Ch09 MDOF and continuous

9.2.1 Courant-Fisher minimax characterization of eigenvalues and the eigenvalue separation property When no orthogonality constraints are imposed on the choice of u such as in the discus

9 MDOF and continuous

at reasonable prices and more and more computationally sophisticatedprocedures are being developed, it is easy to predict that this current state ofaffairs is probably not going to change for many years to come Finite-elementcodes for engineering problem solving were initially developed for structuralmechanics applications, but their versatility soon led analysts to recognizethat this same technique could be applied with profit to a larger number ofproblems covering almost the whole spectrum of engineering disciplines—statics, dynamics, heat transfer, fluid flow, etc Since the essence of the finite-element approach is to establish and solve a (usually very large) set ofalgebraic equations, it is clear that the method is particularly well suited tocomputer implementation and that here, with little doubt, lies the key to itssuccess

However, since their advent, finite-element procedures have taken on alife of their own, so to speak, so that entire books are dedicated to the subject.This makes discussion here impractical for two reasons: first, it would divert

us from the main topic of the book and, second, space limitations wouldnecessarily imply that some important information had to be left out So,although we will occasionally make some comments on FEMs in the course

of the book, the interested reader is urged to refer to specific literature: forexample, Bathe [1], Spyrakos [2] and Weaver and Johnston [3]

As a consequence of the considerations above, this chapter will bededicated to more ‘classical’ approximation methods, basing our treatment

on the fact that in common engineering practice it is often required, as afirst approach to problems, to have an idea of only a few of the first naturalfrequencies—and eventually eigenfunctions—of a given vibrating system

In this light, discrete MDOF systems and continuous systems are consideredtogether

Finally, it must be noted that some of the concepts that will be discussed,despite the possibility to use them as computational tools, have importantimplications and far-reaching consequences that pervade all the field ofengineering vibrations analysis.

9.2 The Rayleigh quotient

In Section 5.5.1 we first encountered the concept of Rayleigh’s quotient.The line of reasoning is based on the consideration that for an undamped(or lightly damped) system vibrating harmonically at one of its naturalfrequencies the stiffness/mass ratio is equal to that particular frequency To

be more specific, consider a n-DOF system with symmetrical mass and stiffness matrices which is vibrating at its jth natural frequency ω j The motion

of the system is harmonic in time so that the displacement vector is written

as where zj is the jth eigenvector The maximum potential and

kinetic energies in this circumstance (since no energy is lost and no energy isfed into the system over one cycle) must be equal and are given by

(9.1)

(9.2a)

where the pj s (j=1, 2,…, n) are the mass orthonormal eigenvectors On the

other hand, a symmetrical continuous system leads to the same result if weconsider the parallel between MDOF and continuous systems outlined inSections 8.7 and 8.7.1 The continuous systems counterpart of eq (9.2a) reads

(9.2b)where the eigenfunctions φj (j=1, 2, 3,…) are chosen to satisfy the condition

We will now consider a discrete n-DOF system and see what happens to

the ratio (9.2a) when the vector entering the inner products at the numeratorand denominator is not an eigenvector of the system under investigation

Let u be a general vector and let be the set of massorthonormal eigenvectors of our system We define the Rayleigh quotient as

meaning that the Rayleigh quotient for an arbitrary vector is always greater

in other words, when u coincides with the lowest eigenvector Furthermore,

if u is chosen in such a way as to be mass-orthogonal to the first m–1

and

Hence

(9.7)

and the equality holds when u coincides with the mth eigenvector By the

same token, we note that in writing the Rayleigh quotient we can factor out

the highest eigenvalue to get

so that

(9.8)

approximation of the kth eigenvector p k , i.e with e small, we have

(9.9)

where the term ex takes into account the (small) contributions to u from all

eigenvectors other than pk Inserting eq (9.9) into eq (9.3) we get

where the denominator can be expanded according to the binomial

(9.12)

