Applied Structural and Mechanical Vibrations 2009 Part 7 pot

However, as shown in Table 4.3, H ω can be a receptance, a mobility or an accelerance or inertance function; in the preceding sections we wrote Rω because, specifically, we have consider

premultiply both sides by P and postmultiply by P to get

which we can write as where we define for brevity of notation

pre- and postmultiplication of both sides by P and PT, respectively, leads to

(7.29)Each equation of (7.29) is a forced SDOF equation with sinusoidalexcitation We assume a solution in the form

where j is the complex amplitude response Following Chapter 4, we arrive

at the steady-state solution (the counterpart of eq (4.42)),

(7.30)

where

By definition, the frequency response function (FRF) is the coefficient H(ω)

of the response of a linear, physically realizable system to the input ; withthis in mind we recognize that

(7.31)

is the jth modal (because it refers to normal, or modal, coordinates) FRF If

we define the n×1 vector of response amplitudes we can

put together the n equations (7.29) in the matrix expression

(7.32)

and the passage to physical coordinates is accomplished by the transformation(7.2), which, for sinusoidal solutions, translates into the relationship betweenamplitudes Hence

(7.33)which must be compared to eq (7.27) to conclude that

(7.34a)

Equation (7.34a) establishes the relationship between the FRF matrix (R) of

receptances in physical coordinates and the FRF matrix of receptances inmodal coordinates This latter matrix is diagonal because in normal (ormodal) coordinates the equations of motion are uncoupled This is not true

for the equations in physical coordinates, and consequently R is not diagonal.

Moreover, appropriate partitioning of the matrices on the right-hand side of

eq (7.34a) leads to the alternative expression for the receptance matrix

(7.34b)

where the term is an (n×1) by (1×n) matrix product and hence results

in an n×n matrix From eq (7.34a) or (7.34b) it is not difficult to determine

that

(7.35)

i.e R is symmetrical; this conclusion can also be reached by inspection of eq

(7.28) where it is evident that This result is hardly surprising In

fact, owing to the meaning of the term R jk (i.e eq (7.25)), it is just a differentstatement of the reciprocity theorem considered in Section 7.2

7.4 Time-domain and frequency-domain response

In Section 7.2, eq (7.7b) represents, in the time domain, the normal coordinateresponse of a proportionally damped system to a general set of applied forces

Since we pass to physical coordinates by means of the transformation u=Py,

and it is evident that or equivalently,

On the other hand, if we take the Fourier transform of both sides of eqs(7.6), we get

(7.38)

where we have called Y j(ω) and Φj(ω) the Fourier transforms of the functions

y j (t) and respectively If we form the column vectors

and

transform of f, we obtain from eq (7.38)

(7.39)

Now, since u(t)=Py(t) it follows that and eq (7.39)leads to

(7.40)which is the frequency-domain counterpart of the time-domain equation(7.36) Summarizing the results above and referring to the discussion ofChapter 5 about impulse-response functions and frequency-responsefunctions, we can say that—as for the SDOF case—the modal coordinates

functions h j (t) and are a Fourier transform pair and fully

define the dynamic characteristics of our n-DOF proportionally damped

system

In physical coordinates, the dynamic response of the same system is

characterized by the matrices h(t) and R(ω) whose elements are given,respectively, by eqs (7.37b) and (7.28) These matrices are also a Fouriertransform pair (Section 5.4), i.e

(7.41)

which is not unexpected if we consider that the Fourier transform is a lineartransformation Also, from the discussion of Chapter 5, it is evident that theconsiderations of this section apply equally well if ω is replaced by the Laplace

operator s and the FRFs are replaced by transfer functions in the Laplace

domain Which transform to use is largely dictated by a matter of convenience

A note about the mathematical notation

In general an FRF function is indicated by the symbol H(ω) and, consequently,

a matrix of FRF functions can be written as H(ω) However, as shown in

Table 4.3, H( ω) can be a receptance, a mobility or an accelerance (or

inertance) function; in the preceding sections we wrote R(ω) because, specifically, we have considered only receptance functions, so that R(ω) is just a particular form of H(ω) Whenever needed we will consider also the other particular forms of H(ω), i.e the mobility and accelerance matrices and we will indicate them, respectively, with the symbols V(ω) and A(ω)

which explicitly show that the relevant output is velocity in the first case

and acceleration in the second case Obviously, the general FRF symbol H(ω) can be used interchangeably for any one of the matrices R(ω), V(ω) or A(ω).

