Applied Structural and Mechanical Vibrations 2009 Part 6 docx

It is evident that—with only minor modifications—the same results apply if we use the z eigenvectors; we only have to take into account the fact that in this case Also, the perturbed p

Now, in order to determine completely the first order perturbation of the

eigenvector, only the coefficient c kk is left; by imposing the normalizationcondition and retaining only the first power of γ, we get

and by virtue of the expansion (6.79) we are finally led to

(6.83)

We can now explicitly write the result of the first-order perturbation

calculation for the ith eigenvalue and the ith eigenvector as

(6.84)

(6.85)

From the expressions above, it may be noted that only the ith unperturbedparameters enter into the calculation of the perturbed eigenvalue, while thecomplete unperturbed eigensolution is required to obtain the perturbedeigenvector Roughly speaking, we could say that the perturbation has the

effect of ‘mixing’ the ith unperturbed eigenvector with all the othereigenvectors for which the term in brackets of eq (6.85) is differentfrom zero Furthermore, a quick glance at the same equation suggests thatthe closer eigenvectors (to ) give a greater contribution, because, for thesevectors, the term is smaller

It is evident that—with only minor modifications—the same results apply

if we use the z eigenvectors; we only have to take into account the fact that in

this case Also, the perturbed p vectors are orthonormal with respect to the new mass matrix, not with respect to M0

Example 6.3 Let us go back to the system of Fig 6.1, whose eigensolutionhas been considered in Example 6.1, and make the following modifications:

we increase the first mass by 0.25m and decrease the second mass by 0.1m The total mass of the system changes from 4.0m into 4.15m, which

corresponds to an increase of 3.75% with respect to the original situation

Also, let us increase the stiffness of the first spring of 0.1k so that the term changes from 5.0k into 5.1k, i.e an increase of 2.0% with respect to

the original situation These modifications can be considered small and weexpect accurate results from our perturbative calculations

The perturbation terms are

We remember from Example 6.1 that:

For the first eigenvector, the expansion coefficients are given by

from which it follows that

(6.87a)

and the same procedure for the second eigenvector leads to

(6.87b)

Because of the simplicity of this example, these results can be compared

to the exact calculation for the modified system, which can be performedwith small effort The exact eigenvalues are and

which corresponds to a relative error of 0.07% on thefirst frequency and a relative error of 0.46% on the second The exacteigenvectors are

and they must be compared, respectively, to eqs (6.87a and b)

Some complications appear in the case of degenerate eigenvalues We willnot deal with this subject in detail but a few qualitative considerations can

be made Suppose, for example, that two independent eigenvectors and correspond to the unperturbed eigenvalue (twofold degeneracy) Ingeneral, the perturbation will split this eigenvalue into two different values,say λi1 and λi2; as the perturbation tends to zero, the eigenvectors will tend

to two unperturbed eigenvectors and , which will be in general twolinear combinations of and The additional problem is, as a matter

of fact, the determination of and : this particular pair—out of theinfinite number of combinations of and —will depend on theperturbation itself

For instance, let be an m-fold degenerate eigenvalue and let

be a possible choice of mass-orthonormal eigenvectors (i.e

a basis in the subspace relative to the ith eigenvalue) We can then write the

expansions

(6.88)and

(6.89)

substitute them in the first-order problem and project the resulting equationsuccessively on the eigenvectors (this is done bypremultiplying, respectively, by ) We obtain, after some

manipulation, a system of m homogeneous equations, which admits nontrivial

solutions if the determinant of the coefficients is equal to zero This condition

results in an algebraic equation of degree m in and its m solutions

represent the first-order corrections to Substitution ofeach one of these values into the homogeneous system allows the calculation

of the zeroth-order coefficients for the relevant

eigenvector We have thus obtained the desired m linear combinations of the

unperturbed eigenvectors (i.e the ); once these are known, the coefficients can be obtained by projecting the first-order equation on differenteigenvectors

It is interesting to note that, in many cases, the effect of the perturbation

is to completely or partially ‘remove the degeneracy’ by splitting thedegenerate eigenvalue into a number of different frequencies that wereindistinguishable in the original system This circumstance can be useful insome practical applications, and it is worth pointing out that similarprocedures apply—with only minor modifications—in the case of distinctbut closely spaced eigenvalues

