Vibration and Shock Handbook 07

Vibration and Shock Handbook 07 Every so often, a reference book appears that stands apart from all others, destined to become the definitive work in its field. The Vibration and Shock Handbook is just such a reference. From its ambitious scope to its impressive list of contributors, this handbook delivers all of the techniques, tools, instrumentation, and data needed to model, analyze, monitor, modify, and control vibration, shock, noise, and acoustics. Providing convenient, thorough, up-to-date, and authoritative coverage, the editor summarizes important and complex concepts and results into “snapshot” windows to make quick access to this critical information even easier. The Handbook’s nine sections encompass: fundamentals and analytical techniques; computer techniques, tools, and signal analysis; shock and vibration methodologies; instrumentation and testing; vibration suppression, damping, and control; monitoring and diagnosis; seismic vibration and related regulatory issues; system design, application, and control implementation; and acoustics and noise suppression. The book also features an extensive glossary and convenient cross-referencing, plus references at the end of each chapter. Brimming with illustrations, equations, examples, and case studies, the Vibration and Shock Handbook is the most extensive, practical, and comprehensive reference in the field. It is a must-have for anyone, beginner or expert, who is serious about investigating and controlling vibration and acoustics.

7 Vibration Modeling and

Differential Formulation † Integral Formulation and Rayleigh–Ritz Discretization † Finite Element Method † Lumped Mass Matrix † Model Reduction

7.3 Vibration Analysis 7-9

Natural Vibration † Harmonic Response † Transient Response † Response Spectrum

7.4 Commercial Software Packages 7-13

ABAQUS † ADINA † ALGOR † ANSYS † COSMOSWorks † MSC.Nastran †

ABAQUS/Explicit † DYNA3D † LS-DYNA

7.5 The Basic Procedure of Vibration Analysis 7-16

Planning † Preprocessing † Solution † Postprocessing † Engineering Judgment

7.6 An Engineering Case Study 7-19

Objectives † Modeling Strategy † Boundary Conditions † Material † Results

7.7 Comments 7-21

Summary

In this chapter, several aspects of vibration modeling are addressed They include the formulation of theequations of motion both in differential form and integral form, the Rayleigh–Ritz method and the finiteelement methods, and model reduction Natural vibration analysis and response analysis are discussed in detail.Several commercial finite element analysis (FEA) software tools are listed and their capabilities for vibrationanalysis are introduced The basic procedure in using the commercial FEA software packages for vibrationanalysis is outlined (also see Chapter 6 and Chapter 9) The vibration analysis of a gearbox housing ispresented to illustrate the procedure

7.1 Introduction

Vibration phenomenon, common in mechanical devices and structures [2,9], is undesirable in manycases, such as machine tools But this phenomenon is not always unwanted; for example, vibration isneeded in the operation of vibration screens Thus, reducing or utilizing vibration is among thechallenging tasks that mechanical or structural engineers have to face Vibration modeling has been usedextensively for a better understanding of vibration phenomena The vibration modeling here implies aprocess of converting an engineering vibration problem into a mathematical model, whereby the majorvibration characteristics of the original problem can be accurately predicted The mathematical model ofvibration in its general sense consists of four components: a mass (inertia) term; a stiffness term; an

excitation force term; and a boundary condition term These four terms are represented in differentialequations of motion for discrete (or, lumped-parameter) systems, or boundary value problems forcontinuous systems A damping term is included if damping effects are of concern Depending on thenature of the vibration problem, the complexity of the mathematical model varies from simple spring–mass systems (see Chapter 1) to multi-degree-of-freedom (DoF) systems (see Chapter 3); from acontinuous system (see Chapter 4) for a single structural member (beam, rod, plate, or shell) to acombined system for a built-up structure; from a linear system to a nonlinear system The success of themathematical model heavily depends on whether or not the four terms mentioned before can representthe actual vibration problem In addition, the mathematical model must be sufficiently simplified inorder to produce an acceptable computational cost The construction of such a representative and simplemathematical model requires an in-depth understanding of vibration principles and techniques,extensive experience in vibration modeling, and ingenuity in using vibration software tools.Furthermore, it also requires sufficient knowledge of the vibration problem itself in terms of workingconditions and specifications.