By the same token, H(s) is a general transfer function and R(s), V(s) or A(s)

are the receptance, mobility and accelerance transfer functions

Finally, it is worth noting that some authors write FRFs as H(iω) in order

to remind the reader that, in general, FRFs are complex functions with a real

and imaginary part or, equivalently, that they contain both amplitude and

phase information We do not follow this symbolism and write simply H(ω).

7.4.1 A few comments on FRFs

In many circumstances, one may want to consider an FRF matrix other than

R(ω) The different forms and definitions are listed in Table 4.3 and it is notdifficult to show that, for a given system, the receptance, mobility andaccelerance matrices satisfy the following relationships:

(7.42)

which can be obtained by assuming a solution of the form (7.23) and notingthat

(7.43)

where we have defined the (complex) velocity and acceleration amplitudes v

and a However, the definitions of Table 4.3 include also other FRFs, namely

the dynamic stiffness, the mechanical impedance and the apparent masswhich, for the SDOF case are obtained, respectively, as the inverse ofreceptance, mobility and accelerance This is not so for an MDOF system

Even if in this text we will generally use only R(ω), V(ω) or A(ω), the

reader is warned against, say, trying to obtain impedance information bycalculating the reciprocals of mobility functions In fact, the definition of a

mobility function V jk , in analogy with eq (7.25), implies that the velocity at point j is measured when a prescribed force input is applied at point k, with

all other possible inputs being zero The case of mechanical impedance isdifferent because the definition implies that a prescribed velocity input is

applied at point j and the force is measured at point k, with all other input

points having zero velocity In other words, all points must be fixed (grounded)except for the point to which the input velocity is applied

Despite the fact that this latter condition is also very difficult (if notimpossible) to obtain in practical situations, the general conclusion is that

(7.44)

where we used for mechanical impedance the frequently adopted symbol Z.

Similar relations hold between receptance and dynamic stiffness and between

accelerance and apparent mass So, in general [1], the FRF formats of dynamicstiffness, mechanical impedance and apparent mass are discouraged becausethey may lead to errors and misinterpretations in the case of MDOF systems.Two other observations can be made regarding the FRF which are ofinterest to us:

• The first observation has to do with the reciprocity theorem Followingthe line of reasoning of the preceding section where we determined (eq(7.35)) that the receptance matrix is symmetrical, it is almoststraightforward to show that the same applies to the mobility andaccelerance matrices

• The second observation is to point out that only n out of the n2 elements

of the receptance matrix R(ω) are needed to determine the naturalfrequencies, the damping factors and the mode shapes

We will return to this aspect in later chapters but, in order to have an idea,suppose for the moment that we are dealing with a 3-DOF system withdistinct eigenvalues and widely spaced modes In the vicinity of a naturalfrequency, the summation (7.28) will be dominated by the term corresponding

to that frequency so that the magnitude can be approximated by(eqs (7.28) and (7.34b))