The subject of sensitivity analysis is much broader than shown in ourdiscussion; in general, we can say that some linear systems are extremelysensitive to small changes in the system, and others are not Sensitive systemsare often said to be ‘ill-conditioned’, whereas insensitive systems are said to

be ‘well-conditioned’

We will see that the generalized eigenvalue problem of eq (6.24) (or (6.29),which is the same) can be transformed into a standard eigenvalue problem

(eq (6–26a)), where A is an appropriate matrix whose form and entries depend

on how the transformation is carried out The key point is that the eigenvalues

are continuous functions of the entries of A, so we have reason to believe

that a small perturbation matrix will correspond to a small change of theeigenvalues But one often needs precise bounds to know how small is ‘small’

in each case

We will not pursue this subject further here for two reasons: first, adetailed discussion is beyond the scope of this book and, second, it wouldlead us too far away from the main topic of this chapter For the moment,

it suffices to say that if A is diagonalizable (see Appendix A on matrix

analysis), it can be shown that it is possible to define a ‘condition number’that represents a quantitative measure of ill-conditioning of the system andprovides an upper bound on the perturbation of the eigenvalues due to aunit norm change in the system matrix; furthermore, it may be of interest

to note that normal matrices are well-conditioned with respect to eigenvaluecomputations, that the condition number is generally conservative and that

a better bound can be obtained if both A and the perturbing matrices are

Hermitian (The interested reader is referred to Horn and Johnson [1] andJunkins and Kim [2].)

6.4.1 Light damping

The free vibration of a damped system is governed by eq (3.101), i.e

(6.90)

As in the undamped case there are 2n independent solutions which can be superposed to meet 2n initial conditions Assuming a trial solution of the

form

(6.91)leads to

(6.92)

which admits a nontrivial solution if the matrix in parentheses on the hand side is singular Equation (6.92) represents what is commonly called acomplex (or quadratic) eigenvalue problem because the eigenvalue and

left-the elements of left-the eigenvector z are, in general, complex numbers; if and

z satisfy eq (6.92), then so also do * and z*, where the asterisk denotes

complex conjugation In general, the complex eigenvalue problem is muchmore difficult than its undamped counterpart and much less attention hasbeen given to efficient numerical procedures for its solution, but we willreturn to these aspects later

For the moment, let us make the following assumptions: the solution ofthe undamped problem is known and the system is lightly damped Thedamping term can then be considered a small perturbation of the originalundamped system and we are in a position to investigate its effect on theeigensolution of the conservative system

Let and pj (j=1, 2, …, n) be the eigenvalues and the mass-orthonormal

eigenvectors of the conservative system (i.e when C=0 in eq (6.92)); under

the assumption of light damping we can write

Substitute eq (6.95) in (6.94) and premultiply the resulting expression by

• Each correction to the unperturbed eigenvalues takes the form of a real

negative part (matrix C is generally positive definite) which transforms

the solution into a damped oscillatory motion and accounts for the factthat the free vibration of real systems dies out with time because there

is always some loss of energy

• The first-order correction involves only the diagonal terms of the matrix

PTCP which is, in general, nondiagonal unless some assumptions are made on the damping matrix (remember that both M and K become diagonal under the similarity transformation PTMP and PTKP) Off-

diagonal terms have only a second-order effect on the unperturbedeigenvalues

When eq (6.96) gives

(6.98a)

Again, a term M kk appears at the denominator on the right-hand side if thecalculation is made with eigenvectors that are not mass-orthonormal; notealso that a minus sign appears on the right-hand side if we start with

The perturbed eigenvector is then

(6.98b)

showing that the perturbation splits the original real eigenvector into a pair

of complex vectors having the same real part as the undamped mode(remember that, in vibration terminology, the term mode is analogous to

eigenvector: more precisely, a mode is a particular pattern of motion which

is mathematically represented by an eigenvector) but having small conjugateimaginary parts

On physical grounds—unless damping has some desirable characteristicswhich will be considered in a later section—this occurrence translates intothe fact that, in a particular mode, each coordinate has a relative amplitudeand a relative phase with respect to any other coordinate In other words,the free vibration of a generally damped system oscillating in a particularmode is no longer a synchronous motion of the whole system: the individualdegrees of freedom no longer move in phase or antiphase and they no longerreach their extremes of motion together

