may include, but are not limited to: neglecting tangential
terms and inertial terms, and the use linear approximations in the characteristic
5
equations. Despite the popularised use of some of the existing cylindrical shell theories,
it is still necessary to specify the theory used when performing analytical work with
cylindrical shells.
Armenàkas, Gazis and Herrmann [2] detail the theoretical eigen-value solutions to the
three dimensional linear theory of elasticity equations for stress free cylindrical
surfaces. Numerical computations were performed in Fortran to find the first six natural
frequencies for a wide range of geometric parameters. In performing the analysis, the
cylinder was assumed to be hollow, isotropic and infinitely long. Their results tabulate
and plot the normalised natural frequencies and corresponding radial, tangential and
axial mode shapes for varied thickness to radius ratios and thickness to length ratios. It
was found that the normalized natural frequencies vary considerably for high thickness
to length ratios and vary much less as the thickness to length ratio decreases. As it
stands, the results offer a satisfactory check on the validity and applicability of a range
of other future and existing simplified shell theories for determining free dynamic
behaviour of cylinders.
Leissa [4] is referred to in many pieces of literature regarding the theory vibration of
cylindrical shells. The publication summarizes a large range of the existing approximate
cylinder theories of the era, including the: Donnell-Mushtari, Love-Timoshenko,
Reissner-Naghdi-Berry, Vlasov, Sanders, Flügge, and Houghton-Johns. Leissa outlines
the principles from which these theories were derived, including strain displacement
shell theory, force and moment resultants and the fundamental equations of motion. The
assumptions made during the derivation of each cylinder theory are also specified.
Comparative tables are provided showing the numerical variations in frequency
6
solutions that exist between the approximate methods and the exact solutions to the
three-dimensional elasticity theory as seen in Armenàkas et al [2]. It was found that a
close agreement was met between theories for shells that were very thin, of moderate
length and with small numbers of circumferential waves. In addition to plane isotropic
cylinders, complicating effects such as rings and stringers, initial stresses, variable
cylinder thickness, large non-linear deflections, shear deformation, rotary inertia,
composite layered materials and the effects of surrounding fluids are discussed. In each
case, the effects of cylinder end boundary conditions such as free, shear diaphragm and
clamped in various combinations are considered.
Numerical analysis and justification of the theories outlined in Leissa [4] has continued
over the years for many different boundary conditions and shell configurations.
Buchanan and Chua [5] recognized the absence of published vibration results for finite
length cylinders under the fixed-free and fixed-fixed boundary conditions. They
performed a finite element analysis involving both a standard isotropic material and
Beryllium to tabulate the non-dimensional natural frequencies and corresponding mode
shapes for various length/radius ratios. It was found that as this ratio increased, the
effect of material characteristics tended to have more influence on the order of natural
frequencies than did the cylinder geometry. It is stated in Buchanan [5] that with enough
information, the effects caused by geometry and those caused by material properties can
be observed in separation.
Various comparative studies regarding cylindrical shell theories have been performed.
One such study was that of El-Mously [6] which compared three explicit formulae that
are used for predicting the natural frequencies and mode shapes of thin cylindrical
7
shells. The formulae considered include: the Weingarten-Soedel, Calladine-Koga and
the Timoshenko-beam-on-Pasternak-foundation. All of these approximations contain
limitations and restrictions for use depending on the certain geometric ratios. The
formulae were numerically compared with the analytical solutions to Flügge’s equations
and with finite element results.
Saijyou and Yoshikowa [7] presents experimental validation for the use of a “modified
bending stiffness” approach to estimate the modes of vibration that are actually excited
in a simply supported cylindrical shell. They found that different modes present
different levels of modified bending stiffness, which in turn effect the magnitude of the
driving force necessary to excite a particular mode. If two modes have similar natural
frequencies, the more flexible mode is likely to be excited.
An approximate approach using superposition of the axial and circumferential standing
waves determined by the wave numbers n, the number of full circumferential waves,
and m, the number of longitudinal waves, is presented in Zhang [8]. The axial wave
number is approximated from an equivalent beam with similar boundary conditions to
represent the cylindrical shell. Finite element modelling was used to validate the method
and found the natural frequencies to be within 2% of each other. These results were
expressed in Zhang to be “reasonably accurate”. Wang and Lai [9] claimed that this
theory was flawed because it neglects the coupling that exists between axial and
circumferential vibration and is only reasonable for relatively long cylindrical shells.
Zhang [10] confirmed that the theory was appropriate as it was validated with other
methods in literature for simply supported/simply supported, clamped/clamped and
clamped/simply supported boundary conditions. A large advantage of this method is
8
that it can be easily extended to include more complex boundary and loading conditions
without the need for intensive computations.
While it is clear that a vast range of theories and approximations exist for idealized
cylinders undergoing free vibration, the application of such theories to structures such
as submarines and aircraft fuselages requires further effort to account for the existence
of complicating effects. This includes studies into the effects of stringer and stiffeners,
illustrated in figures 1.1 and 1.2 respectively, on the natural properties and radiated
acoustic signatures of cylinders.
Figure 1.1 Stringer stiffened cylinder
Figure 1.2 Ring stiffened cylinder
9
Ruotolo [11] outlines that there are two primary ways for determining the effects of
stiffeners on cylindrical shells, which include: treatment of the stiffeners as discrete
elements or by averaging the properties over the shell surface as a smeared approach.
Figure 1.3 demonstrates the effect that ring stiffeners can have on the lengthwise mode
shape of a shear diaphragm - shear diaphragm cylindrical shell for varied
circumferential mode numbers. The net mode shape in this example is obvious which
therefore justifies the use of a smeared approach for low circumferential mode numbers
when an adequate number of ring stiffeners are used. The work undertaken in Ruotolo
[11] compared the use of Love’s, Donnell’s, Sander’s and Flügge’s shell theories under
the smeared stiffener condition for different stiffness scenarios including: only rings,
only stringers and rings plus stiffeners. Analytical results were compared with a finite
element model and it was found that all theories were in close agreement except for
Donnell’s, which gave errors of up to 40% for the natural frequencies of the structure.
Ruotolo [12] goes on to analytically address the influence of structural stiffness theories
on interior noise generation.
10
Figure 1.3 Mode shapes of ring stiffened circular cylindrical shell support by shear
diaphragm [4]
Norwood [13] has written a detailed literature review in regards to cylinders and the
effect of boundary conditions, end closures, stiffeners and external and internal fluid
loadings, as applies to submarine structures. The review summarises the knowledge that
has been developed such as modelling the reduction in modal frequency under external
pressure and water loading and the effects of ring stiffeners causing an increase in
system natural frequencies. The conclusions from this study were that further work is
needed to improve the fluid/structural interaction in finite element analysis and more
11
specific study into the behaviour of internal bulkheads and the deep frame stiffeners of
submarines is required.
Ruzzene and Baz [14] have performed a finite element analysis of the effects of
stiffeners, damped stiffening and water loading on the associated acoustic pressure field.
The results show that stiffening and damping are suitable methods for passively
reducing the vibration and sound radiation from submerged shells. However, low
frequencies are much more difficult to attenuate by passive methods and it stands that
active control is expected to be a much more effect means of vibration in stiffened
cylinders.
1.3.2 Active control
Attenuation of unwanted noise and vibration in mechanical systems is an area of
particular interest to engineers. Occupant discomfort due to the internal noise generation
within cylindrical structures such as submarines and aircraft fuselages, and the advances
in sonar detection of external acoustic signatures in military marine vessels has led to an
increasing need for developments in noise and vibration control. Passive control and
active control are the two most common attenuation techniques. Passive control has
been an area of wide research and is a procedure involving the modification of the
physical parameters of a system such as mass, stiffness and damping. However, this
technique is limited at low frequencies due to the larger energy absorptive mass
requirements creating conflict with lightweight structural necessity and issues with
system stability. As a consequence, active control methods have proven more successful
in the low frequency range.
12
Active vibration control involves actively adding more energy to a system such that
when it is superimposed with the original response, the total combined response is
reduced. In single frequency noise and vibration control this can be achieved by
introducing a secondary disturbance that is 180 degrees out of phase with the original.
Although the basic concept of active control has been known for many decades, recent
advances in control theory and solid-state transducer and microprocessor technology
have allowed the method to become more practically feasible.
In order to achieve anti-phase noise and vibration attenuation, it is important to use an
appropriate control system to ensure that destructive rather than constructive
interference is sustained in a stable manner. Sievers and Andreas [15] outline the control
theory behind many systems that can be applied to reducing narrowband and single
frequency disturbances. They discuss various methods of control including: adaptive,
discrete, analogue, frequency domain and time domain approaches. The discussions
indicate the wide variety of options available for control, the choice of which depends
on the type of disturbance and level of performance desired. Multiple input multiple
output adaptive feedforward systems are discussed and a new compensator design was
put forward to increase the robustness of this control algorithm.
An experimental analysis of active control of vibratory power transmission in a
cylindrical shell is discussed in Pan and Hansen [16]. This analysis involved extending
the Flügge equation to account for the linear inertia of cylinder walls as applies to a
cylinder of semi infinite length, simply supported at one end and anechoically
terminated at the other. The experiment investigated the influence on attenuation control
of variables including; error sensor type and location, control force type and location,
13
cylinder radius and thickness and the excitation frequency of harmonic radial forces
arranged circumferentially around the cylinder rig. The results found that an attenuation
of 30dB in transmitted vibration power could be achieved by 3 or more control forces
and was more effective in the radial direction than the axial. Extensional wave
transmission gives a good approximation to the total power transmission while
acceleration and power transmission cost functions are effective if chosen for just a
single direction due to the wave coupling present in cylinders.
Thomas et al [17,18] conducted research on the active control of sound transmission
through a thin cylindrical shell as applied to aircraft with high-speed turbo props. Their
model was based on previous works of Bullmore et al [19,20] which looked at
producing reductions in low frequency cabin noise related to harmonic propeller blade
tones by using primary and secondary sound sources and comparing the results with
computational modelling. Thomas et al [17,18] used a theoretical expression giving the
total kinetic energy of the cylinder walls to form a control cost function. The optimal
configuration of secondary forces for minimizing the radial component of energy was
set as the control criteria. The results were found to show that large reductions in
vibration energy were difficult to due to the high number of structural modes occurring
as a response to the primary forcing. Control was found to be most effective when there
were fewer modes governing the response at the low frequency modal density regions.
However, Nelson and Elliot outline in numerous sources [17,18,21] that reducing the
vibration energy of a distributed structural system does not necessarily imply a
reduction in radiated sound levels.
14
The behaviour of structural acoustic coupling within cylindrical shells is an area that has
only recently received literary coverage. Kessissoglou [22] describes the effect of active
vibration control of a submerged cylinder on its radiated sound pressure levels. An
analytical cylinder model of length L was set up containing two mass balanced end
plates and two internal dummy bulkheads. A primary axial input excitation was applied
at one end with a secondary axial control force at the opposing end. Both axial and
radial displacements were considered when investigating the radiated sound pressure
levels. To control axial displacements, an error sensor was located at each end of the
cylinder. For control of radial displacements a ring of error sensors was located around
the circumference at selected axial positions along the cylinder. The first two axial
resonance modes were observed while maintaining constant axisymmetric
circumferential mode. The results indicated that axial attenuation achieved much higher
reduction in acoustic pressure level than radial attenuation at all tested resonant
frequencies.
Active control system design requires decisions regarding the optimal location for
control actuators and error sensors within the structure to be controlled. Kessissoglou,
Ragnarsson and Lofgren [23] performed an analytical and experimental study into this
mater on an L-shaped plate with simply supported conditions along the parallel L-
shaped edges and free conditions at the two ends. They found that the level of control is
much more dependent on actuator and error sensor position rather than the quantity or
type of actuators present. Optimised control occurred when the control force was in line
with the primary force and the error sensor midway between in a symmetrical
configuration.
15
In large shell and plate structures, the use of a single error sensor and single actuator to
achieve global control may not produce the best level of attenuation possible. For this
reason it was necessary to enhance the available literature with a study on the use of
multiple actuators and error sensors. Keir, Kessissoglou and Norwood [24] performed a
theoretical and experimental analysis on a T-shaped plate with simply supported
conditions along the parallel T-shaped edges and free conditions elsewhere. The results
showed that multiple error sensors for a single actuator caused deterioration in control
performance. Two error sensors with two dependently driven actuators produced greater
attenuation than the single sensor single actuator arrangement with less importance on
the choice of error sensor location. Use of three sensors and three actuators showed only
slight improvement to the latter. Independently driven actuators were shown to produce
better attenuation levels than dependently driven actuators for arbitrary sensor locations.
Symmetrical arrangements were found to be the most effective for single actuator and
single error sensor control.
Whilst active control of vibrations under single frequency or multiple known frequency
excitations has received considerable literature coverage, very little has been reported
on the application of feedforward broadband structural control. Vipperman, Burdisso
and Fuller [25] discuss the use of adaptive least mean square and recursive least mean
square algorithms in controlling broadband vibration of a simply supported beam.