(7.45)

where j, k=1, 2, 3 Let us suppose further that we obtained an entire column

of the receptance matrix, say the first column, i.e the functions R11, R21 and

R31; a plot of the magnitude of these functions will, in general, show threepeaks at the natural frequencies ω1, ω2 and ω3 and any one function can beused to extract these frequencies plus the damping factors ζ1, ζ2 and ζ3.Now, consider the first frequency ω1: from eq (7.45) we get the expressions

and ω3 leads, respectively, to p2 and p3 and, since the choice of the firstcolumn of the receptance matrix has been completely arbitrary, it is evidentthat any one column or row of an FRF matrix (receptance, mobility oraccelerance) is sufficient to extract all the modal parameters This isfundamental in the field of experimental modal analysis (Chapter 10) inwhich the engineer performs an appropriate series of measurements in order

to arrive at a modal model of the structure under investigation

Kramers-Kronig relations

Let us now consider a general FRF function If we become a little moreinvolved in the mathematical aspects of the discussion, we may note thatFRFs, regardless of their origin and format, have some properties in common.Consider for example, an SDOF equation in the form (4.1) (this simplifyingassumption implies no loss of generality and it is only for our presentconvenience) It is not difficult to see that a necessary and sufficient condition

for a function f(t) to be real is that its Fourier transform F(ω) have the

symmetry property which, in turn, implies that Re[F(ω)] is

an even function of ω, while Im[F(ω)] is an odd function of ω Since H(ω) is

the Fourier transform of the real function h(t), the same symmetry property applies to H(ω) and hence

(7.47)

where, for brevity, we write HRe and HIm for the real and imaginary part of

H, respectively In addition, we can express h(t) as

(7.48)

divide the real and imaginary parts of H( ω) and, since h(t) must be real,

arrive at the expression

(7.49)

where the change of the limits of integration is permitted by the fact that,owing to eqs (7.47), the integrands in both terms on the r.h.s are evenfunctions of ω

If we now introduce the principle of causality—which requires that the

effect must be zero prior to the onset of the cause—and consider the cause

to be an impulse at t=0, it follows that h(t) must be identically zero for

negative values of time The two terms of eq (7.49) are even and odd functions

of time and so, if h(t) is to vanish for all t<0, we have

know HRe(ω), we can compute HIm(ω) and vice versa.

The explicit relations between HRe and HIm can be found by writing therelation

where the lower limit of integration can be set to zero because we assumed

h(t)=0 for t<0 Next, by separating the real and imaginary parts of H(ω) weobtain

(7.51)

In addition, from eq (7.49) we have

which (introducing the dummy variable of integration) can be substituted

in the second of eqs (7.51) to give

and hence, since it can be shown that

we can perform the time integration to obtain the result

(7.52)

where the symbol P indicates that it is necessary to take the Cauchy principal

value of the integral because the integrand possesses a singularity

By following a similar procedure and noting that from eq (7.50) we can

into the first of eqs (7.51) to obtain

(7.53)

Equations (7.52) and (7.53) are known as Kramers-Kronig relations Notethat they are not independent but they are two alternative forms of the same

restriction on H(ω) imposed by the principle of causality

The conclusion is that for any given ‘reasonable’ choice of HRe on the real

axis there exists one and only one ‘well-behaved’ form of HIm The terms

‘reasonable’ and ‘well-behaved’ are deliberately vague because a detaileddiscussion involves considerations in the complex plane and would be out ofplace here: however, the reader can intuitively imagine that, for example, by

‘reasonable’ we mean continuous and differentiable and such as to allow theKramers-Kronig integrals to converge

We will not pursue this subject further because, in the field of our interest,the Kramers-Kronig relations are unfortunately of little practical utility Infact, even with numerical integration, the integrals are very slowly convergent

and experimental errors on, say, HRe may produce anomalies in HIm which

can be easily misinterpreted and vice versa Nevertheless, the significance of

the Kramers-Kronig relations is mainly due to the fact that they exist andthat their very existence reflects the fundamental relation between cause andeffect, a concept of paramount importance in our quest for an increasinglyrefined and complete description of the physical world

7.5 Systems with rigid-body modes

Consider now an undamped system with m rigid-body modes From the

equations of motion

and the usual assumption of a harmonic solution in the form we get

(7.54)whose formal solution is given by

(7.55)where is the receptance matrix of our undamped system

As in Section 7.3, our scope is to arrive at an explicit expression for this FRFmatrix

