Active Vibration Control on Cyllindrical Shell

A

Seminar Reporton

ACTIVE VIBRATION CONTROL ONCYLLINDRICAL SHELL

Submitted in Partial Fulfillment of

the Requirements for the Degree

of

Master of Engineering

in

Machine Design

to

North Maharashtra University, Jalgaon

Submitted by

Miss.Dipmala S.Savale

Under the Guidance of

Prof.P.G.Damle

DEPARTMENT OF MECHANICAL ENGINEERING

SSBT’s COLLEGE OF ENGINEERING AND TECHNOLOGY,

BAMBHORI, JALGAON - 425 001 (MS)December 2014

SSBT’s COLLEGE OF ENGINEERING AND TECHNOLOGY,

BAMBHORI, JALGAON - 425 001 (MS)

DEPARTMENT OF MECHANICAL ENGINEERING

CERTIFICATE

This is to certify that the seminar entitled ACTIVE VIBRATION CONTROL ON

CYLLINDRICAL SHELL, submitted by Miss.Dipmala S.Savale in partial fulfill-

ment of the degree of Master of Engineering in Machine Design has been satisfactorily

carried out under my guidance as per the requirement of North Maharashtra Univer-

sity, Jalgaon.

Date: December 4, 2014

Place: Jalgaon

Prof.P.G.Damle

Guide

Prof. Dr. Dheeraj. S. Deshmukh Prof. Dr. K. S. Wani

Head Principal

SSBT’s College of Engineering and Technology, Bambhori, Jalgaon (MS) i

DECLARATION

I hereby declare that the work presented in this seminar entitled “ACTIVE

VIBRATION CONTROL ON CYLLINDRICAL SHELL”, submitted to the

Department of Mechanical Engineering, SSBT’s College of Engineering and Technol-

ogy, Bambhori, Jalgaon - 425 001 (MS), in partial fulfillment of the degree of Master

of Engineering in Machine Design of North Maharashtra University, Jalgaon, is my

original work.

Wherever contributions of others are involved, every effort is made to indicate this

clearly, with due acknowledgement and reference to the literature.

Date: December 4, 2014

Place: Jalgaon

(Miss.Dipmala S.Savale)

In my capacity as guide of the candidate’s seminar, I certify that the above statements

are true to the best of my knowledge.

(Prof.P.G.Damle)

SSBT’s College of Engineering and Technology, Bambhori, Jalgaon (MS) ii

Acknowledgements

Those who walk the difficult path to success never rest at this destiny they walk ahead

towards greater success. I consider myself lucky to work under guidance of such talented and

experienced people during the preparation of my Seminar report who guided me all through

it. I am indebted to HOD Dr. D. S. Deshmukh for his support at various stages during the

formation of this piece of work. A special mention must go to my guide Prof.P.G.Damle who

supported me with his vast knowledge, experience and suggestion. Only their inspiration has

made this seminar report easy and interesting. I would also like to thank our Principal Dr. K.

S. Wani For his warm support and providing all necessary facilities to us, the student. Last

but not list I am thankful to all the Teachers and Staff members of Mechanical Department

for their expert guidance and continuous encouragement throughout to see that the maximum

bene

t is taken out of this experience. At last, I would like to thanks to my Parents for their

support love and encouragement during the tenure of this Seminar.

Miss.Dipmala S.Savale

SSBT’s College of Engineering and Technology, Bambhori, Jalgaon (MS) iii

Contents

Acknowledgements iii

Abstract 1

1 Introduction 2

2 Vibrations Theory 5

2.1 General structural dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Vibration fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.2 Frequency response function . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.3 Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.4 Modal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 The concept of the active constrained layer damping 11

4 Variation modeling of the shell/ACLD system 13

4.1 Main assumptions of the model . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Kinematic relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.3 Stress-strain relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3.1 Cylindrical shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3.2 Piezo-electric constraining layer . . . . . . . . . . . . . . . . . . . . . 15

4.4 Energies of shell/ACLD system . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4.1 Potential energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4.2 Kinetic energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.4.3 Work done on shell/ACLD system . . . . . . . . . . . . . . . . . . . . 17

4.5 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.6 Boundary control strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.6.2 Control strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.6.3 Implementation of the boundary control strategy . . . . . . . . . . . 21

