Vibration and control

The maximum value at resonance, called the peak resonance, and denoted by M p, can be shown see, for instance, Inman, 2001 to be related to the damping ratio by 2 Also, Figure 1.9 can be

Vibration with Control

Vibration with Control D J Inman

2006 John Wiley & Sons, Ltd ISBN: 0-470-01051-7

Vibration with Control

Daniel J Inman

Virginia Tech, USA

Copyright © 2006 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,

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ISBN-10 0-470-01051-7 (cloth : alk paper)

1 Damping (Mechanics) 2 Vibration I Title.

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viii CONTENTS

Advance-level vibration topics are presented here, including lumped-mass and mass systems in the context of the appropriate mathematics, along with topics from controlthat are useful in vibration analysis and design This text is intended for use in a secondcourse in vibration, or in a combined course in vibration and control This book is alsointended as a reference for the field of structural control and could be used as a text instructural control Control topics are introduced at beginner level, with no knowledge ofcontrols needed to read the book

distributed-The heart of this manuscript was first developed in the early 1980s and published in 1989

under the title Vibration with Control, Measurement and Stability That book went out of

print in 1994 However, the text remained in use at several universities, and all used copiesseem to have disappeared from online sources in about 1998 Since then I have had yearlyrequests for copying rights Hence, at the suggestions of colleagues, I have revised the olderbook to produce this text The manuscript is currently being used in a graduate course atVirginia Tech in the Mechanical Engineering Department As such, presentation materialsfor each chapter and a complete solutions manual are available for use by instructors.The text is an attempt to place vibration and control on a firm mathematical basis andconnect the disciplines of vibration, linear algebra, matrix computations, control, and appliedfunctional analysis Each chapter ends with notes on further references and suggests wheremore detailed accounts can be found In this way I hope to capture a ‘big picture’ approachwithout producing an overly large book The first chapter presents a quick introductionusing single-degree-of-freedom systems (second-order ordinary differential equations) tothe following chapters, which extend these concepts to multiple-degree-of-freedom systems(matrix theory, systems of ordinary differential equations) and distributed-parameter systems(partial differential equations and boundary value problems) Numerical simulations andmatrix computations are also presented through the use of MatlabTM New material hasbeen added on the use of Matlab, and a brief introduction to nonlinear vibration is given.New problems and examples have been added, as well as a few new topics

ACKNOWLEDGMENTS

I would like to thank Jamil M Renno, a PhD student, for reading the final manuscriptand sorting out several typos and numerical errors In addition, Drs T Michael Seigler,

Kaihong Wang, and Henry H Sodano are owed special thanks for helping with the figures.

I would also like to thank my past PhD students who have used the earlier version of the book,

as well as Pablo Tarazaga, Dr Curt Kothera, M Austin Creasy, and Armaghan Salehian whoread the draft and made wonderful corrections and suggestions Professor Daniel P Hess

of the University of South Florida provided invaluable suggestions and comments for which

I am grateful I would like to thank Ms Vanessa McCoy who retyped the manuscript fromthe hard copy of the previous version of this book and thus allowed me to finish writingelectronically

Thanks are also owed to Wendy Hunter of Wiley for the opportunity to publish thismanuscript and the encouragement to finish it I would also like to extend my thanks andappreciation to my wife Cathy Little, son Daniel, and daughters Jennifer and Angela (andtheir families) for putting up with my absence while I worked on this manuscript

Daniel J Inmandinman@vt.edu

Single-degree-of-freedom Systems

In this chapter the vibration of a single-degree-of-freedom system will be analyzed andreviewed Analysis, measurement, design, and control of a single-degree-of-freedom system(often abbreviated SDOF) is discussed The concepts developed in this chapter constitute anintroductory review of vibrations and serve as an introduction for extending these concepts

to more complex systems in later chapters In addition, basic ideas relating to measurementand control of vibrations are introduced that will later be extended to multiple-degree-of-freedom systems and distributed-parameter systems This chapter is intended to be areview of vibration basics and an introduction to a more formal and general analysis formore complicated models in the following chapters

Vibration technology has grown and taken on a more interdisciplinary nature This hasbeen caused by more demanding performance criteria and design specifications for all types

of machines and structures Hence, in addition to the standard material usually found inintroductory chapters of vibration and structural dynamics texts, several topics from controltheory and vibration measurement theory are presented This material is included not totrain the reader in control methods (the interested student should study control and systemtheory texts) but rather to point out some useful connections between vibration and control

as related disciplines In addition, structural control has become an important disciplinerequiring the coalescence of vibration and control topics A brief introduction to nonlinearSDOF systems and numerical simulation is also presented