Except for few special cases that promise exact and explicit analytical solutions, vibration models have

to be studied by means of approximate numerical methods such as the finite element method The finiteelement method has been very successfully used for vibration modeling for the past two decades Itssuccess is attributed to the development of sophisticated software packages and the rapid growth ofcomputer technology

In this chapter, several aspects of the construction of mathematical models of linear vibrationproblems without damping will be addressed The capabilities of the available software packages forvibration analysis are listed and the basic procedure for vibration analysis is summarized As anillustration, an engineering example is given

7.2 Formulation

7.2.1 Differential Formulation

In a majority of engineering vibration problems,

the amplitude of vibrations is very small, so that

the following assumptions hold: (1) a linear form

of strain–displacement relationships, and (2) a

linear form of stress–strain relationships (Hooke’s

Law) If the energy losses are negligible, it is

straightforward to apply Newton’s (second) law

and Hooke’s Law to derive the equations of

motion, which appear as differential equations

Consider a single-DoF spring–mass system, as

shown in Figure 7.1 The two laws are given by

m€uðtÞ ¼ 2f ; Newton’s law

f ¼ kuðtÞ; Hooke’s Law

M€uðtÞ ¼ 2F; Newton’s law

F ¼ KuðtÞ; Hooke’s Law

(

ð7:2Þwhere M is the (diagonal) mass matrix, and K is the stiffness matrix

In the case of continua, the differential equations of motion can be derived by means of Newton’s lawand Hooke’s Law in the same way as above But in this case the boundary conditions have to be specified

in order to make the problem statement complete(see Chapter 4).For example, as a direct consequence

of Newton’s law and Hooke’s Law, the differential equation of bending vibration of a clamped–clampedEuler beam may be given as (Chapter 4)

r›2uðx; tÞdt2 ¼ 2f ; Newton’s law

7.2.2 Integral Formulation and Rayleigh–Ritz Discretization

Besides the approach in which Newton’s law and Hooke’s Law are directly used to establish equations ofmotion, there are alternatives: Hamilton’s principle, the minimum potential energy principle, and thevirtual work principle; all of which appear in integral form From a mathematical standpoint, thedifferential equations and the integral equations are equivalent in that one can be derived from another.However, they are very different in that the integral equations facilitate the application of the discretizationschemes such as the finite element method, an element-wise application of Rayleigh–Ritz method.Therefore, Hamilton’s principle, as one of the integral formulations, and its Rayleigh–Ritz discretizationare briefly introduced here in order to provide a better understanding of the finite element method.Denote T as the system kinetic energy, V the system potential energy, and dW the virtual work done bynonconservative forces Hamilton’s principle [11] states that the variation of the Lagrangian ðT 2 VÞStandard terminology plus the line integral of the virtual work done by the nonconservative forcesduring any time interval must be equal to zero If the time interval is denoted by ½t1; t2; then Hamilton’sprinciple can be expressed as

wherewiðx; y; zÞ is called a Rayleigh–Ritz shape function and qiðtÞ is called a generalized coordinate

In this way, the system kinetic energy and the system potential energy can be, respectively, expressed

as follows:

T ¼ 12

Xn i¼1

Xn j¼1

kijqiqj; 12uTðtÞKuðtÞ ð7:7Þwhere uTðtÞ ; ½q1; q2; …; qn; M ; ½mij ; K ; ½kij :

The virtual work done by the generalized forces is

Because du; the variation of the generalized coordinate vector, is arbitrary and independent, from theabove equation one obtains

which is the vibration equation resulting from a Rayleigh–Ritz discretization

7.2.3 Finite Element Method

In the finite element method (FEM) [7,10,12], a continuum is divided into a number of relatively smallregions called elements that are interconnected at selected nodes This procedure is called discretization.The deformation within each element is expressed by interpolating polynomials The coefficients ofthese polynomials are defined in terms of the element nodal DoF that describe the displacements andslopes of selected nodes on the element By using the connectivity between elements, the assumeddisplacement field can then be written in terms of the nodal DoF by means of the element shapefunction Using the assumed displacement field, the kinetic energy and the strain energy of eachelement are expressed in the form of the element mass and stiffness matrices The energy expressionsfor the entire continua can be obtained by adding the energy expressions of its elements This leads tothe assembled mass matrix and the assembled stiffness matrix, and finally to the finite elementvibration equation

The displacement in the interior of an element e is determined by a polynomial

where the matrix N is called the shape function matrix of the element e; and ue the vector of thenodal DoF.

Based on the element displacement expression Equation 7.13, one can obtain the strain and the stress

in the element e and finally the strain energy

The strain and the stress in the element e are

1 ¼›uðx; y; z; tÞ ¼ ›Nue¼ Bue ð7:14Þand

is called the element stiffness matrix

The velocity at a point ðx; y; zÞ in the element e can be obtained from Equation 7.13 as

So the kinetic energy of the element e is

Te¼ 12

ðr_uT_u dv ¼ 12ð_ueÞTMe_ue ð7:19Þwhere

is called the element mass matrix

The equivalent nodal force Fecorresponding to the force feapplied onto the element e is determined

by equaling the work done by Feto the work done force by fealong any virtual displacement This leads tothe following:

it is necessary to transform the expressions of the kinetic energy, the strain energy, and the equivalentnodal force of the element e from the local coordinate system into the global one