SSBT’s College of Engineering and Technology, Bambhori, Jalgaon (MS) iv

5 Performance of shell with ACLD and PCLD treatments 22

5.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5.2 Performance of the shell/ACLD system . . . . . . . . . . . . . . . . . . . . . 22

6 Conclusion 26

Bibliography 29

SSBT’s College of Engineering and Technology, Bambhori, Jalgaon (MS) v

List of Tables

SSBT’s College of Engineering and Technology, Bambhori, Jalgaon (MS) vi

List of Figures

2.1 a) SDOF system b) and free body diagram . . . . . . . . . . . . . . . . . . . 6

2.2 Frequency response of a forced SDOF system . . . . . . . . . . . . . . . . . . 7

2.3 Example of a time and frequency domain transformation for a vibrating beam 8

2.4 First mode of vibration in a tensioned string. . . . . . . . . . . . . . . . . . . 9

2.5 Mode separation of frequency response function . . . . . . . . . . . . . . . . 10

3.1 The Shell/ ACLD system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.1 Schematic drawing of the structure and geometry of shell/ACLD system. . . 14

5.1 Performance of the ACLD with the boundary controller (a) compliance and

(b) control voltage. Mode (1,0)-1860 Hz, Mode (2,0)-2020 Hz, Mode (3,0)-

2060 Hz, Mode (4,0)-2880Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 Shape of the dominant modes of vibration of the shell/ACLD system . . . . 24

6.1 Performance of the ACLD with proportional controller (a) compliance and

(b) control voltage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6.2 Performance of the ACLD with derivative controller (a) compliance and (b)

control voltage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

SSBT’s College of Engineering and Technology, Bambhori, Jalgaon (MS) vii

Abstract

Cylindrical shell-like structures exist in pipelines, pressure vessels, aircraft fuselages, ship

hulls and submarine hulls. Improved understanding of the dynamic behavior 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 sig-

natures of submarines and passenger discomfort due to both noise and vibrations in aircraft.

Distributed-parameter modeling of thin cylindrical shells which are fully treated with

active constrained layer damping (ACLD) is presented. Hamilton’s principle is utilized to

develop the shell/ACLD model as well as the associated boundary conditions. A globally

stable boundary control strategy is developed to damp out the vibration of the shell/ACLD

system. The devised boundary controller is compatible with the operating nature of the

ACLD treatments where the strain induced, in the active constraining layer, generates a

control force acting at the boundary of the treated shell. As the boundary control strategy

is based on a distributed parameter model of the shell/ACLD system, the classical spillover

problems resulting from using ”truncated” finite element models is eliminated. Also, such

an approach makes the boundary controller capable of controlling all the modes of vibra-

tion of the shell/ACLD and guarantees that the total energy of the system is continuously

decreasing with time. Numerical examples are presented to demonstrate the effectiveness

of the ACLD in damping out the vibration of cylindrical shells. Such effectiveness is deter-

mined for different control gains and compared with the performance of conventional passive

constrained layer damping (PCLD). The results obtained demonstrate the high damping

characteristics of the boundary controller particularly over broad frequency bands.

SSBT’s College of Engineering and Technology, Bambhori, Jalgaon (MS) 1

Chapter 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, con-

struction, aerospace, naval, manufacturing, military and music industries to name a few.

These applications all contain mechanical systems, which can be viewed upon a comprising

of distributed elements with characteristics of mass, stiffness and damping. A vibrating re-

sponse 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 forcing 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 behavior 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 signatures in submarines. In order to reduce these

undesired effects it is necessary to have a knowledge base of the dynamic behavior of cylin-

drical 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 feed forward control loops to detect the unwanted distur-

bance and apply a secondary force to minimize the resulting structural response.

Extensive efforts have been exerted to control the vibration of cylindrical shells using

either passive or active control means. For example, unconstrained passive damping layer

treatments to suppress the axi -symmetric vibrations of thin cylindrical shells. However, for

higher damping characteristics; the passive constrained layer damping (PCLD) treatments

have been successfully employed in various types of cylindrical shells. Recently, several at-

tempts have been made to actively control the vibration of shells using discrete piezo-electric

actuators bonded to the shell surfaces or distributed piezo-electric actuators embedded in

the composite fabric of the shell .In all the above studies, the emphasis is placed on using

separately the passive or the active vibration control actions. In the present study, a rad-

ically different approach is adopted whereby the passive and active control strategies are

combined to operate in unison. In the proposed hybrid configuration, an optimal balance