Simple harmonic motion, or oscillation, is exhibited by structures that have elastic restoringforces Such systems can be modeled, in some situations, by a spring–mass schematic, asillustrated in Figure 1.1 This constitutes the most basic vibration model of a structure and can

be used successfully to describe a surprising number of devices, machines, and structures.The methods presented here for solving such a simple mathematical model may seem to be

Vibration with Control D J Inman

2006 John Wiley & Sons, Ltd ISBN: 0-470-01051-7

If x= xt denotes the displacement (m) of the mass m (kg) from its equilibrium position

as a function of time t (s), the equation of motion for this system becomes [upon summingforces in Figure 1.1(b)]

m¨x + kx + xs− mg = 0where k is the stiffness of the spring (N/m), xs is the static deflection (m) of the springunder gravity load, g is the acceleration due to gravity (m/s2), and the overdots denotedifferentiation with respect to time (A discussion of dimensions appears in Appendix A, and

it is assumed here that the reader understands the importance of using consistent units.) Fromsumming forces in the free body diagram for the static deflection of the spring [Figure 1.1(c)],

mg= kx sand the above equation of motion becomes

This last expression is the equation of motion of a single-degree-of-freedom system and is

a linear, second-order, ordinary differential equation with constant coefficients

Figure 1.2 indicates a simple experiment for determining the spring stiffness by addingknown amounts of mass to a spring and measuring the resulting static deflection, xs Theresults of this static experiment can be plotted as force (mass times acceleration) versus xs,the slope yielding the value of k for the linear portion of the plot This is illustrated inFigure 1.3

Once m and k are determined from static experiments, Equation (1.1) can be solved toyield the time history of the position of the mass m, given the initial position and velocity

of the mass The form of the solution of Equation (1.1) is found from substitution of anassumed periodic motion (from experience watching vibrating systems) of the form

where n=√k/m is the natural frequency (rad/s) Here, the amplitude, A, and the phase

shift, , are constants of integration determined by the initial conditions.

The existence of a unique solution for Equation (1.1) with two specific initial conditions is

well known and is given by, for instance, Boyce and DiPrima (2000) Hence, if a solution of

the form of Equation (1.2) form is guessed and it works, then it is the solution Fortunately,

in this case the mathematics, physics, and observation all agree

To proceed, if x0is the specified initial displacement from equilibrium of mass m, and v0isits specified initial velocity, simple substitution allows the constants A and  to be evaluated.The unique solution is

xt=



2x2

0+ v2 0



(1.3)

Alternatively, xt can be written as

xt=v0

by using a simple trigonometric identity

A purely mathematical approach to the solution of Equation (1.1) is to assume a solution

of the form xt= A et and solve for , i.e.,

m2et+ ket= 0

This implies that (because et= 0, and A = 0)

2+

km

1/2

= ±nj

where j= −11/2 Then the general solution becomes

where A1 and A2 are arbitrary complex conjugate constants of integration to be determined

by the initial conditions Use of Euler’s formulae then yields Equations (1.2) and (1.4) (see,for instance, Inman, 2001) For more complicated systems, the exponential approach is oftenmore appropriate than first guessing the form (sinusoid) of the solution from watching themotion

Another mathematical comment is in order Equation (1.1) and its solution are valid only

as long as the spring is linear If the spring is stretched too far, or too much force is applied

to it, the curve in Figure 1.3 will no longer be linear Then Equation (1.1) will be nonlinear(see Section 1.8) For now, it suffices to point out that initial conditions and springs shouldalways be checked to make sure that they fall in the linear region if linear analysis methodsare going to be used

SPRING–MASS–DAMPER SYSTEM 5

Friction-free Surface

k

y

f k

f c x

N s/m or kg/s in terms of fundamental units

Again, the unique solution of Equation (1.6) can be found for specified initial conditions

by assuming that xt is of the form

xt= A etand substituting this into Equation (1.6) to yield

Equation (1.8) is called the characteristic equation of Equation (1.6) Using simple algebra,

the two solutions for  are

12= − c

2m±12



c2

m2− 4k

The quantity under the radical is called the discriminant and, together with the sign of m c,

and k, determines whether or not the roots are complex or real Physically, m c, and k areall positive in this case, so the value of the discriminant determines the nature of the roots

of Equation (1.8)