Various impulse filters were applied and it was found that a Finite Impulse Response
filter gave significant improvements in control performance with power reductions of
up to 20dB at resonant.
16
The combining of both passive and active control strategies is an area of recent
development expressed in Baz and Chen [26]. This work outlines the modelling of
active constrained layer damping using energy principles to describe the vibratory
behaviour of simply supported cylindrical shells with composite fabric walls. The
cylinder construction uses a combination of passive constrained layer damping
materials with embedded piezo-electric actuators to apply control. Control is achieved
by controlling the strain experienced by the material in the constraining layer. Optimal
balance between the simplicity of passive techniques and the efficiency of active
methods is sought. The advantage of using such a system is that vibrations of large
structures can be controlled without the need for large actuation voltages.
1.4 Thesis Layout
Chapter 2 presents a summary of the background mathematical theories that are most
commonly used to describe vibrations in general and the dynamic behaviour of thin
cylindrical shells.
Chapter 3 outlines the basic theory behind active control using feedforward systems as
applicable to this thesis.
Chapter 4 presents the experimental testing and results for the free and forced vibration
modal characteristics of the cylinder.
Chapter 5 presents the active control experimental procedure and results for attenuating
single frequency oscillatory responses in the cylinder.
17
Chapter 6 contains a detailed discussion of the trends observed in the experimental
results of this thesis.
Chapter 7 summarizes the results and concludes the theoretical developments achieved
in this thesis. An outline for necessary future work is also given.
17
Chapter 2 Vibrations Theory
2.1 General structural dynamics
2.1.1 Vibration fundamentals
A vibration or oscillation is any repeated motion of a physical system. Every
mechanical system can be understood to consist of a continuous distribution of elements
each displaying the characteristics of mass, elasticity and damping. A single-degree-of-
freedom (SDOF) model as shown in figure 2.1 is the most basic unit from which more
complex multi-degree-of-freedom systems can be constructed for vibrations analysis.
The number of degrees of freedom of a system equals the number of independent
coordinates necessary to completely specify the motion of that system. Ideally,
mechanical systems such as thin cylindrical shells would be modelled as continuous
systems with an infinite number of degrees of freedom. However, obtaining the exact
solutions to these systems is often very complicated and sometimes not possible so it is
best to use lumped parameter models to approximate the continuous behaviour. In
general, results of greater accuracy are obtained by increasing the number of degrees of
freedom, however, this comes with the downside of requiring more computations.
Figure 2.1 a) SDOF system b) and free body diagram.
18
On applying force equilibrium to the free body diagram of figure 2.1b for the system in
free vibration, the following homogeneous differential equation is obtained.
0KxxCxM =++ &&& (2-1)
Where M is the elemental mass, C is the damping coefficient, K is the spring constant
and x is the displacement of the mass from its equilibrium position. A single dot above
the x denotes the first derivative of displacement with respect to time, known as
velocity. The double dot above the x denotes the second derivative of displacement with
respect to time, known as acceleration.
The general solution to a SDOF system in free vibration is given by an exponentially
decaying sine function as follows:
)tsin(Ae)t(x d
tn φ+ω=
ζω− (2-2)
Where A is the amplitude, t is the time, φ is the phase angle, ζ is the damping ratio, ωn
is the natural frequency and 2
nd 1 ζωω −= is the damped natural frequency.
The SDOF system shown in figure 2.1 can also be excited by a persistent disturbance
instead of an initial excitation as in the free response case. If a harmonic force or
displacement excitation is applied then the homogeneous equation in equation (2-1) is
modified to include the disturbance and is written as follows:
tsinFKxxCxM 0 ω=++ &&& (2-3)
19
Where F0 is the amplitude of the forcing and ω is the frequency of the applied harmonic
forcing. The general steady state solution to equation (2-3) is given by:
2
n
22
n
0
21
)tsin(
K
F)t(x
+
−
−=
ω
ωζ
ω
ω
φω (2-4)
The non-dimensional frequency response amplitude is shown in figure 2.2. The
important feature to note from this graph is the very high amplitude that occurs when
the driving frequency (ω) is somewhat close to the natural frequency (ωn). Under this
condition, the system is described as being driven at resonance. The increased
amplitudes due to resonance can lead to increased displacements, increased noise
generation and higher stress levels that can accelerate fatigue failure. In order to reduce
these undesired resonant effects, it is important to have the ability to change a systems
natural frequency, adjust the driving frequency or destructively interfere with the
driving signal by wave superposition.
20
Figure 2.2 Frequency response of a forced SDOF system.
2.1.2 Frequency response function
Complex oscillatory behaviour is often very difficult to analyse within the time domain.
It is much simpler to deal with vibrations data in the frequency domain by performing a
Fast-Fourier-Transform (FFT) manipulation. In frequency domain analysis of linear
systems, a frequency response function (FRF) represents the transfer function H(ω) of
the system and is the mathematical relationship between the input X(ω) and output Y(ω)
frequency autospectrums given for a single input/single output set-up as follows [27]:
)(X
)(Y)(H
ω
ω=ω (2-5)
21
The transformation between time domain and frequency domain is shown in figure 2.3
where the top three boxes represent the time and spatial domain, whilst the bottom three
represent the frequency domain for a vibrating cantilever beam.
Figure 2.3 Example of a time and frequency domain transformation for a vibrating
beam.
2.1.3 Coherence
The functions X(ω), Y(ω) and H(ω) apply to ideal linear systems which contain no
noise. In reality the degree of correlation between measured input and measured output
must be checked. This is performed by the coherence function, )(2
xy ωγ which is defined
as follows [27]:
)(G)(G
)(G)(
yyxx
2
xy2
xyωω
ω=ωγ (2-6)
22
Where )(X)(X)(G *
xx ωω=ω , )(Y)(Y)(G *
yy ωω=ω and )(X)(Y)(G *
xy ωω=ω are the
input auto-spectral density, output auto-spectral density and cross-spectral density
respectively. X*(ω) and Y
*(ω) are the complex conjugates of the input X(ω) and output
Y(ω) respectively.
The coherence function has an upper bound of 1 indicating a system with no extraneous
noise and a lower bound of 0 indicating absolutely no correlation between input and
output measurements. The condition 1)(0 2
xy <ωγ< generally occurs due to: extraneous
noise, resolution bias errors, system non-linearity or y(t) caused by additional inputs
apart from x(t).
2.1.4 Modal analysis
A multi degree of freedom (MDOF) system consisting of N degrees of freedom requires
N co-ordinates to completely specify its motion and has N natural frequencies.
Corresponding to each of these natural frequencies is a mode shape, which describes the
expected curvature pattern of system when oscillating at that frequency. An example of
the first mode shape of a vibrating string under tension and its resultant when combined
with its second harmonic is illustrated in figure 2.4.
23
Figure 2.4 First mode of vibration in a tensioned string.
The collective term for the natural frequency and its corresponding mode shape is
called a ‘mode’ of vibration. A continuous system can be described as having an infinite
number of degrees of freedom. This implies an infinite number of modes whereby the
superposition of each simple mode shape will result in the total wave motion of the
structure under vibration. Gade et al [28] explains that it is possible to break down the
FRF of a continuous system into its constituent modes which each have a characteristic
resonant frequency, damping and mode shape. This break down is represented in figure
2.5. In general if there is reasonable separation between resonance points and the
structure is lightly damped, then coupling between mode shapes is minimal. Under this
condition, a system driven at resonance can be considered to behave primarily as a
SDOF system. Thus for a thin cylindrical shell, each peak on the FRF is expected to
have a unique associated mode, assuming that the natural frequencies are reasonably
separated.
24
Figure 2.5 Mode separation of frequency response function [28].
2.2 Theory of cylinder vibration
Cylindrical shell theory is most commonly understood by using the cylindrical
coordinates shown in figure 2.5. The theory in this report is presented in terms of axial
displacement, u, circumferential displacement, v, and radial displacement, w.
Figure 2.6 Cylindrical shell co-ordinate system [1].
The simplest theory used to describe the motion of a cylinder is the Donnell-Mushtari
set of equations. These equations were developed for uniform unstiffened cylindrical
shells of homogenous isotropic linearly elastic material properties undergoing relatively
small displacements.
25
With appropriate boundary conditions, the equations can be solved to obtain the eigen
values, which in turn give the natural frequencies of the system. Leissa [4] outlines the
Donnell-Mushtari equations of cylindrical motion in matrix form as follows:
[ ] [ ]0}u{ i =L (2-7)
Where {ui} is the displacement vector given by
=
w
v
u
}u{ i (2-8)
u, v, and w are the components of displacement in the x, θ, z directions respectively as
shown in figure 1. [L] is a differential matrix operator.
The Donnell – Mushtari matrix system is given by
=
∂
∂−+
∂
∂+
∂
∂+
∂
∂
∂
∂
∂
∂
∂
∂−−
∂
∂+
∂
∂−
∂∂
∂+
∂
∂
∂∂
∂+
∂
∂−−
∂
∂−+
∂
∂
0
0
0
w
v
u
tE
)v1(
saa
1
R
1
xa
v
a
1
tE
)v1(
a
1
x2
)v1(
xa2
)v1(
xa
v
xa2
)v1(
tE
)v1(
a2
)v1(
x
2
22
2
2
2
2
2
2
2
2
2
2
2
22
2
2
22
2
2
2
2
22
2
2
22
2
ρ
θ
β
θ
θρ
θ
θ
θρ
θ
(2-9)
Where ‘a’ is the cylinder radius, ρ is the density, E is the Young’s modulus, ν is
Poisson’s ratio, s = x/a is the non-dimensional length, and β2 = h
2/12a
2 is the non-
dimensional thickness parameter, h is the shell thickness.
26
The boundary conditions applied to the system in equation (2-9) are an adapted form of
the “simply supported” condition from beam and plate theory at both ends of the finite
length cylinder. This condition is generally parametrically described in the following
manner:
L,0x
0z/ww
0wv
0u
=
≠∂∂=′
==
≠
(2-10)
Leissa [4] adopts the term “shear diaphragm” boundary conditions to describe the
simply supported situation that exists for a cylindrical shell. These conditions are used
because the cylinder considered in this thesis is enclosed at both ends by flat, thin plates
that behave in a very similar way to the conditions described in (2-10).
For harmonic motion, the following general solutions for axial, tangential and radial
displacements in terms of circumferential and longitudinal mode numbers (n) and (m)
respectively are given as follows:
∑∑∞
=
ω∞
=
πθ=
0n
tj
1m
nm eL
xmcos)ncos(Uu (2-11)
∑∑∞
=
ω∞
=
πθ=
0n
tj
1m
nm eL
xmsin)nsin(Vv (2-12)
∑∑∞
=
ω∞
=
πθ=
0n
tj
1m
nm eL
xmsin)ncos(Ww (2-13)
On substituting the general solutions in (2-11), (2-12) and (2-13) into the equations of
motion and finding the determinant of the coefficient matrix, the resulting characteristic
equation can be solved to yield the natural frequencies and corresponding modes of
vibration.
27
The mode shapes of a cylindrical shell are described by two integers; the longitudinal
mode number, m, and the circumferential mode number, n. The longitudinal mode
number m represents the number of half sine waves that fit along a cylinders length
whilst the circumferential wave number n represents the number of full sine waves
around the circumference of the cylinder. Figure 2.4 shows the first three cylinder
shapes corresponding to the longitudinal wave numbers m = 1, 2, 3.
Figure 2.7 First three longitudinal mode shapes of a cylinder [4]
Figure 2.8 shows the first four circumferential shapes corresponding to the
circumferential mode numbers n = 0, 1, 2, 3.
Figure 2.8 First 4 circumferential mode shapes of a cylinder [4]
28
Circumferential and longitudinal modes generally occur simultaneously and various
configurations exist for different natural frequencies. Figure 2.9 gives an illustration of
the types of lower order mode shape combinations that can occur in thin cylindrical
shells.
Figure 2.9 Combined longitudinal and circumferential modes [1]
Unlike many other structures, the simplest modes of vibration in cylindrical shells do
not necessarily have the lowest natural frequencies. Solutions to the Donnell - Mushtari
equations yield 3 frequency roots for every set of fixed mode numbers. The lowest
frequency is the most generally reported as it is the strongest sound radiator. If the
natural frequencies are known, the associated mode shape can be classified by the
dominant vibrational form, whether radial, axial or circumferential. The lowest
frequency is usually associated primarily with radial motion [2]. Generally the lowest
29
natural frequency will occur for a circumferential mode greater than 1 and is dependent
on the geometry of the cylinder as recognized by comparison of figures 2.10 and 2.11.
Figure 2.10 Natural Frequencies of unstiffened cylinder, a = 2m, L = 6m, h = 0.02m
Figure 2.11 Natural Frequencies of unstiffened cylinder, a = 2m, L = 3m, h = 0.02m
30
Chapter 3 Active Control Theory
3.1 Basic principles
Active control is a technique encompassing the principle of destructive interference to
actively cancel out any unwanted acoustic or vibration disturbances within a system.