is achieved between the simplicity of the passive damping and the efficiency of the active

control. A preferred embodiment of such hybrid configuration is the active constrained layer

damping (ACLD) treatment which has been recognized as an effective means for damping

the vibration of beams and plates. The ACLD treatments have been controlled using simple

proportional and/or derivative feedback of the transverse deflection or the slope of the de-

flection line. The control gains have generally been arbitrarily selected to be small enough to

avoid instability problems. In 1994, Shen developed the stability bounds for full ACLD treat-

ments and Baz and Ro devised optimal control strategies for selecting the gains. In 1996,the

control gains are selected by Baz using the theory of robust controls to ensure stability in the

presence of parameter uncertainty and to reject the effect of external disturbances .In this

Seminar the focus is placed on extending the use of the ACLD treatments to control the vi-

bration of cylindrical shells with particular emphasis on developing a distributed-parameter

model using Hamilton’s principle to describe the axi-symmetric vibrations of shells which


are fully-treated with ACLD treatments. The variational formulation, being energy-based, is

much simpler than the force equilibrium based shear model of Pan which is used to analyze

the dynamics of circular sandwiched shells treated with PCLD treatments. Also, it directly

provides the boundary conditions associated with the ACLD treatment.

The present model is an extension of the boundary control model developed by Deng,

to control the vibrations of plain and untreated shells. The variational model is utilized to

devise a globally stable boundary control strategy which is compatible with the operating

nature of the ACLD treatments. In this manner, the instability problems associated with the

simple proportional and/or derivative controllers are completely avoided. Furthermore, as

the control strategy is based on a distributed-parameter model, hence the classical spillover

problems resulting from using ”truncated” finite element models are eliminated. Accordingly,

the devised boundary controller will be able to control all the modes of vibration of the

ACLD-treated structures.


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 sys-

tem 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

modeled 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 behav-

ior. 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

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.

(Mx+ Cx+Kx) = 0 (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:

x(t) = Aeωnξt sin(ωdt+ φ) (2.2)

Where A is the amplitude, t is the time, φ is the phase angle, ξ is the damping ratio, ωn

the natural frequency and ωd is the damped natural frequency given by

ωd = ωn√

1− ξ2

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

(Mx+ Cx+Kx) = Fo sinωt (2.3)

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

x(t) =F0

K

sin (ωt− φ)√[1− ( ω

ωn)2]2 + (2ξ ω

ωn)2

(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

Figure 2.2: Frequency response of a forced SDOF system

2.1.2 Frequency response function

Complex oscillatory behavior is often very difficult to analyze 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 the system


and is the mathematical relationship between the input X(ω) and output Y (ω) frequency

auto spectrums given for a single input/single output set-up as follows :

H(ω) =Y (ω)

X(ω)(2.5)

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, γ2xy(ω) which is defined as follows :

γ2xy(ω) =|Gxy(ω)|2

Gxx(ω)Gyy(ω)(2.6)

WhereGxx(ω) = X(ω)X∗(ω), Gyy(ω) = Y (ω)Y ∗(ω), Gxy(ω) = Y (ω)X∗(ω); ,

Are the input auto-spectral density, output auto-spectral density and cross-spectral den-

sity Respectively.

X∗(ω) and Y ∗(ω) are the complex conjugates of the input X(ω) and output Y (ω) re-

spectively.

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 0 < γ2xy(ω) < 1 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

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.

Grade 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 figure2.5. 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.


Figure 2.5: Mode separation of frequency response function


Chapter 3

The concept of the active constrained

layer damping

The ACLD treatment consists of a conventional passive constrained layer damping which

is augmented with efficient active control means to control the strain of the constraining

layer, in response to the shell vibrations as shown in Fig. 3.1. The shear deformation of

the visco-elastic damping layer is controlled by an active piezo-electric constraining layer

which is energized by a control voltage Vc. This control voltages generated based on the

boundary control strategy devised in this study. In this manner, the ACLD when bonded

to the shell acts as a smart constraining layer damping treatment with built-in actuation

capabilities. With appropriate strain control, through proper manipulation of Vc, all the

structural modes of vibration can be damped out. Also, the ACLD provides a practical

means for controlling the vibration of massive structures with the currently available piezo-

electric actuators without the need for excessively large actuation voltages. This is due to the

fact that the ACLD properly utilizes the piezo-electric actuator to control the shear in the

soft visco-elastic core which is a task compatible with the low control authority capabilities

of the currently available piezo-electric materials.


