Active Vibration Control of Finite Thin-Walled … pdfs/thesis_mcgann.pdfActive Vibration Control of Finite Thin-Walled Cylindrical Shells ... pressure vessels, ... cylinder length

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



5.2 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


.................................................................................................................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



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 4.24 Longitudinal mode shape at 546Hz measured along b = 16.5 using 33

point mesh. ..............................................................................................60


point mesh. ..............................................................................................60


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


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



Table A6: Global controlled mode shape data at primary frequency of 478Hz ......94




responses at primary frequency of 478Hz during active control with error

sensor at point (8,8) and accelerometer about circumference x = 8 .......96



sensor at point (8,8) and accelerometer about circumference x = 24 .....97


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


responses at primary frequency of 227Hz under active control using two

error sensors ..........................................................................................100


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.



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


mesh.

227Hz Circumferential

Mode shape

-40

-35

-30

-25

-20

0

90

180

270


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.


58


-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)


mesh.


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.


60


-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.


point mesh.

61


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.


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


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.


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


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

cylindrical shells using wave propagation approach”’, Journal of Sound and

Vibration, 249, 1011-1015.

88

10. Zhang, X. M., (2002), ‘Reply to “Comments on ‘Vibration analysis of thin

cylindrical shells using wave propagation approach’”’, Journal of Sound and

Vibration, 253, 1140-1142.

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-

788.

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)


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


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



Active Control Results Around Cylinder Circumference (error sensor (16, 4))


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




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)


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



Active Control Results Along Cylinder Length (error sensor at (8, 8))


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



sensor at point (8,8) and accelerometer about circumference x = 8

Active Control Results Around Cylinder Circumference x = 8


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



sensor at point (8,8) and accelerometer about circumference x = 24

Active Control Results Around Cylinder Circumference x = 24


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



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


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)


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


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)


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

Documents

Active Vibration Control of Finite Thin-Walled … pdfs/thesis_mcgann.pdfActive Vibration Control of Finite Thin-Walled Cylindrical Shells ... pressure vessels, ... cylinder length