Referring back to Section 6.6, we can expand the vector z on the basis of

the system’s eigenvectors, which now include the m rigid-body modes: the

expansion (whose coefficients must be determined) reads

(7.56)

where we assume all modes to be mass orthonormal Equation (7.56) can besubstituted in eq (7.54) to obtain a somewhat lengthy expression which, inturn, can be premultiplied by to give

(7.57a)and premultiplied by to give

and its (jk)th element is

(7.59b)

Note that the expansion (7.56) on the basis of modes which are not mass

orthonormal results in a term M ii in the denominator of the first sum on the

right-hand side of eqs (7.59a) and (7.59b) and in a term M ii in the denominator

of the second sum

Equations (7.59a) and (7.59b) are, respectively, the counterpart of eqs(7.34b) and (7.28) for an undamped system with rigid-body modes: therigid-body modes contribution is evident and it is also evident that thefunction

is the lth modal FRF H l(ω) of an undamped system In this light, the discussion

of this section can be extended with only little effort to a proportionally

damped system with m rigid-body modes The reader is invited to do so.

As far as unrestrained systems are concerned, it is interesting to note thatthe mode displacement and the mode acceleration methods can also be used

to determine their response The mode displacement method does not presentadditional difficulties due to the presence of rigid-body modes, but theextension of the mode acceleration method is not straightforward In essence,the reason lies in the fact that the stiffness matrix of an unrestrained system

is singular and the method (Section 7.2.1) requires the calculation of K–1.However, this difficulty can be circumvented; we do not pursue this subjecthere and for a detailed discussion the interested reader is referred, for example,

to Craig [2]

7.6 The case of nonproportional viscous damping

The preceding sections have all dealt either with undamped systems or withsystems whose damping matrix becomes diagonal under the transformation

PTCP In these cases, the modal approach for the calculation of their response

properties relies on the possibility to directly uncouple the equations ofmotion, solve each equation independently and superpose the individualresponses

As stated in Section 6.7.1, the assumption of proportional damping is notalways justified and a general damping matrix leads, in the homogeneouscase, to the complex eigenvalue problem (6.92) This, in turn, can either be

solved directly as it is or can be tackled by adopting a state-space formulation,

as shown in Section 6.8 (eqs (6.75a and b) or eqs (6.179))

The nature of the problem itself leads to a complex eigensolution, but theeigenvectors that we obtain in the first case satisfy the ‘undesirable’orthogonality conditions of eqs (6.158) and (6.159) which, in general, are oflimited practical utility By contrast, the state-space formulation results either

in a generalized or in a standard eigenvalue problem—both of which formsare preferred for numerical solution—and in a set of much simplerorthogonality conditions This approach is also more effective in thenonhomogeneous case

Let us first consider the equations

(7.60a)and write them in matrix form as

or

(7.60b)

where we define the matrix q=[f 0]T and the matrices , and x as in eq

(6.175c) We are already familiar with the solution of the homogeneouscounterpart of eq (7.60b); hence we can express the solution of (7.60b) asthe superposition of eigenmodes

(7.61)

which can be substituted in eq (7.60b) and, taking eqs (6.178) into account,

premultiplied by to get the 2n independent first-order equations

or, equivalently

(7.62)where we defined

(7.63)

and took into account the relation Equations (7.62) can beeasily solved by multiplying both sides by and writing the result as

where S is the 2n×2n matrix of eigenvectors and now,

substitute (7.65) in eq (7.60b) and premultiply by ST to obtain

(7.66)Without loss of generality, we can assume and arrive at the matrixequation

Finally, if we remember that it follows that the last n elements

of x are the derivatives of the first n elements; this implies, as we know from

the preceding chapter, that each eigenvector is in the form

(7.70)

By virtue of eq (7.70), the 2n×2n matrices S and S T can be partitioned into

(7.71a)and

(7.71b)

where the orders of Z, ZT and diag( j ) are n×2n, 2n×n and 2n×2n, respectively.