Figure 3.1: The Shell/ ACLD system


Chapter 4

Variation modeling of the shell/ACLD

system

4.1 Main assumptions of the model

Fig. 4.1 shows a schematic drawing of the ACLD treatment of a sandwiched cylindrical

shell. It is assumed that the shear strains in the piezo-electric actuator layer and in the base

shell are negligible. It is also assumed that the longitudinal and tangential stresses in the

visco-elastic core are negligible. The transverse displacements w of all points on any cross-

section of the sandwiched shell are considered to be equal. Furthermore, the piezo-electric

actuator layer and the base shell are assumed to be elastic and dissipate no energy whereas

the core is assumed to be linearly visco-elastic. In addition, it is assumed that the thickness

and modulus of elasticity of the sensor are negligible as compared to those of the base shell.

4.2 Kinematic relationships

From the geometry of Fig. 4.1, the shear strain g in the core is

γ =[hwx + (u1 − u3)]

h2(4.1)


Figure 4.1: Schematic drawing of the structure and geometry of shell/ACLD system.

h = h2 +h12

+h32

(4.2)

In the above equations, u1 and u3 are the longitudinal deflections of the piezo actuator

layer and shell layer, respectively; and w denotes the transverse deflection of the shell system.

Subscript x denotes partial differentiation with respect to x and h1, h2 and h3 define the

thicknesses of the piezo-actuator, the visco-elastic layer, the base shell system, respectively.

4.3 Stress-strain relationships

4.3.1 Cylindrical shell

Using Donnell-Mushtari theory of thin cylindrical shells[26], one can write the longitudinal

strains εix and the tangential strains eiθ in the ith layer as follows:

εix = uix − [Z + (h1 + h2)]wxy (4.3)

And

εiθ = −wR

(4.4)

and R denote the longitudinal strain at the middle of the ith layer and the radius of the

mid-surface of the core layer, respectively. Subscript i is set equal to 1 for the base shell

and 3 for the piezo-electric constraining layer. Hence, the corresponding longitudinal and

tangential stresses six and siq in the ith layer are given by:

σix =Ei

1− V 2i

[εix + Viεiθ] (4.5)


And

σiθ =Ei

1− V 2i

[εixVi + εiθ] (4.6)

Where Ei and nini are Young’s modulus and Poisson’s ratio for the ith layer, respectively.

The longitudinal and tangential forces, Nix and Niq, acting on the ith layer

Nix =

∫ ai

bi

σixdz, andNiθ =

∫ ai

bi

σiθdz (4.7)

Note that when i = 1, ai = −h22

and bi = − h22+h1

, and when i = 3, ai = −h22

and

bi = − h22+h1

,

Equations (4.3)-(4.7) gives

Nix = Ki[uix + Viw

R], andNiθ = −Ki[Viuix +

w

R], (4.8)

Where

Ki =Eihi

1− V 2i

4.3.2 Piezo-electric constraining layer

The strain εp induced by the piezo-electric layer due to the application of a control voltage

Vc is

εp =[d31 d32 d33

]T Vch1

(4.9)

where d31, d32 and d33 are the piezo-electric strain coefficients. Hence, the corresponding

induced stresses σp are obtained from:

σp =Ep

1− V 2p

1 vp 0

vp 1 0

0 0 1−vp2

εp (4.10)

Where Ep and vp are Young’s modulus and Poisson’s ratio for the piezo-electric actuator

layer respectively.

Integrating the piezo-stresses over the cross section of the actuator gives the control forces

and moments generated by the actuator. It is important to note here that the control forces

Nxp along the tangential direction and the associated control moments vanish because of the

axi-symmetric nature of the vibration of the shell. Only the longitudinal control forces, Nxp

generated along the x-axis exist and are given by


Nxp =

∫ ap

bp

σpxdz, (4.11)

Where

ap = −h22

and

bp = − h22 + h1

Also, σpx is the x component of the piezo stress. Equations (4.9)-(4.11) yield the following

expression for the control force Nxp

Nxp = K1(d31 + vpd32)Vch1

(4.12)

The expressions, given by Equation 4.8, for the longitudinal and tangential forces, Nix

and Niθ, acting on the different layers of the shell as well as the piezo-actuator control

force, given by Equation 4.12, are used to compute the potential and control energies of the

shell/ACLD system.