Take the simple case of a single frequency, fixed amplitude, sinusoidal disturbance as
represented in figure 3.1 by the blue line. The principle of superposition suggests that a
secondary control signal, shown by the red line, 180 degrees out of phase with
disturbance will provide complete cancellation, provided that the frequency and
magnitudes are equal at the point interest.
Figure 3.1 Basic principle of superposition
Paul Lueg established the concept of active noise control in his 1936 patent [29]. He
considered the use of a microphone and loudspeaker to apply noise cancellation to a
one-dimensional propagating sound wave in a duct. This is shown in figure 3.2,
whereby a microphone detects an upstream pressure disturbance of sinusoidal form and
a control circuit measures then sends the same noise in anti-phase through a
31
loudspeaker. While Lueg’s concept was valid, the available technology at the time used
for detection, processing and generation of sound was not available.
Figure 3.2 First patented active noise control concept
Over the past 70 years there has been substantial advancement in microprocessor
technology, which has lead to an increased academic interest in the use of active control
systems. The control of vibrations in structures can take either passive or active forms.
The former uses spring/damper systems to isolate the system from its vibratory source.
Although this system is proven to work effectively for a large frequency bandwidth, the
general rule of thumb is that the lower the frequency, the more spring and damper
material is required for energy dissipation. This presents an issue in areas such as
aircraft design and submarine operations, where system weight and size consideration is
of high importance. Active vibration control provides a more suitable lightweight
solution for the attenuation of low frequency vibrations than the existing passive control
methods.
32
3.2 Adaptive feedforward control
The method for actively controlling vibrations throughout the experiments in this thesis
involves the use of an adaptive feedforward control loop illustrated in figure 3.3. This
system incorporates a slight addition to the basic system envisaged by Lueg in figure
3.2. A reference signal of the incoming disturbance is measured and the controller
predicts an output that is expected to attenuate the disturbance signal. In addition to the
feedforward set-up seen in Lueg’s system, an error sensor is used to monitor the output
at a specified location, allowing for the control system to adapt and improve its
cancellation in an iterative process.
Figure 3.3 Adaptive feedforward vibration control system
An adaptive feedforward control loop has many advantages over a feedback loop in that
it can offer prevention of a disturbance by producing a cancellation signal prior to the
disturbance taking effect. In a feedback set-up, the disturbance will have already passed
through the system and this is often very undesired, especially in applications such as
submarine sonar stealth. Adaptive feedforward control, when stably converging, also
has superior attenuation performance over feedback control, rendering it more suitable.
33
The main disadvantage of adaptive feedforward control is the requirement of a reference
measurement to accurately predict an impending disturbance. While this might be quite
simple for a tonal disturbance, a random broadband excitation is much more difficult to
‘predict’. Suitable techniques may need to be implemented in practice to rapidly
determine the incoming disturbance before it propagates through the system prior to
application of control.
3.3 Active control system design
The three basic hardware components required for an active vibration control system are
a sensor, controller and an actuator. Various sensors can be used including
accelerometers, proximity detectors, fibre optics and piezo-electric materials. Actuators
generally consist of inertial shakers or piezo-electric crystals. Control systems can come
as pre-packaged multi-channel units with single input/single output (SISO) and multiple
input/multiple output (MIMO) capabilities, single channel controllers, or derived from
scratch depending on the users’ preference.
When designing an active control system there are a few preliminary tasks that should
be performed to achieve the best and most stable level of vibration control. The first
task is to select the location for the error sensor and control actuator within the system.
While there are algorithms and functions that can be used to calculate the optimum
location for these components, commonsense was found to be appropriate during this
thesis. The error sensor location was generally chosen to be at the anti-nodal points of
the uncontrolled mode shape. All system disturbances requiring control in this thesis
were generated using single frequency sine wave input signals. The second task is to set
34
the input and output signal gains to ensure that there is adequate energy to achieve
control and to also ensure that the system was not driven beyond its linear behaviour.
The controller consists of two fundamental parts. The first is the digital control filter,
which is responsible for generating a control output signal in real time based on filter
weight parameters. The second part is the adaptive algorithm that plays the role of
tuning the filter weights based on three inputs. These inputs are the reference signal and
the error signal, both of which update the algorithm in real time, and the cancellation
path identification transfer function. Prior to performing control, it is necessary to create
a model to define the transfer path between the control actuator and the corresponding
error sensor. This is because the control excitation signal takes time to transmit through
the physical system and the predicted outputs would vary for each sensor/actuator
configuration. The function of the error signal is to indicate any residual vibration
disturbance that may still exist while control is being applied. The adaptive algorithm
applies a least mean square function to these residual vibration magnitudes in order to
continually update the control filter weights. Figure 3.4 provides a block diagram to
assist the understanding of the adaptive control systems’ internal functions.
35
Figure 3.4 Adaptive feedforward control system functional diagram [30].
One of the most influential parameters in determining control system stability is the
convergence coefficient. Analogous to finite element and numerical methods, a smaller
convergence coefficient corresponds to a more refined mesh size which is often less
likely to diverge and become unstable in the iterative loop. However, the adaptive
process becomes drastically slowed and often stagnant if the selected convergence
coefficient is too small. So a compromise is to be found for the quickest and most
effective control solution. Finding such a compromise generally requires a large amount
of time spent performing trial and error convergence tests. It can be concluded from this
chapter that whilst active control appears simple in theory, a significant amount of time
and preliminary thought must be spent in order to configure a system to achieve optimal
control.
36
Chapter 4 Experimental Testing and Results
4.1 Introduction
This chapter provides details on the cylindrical rig and experimental instrumentation
used during the determination of system natural frequencies and the plotting of mode
shapes. The testing procedures used and their corresponding results are also included.
4.2 Experimental arrangement
4.2.1 Experimental rig
The testing for this thesis was conducted on a suspended cylindrical tube arrangement
as constructed by Goodwin [1]. The cylinder is a 1100mm length mild steel tube of
mean diameter 148.8mm and shell thickness of 1.6mm providing reasonable dynamic
flexibility. Welded to each end of the cylinder is a 6mm thick circular end-plate to
provide a rigid support through which the disturbance and control vibration signals can
be transmitted. The configuration seen in figure 4.1 shows the assembly arrangement
consisting of removable fasteners and adapter-plates to allow for easy removal of
components if necessary. The configuration shown is repeated at both ends of the
cylinder.
37
Figure 4.1 Cylinder end-plate and shaker assembly sketch
The cylinder was suspended from an A-frame by elastic ockie straps, as shown in figure
4.2, to simulate free-free boundary conditions. The low stiffness of these straps ensured
that rigid body motion in any of the six degrees of freedom was of very low frequency,
causing negligible interference with the responses generated by higher experimental
frequency excitations. Such rigid body motions that occurred throughout testing
include: translation in the axial and transverse directions by swing, translation in the
vertical direction by bounce and some rotational motion.
38
Figure 4.2 A-frame assembly
4.2.2 Design Modifications to Existing Rig
A decision was made to introduce a force transducer between the inertial shaker and the
square adapter plate at one end of the cylinder to monitor the coherence between the
primary forcing input and accelerometer output. Threaded studs on both ends of the
force transducer were chosen as the preferred option for attachment. These allow for
easy disassembly of the system should a new configuration be required in future work.
The studs required unified course thread (UNC) tapped holes in the components on
either side of the transducer. However, one of these components was the inertial shaker,
which could not be modified. Consequently a small circular adapter-plate machined
from a piece of standard aluminium bar stock was designed to accommodate the tapping
on the shaker side as shown in figure 4.3(b). A tapping was also added to the centre of
39
the existing square adapter plate. The new cylinder, force transducer and shaker
assembly is shown in figure 4.4.
Figure 4.3 (a) Original surface of inertial shaker. (b) Inertial shaker with adapter-
plate attachment
Figure 4.4 Modified assembly with force transducer
40
4.2.3 Description of equipment
A large selection of common electronic equipment was used to obtain vibration
measurements throughout the testing procedures. For each testing stage including:
impact testing, broadband excitation, mode shape plotting and active control there were
different set-up configurations and instrumentation requirements. A list of the common
equipment used to perform the experiments is given in table 4.1.
Table 4.1: List of components used during experimentation
Equipment Name Type of Model
Personal Computer Laptop/PC DELL Latitude D800
Signal analyser and
generator Pulse
Bruel & Kjær Pulse Front-
End Type 3560C
Active Controller EZ-ANC Active Controller EZ-ANCII
Shaker Inertial Actuator Ultra Electronics D/L2
Amplifier Charge Amplifier Bruel & Kjær Type 2635
Accelerometer Piezoelectric Accelerometer Bruel & Kjær Type 4393
Hammer Impact Hammer Bruel & Kjær Type 8202
Force Transducer Dynamic Force Transducer Bruel & Kjær Type 8200
Power Pack Laboratory Power Supply Powertech MP-3084
Inertial Shaker Signal
Conditioner Signal Conditioner Ultra Electronics (D/L2)
The Pulse Front-End is a multi-channel device capable of generating and analysing
noise and vibrations signals. The device provides an interface between transducer and
actuator instrumentation and the Pulse software program. The Pulse software contains a
database of calibration data for common Bruel & Kjær instrumentation from which the
components in use can be selected. In taking measurements an appropriate trigger level
was set such that excessive loading was not required to initiate the measuring process. A
41
number of functions are available in Pulse for data analysis and comparison including
frequency response spectra, coherence plots and time domain functions.
The EZ-ANCII active noise controller is a multi-channel device capable of generating a
primary disturbance signal from which a reference and error signal can be used in
determining a control output. The system can be used as either a SISO or MIMO control
loop depending on the requirements. The adaptive feedfoward control algorithm used in
the EZ-ANCII is based on a filtered least mean square algorithm applied to the error
signal. Optimal control convergence can be achieved by modifying the appropriate
algorithm parameters through a software interface.
Piezoelectric accelerometers were used throughout experimentation to convert
mechanical movements into charge pulses. A mass bonded to a piezoelectric crystal
inside the accelerometer casing generates a compressive force on the crystal when
accelerations are experienced, which produces a charge. The transducers are light-
weight and can be easily moved and reattached to measurement locations by a magnetic
mount. The component has a high natural frequency that is well above the range seen in
this thesis.
The impact hammer was used to generate an impulse force on the test structure. The
hammer tip was chosen from a range of materials with different stiffness depending on
the frequency band chosen for analysis. Softer tips such as nylon are easier to control
and more suited to low frequency range, while steel tips are better suited to higher
frequency ranges. A force transducer is fixed within the hammer to transform the
impulse into a charge signal by the same means as the piezoelectric accelerometer.
42
Inertial shakers utilise the oscillation of a permanent magnet inside an energised AC
wire coil. The magnet shakes according to the solenoid coils’ input electric signal. The
supports constraining the magnet have a significant influence on the resonant frequency
of the actuator. Driving at this frequency should be avoided to prevent unexpected
dynamic behaviour.
The charge amplifier is required to enhance the charge signals received from
accelerometer and force transducer equipment to be more easily read by the analysers.
The amplifiers can be used to adjust transducer sensitivity, voltage gain and upper and
lower band pass frequency levels.
4.3 Determination of natural frequencies
4.3.1 Free response experimental procedure
When a system is harmonically forced at one of its natural frequencies, instabilities can
arise causing significant noise and vibration amplitudes. For this reason the natural
frequencies of the cylinder were chosen as the driving frequencies for the primary
forcing during active control. This is based on the assumption that if large amplitude
responses can be controlled then the smaller amplitudes at other frequencies will also be
reduced. In order to determine the free response natural frequencies of the cylindrical
shell it was desired to generate a frequency domain response function of the cylinder
resulting from an initial impulse excitation. This was achieved by using an impact
hammer to strike the rig in the radial direction and an accelerometer to map the output
response. The equipment configuration for this procedure is shown in figure 4.5. Unlike
Goodwin [1] who carried out free response testing with only a single control shaker
mounted to the rig, the free response testing in this thesis had both primary and
43
secondary shakers attached. This minimised any effects caused by adding or removing
lumped masses from the system that may have shifted the natural frequencies between
free response testing and active control testing.
Figure 4.5 Equipment Configuration for free response testing.
Before the test was conducted, it was necessary to choose an appropriate hammer tip for
exciting a consistent energy level across the impulse frequency spectrum. A hard plastic
tip was chosen as the impact surface for the hammer. Other tip choices included rubber,
nylon and hardened steel. The tip hardness is an important factor in determining the
input frequency band. A harder tip will excite higher frequencies with greater energy
than a soft tip, which provides better energy to the low-end frequencies. The desired
frequency band for this thesis was chosen to be up to 800Hz. Autospectrum testing
showed that the hard plastic tip had a 3dB roll off over this frequency range indicating a
suitable energy level at all frequencies.