For obvious reasons, this pattern of motion is usually called a ‘complexmode’, as opposed to the ‘real mode’ of the undamped system where eachcoordinate does have an amplitude, but a phase angle which is either 0° or180° and real numbers suffice for a complete description

6.5 Structure and properties of matrices M, K and C: a few considerations

A fundamental part of the analysis of MDOF systems—and of any physicalphenomenon in general—is the solution of the appropriate equations ofmotion However, as we stated in Chapter 1, the first step in any investigation

is the formulation of the problem; this step involves the selection of amathematical model which has to be both effective and reliable, meaningthat we expect our model to reproduce the behaviour of the real physicalsystem within an acceptable degree of accuracy and, possibly, at the leastcost We always must keep in mind that, once the mathematical model hasbeen chosen, we solve that particular model and the solution can never givemore information than that implicitly contained in the model itself Theseobservations become more important when we consider that:

• Numerical procedures implemented on digital computers play a centralrole (think, for example, to the finite-element method) in the analysis ofsystems with more than three or four degrees of freedom

• Matrix algebra is the ‘natural language’ of these procedures

• The effectiveness and reliability of numerical techniques depend on thestructure and properties of the input matrices

• Continuous systems (i.e systems with an infinite number of degrees offreedom) are very often modelled as MDOF systems

As in the case of an SDOF system, the principal forces acting on an MDOFsystem are (1) the inertia forces, (2) the elastic forces, (3) the damping forcesand (4) the externally applied forces We will not consider, for the moment,the forces of type (4) Under the assumption of small amplitude vibrations,

we have seen in Chapter 3 that matrices M, K and C are symmetrical

Symmetry is a desirable property and results in significant computational

advantages In essence, the symmetry property of M and K depends on the form of the kinetic and potential energy functions and the symmetry C of

depends on the existence of the Rayleigh dissipation function

Unfortunately, for most systems the damping properties are very difficult,

if not impossible, to define For this reason the most common choices for thetreatment of its effects are (1) neglect damping altogether (this is often abetter assumption than it sounds), (2) assume ‘proportional damping’ (Section6.7) or (3) use available experimental information on the dampingcharacteristics of a typical similar structure or on the structure itself

We know from Chapter 3 that both kinetic and potential energies can bewritten as quadratic forms and we know from basic physics that they areessentially positive quantities If none of the degrees of freedom has zeromass, (eq (3.95)) is never zero unless is a zero vector and hence M,

besides being symmetrical, is also positive definite; if some of the degrees of

freedom have zero mass then M is a positive semidefinite matrix.

Similar considerations apply for the stiffness matrix; unless the system is

unrestrained and capable of rigid-body modes, K is a positive definite matrix.

When this is not the case, i.e when rigid-body modes are possible (Section6.6), the stiffness matrix is positive semidefinite It is worth pointing out

that if a matrix A is symmetrical and positive definite, then A–1 always exists

(i.e A is nonsingular) and is a symmetrical positive definite matrix itself The fact that either M, or K, or both, are nonsingular is useful when we

want to transform the generalized eigenvalue problem (eq (6.29)) into astandard eigenvalue problem (eq (6.26a)), which is the form required bysome numerical eigensolvers (section 6.8)

6.5.1 Mass properties

The simplest procedure for defining the mass properties of a structure is byconcentrating, or lumping, its mass at the points where the displacementsare defined This is certainly not a problem for a simple system such as theone in Fig 6.1 where mass is, as a matter of fact, localized, but a certaindegree of arbitrariness is inevitable for more complex systems In any case,whatever the method we use to concentrate the masses of a given structure,

if we choose the coordinates as the absolute displacement of the masses weobtain a diagonal mass matrix In fact, the off-diagonal terms are zero because

an acceleration at one point produces an inertia force at that point only; this

is not strange if we consider that m ij is the force that must be applied at

point i to equilibrate the inertia forces produced by a unit acceleration at point j, so that m ii =m i and m ij=0 for

A diagonal matrix is certainly desirable for computational purposes, but aserious disadvantage of this approach is the fact that the mass associated withrotational degrees of freedom is zero because a point has no rotational inertia