4.4 Energies of shell/ACLD system

4.4.1 Potential energies

The potential energies associated with the extension U1, bending U2 and shearing U3 of the

different layers of the shell/ACLD system are given by

U1 = πR

3∑i=1

{∫ L

0

Ki[uix + viw

R]uixdx+

∫ L

0

Ki[viuix +w

R]uixdx}, (4.13)

U2 = πRDt

∫ L

0

w2xxdx, (4.14)

and

U3 = πRG2h2

∫ L

0

γ2dx, (4.15)

Where Dt =∑3

n=1Eih3i with Eih

3i Denoting the flexural rigidity of the ith layer and G29

is the shear modulus of the visco-elastic layer.


4.4.2 Kinetic energies

The kinetic energy T associated with the transverse deflection w of the shell/ACLD system

is given by

T = πRm

∫ L

0

w2t dx, (4.16)

Where m is the mass per unit perimeter length of the sandwiched shell system.

4.4.3 Work done on shell/ACLD system

The work done W1 by the external transverse loads q acting on the shell/ACLD system per

unit perimeter length of the sandwiched shell is given by:

W1 = πR

∫ L

0

qwdx, (4.17)

and the work done W2 by the piezo-electric control forces is given by:

W2 = πR

∫ L

0

Nxpuixdx, (4.18)

In this , Nxp is assumed constant over the entire length of the constraining layer in order

to maintain and emphasize the simplicity and practicality of the ACLD treatment.

The work W3 dissipated in the visco-elastic core is given by:

W3 = −πRh2∫ L

0

τdγdx, (4.19)

Where τd is the dissipative shear stress developed by the visco-elastic core. It is given by:

τd = (G2ηvω

)γt = (G2ηv)γt, (4.20)

Where ηv , ω and i denote the loss factor of the visco-elastic core, the frequency and

(−1)1/2 respectively.

In Equation 4.20, the behavior of the visco-elastic core is modeled using the common

complex modulus approach which is a frequency domain-based method . Adoption of his

approach results in a variational model of the ACLD which can be easily reduced to the

classical model of Pan when the piezoelectric strain is set equal to zero.


4.5 The model

The equations and boundary conditions governing the operation of the shell/ACLDsystem

are obtained by applying Hamilton’s principle which is an integral principle considering the

entire motion of the system between two instants:

∫ r2

r1

δ(T −3∑i=1

Ui)dt+

∫ r2

r1

δ(3∑i=1

Wi)dt = 0, (4.21)

Where δ0 denotes the first variation in the quantity inside the parentheses, t denotes the

time and t1 and t2 define the bounds of the time interval where the shell/ACLD dynamics

are considered.

The resulting equations of the shell/ACLD system are:

K1u1xx +K1v1wxR− G2

h2(u1 − u3 + hwx) = 0, (4.22)

K1u3xx +K3v3wxR

+G2

h2(u1 − u3 + hwx) = 0, (4.23)

And

Dtwxxxx+mwtt−G2h

h2(u1x−u3x +hwx)+

K1 +K3

R2w+

K1v1R

u1x +K3v3R

u3x−q = 0, (4.24)

Where G2 = G2′(1+ηvi) is the complex modulus of the visco-elastic material. The above

equations are subject to the following boundary conditions:

[u1x +v1Rw] δu1|L0 = (d31 + v1d32)

v

h1

∣∣∣∣L0

, (4.25)

K1[u3x +v3Rw] δu3|L0 = 0, (4.26)

Dtwxx δwx|L0 = 0, (4.27)

And

[Dtwxxx +G2h

h2(u1 − u3 + hwx)δw]

∣∣∣∣L0

= 0, (4.28)

Eliminating u1 and u3 from Equations 4.22- 4.24 yields the following sixth order partial

differential equation in the transverse deflection w of the shell/ACLD system:


Dtwxxxx − g(1 + Y )wxxxx + [(1− v21R2

)K1 + (1− v23R2

)K3 +2(v1 − v3)Rh2

G2h]wxx

− [g(1− v21R2

)K1 + (1− v23R2

)K3 + (v1 − v3)2G2

R2h2]w +mwxxtt −mgwtt − qxx + gq = 0,

(4.29)

Where

g =G2

h2[(K1 +K3)