44
Trigger settings in the Pulse unit were configured to initiate data acquisition when the
rig was struck. An exponential window was introduced to the accelerometer response
time spectrum to prevent leakage to be sure that the signal finished decaying to zero
within the sample time. Prior to taking measurements, correct use of the hammer was
practiced to ensure unwanted occurrences, such as double impacts due to rebound, did
not take place during measurement.
4.3.2 Free response experimental results
Figure 4.6 Hammer impact locations and accelerometer location.
Free response results were collected for different locations of the hammer impact and a
fixed accelerometer location. The locations of these impact points are indicated by the
red points shown in figure 4.6 along the length of the cylinder. The 33 nodal points
were previously marked out by Goodwin [1] and were used as a reference in this thesis.
The accelerometer was fixed at point 4, indicated by the blue dot, during each impact
test. A linear average of ten impact measurements per hammer impact location was set
in the Pulse FFT analyser settings. For each impact location, a frequency response
function and corresponding coherence function were collected. As discussed in chapter
2, the coherence function is a measure of linearity of the data between input and output
responses of the system. A coherence of 1 indicates a high correlation between input
and output measurements. Figures 4.7 and 4.8 show the results collected from all
45
impact test points whilst figures 4.9 and 4.10 indicate a single result from the point 13
for greater clarity.
FRF Hammer Test
-100
-80
-60
-40
-20
0
20
40
0 100 200 300 400 500 600 700 800
Frequency (Hz)
Accele
rati
on
(d
B)
point7
point10
point13
point16
point19
point22
point25
point28
point31
Figure 4.7 Frequency response functions of the cylinder from an impulse excitation
at different locations.
Hammer Coherence Test
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 100 200 300 400 500 600 700 800
Frequency (Hz)
Co
here
nce
point7
point10
point13
point16
point19
point22
point25
point28
point31
Figure 4.8 Coherence functions of the cylinder from an impulse excitation at
different locations.
46
FRF - Point 13
-100
-80
-60
-40
-20
0
20
40
0 100 200 300 400 500 600 700 800
Frequency (Hz)
Ac
ce
lera
tio
n (
dB
)
Figure 4.9 Frequency response function of the cylinder from an impulse excitation
at point 13.
Coherence - Point 13
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 100 200 300 400 500 600 700 800
Frequency (Hz)
Co
he
ren
ce
Figure 4.10 Coherence function of the cylinder from an impulse excitation at point
13.
47
4.3.3 Forced response experimental procedure
A second test was carried out to determine the natural frequencies of the cylinder. A
pseudo-random noise signal was generated by the Pulse system to drive the primary
shaker. The input broadband signal spanned from 0 to 800Hz. Figure 4.11 shows the
experimental set-up for this test. A force transducer measured the input signal while an
accelerometer obtained the dynamic response of the cylindrical rig. Multiple
accelerometer points were measured to account for the effects of various mode shape
nodal and anti-nodal points that may exist within the frequency range of interest. This
allowed all natural frequencies to be identified and then compared with those obtained
from the free response hammer test.
Figure 4.11 Equipment configuration for forced response testing.
4.3.4 Forced response experimental results
The frequency response function and coherence function were generated by the Pulse
front-end and are displayed in figure 4.12 through to 4.15.
48
Forced Broadband FRF
-20
0
20
40
60
80
100
120
0 100 200 300 400 500 600 700 800
Frequency (Hz)
Accele
rati
on
(d
B)
p7
p10
p13
p16
p19
p22
p25
p28
p31
Figure 4.12 Frequency response functions of the cylinder at multiple accelerometer
locations using a forced broadband excitation.
Forced Broadband Coherence
0
0.2
0.4
0.6
0.8
1
1.2
0 100 200 300 400 500 600 700 800
Frequency (Hz)
Co
here
nce
p7
p10
p13
p16
p19
p22
p25
p28
p31
Figure 4.13 Coherence functions of the cylinder at multiple accelerometer locations
using a forced broadband excitation.
49
Forced Broadband FRF - Point 19
0
10
20
30
40
50
60
70
80
90
0 100 200 300 400 500 600 700 800
Frequency (Hz)
Ac
ce
lera
tio
n (
dB
)
Figure 4.14 Frequency response function of the cylinder at point 19 using a
broadband excitation.
Forced Broadband Coherence - Point 19
0
0.2
0.4
0.6
0.8
1
1.2
0 100 200 300 400 500 600 700 800
Frequency (Hz)
Co
he
ren
ce
Figure 4.15 Coherence function of the cylinder at point 19 using a broadband
excitation.
50
4.3.5 Natural frequencies
The system natural frequencies were identified from the peaks of both the free response
FRF and the forced FRF. These results have been compared in tables 4.2 and 4.3 and
with those obtained by Goodwin [1] for validation.
Table 4.2: Natural frequency comparison from impact testing
Free Response Natural Frequencies
(Hz)
Goodwin [1] - Free Response
Natural Frequencies (Hz)
227.2 225.1
478.3 476.2
N/A 522.1
546.6 546.2
591.0 590.8
735.1 732.4
Table 4.3: Natural frequency comparison from forced testing
Forced Response Natural
Frequencies (Hz)
Goodwin [1] – Forced Response
Natural Frequencies
129.0 N/A
226.9 224.9
477.6 476.3
N/A 522.1
546.0 546.1
591.2 591.0
734.6 732.5
In general, all natural frequencies obtained from the experiments in this thesis are very
similar to those obtained in Goodwin [1]. Any minor differences can be attributed to the
slight modification in the set-up of the experimental rig, whereby both shaker masses
were fixed to the rig during testing. In contrast, Goodwin only had the primary shaker
attached. The similarity between the results of this thesis and Goodwin’s results provide
51
adequate validation for the data. It was decided that any slight variation between the
results obtained between the forced and free response testing can be neglected and the
nearest whole number frequency chosen for use during mode shape plotting and active
control. The selected natural frequencies for control in this thesis include 227Hz, 478Hz
and 546Hz.
4.3 Determination of mode shapes
A mode shape is a natural property of an oscillatory system which describes the pattern
of vibration amplitude across the geometry of the system. Each natural frequency has its
own mode shape. The mode shapes of the cylinder were measured to obtain a greater
understanding of the vibration levels and to determine the nodal and anti-nodal regions
for the specified resonant frequencies of 227Hz, 478Hz and 546Hz. This information
was used for choosing the optimum error sensor location during active control. For
example, an error sensor placed on a nodal line often produced poor attenuation and
created difficulties in achieving a stable control algorithm. In theory, axial excitation
should excite only the ‘breathing’ modes of a cylinder. Hence, both the circumferential
and longitudinal mode shapes were measured and a three dimensional map of the
vibration levels across the cylinder was obtained.
4.3.1 Mode shape mapping procedure
To map the acceleration magnitudes over the entire cylinder, a mesh was marked out on
the outer surface using a marker pen. As shown in figure 4.16, the mesh consisted of 11
equally spaced points along the cylinder length and 16 around the circumference, to
give a total of 176 points. The longitudinal mesh was selected to be relatively coarse
(element size = 110mm) because pre-testing evaluations confirmed that longitudinal
52
mode numbers ‘m’ of up to only m = 3 existed for the chosen driving frequencies. The
circumferential plot required a slightly more refined mesh for better mode definition. A
coordinate system was also introduced to track the mesh node positions. The x-axis in
figure 4.16 refers to the longitudinal location starting from 0 at the primary shaker end
of the cylinder. The b-axis refers to the circumferential location starting from 1 at the
datum line.
Figure 4.16 Experimental mesh definition for 11 x 16 point mesh.
53
The following procedure was followed to experimentally determine the mode shapes for
each of the chosen resonant frequencies of 227Hz, 478Hz and 564Hz:
1. The instrumentation was set up as shown in figure 4.11
2. A signal was generated by the Pulse system to drive the primary shaker using a
sinusoidal disturbance.
3. An accelerometer was used to measure the magnitude of radial acceleration at
each of the 176 node points by traversing lengthwise along the cylinder for each
circumferential co-ordinate b. At each location, the Pulse system generated a
FRF based on the linear average of 10 data measurements from which the
magnitude at the driven frequency was displayed. A period of 30seconds was
allowed between each mesh point measurement. This was to allow for the decay
of any unwanted transients created by the magnetic snapping force between the
accelerometer and the steel cylinder.
4. The procedure was run for each of the frequencies and the results collected as a
matrix of data (see Appendix A) to be plotted in a three-dimensional mesh
surface cylinder format in MatLab.
5. To obtain a clearer view of each of the mode shapes, and to check for effects
such as symmetry that are not necessarily obvious from the 3-dimensional plots
a second procedure was run. The lengthwise mode shape was measured and
plotted using a 33 point mesh (element size = 34mm) passing through an anti-
nodal point of the circumferential mode. The circumferential mode shape was
measured and plotted using a 32 point mesh passing through the anti-nodal
point/s of the longitudinal mode.
54
6. The separate longitudinal and circumferential mode plots provided a more
accurate indication of where on the cylinder control error sensors could be
placed for optimum attenuation.
4.3.2 Mode shape results
When a 227Hz sine wave signal was sent axially into the cylinder rig, the three
dimensional response was measured as shown in figure 4.17 as a series of acceleration
magnitudes. The results show a half sine wave along the cylinder length and two full
sine waves (magnitude) around the circumference indicating the (n, m) = (2, 1) mode
shape. The lengthwise mode shape shown in figure 4.18 was measured along the anti-
nodal line b = 4 (see figure 4.16). The circumferential mode shape shown in figure 4.19
was measured at the centre of the cylinder about the line x = 16.
55
Figure 4.17 3-D cylinder mesh plot of uncontrolled 227Hz mode shape.
56
227Hz Lengthwise mode shape
-55
-50
-45
-40
-35
-30
-25
-20
0 100 200 300 400 500 600 700 800 900 1000 1100
Distance along cylinder (mm)
Ac
ce
lera
tio
n (
dB
)
Un-controlled
Figure 4.18 Longitudinal mode shape at 227Hz measured along b = 4 using 33 point
mesh.
227Hz Circumferential
Mode shape
-40
-35
-30
-25
-20
0
90
180
270
Figure 4.19 Circumferential mode shape at 227Hz measured about x = 16 using 32
point mesh.
57
When a 478Hz sine wave signal was sent axially into the cylinder rig, the three-
dimensional response was measured as shown in figure 4.20 as a set of acceleration
magnitudes. The results show two half sine waves along the cylinder length and two full
sine waves (magnitude) around the circumference indicating the (n, m) = (2,2) mode
shape. The lengthwise mode shape shown in figure 4.21 was measured along the anti-
nodal line b = 8 (see figure 4.16). The circumferential mode shapes shown in figure
4.22 (a) and figure 4.22 (b) were measured about the lines x = 8 and x = 24 which
correspond to locations ¼ and ¾ along the cylinder length respectively.
Figure 4.20 3-D cylinder mesh plot of uncontrolled 478Hz mode shape.
58
478Hz Lengthwise mode shape
-30
-25
-20
-15
-10
-5
0
0 100 200 300 400 500 600 700 800 900 1000 1100
Distance along cylinder (mm)
Acele
rati
on
(d
B)
Figure 4.21 Longitudinal mode shape at 478Hz measured along b = 8 using 33 point
mesh.
Figure 4.22 (a) Circumferential mode shape at 478Hz measured about x = 8 using 32
point mesh. (b) Circumferential mode shape at 478Hz measured about x
= 24 using 32 point mesh.
59
When a 546Hz sine wave signal was sent axially into the cylinder rig, the three-
dimensional response was measured as shown in figure 4.23 as a series of acceleration
magnitudes. The lengthwise mode shape shown in figure 4.24 was measured along the
anti-nodal line b = 16.5 (see figure 4.16). The circumferential mode shapes were
measured about the lines x = 8 and x = 16 and x = 24 as these showed varied
circumferential behaviour. The longitudinal mode number m is identified as m = 3 [3]
whilst the circumferential mode is undecided as figures 4.25 and 4.26 both indicate
characteristics of n = 1 and n = 3.
Figure 4.23 3-D cylinder mesh plot of uncontrolled 546Hz mode shape.
60
546Hz Lengthwise mode shape
-30
-25
-20
-15
-10
-5
0 100 200 300 400 500 600 700 800 900 1000 1100
Distance along cylinder length (mm)
Accele
rati
on
(d
B)
Figure 4.24 Longitudinal mode shape at 546Hz measured along b = 16.5 using 33
point mesh.
Figure 4.25 Circumferential mode shape at 546Hz measured about x = 16 using 32
point mesh.
61
Figure 4.26 (a) Circumferential mode shape at 546Hz measured about x = 8 using 32
point mesh (b) Circumferential mode shape at 546Hz measured about x =
24 using 32 point mesh.
62
Chapter 5 Active Control
5.1 SISO control method
A discussion on how active control works and the basic system parameter set-up is
given in chapter 3.3. Figure 5.1 shows the hardware configuration used to apply single
input/single output (SISO) control to the cylindrical rig. Four channels out of the twenty
available in the EZ-ANCII controller were used for generator output, control signal
output, reference signal input and error signal input.