This means that—when rotational degrees of freedom must be considered in

a specific problem—the mass matrix is singular In principle, the problem could

be overcome by assigning some rotational inertia to the masses associatedwith rotational degrees of freedom (in which case the diagonal mass coefficientwould be the rotational inertia of the mass), but this is easier said than done.The general conclusion is that the lumped mass matrix is a diagonal matrixwith nonzero elements for each translational degree of freedom and zerodiagonal elements for each rotational degree of freedom

A different approach is based on the assumed-modes method, a

far-reaching technique developed along the line of reasoning of Section 5.5 (seealso Chapter 9) In that section, a distributed parameter system was modelled

as an SDOF system by an appropriate choice of a shape, or trial, functionunder the assumption that only one vibration pattern is developed during

the motion This basic idea can be improved by superposing n shape functions

so that

(6.99)

where the z i (t) constitute a set of n generalized time-dependent coordinates.

(Note that we considered the trial functions in eq (6.99) to depend on onespatial coordinate only, thus implying a one-dimensional problem (forexample, an Euler-Bernoulli beam); this is only for our present convenience,and the extension to two or three spatial coordinates is straightforward.)

In essence, eq (6.99) represents an n-DOF model of a continuous system,

and since the kinetic energy of a continuous system is an integral expressiondepending on the partial derivative we can substitute eq (6.99) intothis expression to arrive at the familiar form

which is formally equivalent to eq (6.100) when we define the coefficients

m ij as

(6.103)

and it is evident that m ij =m ji , i.e the mass matrix is symmetrical Note that

if we define the row vector of shapes and the column vector

of generalized coordinates we can write eqs (6.99) and (6.102)and the mass matrix, respectively, as

(6.104)

For purposes of illustration, let us consider a clamped-free bar in axialvibration: we can model this continuous system as a 2-DOF system andexpress the displacement by means of the two shape functions and

influence coefficients which express the relation between the displacement

at a point and the forces acting at that and other points of the system The

flexibility influence coefficient a ij is defined as the displacement at point x=x i due to a unit load applied at point x=x j with all other loads equal to zero.The principle of superposition for linearly elastic systems allows one to write

the displacement u i at point i as

(6.106)

where f j is the force applied at x=x j The units of the flexibility coefficientsare metres per newton when the relation (6.106) is between lineardisplacements and forces, if angular displacements and torques areconsidered, the units change accordingly In the case of a single spring it is

evident that the flexibility coefficient is simply a=1/k, where k is the spring

constant

Equation (6.106) can be written in matrix form as u=Af, where A is called

the flexibility matrix and the other symbols are obvious

The stiffness influence coefficient k ij is defined as the load required at

x=x i to produce a unit displacement at x=x j when all other displacements areheld to zero This definition is more involved than the definition of theflexibility coefficient; nevertheless the stiffness coefficients are sometimeseasier to determine than the flexibility coefficients Consider, for example a

one-dimensional 3-DOF system, if x=x1 is given a unit displacement (i.e

u1=1) and u j=0 for the forces at points 1, 2 and 3 required to maintain

this displacement configuration are exactly k11, k21 and k31, i.e the first column

of the stiffness matrix; moreover, these coefficients must be considered withthe appropriate sign: positive in the sense of positive force and negative

otherwise By a line of reasoning parallel to that used for a ij , we can write

(6.107)

or, equivalently, f=Ku, where K is the stiffness matrix The conservative nature

of the linearly elastic systems we consider (meaning that the sequence ofload applications is unimportant) leads to Clapeyron’s law for the total strainenergy which reads

(6.108)

and is valid for a structure which is initially stress free and not subjected totemperature changes Furthermore, Maxwell’s reciprocity theorem holds and

it can be invoked to prove that and or, in matrix form

(6.109)

i.e the flexibility matrix and the stiffness matrix are symmetrical

The structure of eqs (6.106) and (6.107) suggests that the two matricesshould be related; it is so, and it is not difficult to prove that

(6.110)

The interested reader can find excellent discussions of this topic in many

textbooks (e.g Bisplinghoff et al [3]).

Depending on the specific problem, in some cases it may be easier toobtain directly the stiffness matrix whereas in some other cases it may bemore convenient to obtain the flexibility matrix and then invert it We mustremember, however, that for an unrestrained system—where rigid body

motions are possible—the stiffness matrix K is singular, hence it has no inverse.