Let L be the transformation matrix from the global coordinate system to the local coordinate system.Then the nodal displacement vector uein the local coordinate system is related to the nodal displacement

vector uein the global coordinate system by the following:

Similarly, the equivalent nodal force vector Fein the local coordinate system is related to the equivalentnodal force vector Fein the global coordinate system by

Substituting Equation 7.23 into Equation 7.16 and Equation 7.19, and noting that L is a normal matrix

ðLT¼ L21Þ; the element stiffness and mass matrices in the global coordinate system can be, respectively,expressed as

is called the global mass matrix The vector _u is the global nodal velocity vector

The total virtual work done by the external forces is

dW ¼X

e ðdueÞTFe¼ ðduÞTF ð7:32Þwhere the vector

F ¼X

is a global generalized force vector

Substituting Equation 7.28, Equation 7.30, and Equation 7.32 into Hamilton’s principle (Equation 7.4)and conducting the routine variation operation, one has

which is the vibration equation resulting from the finite element discretization

7.2.4 Lumped Mass Matrix

The element mass matrix given by Equation 7.20 is normally a full symmetric matrix, because theelement shape functions are not orthogonal with each other It is desirable to reduce this full matrix into adiagonal matrix In practice, this is achieved by lumping the element mass at its nodes For example, theconsistent element mass matrix of a beam element is

Me¼ rAl420

37775

ð7:35Þ

When the inertia effect associated with the rotational DoF is negligible, the element lumped mass matrixcan be obtained by lumping one half of the total beam element mass at each of the two nodes along thetranslation DoF:

Me¼ rAl2

37775

ð7:36Þ

When the inertia effect associated with the rotational DoF is not negligible, the mass moment of inertia ofone half of the beam element about each node can be computed and included at the diagonal locationscorresponding to the rotational DoF:

Me¼ rAl2

37775

ð7:37Þ

7.2.5 Model Reduction

The finite element discretization of an engineering vibration problem usually generates a very largenumber of DoF In particular, when automatic meshing schemes are not properly applied, or three-dimensional elements must be used, the number of elements created could become too great to be cost-effectively handled with limited computer capabilities To solve this problem, modelers have to payclose attention to how the meshing is done in commercial software packages Very often, simplificationand idealization based on the nature of the problem of concern can tremendously reduce the number

of elements For example, there could be two ways of generating the finite elements of a clamped-freesteel beam with a metal block attached to its free end One way is to mesh both the beam and the blockusing three-dimensional elements; the other way is to mesh the beam with one-dimensional beamelements and treat the block as a lumped mass, zero-dimensional element It is obvious that the firstapproach will result in many more elements than the second approach However, both approaches willgive very similar results for the first several natural frequencies and the associated mode shapes Anothertechnique for reducing the number of elements comes from deleting the detailed features The detailedfeatures here imply those geometrical details, such as filets, chamfers, small holes, and so on, which donot have significant contributions to the vibration behavior of the entire structure, but increase thenumber of elements Generally these detailed features can be deleted without any visible effect on theresults, if the global behavior of the vibration problem is of concern Note that such detailed features mayhave to be kept if the localized behavior such as fatigue (stress) induced by vibration is to be evaluated.When further model reduction is necessary, Guyan reduction [3] is considered It was proposed twodecades ago when computer capabilities were much more limited than today In fact, Guyan reduction isstill in use today and has been cast into many commercial software packages In Guyan reduction, themodel scale is reduced by removing those DoF (called slave DoF) that can be approximately expressed bythe rest of the DoF (called master DoF) through a static relation The DoF associated with zero mass orrelatively small mass are likely candidates for slave DoF.

By rearranging the DoF u so that those to be removed, denoted by u2; appear last in the vector, andpartitioning the mass and the stiffness matrices accordingly, one obtains

and I is the unit (identity) matrix

Substituting Equation 7.40 into Equation 7.38 and premultiplying the resulting equation by QT; oneobtains a new reduced-order model

where

QTMQ ¼ M112 M12K2122K21þ KT21K2122M22K2122K21 ð7:43Þand

QTKQ ¼ K112 K12K21

7.3.1 Natural Vibration

As noted in previous chapters, the natural vibration frequencies (or simply natural frequencies) and theassociated mode shapes of a vibrating system are independent of excitation forces In other words, they areintrinsic characteristics of the vibration problem Therefore, they constitute an important part of vibrationtheory and vibration engineering When vibration engineers specify design requirements in terms ofvibration, they normally do so by restricting natural frequencies, and sometimes restricting mode shapes

as well For instance, in order to enhance the passenger comfort, vehicle designers have to ensure that thefirst few natural frequencies of the vehicle are not within a certain range; in order to avoid vibrationresonance, the natural frequencies of a transmission shaft should be designed not to be identical or evenclose to the rotating speeds of the shaft; in order to effectively control vibration, vibration sensors andactuators have to be located at those places where the dominant mode shapes have large displacements