Figure 5.1 SISO Active control hardware configuration.
63
The following procedure was used for single error sensor control at the chosen
frequencies of 227Hz, 478Hz and 546Hz:
1. The error sensor location was selected based on the point of maximum
acceleration amplitude obtained from the plotted mode shape results.
2. Once the error sensor was attached to the cylinder it’s charge amplifier was set
to produce an amplitude between 0.5 and 0.75 on the EZ-ANCII software
interface display. Input gain settings were also adjusted to achieve this.
3. The EZ-ANCII signal generator was set to produce a sine wave signal output of
the desired frequency.
4. Filtering, adaptive algorithm and system cancellation path identification
variables were adjusted until stable control was achieved in the system.
5. While the generator was left running, the Pulse front-end system was used to
obtain the magnitude of vibration of the uncontrolled signal prior to control.
6. The active control mode was then switched on and left until the error signal had
converged to a stable value. This was generally close to 10 % of the original
amplitude. The pulse unit was again used to obtain the controlled magnitude of
vibration. The active control mode was then switched off.
7. Steps 5 and 6 were repeated for each nodal point in the selected 176 point
cylindrical mesh and for each of the selected driving frequencies.
8. After the three-dimensional control data was obtained, a more refined set of data
was collected by repeating steps 5 and 6 along a 33 point longitudinal mesh and
around a 32 point circumferential mesh.
64
5.1.1 SISO control mode 1 results
The error sensor was placed at the point (16,4) corresponding to halfway along the
cylinder length and a quarter of the way around the circumference from the reference
line (see figure 4.16). Figure 5.2(a) and 5.2(b) display the uncontrolled and controlled
vibration magnitude plots at a primary 227Hz sine wave shaker excitation. The
lengthwise and circumferential control plots are shown in figure 5.3 and figure 5.4
respectively.
Figure 5.2 (a) 3-D cylinder mesh plot of uncontrolled magnitudes at 227 Hz. (b) 3-
D cylinder mesh plot of controlled magnitudes at 227Hz.
65
Figure 5.3 Controlled and uncontrolled magnitudes at 227Hz measured along the
cylinder length through b = 4 using a 33 point mesh.
Figure 5.4 Controlled and uncontrolled response at 227Hz measured around the
circumference through x = 16 using a 32 point mesh.
66
5.1.2 SISO control mode 2 results
The error sensor was placed at the point (8,8) corresponding to a quarter of the way
along the cylinder length and halfway around the circumference from the reference line
(see figure 4.16). Figure 5.5(a) and 5.5(b) display the uncontrolled and controlled
vibration magnitude plots at a primary 478Hz sine wave shaker excitation. The
lengthwise and circumferential control plots are shown in figure 5.6 and figure 5.7
respectively.
Figure 5.5 (a) 3-D cylinder mesh plot of uncontrolled magnitudes at 478 Hz. (b) 3-
D cylinder mesh plot of controlled magnitudes at 478Hz.
67
Figure 5.6 Controlled and uncontrolled magnitudes at 478Hz measured along
cylinder length through b = 8 using a 33 point mesh.
Figure 5.7 (a) Controlled and uncontrolled response at 478Hz measured around the
circumference through x = 8 using a 32 point mesh. (b) Controlled and
uncontrolled response around measured around the circumference
through x = 24 using a 32 point mesh.
68
5.1.3 SISO control mode 3 results
Application of active control to the 546Hz mode shape was unsuccessful. The error
sensor was located at the point of maximum acceleration as shown in figure 5.8.
However, convergence of the adaptive control algorithm was never achieved. The
system cancellation path identification was defined on multiple occasions and a large
range of convergence coefficient values were tried. Reasons for the lack of success in
controlling this mode shape are discussed in chapter 6.
Figure 5.8 Location of error sensor in 546Hz active control attempt.
69
5.2 Dual error sensor control method
The single input/single output control results all show that the maximum attenuation
levels occur in and around the error sensor location. This leads to the assumption that
the use of multiple error sensors throughout the cylinder will produce improved levels
of attenuation. However, Kessissoglou et al [23] states that while performance can
improve if a second error sensor is placed on the same anti-nodal line as an optimally
located single error sensor, arbitrarily locating the second sensor will in fact deteriorate
the performance. This statement was based on results obtained from the control of
rectangular plate mode shapes. It was decided to test this theory for cylindrical shells.
The active control unit and instrumentation were therefore set-up as shown in figure 5.9.
The same procedure as per single error sensor control was used.
Figure 5.9 Dual sensor active control configuration.
70
5.2.1 Dual error sensor control results
The use of two error sensors was tested for the 227Hz mode shape (n, m) = (2, 1). As
there is only one lengthwise anti-nodal point, the two sensors were located around the
circumference of the cylinder about coordinate a = 16. It was found while setting up the
control system, that locating the second error sensor away from circumferential anti-
nodes drastically reduced the chance of adaptive algorithm convergence. A working
solution was found for two error sensors located at coordinates (16,4) and (16,12). The
lengthwise and circumferential results are shown in figures 5.10 and 5.11.
Figure 5.10 Controlled and uncontrolled responses at 227Hz using two error sensors
at (16, 4) and (16, 12) and measured along cylinder length through b = 4
using a 33 point mesh.
71
Figure 5.11 Controlled and uncontrolled response at 227Hz using two error sensors
at (16, 4) and (16, 12) and measured around the circumference through x
= 16 using a 32 point mesh.
72
Chapter 6 Discussion
6.1 Mode 227Hz
6.1.1 Mode shape
The (m, n) = (1, 2) mode shape classification corresponding to a sinusoidal input
frequency of 227Hz is confirmed by the results in Goodwin [1]. However, it is
important to notice that only the magnitudes of vibration were measured on the cylinder
in this thesis. Graphical results may appear inverted in some regions because the phase
of the frequency response function was not taken into account. Figure 6.1 is shown
below to demonstrate this difference for the 227Hz circumferential mode shape.
227Hz Circumferential mode shape comparison
-45
-40
-35
-30
-25
-200
90
180
270
Magnitude Imaginary
Figure 6.1 The (m, n) = (1, 2) mode shape found by magnitude measurements
compared with the measurement of imaginary components in Goodwin
[1].
73
6.1.2 Single error sensor control
The active control results along the cylinder length and about the circumference are
very similar in shape and magnitude to those obtained by Goodwin [1]. An attenuation
of up to 34.4dB about the circumference was achieved, which is a large improvement
over the maximum of 22.7dB seen in Goodwin [1]. However, the attenuation levels
along the cylinder length were not quite as substantial. A maximum attenuation of
22.8dB was obtained in this direction compared to Goodwin’s [1] 32.3dB. The
significant features of the controlled cylinder including: the symmetry of control, anti-
nodes of the uncontrolled mode shape becoming nodes of the controlled shape and
nodes of the uncontrolled mode shape becoming anti-nodes of the controlled shape as
discussed in Goodwin [1], were also present.
On observing the matrix of data for global control, it was found that 33 points out of the
total 176 points produced negative attenuation during control. This indicates that
control was 81.25% effective across the entire cylinder. The existence of negative
attenuations can be attributed to the following list of reasons:
• The level of control input energy was slightly too high.
• Inaccuracy in marking out mesh points and or inaccurate accelerometer
placement in mapping both controlled and uncontrolled responses may have lead
to errors.
• The relative coarseness of the mesh size used for accelerometer readings may
have lead to mesh points not coinciding exactly with predicted nodal and anti-
nodal points.
• Excitation of other modes during control, causing an input of energy at the nodal
points of the uncontrolled response.
74
Despite some small errors and imperfections, active control using a single error sensor,
single actuator set-up has been shown to be an effective method of globally controlling
cylinder vibrations at a 227Hz sinusoidal disturbance.
6.1.3 Dual error sensor control
Active control using two error sensors achieved a maximum attenuation of 25.3dB
about the central circumference and 20.3dB along the cylinder length. The number of
negative attenuation points in the data was 15 out of a total of 65, indicating that control
was 77% effective. Although these values are not quite as significant as control by using
a single error sensor the results were found to follow the same graphical trend. Figure
6.2 shows the longitudinal results comparison between single and dual error sensor
control. It is clear from this figure that the majority of the dual error sensor controlled
data lies above the single error sensor data for the majority of the cylinder length. This
implies a reduction in attenuation levels. As such, active control was slightly less
effective using dual sensors as opposed to a single sensor.
75
227Hz Active control along length - 2 sensors vs 1 sensor
-60
-50
-40
-30
-20
-10
0
0 100 200 300 400 500 600 700 800 900 1000 1100
Distance along length (mm)
Accele
rati
on
(d
B)
Un-controlled 2 Sensor Controlled 1 Sensor Controlled
Figure 6.2 Comparison of lengthwise control results between the use of 2 error
sensors and a single error sensor.
Figure 6.3 shows the active control trend about the central circumference of the
cylinder. The results indicate that similar vibration attenuation was achieved all round
for both single and dual sensor control. However, the nodes of the controlled response
using a single error sensor show significantly lower magnitudes. This again implies that
active control was slightly less effective using dual sensors as opposed to a single
sensor.
76
227Hz Active control around circumference - 2 sensors vs 1
sensor
-60.0
-50.0
-40.0
-30.0
-20.00
90
180
270
Un-controlled 2 Sensor controlled 1 Sensor controlled
Figure 6.3 Control results about the circumference comparing the use of 2 error
sensors and a single error sensor.
It was expected according to Keir et al [24] that introducing more than one error sensor
would improve the performance of control. However, this did not occur with the chosen
configuration. Reasons for this apparent discrepancy can be attributed to redundancy in
the choice of second error sensor location. It is known from the single error sensor
results that control of the (m, n) = (1, 2) mode about the circumference occurs
symmetrically. The anti-nodal point at which the second error sensor was placed was
expected to become a node in the controlled response with the use of only a single
sensor. The dual sensor configuration was chosen based on statements made in
references [1] and [23] which claim ‘active control results may be improved if multiple
error sensors are used on the same anti-nodal line’. The anti-nodal line was assumed to
be that created by the peak of the m = 1 lengthwise mode shape rotated about the
cylinder. According to this statement it may still be possible to attain improved control
if multiple error sensors are located lengthwise along an anti-nodal line of the n = 2
77
mode shape. However, it must be noted that the use of multiple error sensors creates a
significant reduction in the active control algorithm stability, especially whilst only
using a single control actuator. The work of the control actuator must be split among the
two or more error sensors, of which the compromise is sub optimal. This situation is
described in Keir [24] as a cause of deterioration in control performance. Further
investigation into the use multiple sensors and their optimal location is necessary to
make sound judgements over its feasibility in comparison to single error sensor control
of the cylindrical shell in this thesis.
6.2 Mode 478Hz
6.2.1 Mode shape
The shape that occurs as a result of the 478Hz driving frequency is classified as the (m,
n) = (2, 2) mode. This is due to the two half sine waves seen along the cylinder length
and the two full sine waves observed about the circumference. However, it is observed
from the three dimensional MatLab data plot that the circumferential mode shape is not
as well defined as that seen in the 227Hz mode shape. The circumferential mode shapes
observed at the beginning, middle and end of the cylinder do not resemble the n = 2
mode number, but more closely resemble that of n = 1 as shown in figure 6.4. However,
figure 6.5 shows the circumferential mode shapes about the anti-nodal lines at a quarter
and three quarters of the cylinder length which do resemble the n = 2 mode number. The
conclusions drawn from this contradiction are that the global mode shape must be the
result of the coupling of two or more modes. The relative closeness of the 478Hz
natural frequency to its succeeding two natural frequencies is also suggestive of such
coupling.
78
478Hz Circumferential mode shape
-45
-40
-35
-30
-25
-200
90
180
270
Primary End Middle Secondary End
Figure 6.4 Cylinder beginning, middle and end circumferential mode shapes at
478Hz
Figure 6.5 n =2 circumferential mode shapes measured at ¼ and ¾ along the
cylinder length at 478Hz
79
6.2.2 Single error sensor control
The application of active control to the 478Hz mode shape was found to be extremely
effective along the length of the cylinder and reasonably effective about the two
measured circumferential locations. The maximum attenuation achieved in controlling
the lengthwise mode shape was 30.9dB whilst the maximum attenuation about the
circumference was found to be 31.1dB. Note that both of these values were obtained at
the location of the error sensor. However, a maximum of 19.8dB was still achieved
about the second circumferential location x = 24, which contained no error sensors.
Negative attenuations were found at 9 points out of the 176 total, confirming the global
control to have been 95% effective.
Symmetry of control is seen to occur both lengthwise and circumferentially for the
478Hz frequency as it did at the 227Hz frequency. While the error sensor was placed at
one-quarter length corresponding to a nodal point in the lengthwise control shape, there
was a mirrored nodal point at approximately three quarters down the cylinder length.