The symbol o(ε3) means terms of order ε3 or smaller This result can be

stated in words by saying that when the ‘trial vector’ u used in forming the

Rayleigh quotient is an approximation of order ε of the kth eigenvector,

then the Rayleigh quotient approximates the kth eigenvalue k with an error

of order ε2 Alternatively, we can put it in more mathematical terms and say

that the functional R(u) has stationary values in the neighbourhood of

eigenvectors: the stationary values are the eigenvalues, while the eigenvectorsare the stationary points To answer the question of whether the stationarypoints are maxima, minima or saddle points we must rely on some previousconsiderations and a few others that will follow

9.2.1 Courant-Fisher minimax characterization of eigenvalues and the eigenvalue separation property

When no orthogonality constraints are imposed on the choice of u (such as

in the discussion that leads to eq (9.5)) we may note that, as our trial vectorranges over the vector space, eqs (9.6) and (9.8) always hold This leads to

the important conclusions that Rayleigh quotient has a minimum when u=p 1 and a maximum when u=pn, so that we can write

(9.13)

and it is understood that u can be any arbitrary vector in the n-dimensional

Euclidean space of the system’s vibration shapes On the other hand, thefollowing heuristic argument can give us an idea of what happens at astationary point other than 1 and n , say at m , when u is completely

arbitrary First, we write the obvious chain of inequalities

and then we note that the Rayleigh quotient is a continuous functional of u Suppose now that, in ranging over the vector space, u finds itself in the

vicinity of pm; continuity considerations imply that Then, if our

trial vector moves toward pm–1 the value of the Rayleigh’s quotient will tend

to decrease while it will tend to increase if u moves toward pm+1 Theconclusion is that the stationary point at m is a saddle point, i.e thecounterpart of a point of inflection with horizontal tangent when we look

for the extremum points of a function f(x) in ordinary calculus.

The situation is different if the trial vector is not completely arbitrary but

satisfies a number of orthogonality constraints In this case u is not free to

range over the entire vector space and, referring back to the discussion thatled to eq (9.7), we can write

(9.14)

meaning that Rayleigh’s quotient has a minimum value of m (which occurs

when u=pm ) for all trial vectors orthogonal to the first m–1 eigenvectors

If we turn now to the utility of the considerations above we note thatRayleigh’s quotient may provide a method for estimating the eigenvalues of

a given system In practice, however, this possibility is often limited to thefirst eigenvalue because the calculation procedure (see also Section 5.5.1)must start with a reasonable guess of the eigenshape that corresponds to theeigenvalue we want to estimate This means that—unless we are dealingwith a very simple system, in which case we can attack the problem directly—only the first eigenshape can generally be guessed with an acceptable degree

of confidence Moreover, the deflection produced by a static (typically gravity)load often proves to be a good trial function for the estimate of 1, while nosuch intuitive hints exist for higher modes

So, as far as the first eigenvalue is concerned, the method is very usefuland can also be improved by forming a sequence of trial vectors designed

to minimize the value of the functional R(u) which, owing to eq (9.6), willtend to 1; this is exactly the procedure we followed in Section 5.5.1 andidentified under the name of ‘improved Rayleigh method’ It goes withoutsaying that the lowest eigenvalue is the most important in a large number

of applications

By contrast, eq (9.14) is of little practical utility because we usually have

could ask whether it is possible to obtain some information on theintermediate eigenvalues without any previous knowledge of the lowereigenvectors This is precisely the result of the Courant-Fisher theorem Itmust be pointed out that the importance of the theorem itself and of itsconsequences is not so much in the possibility of estimating eigenvaluesindependently, but in its fundamental nature; in fact, it provides a rigorousmathematical basis for a large number of developments in the solution ofeigenvalue problems (e.g Wilkinson [4], Bathe [1] and Meirovitch[5])

The Courant-Fisher theorem, which we state here without proof, isgenerally given in terms of a single Hermitian (symmetrical, if all its entriesare real) matrix in the following form:

Theorem 9.1 Let A be a Hermitian matrix with eigenvalues

(9.15)

and

(9.16)

where the wis (in the appropriate number to satisfy eq (9.15) or (9.16)) are

a set of (mutually independent) given vectors of the vector space

A few comments are in order at this point

First of all, we may note that the typical eigenvalue problem of vibrationanalysis involves two symmetrical matrices, while the theorem above iswritten for a single matrix alone However, this is only a minor inconvenience,because we have shown in Chapter 6 (Section 6.8, eqs (6.165) and (6.166))that the generalized eigenvalue problem can be transformed into a standardeigenvalue problem in terms of a single symmetrical matrix Obviously, when

we are dealing with this single matrix, which we call, A the Rayleigh quotient

is defined as

Second, when m=1 or m=n, the theorem reduces to the statements

of eqs (9.13)

In general, the statement of greatest interest to us is given by eq (9.15),because the attention is usually on lowest order eigenvalues With this inmind, let us look more closely at this statement of the theorem For example,suppose that we are trying to estimate 2; we can choose an arbitrary n×1

vector w and constrain our trial vector u to be orthogonal to w, i.e to satisfy

the constraint equation

(9.17)

Now, under the mathematical constraint expressed by eq (9.17), if u and

w are allowed to vary within the vector space, the maximum value that can

be obtained among the values is exactly 2 If the eigenvalue

we are trying to estimate is 3, two mathematical constraints are needed,

meaning that we choose two vectors w1, w2 and our trial vector must satisfy

Courant-Fisher theorem can also be looked upon as an optimizationprocedure to estimate eigenvalues

On more physical grounds, we may summarize the evaluation of, say, 2

by noting that enforcing the vibration shape u on our system—unless ucoincides with one of the eigenshapes—necessarily increases the stiffness ofour system, the mass being fixed In practice, we are dealing with a newsystem whose first eigenvalue satisfies the obvious inequality butalso, owing to the constraint (9.17), (this inequality is less obvious,but it is not difficult to prove; the proof is left to the reader) Then, thetheorem states that the maximum value of that can be obtained underthese conditions is 2 Likewise, the evaluation of m implies m–1

mathematical constraints of the form (9.17)

There are a number of important consequences of the Courant-Fishertheorem; for our purposes, one that deserves particular attention is the so-called separation property of the eigenvalues (or interlacing property), which

we state here without proof in the form of the following theorem

Theorem 9.2 Let A be a given Hermitian n×n matrix with eigenvalues λj , j=1, 2,…n If we consider the eigenproblems

(9.18)

where A(k ) is obtained by deleting the last k rows and columns of A, we

have the eigenvalue separation property

(9.19)

where the index k may range from 0 to n–2.

In other words, if, for example, we turn our eigenvalue problem of order

n into an eigenproblem of order n–1 by deleting the last row and column

from the original matrix, the eigenvalues of the n–1 eigenproblem are

‘bracketed’ by the eigenvalues of the original problem Conversely, if A is a

n×n Hermitian matrix, v a given n×1 vector and b is a real number, the

eigenvalues of the (n+1)×(n+1) matrix

satisfy the inequalities The extension

of Theorem 9.2 to the case of two real, positive definite n×n matrices is not

difficult and it can be shown that the eigenvalues of the two eigenproblems

in which the n×n matrices K and M are obtained by bordering and (of order (n–1)×(n–1)) with the (n–1)×1 vectors k and m and the scalars k and

m, respectively, satisfy the separation (interlacing) property.

9.2.2 Systems with lumped masses—Dunkerley’s formula

In the preceding sections, we pointed out that, for a given system, the Rayleighquotient provides an approximation of its lowest eigenvalue which satisfiesthe inequality This means that, unless the choice of the trial vector

is particularly lucky, R(u) always overestimates the value of 1 For a limited(but not small) class of systems, we will now show that Dunkerley’s formulaprovides a different method to estimate 1 Furthermore, the value that weobtain in this case is always an underestimate of 1

Suppose that we are dealing with a positive definite n-DOF system in which the masses are localized (lumped) at n specific points Then, if we

choose the coordinates as the absolute displacements of the masses, the massmatrix is diagonal (Section 6.5)

The generalized eigenproblem for this system is written in the usual form

terms of the flexibility matrix (whose existence is guaranteed

by positive definiteness), i.e

(9.20)