K1K3

]andY =K1K3h

2

(K1 +K3)Dt

(4.30)

For simply-supported shell/ACLD system, the eight boundary conditions given by Equa-

tions 4.25 and 4.28 reduce to the following six boundary conditions: At x=0 and L

mwn +Dtwxxxx + (K1v1R− G2h

h2)(1 +

d32d31

)V

h1= q, (4.31)

w = 0, (4.32)

And

wxx = 0, (4.33)

Similar expressions can be easily obtained for other boundary conditions. It is important

here to note that the sixth order partial differential equation describing the shell/ACLD

system Equation 4.26 is the same as that describing a shell treated with conventional Passive

Constrained Layer Damping (PCLD) as obtained by Pan. However, the boundary condition

given by Equation 4.31 is modified to account for the control action generated by the control

voltage Vc applied to the Active Constraining Layer at the free ends of the shell (that is at

x=0 and L).Therefore, the particular nature of operation of the shell/ACLD system implies

the existence of boundary control action at x=0 and L. In Section 5, a boundary control

strategy is devised to capitalize on this inherent operating nature of the shell/ACLD system

in such a manner that ensures global stability of all the vibration modes of the system.

4.6 Boundary control strategy

4.6.1 Overview

Distributed-parameter control theory is used to devise a boundary control strategy that

generates the boundary control action in order to ensure global stability of all the vibration

modes of the shell/ACLD system. The control strategy is devised to ensure that the total

energy of the shell/ACLD system is a strictly non increasing function of time.


4.6.2 Control strategy

The total energy En of the shell/ACLD system is obtained using Equations 4.13-4.19 as

follows:

En = U1 + U2 + U3 + T, (4.34)

Or

En = πR3∑i=1

∫ L

0

Ki[uix + viw

R]uixdx+

∫ L

0

Ki[viuix +w

R]uixdx

+ πRDt

∫ L

0

w2xxdx+ πRG2h2

∫ L

0

γ2dx+ πRm

∫ L

0

w2t dx, (4.35)

or

En = πR∑i=1,3

∫ L

0

Ki[uix + viw

R] + (1− v2i )(

w

R

2

)dx+ πRDt

∫ L

0

w2xxdx

+ πRG2h2

∫ L

0

γ2dx+ πRm

∫ L

0

w2t dx, (4.36)

Equation 4.36 gives the energy norm of the shell/ACLD system which is quadratic and

strictly positive. This norm is equal to zero if, and only if, u1, u3, w, wx, wxx and wt

are all zero for all the points on the shell between [0,L]. This condition is ensured only

when the shell/ACLD system reverts back to its original un deflected equilibrium position.

Differentiating the different components of Equation 4.36 with respect to time, integrating

by parts and imposing the boundary conditions gives:

dEndt

= Nxp [u1t(L)− u1t(0)]− G2ηvh2nω

∫ L

0

γ2t dx (4.37)

As the second term is strictly negative, hence with a continuously decreasing energy norm

(that is dEn/dt < 0) is obtained when the control action Nxp takes the following form:

Nxp = −Kg[u1t(L)− u1t(0)] (4.38)

where Kg is the gain of the boundary controller. Equation 4.38 indicates that the control

action is a velocity feedback of the longitudinal displacement of the piezo electric constraining

layer. It is also important to note that when the active control action Nxp ceases or fails

to operate for one reason or another (that is when the control voltage Vc =0 as indicated

in Equation 4.12), the shell system remains globally stable as indicated by Equation 4.37.

Such inherent stability is attributed to the second term in the equation which quantifies the


contribution of the Passive Constrained Layer Treatment(PCLD). Hence, the two terms of

Equation 4.37 provide quantitative means for weighing the individual contributions of the

ACLD and the PCLD to the total rate of energy dissipation of the shell system.