6.2.3 Comparison to theory
Numerous theoretical models exist which predict the behaviour during active control,
however, there is very little experimental evidence available in literature. For this
reason, the experimental results obtained in this thesis have been compared with the
theoretical expected results. The plot of an expected controlled and un-controlled
submerged cylinder containing two internal bulkheads at the mode number m = 2 is
shown in figure 6.6. The results are for error sensors placed at 0.22L and 0.78L along
the hull, where L is the total hull length.
80
Figure 6.6 Expected control results for the m = 2 lengthwise mode shape [22]
The theory was adapted to the results of the m = 2 mode shape measured in this thesis
and the comparison is shown in figure 6.7. The uncontrolled vibrations plot closely
follows the theoretical trend while the controlled response tends to have good
correlation with the theory around the error sensor region. However, between 400 and
1000mm along the cylinder length there is a drastic deviation from the expected values.
Despite this variation, the general trend follows the same basic shape as that expected,
and control is considered effective for the majority of the lengthwise measurement
locations.
81
478Hz Active Control Results along Length
-60
-50
-40
-30
-20
-10
0
0 100 200 300 400 500 600 700 800 900 1000 1100
Distance along cylinder length (mm)
Ac
ce
lera
tio
n (
dB
)
Un-controlled Controlled Expected un-controlled Expected controlled
Figure 6.7 Active control results comparison to expected theoretical control at
478Hz
6.3 Mode 546Hz
6.3.1 Mode shape
The 546Hz sinusoidal driving frequency produced a mode shape that contains elements
of the (m, n) = (3, 1) and (m, n) = (3, 3) modes of vibration. This is most likely a
consequence of modal coupling due to the relative proximity of the 478Hz and 592Hz
natural frequencies to the 546Hz peak in the free and forced FRFs.
6.3.2 Failed control
There are a number of possible reasons for the failure to achieve control of the 546Hz
mode. It is possible that the mode shape has been misinterpreted as a consequence of
only taking measurements of the acceleration magnitudes. This neglects the existence of
phase difference between adjacent mesh points. As a result the location of node and
82
anti-nodal points may not have been adequate. Future work should involve a
comparison of the mode shape to the results of a finite element computer simulation.
While a resolution of 0.1Hz per line was used when measuring the system natural
frequencies, the nearest whole number frequencies were chosen to overcome the
discrepancies between the forced and free response values obtained. This crude
approximation can have a significant effect on the vibration energy levels throughout
the system causing dramatic inaccuracies in the plotted mode shape. Enhancing the
resolution of the measured natural frequencies and maintaining their precision when
used to generate the system disturbance input, will greatly improve the chances of
actively controlling the 546Hz natural frequency.
83
Chapter 7 Conclusions and Future Work
7.1 Conclusions
The intentions of this thesis were to further the experimental research initiated in
Goodwin [1] on the effectiveness of using active vibration control to attenuate
unwanted vibrations within a thin-walled cylindrical shell. Improvements into this
research were focused on validating the results of Goodwin [1], applying active control
to higher order modes and investigating the use of multiple error sensors in controlling.
In order to achieve this, a large amount of time was spent familiarising with; the
cylinder rig constructed by Goodwin [1], the Bruel & Kjær Pulse system, Casual
Systems EZ-ANCII active control unit, accompanying software and the common testing
procedures for determining natural frequencies and applying control. Minor
modifications were made to the existing cylinder construction and assembly to
accommodate a force transducer for improved data measurement capabilities.
System natural frequencies were determined from the separate frequency response
functions generated by impulse excitation and broadband excitation. The natural
frequencies were used to create a single frequency sine wave disturbance signal for
active control testing. The natural frequencies selected for the input disturbance signal
during active control were: 227Hz, 478Hz and 546Hz.
In order to observe the active control of vibrations within the cylinder, the global
response to the input disturbance was mapped using an accelerometer located at a series
of predefined points. A mesh grid was plotted around the entire cylinder and a
coordinate system was introduced to keep track of each accelerometer measurement
point. The data along the cylinder length and about its circumference was plotted to
84
obtain the mode shape at each of the chosen natural. The mode shapes were used to
determine the anti-nodal points at which an error sensor could be placed to achieve
optimal control. The following conclusions have been made from the active control
experiments in this thesis:
1. Active vibration control has been found to globally attenuate vibration levels for
the first two selected natural frequencies of 227Hz and 478Hz.
2. Using single error sensor control, symmetry has been found to exist for both the
227Hz and 478Hz excitation frequencies.
3. The use of two error sensors located symmetrically on the anti-nodal points
about the central circumference has shown slight deterioration in control
performance for the 227Hz frequency. Further investigation into multiple error
sensors is necessary to confirm its feasibility in comparison to single error
sensor control.
4. Minimum attenuation levels occurred at nodal points for: single and dual error
sensor control of the 277Hz frequency and for the single error sensor control of
the 478Hz frequency.
5. Maximum attenuation levels occurred at the error sensor location for: single and
dual error sensor control of the 277Hz frequency and for the single error sensor
control of the 478Hz frequency.
6. Active vibration control of the 546Hz natural frequency has been unsuccessful.
Further improvements in; control parameters, error sensor location and mode
shape accuracy are expected to yield a more stable control algorithm for this
mode.
85
7.2 Future Work
Despite the large growth in theoretical research and development of using active control
as an effective means of controlling vibrations in thin-walled cylindrical structures,
there remains a gap in regards to the experimental validation of this work. This thesis
and Goodwin [1] have covered only a small portion of the variety and complexity of
experimental work that should be conducted in future.
It is recommended that future work incorporate improvements to the results of this
thesis through; finite element mode shape comparisons, analytical predictions using
cylinder theories, improved resolution and confirmation of the cylinders’ natural
frequencies and improved learning of the EZ-ANCII control system.
It is also recommended that the active control of higher order modes be investigated for
both single error sensor control and multiple error sensor control. The results of Keir et
al [24] suggest, however, that an increase in error sensors without a corresponding
increase in control actuators can deteriorate control performance. Thus it is
recommended that if multiple error sensor control is found to be ineffective, then
multiple control actuators should be introduced. The performance of radially exciting
the cylinder to control an initial axial disturbance can then also be considered for
investigation. It is known that the acoustic signature of submarines and other cylindrical
vessels is predominately generated by radial displacements. As a result, control
actuation in the radial direction may prove more valuable than the current arrangement.
86
In practice it is rare that a pure harmonic disturbance to the cylinder system should
occur. Generally the input excitation is of a broad frequency range, which excites
multiple system modes simultaneously. In order to improve the relevance of using
active vibration control to attenuate cylinder vibrations in realistic applications such as
submarines, aircraft, pressure vessels and pipelines, it is recommended that experiments
be conducted to control random broadband excitations.
87
References
1. Goodwin, W., 2007, Active vibration control of a finite thin-walled cylindrical
shell, B.E. Thesis, School of Mechanical and Manufacturing Engineering, The
University of New South Wales.
2. Armenàkas, A. E., Gazis, D. C. and Hermann, G., 1969, Free vibrations of
cylindrical shells, Pergamon Press, Oxford.
3. Bradshaw, S., 2003, Finite element analysis of the low frequency modes for a
submarine hull. B. E. Thesis, School of Engineering, James Cook University.
4. Leissa, A. W., 1993, Vibration of shells, American Institute of Physics, New
York.
5. Buchanan, G. R. and Chua, C. L., (2001), ‘Frequencies and mode shapes for
finite length cylinders’, Journal of Sound and Vibration, 246, 927-941.
6. El-Mously, M., (2003), ‘Fundamental natural frequencies of thin cylindrical
shells: A comparative study’, Journal of Sound and Vibration, 264, 1167-1186.
7. Saijyou, K. and Yoshikowa, S., (2002), ‘Analysis of flexural wave velocity and
vibration mode in thin cylindrical shell’, Journal of the Acoustical Society of
America, 112, 2808-2813.
8. Zhang, X. M., Liu, G.R. and Lam, K.Y., (2001), ‘Vibration analysis of thin
cylindrical shells using wave propagation approach’, Journal of Sound and
Vibration, 239, 397-403.
9. Wang, C. and Lai, J. C. S., (2002) ‘Comments on “Vibration analysis of thin
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10. Zhang, X. M., (2002), ‘Reply to “Comments on ‘Vibration analysis of thin
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11. Ruotolo, R., (2001), ‘A comparison of some thin shell theories used for the
dynamic analysis of stiffened cylinders’, Journal of Sound and Vibration, 243,
847-860.
12. Ruotolo, R., (2002), ‘Influence of some thin shell theories on the evaluation of
the noise level in stiffened cylinders’, Journal of Sound and Vibration, 255, 777-
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13. Norwood, C. J., (1995), ‘The free vibration behaviour of ring stiffened cylinders
– A critical review of the unclassified literature’, DTSO Report TR-0200.
14. Ruzzene, M. and Baz, A., (2000), ‘Finite element modelling of vibration and
sound radiation from fluid-loaded damped shells’, Thin Walled Structures, 36,
21-46.
15. Sievers, L. A. and von Flowtow, A. H., (1990), ‘Comparison and extensions of
control methods for narrowband disturbance rejection’, Active noise and
vibration control, 1990: presented at the Winter Annual Meeting of the
American Society of Mechanical Engineers, NCA-Vol 8, 11-22.
16. Pan, X. and Hansen, C. H., (2004), ‘Active control of vibration transmission in a
cylindrical shell’, Journal of Sound and Vibration, 203, 409-434.
17. Thomas, D. R., Nelson, P. A. and Elliot, S. J., (1993), ‘Active control of the
transmission of sound through a thin cylindrical shell, part I: the minimization of
vibrational energy’, Journal of Sound and Vibration, 167, 91-111.
89
18. Thomas, D. R., Nelson, P. A. and Elliot, S. J., (1993), ‘Active control of the
transmission of sound through a thin cylindrical shell, part II: the minimization
of acoustic potential energy’, Journal of Sound and Vibration, 167, 113-128.
19. Bullmore, A. J., Nelson, P. A., Curtis, A. R. D. and Elliot, S. J., (1987), ‘Active
minimization of harmonic enclosed sound fields, part I: theory’, Journal of
Sound and Vibration, 117, 1-13.
20. Bullmore, A. J., Nelson, P. A., Curtis, A. R. D. and Elliot, S. J., (1987), ‘Active
minimization of harmonic enclosed sound fields, part II: A computer
simulation’, Journal of Sound and Vibration, 117, 15-33.
21. Fuller, C. R., Elliot, S. J. and Nelson, P. A., 1996, Active control of vibration,
Academic Press, London.
22. Kessissoglou, N. J., Tso, Y. and Norwood, C. J., ‘Active control of a fluid
loaded cylindrical shell, part 2: active modal control’, Proceedings of the 8th
Western Pacific Acoustics Conference (Westpac8), 7-9 April 2003, Melbourne,
Australia.
23. Kessissoglou, N. J., Ragnarsson, P. and Lofgren, A., (2002), ‘An analytical and
experimental comparison of optimal actuator and error sensor location for
vibration attenuation’, Journal of Sound and Vibration, 260, 671-691.
24. Keir, J., Kessissoglou, N. J. and Norwood, C. J., (2005), ‘Active control of
connected plates using single and multiple actuators and error sensors’, Journal
of Sound and Vibration, 281, 73-97.
25. Vipperman, J. S., Burdisso, R. A. and Fuller, C. R., (1993), ‘Active control of
broadband structural vibration using the LMS adaptive algorithm’, Journal of
Sound and Vibration, 166, 283-299.
90
26. Baz, A. and Chen, T., (2000), ‘Control of axi-symmetric vibrations of
cylindrical shells using active constrained layer damping’, Thin Walled
Structures, 36, 1-20.
27. Norton, M. and Karczub, D., 2003, Fundamentals of noise and vibration
analysis for engineers, 2nd
edn, Cambridge University Press, 2003.
28. Gade, S., Herlufsen, H. and Konstantin-Hansen, H., 2005, ‘Application note:
How to determine the modal parameters of simple structures’, Bruel &Kjaer
Application Note 3560 (Bo0428), Bruel & Kjær, Denmark.
29. Lueg, P., (1936), ‘Process of silencing sound oscillations’, US Patent 2 043 416.
30. Snyder, S. D. and Vokalek, G., 1994, EZ-ANC Users Guide, Casual Systems Pty
Ltd, Adelaide.