This is consistent with the fact that for an unrestrained system we cannotdefine flexibility influence coefficients

Example 6.4 Using the definitions given above, the reader is invited to

determine that the flexibility and stiffness matrices for the one-dimensional3-DOF system of Fig 6.7 are, respectively

It is then straightforward to show that KA=I.

Example 6.5 Consider now the system of Fig 6.8 The mass m is connected

to a base of mass M which can undergo translational and rotational motion.

at one end and free at the other end; we choose a 2-DOF model by choosingthe ‘reasonable’ functions and then

is ‘consistent’ with the stiffness matrix of eq (6.115b) Strictly speaking, theterm applies to the finite-element approach, but the latter can be considered

as an application of the assumed-modes method where the shape functionsrepresent deflection patterns of limited portions (the so-called finite elements)

of a given structural system In the end, in order to construct a mathematicalmodel of the whole structure, these elements are assembled together in acommon (or global) frame of reference

The assumed-modes method leaves open the question of what constitutes

a judicious choice of the shape functions We have already faced this problem

in Chapter 5 where we stated the importance of boundary conditions forcontinuous systems and we defined, for a given system, the classes ofadmissible and comparison functions (incidentally, note that and

are admissible functions for the clamped-free bar in axialvibration) We will not pursue this subject here; for the moment we adhere

to the considerations of Section 5.5 with only one additional observationdue to the fact that now we must choose more than one function: the trialfunctions should be linearly independent and form a complete set We will

be more specific about this and about finite-element modelling wheneverneeded in the course of the discussion

6.5.3 More othogonality conditions

The conditions of eqs (6.42b) and (6.44) are only a particular case of a broader

class of othogonality properties involving the eigenvectors pi and the matrices

M and K If pi is an eigenvector, the eigenvalue problem isidentically satisfied Let us premultiply both sides of the eigenvalue problem

by we get, by virtue of (6.42b) and (6.44)

(6.116)Premultiplication of the eigenvalue problem by yields

(6.117)where the result of eq (6.116) has been taken into account The process can

be repeated to give

(6.118)which can be rewritten in the equivalent form

(6.119)just by premultiplying the term in parentheses on the left-hand side of eq

(6.118) by MM–1 The cases b=0 and b=1 in eq (6.119) correspond,

respectively, to eqs (6.42b) and (6.44)

An analogous procedure can be started by premultiplying both sides ofthe eigenvalue problem by to give

which means, provided that

(6.120)

and the term in the centre is due to the fact that

Now, premultiplying both sides of the eigenvalue problem by

and taking eq (6.120) into account we get

Repeated application of this procedure leads to

(6.121)

Equations (6.119) and (6.121) can be put together to express the family

of orthogonality conditions

(6.122)

6.6 Unrestrained systems: rigid-body modes

Suppose that, for a given system, there is a particular vector forwhich the relationship

(6.123)

holds First of all, this occurrence implies that the matrix K is singular and

that the potential energy is now a positive semidefinite quadratic form.Furthermore, this vector can be substituted in the eigenvalue problem of eq(6.29) to give i.e and hence, since M is generally

positive definite, It follows that r can be considered an eigenvector

corresponding to the eigenvalue At first sight, an oscillation at zerofrequency may appear surprising, but the key point is that this solution doesnot correspond to an oscillatory motion at all: we speak of oscillatory motionbecause eq (6.29) has been obtained by assuming a time dependence of theform However, if we assume a more general solution of the form

(6.124)and we substitute it in eq (6.20) we get which corresponds to a

uniform motion f(t)=at+b where a and b are two constants In practice, the

system moves as a whole and there is no change in the potential energy becausesuch rigid displacement does not produce any elastic restoring force In otherwords, we are dealing with a system with a neutrally stable equilibriumposition Some examples could be: an aeroplane in rectilinear flight, a shaftsupported at both ends by frictionless sleeves or, in general, any structuresupported on springs that are very soft compared to its stiffness This lattersituation is of considerable importance in the field of dynamic structural testingand it is usually referred to as the ‘free’ condition (see also Chapter 10).The maximum number of eigenvectors which correspond to the eigenvalue

(the so-called rigid-body modes; this is why we used the letter r) is six

because a three-dimensional body has a maximum of six rigid-body degrees

of freedom (three translational and three rotational): moreover, for a givenproblem, it is generally not difficult to identify the rigid-body modes byinspection

For example, suppose that we model an aeroplane body and wings as a

flexible beam with three lumped masses, M being the mass of the fuselage and m being the mass of each wing: if we only consider motions in the plane

of the page, the first two modes occur at zero frequency and they correspond

to a rigid-body translation and a rigid-body rotation as shown in Fig 6.9.The higher modes are elastic modes representing flexural deformations ofour simple structure.