It is convenient to define the dimensionless damping ratio, , as

2√km

In addition, let the damped natural frequency,  d, be defined (for 0 <  < 1) by

A=v0+

+2− 1 nx02n

2− 1

B= −v0+

−2− 1 nx02n

2− 1The overdamped response has the form given in Figure 1.6 An overdamped system doesnot oscillate, but rather returns to its rest position exponentially

= 1

so that the discriminant in Equation (1.12) is zero and the roots are a pair of negative realrepeated numbers The solution of Equation (1.11) then becomes

0.5 0.0 0.2 0.4 0.6 0.8

The critically damped response is plotted in Figure 1.7 for various values of the initialconditions v0and x0

It should be noted that critically damped systems can be thought of in several ways First,they represent systems with the minimum value of damping rate that yields a nonoscillatingsystem (Problem 1.5) Critical damping can also be thought of as the case that separatesnonoscillation from oscillation

The preceding analysis considers the vibration of a device or structure as a result of someinitial disturbance (i.e., v0 and x0) In this section, the vibration of a spring–mass–dampersystem subjected to an external force is considered In particular, the response to harmonicexcitations, impulses, and step forcing functions is examined

FORCED RESPONSE 9

x (t)

y x

surface and (b) free body diagram of the system of part (a)

In many environments, rotating machinery, motors, and so on, cause periodic motions

of structures to induce vibrations into other mechanical devices and structures nearby It iscommon to approximate the driving forces, Ft, as periodic of the form

Ft= F0sin twhere F0 represents the amplitude of the applied force and  denotes the frequency of theapplied force, or the driving frequency (rad/s) On summing the forces, the equation for theforced vibration of the system in Figure 1.8 becomes

Recall from the discipline of differential equations (Boyce and DiPrima, 2000), that thesolution of Equation (1.17) consists of the sum of the homogeneous solution in Equation (1.5)

and a particular solution These are usually referred to as the transient response and the

steady state response respectively Physically, there is motivation to assume that the steady

state response will follow the forcing function Hence, it is tempting to assume that theparticular solution has the form

where X is the steady state amplitude and is the phase shift at steady state cally, the method is referred to as the method of undetermined coefficients Substitution ofEquation (1.18) into Equation (1.17) yields



1− m2/k2+ c/k2or

Xk

xt= e− n tA sin dt+ B cos dt+ X sint −  (1.21)

Here, A and B are constants of integration determined by the initial conditions and theforcing function (and in general will be different from the values of A and B determined forthe free response)

Examining Equation (1.21), two features are important and immediately obvious First,

as t becomes larger, the transient response (the first term) becomes very small, and hencethe term steady state response is assigned to the particular solution (the second term) Thesecond observation is that the coefficient of the steady state response, or particular solution,becomes large when the excitation frequency is close to the undamped natural frequency,i.e., ≈ n This phenomenon is known as resonance and is extremely important in design,

vibration analysis, and testing



−9X1− 1 2X2+ 4X1−√1

2

sin 3t+ −9X2+ 1 2X1+ 4X2 cos 3t= 0

FORCED RESPONSE 11

Since the sine and cosine are independent, the two coefficients in parentheses must vanish, resulting

in two equations in the two unknowns X1and X2 This solution yields

xpt= −0 134 sin 3t − 0 032 cos 3tNext, consider adding the free response to this From the problem statement

n= 2 rad/s = 0 4

2n= 0 1 < 1 d= n



1− 2= 1 99 rad/sThus, the system is underdamped, and the total solution is of the form

xt= e− n t

A sin dt+ B cos dt+ X1sin t+ X2cos tApplying the initial conditions requires the following derivative

˙xt = e− n t

dA cos dt− dB sin dt+ X1cos t

− X2sin t− ne−n tA sin dt+ B cos dt

The initial conditions yield the constants A and B:

xt= −e−0 2t0 008 sin 1 ... AND CONTROL OF VIBRATIONS 25

1984, or Rivin, 2003) Another possibility is to use active vibration control and feedbackmethods Both of these approaches are discussed in Chapters and. .. active control

There are many different types of active control methods, and only a few will be considered

to give the reader a feel for the connection between the vibration and control. .. the response of the system, and passive control does not Active control requires anexternal energy source, and passive control typically does not

Feedback control consists of measuring