91
Appendix A Tabulated Experimental Data
Table A1: Global uncontrolled mode shape data at primary frequency of 227Hz
227Hz Global Uncontrolled Vibration Magnitudes (dB)
Longitudinal Coordinate
1 4 7 10 13 16 19 22 25 28 31
1 -26.0 -10.9 -6.1 -3.4 -2.5 -1.4 -1.7 -3.5 -6.4 -11.4 -22.2
2 -28.9 -12.5 -8.0 -5.7 -5.0 -5.0 -4.9 -6.6 -9.4 -14.8 -36.2
3 -21.6 -16.1 -9.1 -7.7 -6.6 -6.0 -6.5 -7.8 -10.3 -14.5 -37.4
4 -37.4 -12.8 -7.1 -4.7 -3.7 -2.0 -3.1 -4.1 -7.3 -11.6 -28.3
5 -30.7 -11.7 -6.6 -4.3 -3.2 -2.5 -3.4 -4.6 -7.3 -12.7 -28.9
6 -29.3 -13.6 -9.6 -6.6 -5.8 -5.8 -5.9 -7.5 -10.7 -16.4 -32.0
7 -27.9 -14.0 -9.0 -7.9 -7.4 -7.0 -7.7 -9.8 -12.0 -18.6 -34.1
8 -26.0 -12.1 -6.5 -4.1 -2.5 -2.9 -3.8 -4.9 -7.9 -13.1 -30.9
9 -25.5 -12.4 -7.1 -4.0 -2.9 -2.6 -3.4 -4.4 -7.0 -13.2 -26.4
10 -22.9 -18.1 -11.1 -8.6 -6.2 -5.1 -4.7 -7.0 -8.6 -12.9 -33.3
11 -37.3 -19.2 -11.1 -9.6 -8.8 -9.2 -8.8 -9.8 -12.4 -18.7 -30.0
12 -34.5 -12.2 -6.9 -5.0 -3.8 -3.1 -3.4 -5.8 -8.0 -14.9 -29.6
13 -26.3 -11.2 -6.1 -3.3 -1.9 -1.7 -2.7 -3.9 -6.8 -12.2 -25.6
14 -33.2 -15.8 -10.4 -7.1 -5.6 -5.0 -5.7 -6.8 -9.7 -15.0 -21.5
15 -19.5 -16.2 -11.8 -9.1 -8.4 -7.9 -7.2 -8.7 -10.7 -15.1 -23.2
Cir
cu
mfe
ren
tial
Co
ord
inate
16 -26.4 -12.2 -7.0 -4.5 -3.5 -3.1 -3.1 -4.5 -7.5 -12.7 -25.7
Table A2: Global controlled mode shape data at primary frequency of 227Hz
227Hz Global Controlled Vibration Magnitudes (dB)
Longitudinal Coordinate
1 4 7 10 13 16 19 22 25 28 31
1 -33.4 -21.5 -17.0 -13.5 -13.2 -12.6 -13.7 -15.3 -19.8 -24.7 -37.9
2 -33.5 -16.1 -10.9 -8.4 -7.5 -5.5 -6.5 -7.5 -10.8 -16.7 -23.1
3 -34.7 -15.5 -10.6 -8.0 -6.6 -5.9 -6.4 -8.1 -10.7 -16.2 -32.1
4 -35.0 -20.8 -17.8 -12.7 -14.2 -12.6 -13.7 -16.7 -18.0 -24.1 -37.0
5 -38.6 -25.8 -17.4 -13.6 -14.0 -11.6 -15.3 -15.5 -19.5 -23.3 -34.5
6 -33.5 -15.3 -10.4 -7.8 -7.2 -6.1 -6.9 -8.6 -10.3 -16.5 -35.2
7 -28.4 -15.4 -10.4 -7.7 -6.6 -5.4 -5.9 -7.8 -10.4 -16.4 -34.2
8 -27.1 -21.2 -18.3 -15.5 -14.0 -14.3 -14.9 -14.0 -17.8 -23.8 -42.2
9 -30.2 -27.1 -19.7 -17.5 -15.2 -15.6 -17.2 -17.5 -19.5 -25.4 -25.4
10 -24.3 -16.6 -12.6 -7.4 -6.4 -6.3 -5.5 -7.8 -10.7 -17.2 -35.3
11 -31.0 -12.1 -6.5 -5.3 -3.4 -2.0 -2.2 -5.5 -6.6 -11.6 -36.0
12 -41.8 -26.3 -22.9 -16.3 -12.8 -10.8 -13.4 -16.7 -15.6 -23.0 -35.0
13 -38.6 -22.2 -17.2 -16.9 -16.4 -15.2 -13.7 -19.5 -19.5 -28.3 -40.0
14 -30.7 -16.1 -10.5 -8.2 -7.0 -6.8 -7.2 -9.0 -11.7 -17.4 -34.8
15 -42.3 -17.2 -11.7 -8.8 -7.4 -6.7 -7.5 -9.0 -11.9 -17.2 -29.7
Cir
cu
mfe
ren
tial
Co
ord
inate
16 -23.4 -28.1 -22.1 -19.1 -14.9 -12.8 -16.1 -15.9 -19.4 -24.3 -33.0
92
Table A3: Lengthwise mode shape data for both controlled and uncontrolled
responses at primary frequency of 227Hz during active control
Active Control Results Along Cylinder Length (error sensor at (16,4))
Point Uncontrolled (dB) Controlled (dB) Attenuation (dB)
0 -47.6 -46.3 -1.3
1 -45.5 -44.4 -1.1
2 -41.4 -40.1 -1.3
3 -37 -41.7 4.7
4 -33.5 -43.9 10.4
5 -31.6 -43.1 11.5
6 -30.8 -43.9 13.1
7 -29.3 -37.1 7.8
8 -28.3 -39.9 11.6
9 -27.7 -41 13.3
10 -27.1 -40 12.9
11 -26.3 -38.9 12.6
12 -26.1 -39.1 13
13 -25.7 -39.6 13.9
14 -25.7 -39.8 14.1
15 -25.4 -40.8 15.4
16 -25.3 -43.7 18.4
17 -25.5 -43.4 17.9
18 -25.2 -48 22.8
19 -25.9 -41.3 15.4
20 -26 -43.8 17.8
21 -26.8 -44.2 17.4
22 -27.1 -42 14.9
23 -27.5 -41.2 13.7
24 -28.2 -45.1 16.9
25 -29.3 -45.7 16.4
26 -30.6 -46.5 15.9
27 -31.9 -46.5 14.6
28 -34.1 -48.5 14.4
29 -36.7 -50.2 13.5
30 -40.7 -50 9.3
31 -46.2 -46.9 0.7
32 -51.7 -40.7 -11
93
Table A4: Circumferential mode shape data for both controlled and uncontrolled
responses at primary frequency of 227Hz during active control
Active Control Results Around Cylinder Circumference (error sensor (16, 4))
Point Uncontrolled (dB) Controlled (dB) Attenuation (dB)
1 -27.9 -35.8 7.9
1.5 -34.2 -33.5 -0.7
2 -36.2 -32.8 -3.4
2.5 -31.6 -31.9 0.3
3 -27.3 -34.3 7
3.5 -25.8 -38.1 12.3
4 -25.5 -59.9 34.4
4.5 -26.7 -39.8 13.1
5 -29.5 -36.2 6.7
5.5 -33.1 -34.1 1
6 -33.9 -33.3 -0.6
6.5 -29 -32.8 3.8
7 -26.1 -35.2 9.1
7.5 -24.7 -42.6 17.9
8 -24.3 -57.1 32.8
8.5 -26 -41.7 15.7
9 -29.7 -36.4 6.7
9.5 -35.3 -34.6 -0.7
10 -35.3 -34.2 -1.1
10.5 -30.2 -33.3 3.1
11 -27 -34.7 7.7
11.5 -25.2 -38.6 13.4
12 -24.5 -54.6 30.1
12.5 -25.2 -41.3 16.1
13 -27.4 -36.5 9.1
13.5 -31.7 -34.4 2.7
14 -37.9 -32.8 -5.1
14.5 -32.5 -32.4 -0.1
15 -28.3 -33.4 5.1
15.5 -26.1 -37.7 11.6
16 -25.3 -52.3 27
16.5 -26.1 -43 16.9
94
Table A5: Global uncontrolled mode shape data at primary frequency of 478Hz
478Hz Global Uncontrolled Vibration Magnitudes (dB)
Longitudinal Coordinate
1 4 7 10 13 16 19 22 25 28 31
1 -26.5 -24.7 -23.1 -22.9 -24.0 -22.3 -18.1 -17.2 -20.0 -28.9 -24.5
2 -31.0 -21.5 -17.9 -16.7 -18.6 -26.0 -18.2 -16.5 -17.0 -20.8 -27.8
3 -40.9 -20.1 -17.2 -17.1 -19.4 -34.8 -18.1 -16.2 -16.7 -20.1 -41.2
4 -31.9 -21.4 -21.0 -22.2 -25.6 -29.9 -17.7 -16.7 -18.1 -23.1 -32.0
5 -28.0 -25.2 -21.6 -20.9 -23.3 -24.2 -19.5 -18.9 -20.2 -26.7 -25.7
6 -25.3 -21.7 -16.3 -14.9 -15.6 -21.0 -19.5 -17.1 -17.4 -20.3 -22.1
7 -24.0 -21.5 -16.6 -14.4 -14.8 -20.2 -20.5 -17.5 -17.3 -18.4 -21.6
8 -24.3 -25.6 -17.2 -15.2 -15.4 -20.7 -27.6 -21.1 -19.2 -19.5 -22.0
9 -25.9 -25.8 -18.8 -16.2 -17.0 -22.4 -24.5 -19.3 -21.0 -22.9 -23.2
10 -29.4 -20.7 -17.1 -16.5 -17.9 -25.9 -18.2 -16.7 -17.2 -21.6 -27.2
11 -39.2 -20.8 -17.2 -16.4 -17.8 -32.2 -18.1 -16.7 -16.5 -19.7 -33.9
12 -37.2 -22.7 -17.7 -16.9 -18.1 -29.5 -22.7 -18.3 -18.2 -20.7 -32.2
13 -27.7 -25.9 -19.7 -18.0 -19.5 -24.5 -23.7 -21.3 -21.2 -24.2 -25.4
14 -24.2 -18.9 -17.4 -17.3 -19.3 -21.4 -16.7 -16.1 -17.7 -22.5 -23.9
15 -23.2 -19.1 -17.6 -17.8 -20.5 -20.0 -15.2 -14.9 -16.3 -21.6 -22.6
Cir
cu
mfe
ren
tial
Co
ord
inate
16 -23.7 -21.2 -21.3 -23.6 -27.0 -19.8 -15.7 -15.6 -18.4 -25.7 -22.8
Table A6: Global controlled mode shape data at primary frequency of 478Hz
478Hz Global Controlled Vibration Magnitudes (dB)
Longitudinal Coordinate
1 4 7 10 13 16 19 22 25 28 31
1 -30.6 -24.6 -22.5 -22.9 -27.8 -26.3 -19.7 -18.4 -20.2 -30.3 -29.0
2 -33.6 -23.5 -21.3 -21.2 -24.2 -30.3 -19.7 -18.7 -19.7 -27.5 -35.0
3 -43.1 -28.0 -23.3 -23.9 -27.4 -38.7 -26.8 -24.1 -25.1 -29.2 -50.1
4 -39.7 -37.7 -33.9 -33.6 -30.1 -34.6 -40.0 -31.9 -36.0 -37.0 -36.9
5 -32.7 -25.8 -20.7 -20.4 -24.1 -28.0 -21.0 -19.2 -21.0 -27.1 -31.2
6 -29.0 -23.2 -20.5 -20.7 -24.8 -25.6 -19.3 -17.6 -18.2 -25.8 -27.1
7 -28.3 -26.0 -27.5 -31.4 -32.8 -24.6 -21.1 -21.2 -22.6 -31.3 -26.7
8 -28.2 -31.9 -24.4 -26.9 -24.6 -24.9 -25.7 -27.2 -28.9 -30.5 -26.8
9 -31.4 -25.8 -19.3 -17.6 -18.8 -27.0 -26.9 -21.0 -21.3 -24.3 -27.6
10 -36.0 -24.8 -20.3 -18.1 -19.7 -24.6 -20.0 -20.8 -20.1 -23.8 -30.9
11 -40.8 -31.3 -25.5 -25.5 -23.4 -38.3 -30.1 -24.1 -25.6 -26.4 -36.7
12 -35.8 -36.7 -28.6 -34.5 -37.0 -36.7 -33.0 -31.3 -33.5 -43.5 -36.6
13 -31.8 -26.6 -20.5 -19.2 -21.4 -29.0 -25.1 -21.7 -22.1 -25.6 -30.2
14 -29.2 -25.1 -19.3 -17.6 -19.3 -25.8 -25.0 -21.1 -20.0 -23.7 -26.7
15 -29.0 -30.1 -23.9 -20.6 -19.4 -24.2 -35.0 -32.5 -22.5 -28.6 -26.2
Cir
cu
mfe
ren
tial
Co
ord
inate
16 -28.1 -33.1 -28.0 -29.2 -25.2 -24.8 -25.7 -24.5 -29.8 -31.5 -26.8
95
Table A7: Lengthwise mode shape data for both controlled and uncontrolled
responses at primary frequency of 478Hz during active control
Active Control Results Along Cylinder Length (error sensor at (8, 8))
Point Uncontrolled (dB) Controlled (dB) Attenuation (dB)