The arguments used to arrive at the orthogonality relationships retain

their validity; hence, the rigid-body modes are M- and K-orthogonal to the

elastic eigenvectors (because they are associated with distinct eigenvalues)

and the rigid-body modes can always be assumed mutually M-orthogonal

(because they just represent a particular case of eigenvalue degeneracy) Theonly difference is the stiffness orthogonality condition for rigid-body modesthat now reads, because of eq (6.123)

(6.125)

for every index i and j.

As in the case of the ‘usual’ orthogonality conditions, rigid-body modes

do not represent a difficulty in all the aspects of the foregoing discussions In

order to be more specific, we can consider an n-DOF system with m

rigid-body modes and write eq (6.56a) as

(6.126)

Fig 6.9 (a) Simple aeroplane model (b) Rigid-body translation (c) Rigid-body

rotation.

where we introduced the rigid-body n×m matrix and the

rigid-body modes The matrices P and y are associated with elastic modes and

retain their original meaning; however, their dimensions are now n×(n–

m) and (n–m)×1, respectively Substitution of eq (6.126) into

where L is the (n–m)×(n–m) diagonal matrix of eigenvalues different

from zero The expressions (6.128) show that the equation for the elasticmodes remains unchanged, while the rigid-body normal equations havesolutions of the form The general solution, forexample in the form of eq (6.27c), can then be written as the eigenvectorexpansion

(6.129)

where the 2n constants are determined by the initial conditions

By using once again the orthogonality conditions, we arrive at the explicitexpression

(6.130a)

or, equivalently

(6.130b)

Equations (6.130a) and (6.130b) are the counterpart of eqs (6.51) and

(6.53) when the system admits m rigid-body modes Furthermore, by virtue

of eq (6.126), it is not difficult to see that the potential and kinetic energyare now given by

(6.131)

where L is the (n–m)×(n–m) matrix of eq (6.128).

The expressions above show that the elastic motion and the rigid-bodymotion are completely uncoupled and that the rigid-body modes, as expected,give no contribution to the potential energy Besides the fact that the generalsolution must take rigid-body modes into account, another aspect that deserves

attention is the fact that K is now singular This is mostly a problem of a

computational nature because some important numerical techniques requirethe inversion of the stiffness matrix The highly specialized subject of thenumerical solution of the eigenproblem is well beyond the scope of the bookbut it is worth knowing that there are simple ways to circumvent this problem.One solution is the addition of a small fictitious stiffness along an adequatenumber of degrees of freedom of the unrestrained system This is generallydone by adding restraining springs to prevent rigid-body motions and makethe stiffness matrix nonsingular If the additional springs are very ‘soft’ (that

is, they have a very low stiffness) the modified system will have frequenciesand mode shapes that are very close to those of the unrestrained system It

is apparent that this procedure involves a certain approximation because, as

a matter of fact, the original system has been modified In practice, however,

a satisfactory degree of accuracy can be achieved in most cases

A second possibility, extensively used by eigensolvers, is called shifting.

We calculate the shifted matrix

(6.132)

where ρ is the shift, and solve the eigenproblem

(6.133)Since the solution of the eigenproblem is unique, it is not difficult to see thatthe original eigenvalues are given by

(6.134)and that the original eigenvectors are left unchanged by the shifting process.Suppose, for example, that a given system leads to the eigenproblem

with eigenvalues and mass-normalized eigenvectors

Imposing a shift of, say, leads to the shifted eigenproblem of eq (6.133)

and to the characteristic equation which admits thesolutions and Equation (6.134) is verified and it is easy todetermine that the eigenvectors are the same as before

This procedure does not involve any approximation, so that, in principle,

it may be sufficient to have solution algorithms for eigenvalues differentfrom zero In fact, it is always possible to operate on the shifted matrix which, in turn, can always be made nonsingular by an appropriate choice ofthe shifting value ρ

A third possibility lies in the fact that, as stated above, rigid-body modescan often be identified by inspection Thus, by imposing the condition oforthogonality between rigid-body modes and the elastic modes we can obtain

as many constraint equations as there are rigid-body modes The constraintequations are then used to perform a coordinate transformation between

(n–m) independent coordinates and the original n coordinates This leads to

a reduced eigenvalue problem where the rigid-body modes have beeneliminated; the reduced system is positive definite and can be solved by means

of any standard eigensolver This procedure, for the reasons explained above,

is often called sweeping of rigid-body modes.