4.6.3 Implementation of the boundary control strategy

The globally stable boundary controller can be implemented using Equations 4.12 and 4.38

to generate the control voltage Vc as follows:

Vc = [h1K1

(d31 + v1d32)]Nxp (4.39)

Vc = −[Kg[h1

K1(d31 + v1d32)]][u1t(L)− u1t(0)]

Vc = −Kg[u1t(L)− u1t(0)]

where KG denotes the equivalent gain of the boundary controller such that KG =

Kg[h1

K1(d31+v1d32)] . Such an equivalent gain combines the control gain Kg and the piezo-

actuator parameters (h1, K1, d31, d32 and v1) which are generally unknown constants. Im-

plementation of the above control strategy requires that the actuator must be designed as

an actuator with self-sensing capabilities using the approaches suggested by Dosch, Inman

and Garcia to measure u1. It is important to note that the temporal derivatives of u1 can

be determined by monitoring the current of the piezo sensor rather than its voltage as de-

scribed, for example, by Miller and Hubbard. The effectiveness of the boundary controller,

given by Equation 4.38, in suppressing the vibration of a shell treated with ACLD treatment

is determined in Section 5when the shell system is subjected to axi-symmetric sinusoidal

transverse load acting uniformly over the entire span of the shell.


Chapter 5

Performance of shell with ACLD and

PCLD treatments

5.1 Materials

The effectiveness of the ACLD treatment is demonstrated using a simply-supported alu-

minum shell which is 0.30 m long, 0.005 m thick and 0.30 m in outside diameter. The shell is

treated with an acrylic base visco-elastic material which is 0.005 m thick and has a complex

shear modulus G2 =20 (1+1.0i) MN/m2.The visco-elastic core is constrained by an active

polymeric piezo-electric film(PVDF) whose thickness h1 and Young’s modulus E1 are 0.005

m and 2.25 GN/m2,respectively. The piezo-electric strain constants d31 and d32 are 23x 10−21

and 3x 10−12 m/V, respectively.

5.2 Performance of the shell/ACLD system

The effectiveness of the devised boundary controller in damping out the vibration of the alu-

minum shell under consideration is determined by subjecting the shell to sinusoidal transverse

loading which is uniformly distributed over the entire span of the shell. The compliance is

calculated at the mid-span using the mechanical compliance approach. Figure 5.1 (a) shows

the compliance of the shell/ACLD system for the gain Kg of the boundary control set to

105 and 3x105 N/m/s. Also shown in the figure is the compliance of the uncontrolled shell

which is treated with the PCLD treatment. In that case, the control loop that regulates the

interaction between the piezo-sensor and the piezo-actuator is maintained open, that is Kg

= 0. It is evident that the ACLD treatment has effectively attenuated the vibration of the

shell over the frequency band under consideration as compared to the conventional PCLD

treatment. The corresponding control voltage used to activate the piezo-constraining layer is


shown in Figure 5.1(b) for different levels of the control gains. Note that effective vibration

attenuation can be achieved by the devised boundary control strategy without the need for

excessively high control voltages. Figure 5.1 shows also that the dominant modes of vibration

in the frequency response occur at 1962 and 2800 Hz. These two modes correspond to modes

(1,0) and (4,0)as shown by the mode shapes displayed in Figure 5.2. These mode shapes are

obtained using classical finite element methods and are validated against the shapes of the

transverse deflection lines at the corresponding modes.

Figure 5.1: Performance of the ACLD with the boundary controller (a) compliance and

(b) control voltage. Mode (1,0)-1860 Hz, Mode (2,0)-2020 Hz, Mode (3,0)-2060 Hz, Mode

(4,0)-2880Hz.


Figure 5.2: Shape of the dominant modes of vibration of the shell/ACLD system

The effectiveness of the boundary controller can also be emphasized when its performance

is compared with the performance of conventional Proportional (P) and Derivative (D)

control laws.

Figure 6.1 shows the corresponding performance characteristics when a conventional Pro-

portional (P) control law is used such that the control action Nx = −Kpw(L/2) where Kp

and w(L/2) are the proportional control gain and the transverse deflection of the shell at

its mid-span. It is evident from Figure 6.1(a) that the P controller is only effective at low

excitation frequencies especially when the control gain is low (for example, when Kp=21010

N/m).Increasing the control gain to Kp=51010 N/m improves the performance over a wider

low frequency band. However, it results in control spillover into the high frequency modes of

vibration. This results in excessive vibrations, as shown in Figure 6.1(a), and high control

voltage as shown in Figure 6.1(b).The effect of using a Derivative (D) feedback controller

on the performance of the ACLD treatment is shown in Figure 6.2. For this controller, the

control action Nx = −Kdwt(L/2) where Kd is the derivative control gain. Figure 6.2(a)


demonstrates clearly the effectiveness of this controller at high excitation frequencies. How-

ever, it is ineffective at low frequencies. Figure 6.2(b) shows the corresponding control

voltage. Hence, a comparison between the characteristics shown in Figure 5.1, Figure 6.1

and Figure 6.2 indicates that the boundary controller, is more effective in damping out the

vibrations of cylindrical shells and requires less control voltage than the conventional P and

D controllers.