Newton’s law

M€uðtÞ ¼ 2FHooke’s Law

F ¼ KuðtÞHamilton’s principle

From a mathematical standpoint, the natural vibration analysis of a multi-DoF system requires thesolution of a matrix eigenvalue problem According to the theory of second-order ordinary differentialequations, the solution of Equation 7.34, when F ¼ 0; can be expressed as u ¼ v ei v t: By substituting

u ¼ v ei v tinto Equation 7.34 and letting F ¼ 0; one can obtain

When conducting vibration modeling, modelers need to understand how idealization andsimplification will affect the resulting natural frequencies and the associated mode shapes Idealizationand simplification cause a difference between the actual mass matrix M and the resulting mass matrix Mr

ðM ¼ Mrþ DMÞ; and a difference between the actual stiffness matrix K and the resulting stiffness matrix

KrðK ¼ Krþ DKÞ: Rayleigh’s quotient [9,11] can be used to determine the effect of DK and DM on aparticular natural frequency Rayleigh’s quotient is defined as

RðxÞ ¼ xTKx

Note that Rayleigh’s quotient RðxÞ becomes the square of the ith natural vibration frequency, RðxÞ ¼v2

i;when x ¼ vi: Thus, Rayleigh’s quotient can be expressed as

i and Dviare the increase of the ith natural frequency and the variation of the ith mode shape,respectively, induced by DK and DM: Because of the fact that RðxÞ reaches the stationary value when x isequal to the eigenvector vi; Equation 7.47 can be simplified as [1,4]

Dv2

i ¼ vT

Equation 7.48 indicates that an increase in stiffness leads to a rise in a natural frequency, but an increase

in mass causes a decrease in a natural frequency, as is intuitively clear

7.3.2 Harmonic Response

Harmonic response analysis determines the response of a vibration system (model) to harmonicexcitation forces A typical output is a plot showing response (usually displacement of a certain DoF)versus frequency This plot indicates how the response at a certain DoF, as a function of excitationfrequency, changes with excitation frequency The harmonic response can also be used to calculate theresponse to a general periodic excitation force, if it can be satisfactorily approximated by a summation ofits major harmonic components

Consider a harmonic excitation force, FðtÞ ¼ F0ei v t: Substituting it into Equation 7.34, we have

According to the theory of differential equations, its steady solution can be written as uðtÞ ¼ ei v tU: Aftersubstitution of uðtÞ ¼ ei v tU into Equation 7.49, one obtains

Harmonic response analysis will solve Equation 7.50 for U againstv: There are many numerical methodsavailable for solving Equation 7.50 The most efficient one is the modal superposition method.

In the modal superposition method, the response is expressed as a linear combination given by

Equation 7.52 represents a set of decoupled modal equations with a much smaller dimensionthan Equation 7.49 After solving Equation 7.52 for u_ðtÞ and transforming u_ðtÞ back to uðtÞ throughEquation 7.51, we obtain uðtÞ:

7.3.3 Transient Response

Transient response analysis (sometimes called time-history analysis) determines the dynamic response of

a structure under the action of time-varying excitation Excitation forces are explicitly defined in the timedomain The computed response usually includes the time-varying displacements, accelerations, strains,and stresses Consider Equation 7.34 in its general form

M€uðtÞ þ KuðtÞ ¼ FðtÞuð0Þ ¼ u0; _uð0Þ ¼ _u0

(

ð7:53Þ

where FðtÞ is the excitation force, u0is the initial displacement, and _u0is the initial velocity As in theharmonic response analysis, Equation 7.53 can be solved by the modal superposition method.Substituting Equation 7.51 into Equation 7.53, premultiplying the result of the first equation by FT;and premultiplying the result of the initial condition by FTM; we obtain

u

_€ ðtÞ þ Lu_ðtÞ ¼ FTFðtÞu

To implement the numerical integration techniques, the overall time period being studied has to bedivided into a number of smaller time steps If the time step is too large, portions of the response (such asspikes) could be missed or truncated On the other hand, if the time step is too small, the analysis willtake an excessively long time or even a prohibitive amount of time

7.3.4 Response Spectrum

The excitation forces, resulting from earthquakes, winds, ocean waves, jet engine thrust, uneven roads,and so on, do not have repeated patterns, for a variety of reasons, and thus it is difficult to describe themusing a deterministic time history Such excitations are normally treated as random excitations Theassumption that such excitation forces are random is recognition of our lack of knowledge of the detailed