For example, we can consider the system of Fig 6.10 where the three

masses can only move along the x-axis Choosing the coordinates shown in

the figure, we obtain the following mass and stiffness matrices:

The rigid-body mode represents a translation of the whole system in the

x direction and, besides normalization, it can be written as r=[1 1 1] T Anyelastic mode must be orthogonal to r, i.e.

(6.135)Equation (6.156), as a matter of fact, is a holonomic constraint ensuringthat the centre of mass remains fixed at the origin; we can use this equation

to reduce by one the number of degrees of freedom of the system by expressingone of the coordinates as a function of the other two Which coordinate to

eliminate is only a matter of choice; for example we can eliminate x1 andwrite the coordinate transformation

where the first equation is obtained from the constraint (6.135) and theother two are simple identities In matrix form the transformation abovecan be written as

(6.136)

where we call z the ‘constrained’ 3×1 vector and the 2×1 vector of

independent coordinates At this stage we can form the reduced eigenproblem

(6.137a)

Fig 6.10 Unrestrained 3-DOF system.

where and are the 2×2 matrices

our simple example above the mass matrix M is diagonal while is not).

6.7 Damped systems: proportional and nonproportional

damping

6.7.1 Proportional damping

In Section 6.4.1 we considered the inclusion of a small amount of viscousdamping in the equations of motion as a perturbative term of an originally

undamped n-DOF system Our intention was twofold: to have a general

idea of what to expect when some dissipative effects are taken into accountand to investigate the behaviour of lightly damped structures, a frequentlyoccurring situation in practical vibration analysis The starting point was eq(6.90), which we rewrite here for our present convenience:

(6.138)When the undamped solution is known it is always possible to perform

the coordinate transformation (6.56a), premultiply by the modal matrix PT

and arrive at

(6.139)

which is not a set of n uncoupled equations unless the matrix P TCP is

diagonal When this is the case, the n uncoupled equations of motion can be

written as

(6.140)

where ζj is the jth modal damping ratio, in analogy with the SDOF equation

(4.18) The solution of eq (6.140) is given in Chapter 4 and for weare already familiar with its oscillatory character at the frequency

and its exponentially decaying amplitude At this point the usefulness ofinvestigating the condition under which the damping matrix can bediagonalized by the coordinate transformation (6.56a) is evident Somecommon assumptions are as follows:

justifications can be given for the assumptions above; for example:

• The first case may represent a situation in which each mass is connected

to a viscous damper whose other end is connected to ‘the ground’ and

every coefficient c ij is in the same proportion a to the mass coefficient m ij

• A damping element in parallel with each spring element with a constantratio can be invoked for the second case, but the use of eqs(6.141) is mostly a matter of convenience which turns out to be adequate

in many practical situations In these circumstances the dampedeigenvectors are the same as the undamped ones and it is evident thateqs (6.141) take advantage of their orthogonality properties to arrive at(eqs (6.139) and (6.140))

(6.142)which allows us to determine the proportionality coefficient(s) whenone (or two in the third case of (6.141)) damping ratio(s) has beenspecified or measured for the system under investigation

For example, suppose we assumed mass proportional damping and wemeasured ζk from a free-vibration amplitude decay test performed byimposing appropriate initial conditions Then, provided that we know thevalue of ωk, we get

and the other damping ratios can be obtained from (6.142) as

A greater degree of control on the damping ratios can be achieved if weassume Rayleigh damping, where we can specify the damping ratios for any

two modes, say the kth and the mth, to get

depending on the specific values of a and b; the reader is invited to consider

a few reasonable cases and draw a graph of the function for eachcase

The foregoing procedure can be extended if we take into account theadditional orthogonality conditions of Section 6.5.1 In fact, assume for