Chapter 6

Conclusion

A variational formulation of the dynamics of shells which are fully-treated with Active Con-

strained Layer Damping treatments

Figure 6.1: Performance of the ACLD with proportional controller (a) compliance and (b)

control voltage.


The equations and the boundary conditions governing the performance of this class of

surface treatment are presented using Hamilton’s principle. These equations are used to

devise a globally stable boundary control strategy which is compatible with the operating

nature of the ACLD treatment. The developed control strategy ensures global stability for

all the vibration modes of the shell/ACLD system and guarantees that the total energy norm

of the system is continuously decreasing with time. Implementation of the boundary control

strategy requires the measurement of the longitudinal displacements of the piezo-constraining

layer.

Figure 6.2: Performance of the ACLD with derivative controller (a) compliance and (b)

control voltage

The performance of the boundary controller is shown to exhibit effective vibration atten-

uation for all the modes of a simply-supported shell subjected to uniform transverse loading,

over a broad frequency band as compared to conventional passive constrained layer damping

treatments and the classical P and D controllers. Such effectiveness stems from treatment


to combine the ability of the ACLD treatment to combine the attractive attributes of both

passive and active controls to produce lower amplitudes of vibration with lower control volt-

ages. It is important here to note that although the analysis, control strategies and the

numerical results presented are for simply-supported shells, the procedures developed in this

chapter can be readily extended to shells subject to other boundary conditions. However, the

proposed boundary controllers become inappropriate for shells with fixed-fixed boundaries

because of the controllability and observability issues raised by Miller and Hubbard. The

control of the vibration of general shells, treated with ACLD treatments, using boundary

controllers is a natural extension of the present work. It is worth also mentioning here that

although the boundary controller presented is shown to be theoretically stable for all the

modes of vibrations, the stability bounds are practically not infinite because of the actuator

and sensor dynamics. An attempt to address these issues in using dynamic controllers is

recommended for future studies.


Bibliography

[1] Markus S. Damping properties of layered cylindrical shells vibrating in axially symmetric

modes. J Sound Vibr 1976;48(4):511-24.

[2] Markus S. Refined theory of damped ax symmetric vibration of double-layered cylin-

drical shells. JMech Engng Sci 1979; 21(1):33-7.

[3] Jones IW, Salerno VL. The effect of structural damping on the forced vibration of

cylindrical sandwich shells. Trans Am Soc Mech Eng J Engng Indust 1966;88:318-24.

[4] Pan HH. Axisymmetric vibrations of a circular sandwich shell with a visco-elastic core

layer. J Sound Vibr 1969;9(2):338-48.

[5] Lu YP, Douglas BE, Thomas EV. Mechanical impedance of damped three-layered sand-

wich rings. AIAA J 1973;11(3):300-4.

[6] DiTaranto RA. Free and forced response of a laminated ring. J Acoust Soc Am

1972;53(3):748-57.20

[7] A. Baz, T. Chen / Thin-Walled Structures 36 (2000) 1-20 Vibr 1981;77(1):1-10.

[8] Alam N, Asnani NT. Vibration and damping analysis of a multi-layered cylindrical shell,

part I: theoretical analysis. AIAA J 1984;22(6):803-10.

[9] Ramesh TC, Ganesan N. Finite element analysis of cylindrical shells with a constrained

viscoelastic layer. J Sound Vibr 1994;172(3):359-70.

[10] Ramesh TC, Ganesan N. Vibration and damping analysis of cylindrical shells with con-

strained damping treatment-a comparison of three theories. J Vibr Acoust 1995;117:213-

9.

[11] Forward RL. Electronic damping of orthogonal bending modes in a cylindrical mast

experiment. JSpacecraft 1981;18:11-7.


[12] Lester HC, Lefebvre S. Piezoelectric actuator model for active sound and vibration

control cylinders. J Intell Mater Sys Struct 1993;4:295-306.

[13] Zhou S, Liang C, Rogers CA. Impedance modeling of two-dimensional piezoelectric

actuators bonded on a cylinder. Adapt Struct Mater Sys ASME 1993;35:247-55.


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Active Vibration Control on Cyllindrical Shell