0 -47.6 -46.3 -1.3
1 -45.5 -44.4 -1.1
2 -41.4 -40.1 -1.3
3 -37 -41.7 4.7
4 -33.5 -43.9 10.4
5 -31.6 -43.1 11.5
6 -30.8 -43.9 13.1
7 -29.3 -37.1 7.8
8 -28.3 -39.9 11.6
9 -27.7 -41 13.3
10 -27.1 -40 12.9
11 -26.3 -38.9 12.6
12 -26.1 -39.1 13
13 -25.7 -39.6 13.9
14 -25.7 -39.8 14.1
15 -25.4 -40.8 15.4
16 -25.3 -43.7 18.4
17 -25.5 -43.4 17.9
18 -25.2 -48 22.8
19 -25.9 -41.3 15.4
20 -26 -43.8 17.8
21 -26.8 -44.2 17.4
22 -27.1 -42 14.9
23 -27.5 -41.2 13.7
24 -28.2 -45.1 16.9
25 -29.3 -45.7 16.4
26 -30.6 -46.5 15.9
27 -31.9 -46.5 14.6
28 -34.1 -48.5 14.4
29 -36.7 -50.2 13.5
30 -40.7 -50 9.3
31 -46.2 -46.9 0.7
32 -51.7 -40.7 -11
96
Table A8: Circumferential mode shape data for both controlled and uncontrolled
responses at primary frequency of 478Hz during active control with error
sensor at point (8,8) and accelerometer about circumference x = 8
Active Control Results Around Cylinder Circumference x = 8
Point Uncontrolled (dB) Controlled (dB) Attenuation (dB)
1 -22.6 -20.5 -2.1
1.5 -18.5 -17.5 -1.0
2 -16.7 -16.3 -0.4
2.5 -16.8 -17.0 0.2
3 -17.2 -18.4 1.2
3.5 -17.8 -21.1 3.3
4 -20.8 -27.3 6.5
4.5 -22.8 -24.3 1.5
5 -20.2 -18.9 -1.3
5.5 -17.0 -16.3 -0.7
6 -15.6 -15.9 0.3
6.5 -15.3 -17.6 2.3
7 -15.4 -20.6 5.2
7.5 -15.5 -26.4 10.9
8 -15.5 -46.6 31.1
8.5 -15.6 -21.1 5.5
9 -15.6 -16.4 0.8
9.5 -15.6 -14.7 -0.9
10 -15.7 -14.5 -1.2
10.5 -16.4 -16.4 0.0
11 -16.9 -18.5 1.6
11.5 -16.5 -22.6 6.1
12 -16.9 -34.5 17.6
12.5 -17.6 -25.0 7.4
13 -17.2 -18.5 1.3
13.5 -16.8 -15.8 -1.0
14 -16.3 -14.6 -1.7
14.5 -16.7 -15.4 -1.3
15 -17.4 -16.8 -0.6
15.5 -19.7 -20.1 0.4
16 -21.5 -24.5 3.0
16.5 -25.5 -27.2 1.7
97
Table A9: Circumferential mode shape data for both controlled and uncontrolled
responses at primary frequency of 478Hz during active control with error
sensor at point (8,8) and accelerometer about circumference x = 24
Active Control Results Around Cylinder Circumference x = 24
Point Uncontrolled (dB) Controlled (dB) Attenuation (dB)
1 -16.5 -16.6 0.1
1.5 -15.9 -14.2 -1.7
2 -15.5 -14.4 -1.1
2.5 -15.8 -15.3 -0.5
3 -16.4 -18.1 1.7
3.5 -16.6 -23.5 6.9
4 -16.8 -36.6 19.8
4.5 -17.0 -23.6 6.6
5 -16.6 -16.7 0.1
5.5 -16.1 -14.7 -1.4
6 -16.2 -14.4 -1.8
6.5 -16.3 -14.9 -1.4
7 -17.4 -17.1 -0.3
7.5 -18.7 -20.3 1.6
8 -20.1 -24.9 4.8
8.5 -22.6 -24.3 1.7
9 -20.8 -18.8 -2.0
9.5 -17.6 -16.2 -1.4
10 -16.4 -15.3 -1.1
10.5 -16.5 -16.5 0.0
11 -17.0 -18.2 1.2
11.5 -17.5 -21.4 3.9
12 -18.9 -28.1 9.2
12.5 -21.5 -26.2 4.7
13 -20.1 -18.9 -1.2
13.5 -17.4 -16.2 -1.2
14 -15.9 -15.5 -0.4
14.5 -15.7 -16.9 1.2
15 -15.7 -19.6 3.9
15.5 -16.3 -23.9 7.6
16 -16.5 -36.1 19.6
16.5 -16.7 -24.2 7.5
98
Table A10: Global uncontrolled mode shape data at primary frequency of 546Hz
546Hz Global Uncontrolled Vibration Magnitudes (dB)
Longitudinal coordinate
1 4 7 10 13 16 19 22 25 28 31
1 -13.9 -20.8 -29.8 -20.4 -16.1 -14.3 -15.2 -16.9 -23.2 -27.6 -16.3
2 -16.0 -21.7 -36.7 -25.6 -22.0 -18.5 -19.0 -21.8 -29.2 -27.0 -18.5
3 -21.9 -24.7 -25.2 -24.0 -22.9 -22.1 -21.1 -22.5 -25.0 -26.1 -24.4
4 -47.9 -33.3 -31.6 -28.1 -24.6 -22.7 -23.0 -24.4 -26.8 -36.5 -35.4
5 -21.7 -22.1 -27.3 -25.3 -23.7 -22.2 -22.0 -25.1 -29.7 -27.9 -23.1
6 -15.4 -19.2 -25.9 -25.0 -22.0 -20.1 -21.2 -22.8 -28.4 -24.6 -18.7
7 -13.0 -18.9 -39.5 -24.5 -19.3 -16.2 -15.7 -21.1 -30.6 -24.3 -16.4
8 -13.2 -19.7 -26.9 -18.5 -14.9 -13.2 -14.0 -16.6 -27.4 -25.9 -15.8
9 -14.4 -20.0 -31.0 -19.1 -16.0 -15.3 -15.6 -17.5 -24.2 -27.0 -19.1
10 -17.1 -22.5 -30.6 -25.0 -21.3 -18.4 -21.0 -22.5 -30.0 -28.0 -19.1
11 -21.2 -26.6 -25.5 -23.7 -23.0 -22.4 -23.1 -25.1 -28.4 -28.5 -24.3
12 -41.0 -43.6 -31.0 -26.9 -25.2 -25.3 -27.0 -28.4 -32.4 -37.8 -43.4
13 -19.0 -24.7 -29.1 -27.9 -25.6 -25.0 -25.3 -26.1 -31.0 -29.3 -23.5
14 -14.1 -19.1 -25.5 -25.6 -22.5 -21.4 -21.4 -23.6 -29.1 -24.9 -18.8
15 -12.8 -18.2 -36.2 -24.8 -19.8 -16.6 -17.6 -19.8 -29.7 -25.4 -16.5
Cir
cu
mfe
ren
tial
Co
ord
inate
16 -12.6 -19.1 -28.5 -18.6 -14.6 -13.2 -13.3 -15.3 -21.7 -26.3 -15.3
Table A11: Lengthwise mode shape data for uncontrolled response at 546Hz
Uncontrolled Longitudinal Response (546Hz)
Point Magnitude (dB) Point Magnitude (dB)
0 -10.5 17 -10.7
1 -12.5 18 -10.7
2 -15.0 19 -10.8
3 -17.6 20 -11.9
4 -21.6 21 -11.5
5 -26.5 22 -12.7
6 -26.6 23 -13.4
7 -22.0 24 -14.8
8 -18.5 25 -16.6
9 -16.4 26 -19.8
10 -14.8 27 -23.7
11 -13.4 28 -29.1
12 -12.7 29 -24.5
13 -11.7 30 -19.4
14 -11.5 31 -15.9
15 -10.8 32 -13.8
16 -10.7
99
Table A12: Circumferential mode shape data for uncontrolled response at 546Hz
Uncontrolled Circumferential Magnitudes (dB) at 546Hz
Longitudinal coordinate
8 16 24
1 -15.5 -12.6 -14.1
1.5 -18.5 -14.5 -16.7
2 -21.5 -16.6 -18.4
2.5 -18.2 -19.4 -16.8
3 -15.0 -20.9 -15.1
3.5 -17.6 -21.5 -16.5
4 -20.2 -21.2 -17.8
4.5 -18.4 -21.2 -18.5
5 -16.7 -19.2 -19.6
5.5 -16.2 -18.0 -18.5
6 -15.8 -17.2 -17.7
6.5 -19.1 -16.0 -18.4
7 -22.4 -13.9 -19.2
7.5 -17.7 -12.2 -17.7
8 -13.1 -11.3 -16.3
8.5 -14.2 -11.4 -15.1
9 -15.4 -13.1 -14.0
9.5 -16.8 -14.2 -16.6
10 -18.2 -17.4 -19.2
10.5 -16.6 -19.2 -18.8
11 -15.0 -20.5 -18.3
11.5 -17.1 -21.7 -20.6
12 -19.3 -23.1 -22.8
12.5 -19.1 -24.5 -21.7
13 -18.9 -21.5 -20.6
13.5 -17.4 -19.4 -19.5
14 -15.9 -18.2 -18.4
14.5 -18.4 -16.8 -18.4
15 -20.9 -14.8 -18.8
15.5 -17.4 -12.2 -15.0
16 -13.9 -11.2 -11.7
Cir
cu
mfe
ren
tial
Co
ord
inate
16.5 -14.7 -10.9 -12.4
100
Table A13: Lengthwise mode shape data for both controlled and uncontrolled
responses at primary frequency of 227Hz under active control using two
error sensors
Active control results along cylinder length (error sensors at (16, 4) and (16, 12)
Point Uncontrolled (dB) Controlled (dB) Attenuation (dB)
0 -46.2 -44.2 -2.0
1 -44.8 -42.1 -2.7
2 -43.0 -39.3 -3.7
3 -37.8 -40.9 3.1
4 -34.2 -42.6 8.4
5 -31.5 -43.6 12.1
6 -31.0 -48.7 17.7
7 -29.5 -46.4 16.9
8 -28.3 -41.7 13.4
9 -27.9 -37.4 9.5
10 -26.9 -38.6 11.7
11 -25.9 -34.9 9.0
12 -25.8 -35.3 9.5
13 -26.2 -39.2 13.0
14 -25.8 -39.1 13.3
15 -25.4 -36.2 10.8
16 -26.0 -43.2 17.2
17 -25.7 -38.7 13.0
18 -25.8 -43.3 17.5
19 -26.2 -46.5 20.3
20 -26.1 -39.1 13.0
21 -26.3 -38.9 12.6
22 -26.6 -34.4 7.8
23 -26.9 -34.3 7.4
24 -28.1 -39.0 10.9
25 -28.7 -39.1 10.4
26 -30.7 -44.6 13.9
27 -31.3 -42.1 10.8
28 -34.7 -44.6 9.9
29 -35.6 -43.1 7.5
30 -38.8 -46.3 7.5
31 -46.0 -49.1 3.1
32 -45.2 -39.8 -5.4
101
Table A14: Circumferential mode shape data for both controlled and uncontrolled
responses at a primary frequency of 227Hz under active control using
two error sensors
Active control about circumference x = 16 (error sensors at (16,4) and (16,12)
Point Uncontrolled (dB) Controlled (dB) Attenuation (dB)
1 -28.4 -34.4 6.0
1.5 -31.5 -32.8 1.3
2 -36.1 -32.2 -3.9
2.5 -37.5 -32.9 -4.7
3 -32.7 -35.5 2.8
3.5 -29.0 -42.0 13.0
4 -27.5 -52.2 24.7
4.5 -27.7 -39.3 11.6
5 -27.0 -35.2 8.2
5.5 -32.3 -33.5 1.2
6 -37.2 -32.1 -5.0
6.5 -34.6 -32.7 -1.9
7 -28.4 -36.5 8.0
7.5 -25.2 -42.0 16.8
8 -24.3 -45.3 21.0
8.5 -24.9 -36.0 11.1
9 -27.1 -33.2 6.1
9.5 -32.6 -31.7 -1.0
10 -36.3 -31.6 -4.7
10.5 -35.3 -32.8 -2.5
11 -32.9 -34.7 1.8
11.5 -27.2 -40.6 13.4
12 -25.6 -50.9 25.3
12.5 -26.0 -39.4 13.5
13 -28.5 -34.5 6.1
13.5 -32.5 -32.8 0.4
14 -37.9 -32.3 -5.6
14.5 -37.4 -33.1 -4.3
15 -31.1 -35.7 4.6
15.5 -27.2 -41.4 14.2
16 -25.9 -47.5 21.5
16.5 -26.4 -38.1 11.6
102
Appendix B Engineering Drawings
103
Transducer Plate
5/8/08 5/8/08