If the system has lumped masses and hence M=diag(m j ), the matrix AM

has the particularly simple form

so that—by virtue of a well known result of linear algebra stating that thetrace (sum of its diagonal elements) of a matrix is equal to the sum of its

eigenvalues—we can write

to evaluate and that, once the lumping of masses has been decided, the m j

are all known As opposed to the Rayleigh quotient, the main drawbacks ofDunkerley’s formula are that the method does not apply to unrestrainedsystems and that it is not possible to have an ‘equals’ sign in eqs (9.22a andb), meaning that, in other words, Dunkerley’s formula always yields anapproximate value

9.3 The Rayleigh-Ritz method and the assumed modes

method

The Rayleigh-Ritz method is an extension of the Rayleigh method suggested

by Ritz In essence, the Rayleigh method allows the analyst to calculateapproximately the lowest eigenvalue of a given system by appropriately

choosing a trial vector u (or a function for continuous systems) to insert in

the Rayleigh quotient The quality of the estimate obviously depends on thischoice, but the stationarity of Rayleigh quotient—provided that the choice

is reasonable—guarantees an acceptable result Moreover, if the assumedshape contains one or more variable parameters, the estimate can be improved

by differentiating with respect to this/these parameter(s) to seek the minimum

value of R(u) The Rayleigh-Ritz method depends on this idea and can be

used to calculate approximately a certain number of undamped eigenvaluesand eigenshapes of a given discrete or continuous system

Consider for the moment a n-DOF system, where n is generally large Our main interest may lie in the first m eigenvalues and eigenvectors, with

In this light, we express the displacement shape of our system as the

superposition of m independent Ritz trial vectors z j, i.e.

(9.23)

where the generalized coordinates c j are, as yet, unknown and in the matrix

Evidently, the closer the Ritz vectors are to the truevibration shapes, the better are the results

The displacement shape (9.23) is then inserted in the Rayleigh quotient togive

(9.24)

so that the coefficients c j can be determined by making R(u) stationary The

m×m matrices and in eq (9.24) are given in terms of the stiffness and

mass matrix of the original system as

(9.25)

Before proceeding further, we may note that the assumption (9.23) consists

of approximating our n-DOF by a m-DOF system, meaning that, in essence,

we impose the constraints

(9.26)

on the original system Since constraints tend to increase the stiffness of a

system, we may expect two consequences: the first is that the m eigenvalues obtained by this method will overestimate the lowest m ‘true’ eigenvalues and the second is that an increase of m will yield better estimates because,

by doing so, we just eliminate some of the constraints (9.26)

The necessary conditions to make R(u) stationary are

(9.27)which, taking eq (9.24) into account, become

(9.28)

Now, owing to the symmetry of and , the calculation of the derivatives

in eq (9.28) leads to a set of equations that can be put together into thesingle matrix equation

(9.29)

problem of order m This result shows that the effect of the Rayleigh-Ritz

method is to reduce the number of degrees of freedom to a predetermined

value m In this regard, it is important to note that the number of eigenvalues

and eigenvectors that can be obtained with acceptable accuracy is generallyless than the number of Ritz vectors; in other words, if our interest is in the

first m eigenpairs, it is advisable to include s Ritz shapes in the process,

where, let us say,

The eigenproblem (9.29) can be solved by means of any standardeigensolver and the result will be a set of eigenvalues with thecorresponding eigenvectors the eigenvalues are approximations ofthe true lower eigenvalues of the original system, while the eigenvectors are

not the mode shapes of the original system The cjs are orthogonal withrespect to the matrices and and can be normalized by any appropriate

normalization procedure If we call these normalized eigenvectors cj, we can

obtain the approximations of the m mode shapes of the original system from

eq (9.23), i.e by writing

(9.30)and note that these approximate eigenvectors are orthogonal with respect tothe matrices of the original system: that is, by virtue of eq (9.25), we have

(9.31a)

where we called and the jth generalized stiffness and mass of the

reduced system, respectively (their values obviously depend on how we decide

to normalize the vectors cj) The natural consequences of eq (9.31a) are that

(9.31b)

and that these approximate vectors can be used in the standard modesuperposition procedure for dynamic analysis

From the above considerations it appears that the choice of the Ritz shapes

is probably the most difficult step of the whole method In general, this is so;however, we may note that the line of reasoning adopted in the improved