With this in mind, noting that

we can recover the displacement solution from eq (7.69) as

(7.72)which represents the response of our system to an arbitrary excitation

7.6.1 Harmonic excitation and receptance FRF matrix

The solution for a harmonic excitation can be worked out as a particular

case of eq (7.64) The jth participation factor is now

(7.73)

where Without loss of generality we can assume zero initialconditions and the normalization condition then, eq (7.64) becomes

(7.74)

Since we are mainly interested in the steady-state solution, we can drop thesecond term on the right-hand side which (if the system is stable and alleigenvalues have negative real parts) dies away as and arrive at the solution

(7.75a)

or, alternatively

(7.75b)

Next, once again by virtue of eq (7.70), we can partition the matrices S and

ST as in eqs (7.71a and b) and obtain

from which it follows that

(7.76a)

or, equivalently

(7.76b)

From the definition of receptance matrix and from eqs (7.76a) and (7.76b)

we get the n×n matrix

(7.77)

whose (jk)th element is obtained as

(7.78a)

Furthermore, for the case in which we are mainly interested—i.e.underdamped systems—we know that both eigenvalues and eigenvectorsappear in complex conjugate pairs; this implies that eq (7.78a) can be written

as the sum of n terms

(7.78b)where the last expression was written by taking eq (6.160) into account

So, as in the other cases, we have obtained an explicit expression for thereceptance FRF matrix This is precisely the response model for the systemunder study and, once again, we can see that the general element of thereceptance matrix is the sum of the contributions of the different modes ofvibration

At this point, it is worth pointing out that in modal analysis terminologythe eigenvalues m are often called the poles and the term z jm z km—referred to

as the residue for mode m—is given a symbol in its own right: for example,

the reader may find in current literature the symbols m A jk or r jk,m , both of which stand for z jm z km

At this point it may be instructive to follow a similar line of reasoning asabove to work out a response model (and an explicit expression for thereceptance FRF matrix) by starting from the state-space formulation of eqs(6.179) As a useful—and not trivial—exercise, the reader is urged to do so

by taking advantage of the guidelines that follow

1 the homogeneous case leads to a standard 2n eigenvalue problem

where, in general, the matrix A is not symmetrical.

2 The eigenvalues and eigenvectors occur in complex conjugate pairs(underdamped case) and the eigenvectors have the form

3 If the matrix A is nondefective (which we assume to be the case), we can

form the 2n×2n matrix S of column eigenvectors so that

(7.79)

4 The forced vibration equations can be cast in the form

(7.80a)

where now we define

(7.80b)

5 The transformation to normal coordinates x=Sy can be substituted in eq

(7.80a) in order to arrive at

(7.81)which is formally similar to eq (7.67) and leads, in the end, to

(7.84)and hence

(7.85)

which is the displacement response of our system to an arbitrary excitation

7 Again, the case of harmonic excitation can be obtained as a particular

case of eq (7.85) The jth participation factor is now

(7.86)

where we called the left jth eigenvector of A (Appendix A) Note that

is a row 1×2n vector (this is why we write the superscript T for

transpose: because in our notation, as it is customary, vectors are arranged

as columns) and its components form the jth row of matrix S–1 just like

the components of sj (the jth right eigenvector of A) form the jth column

9 The expression of a single receptance FRF function R jk(ω) in terms of

individual components is a bit involved; however, if we call b rs the general

(r, s)th element of the 2n×n matrix it is not difficult todetermine that

(7.89)

7.7 MDOF systems with hysteretic damping

We stated in Section 6.7 that this type of damping does not lend itself easily

to a rigorous free-vibration analysis because, strictly speaking, the concept

of hysteretic (or structural) damping is based on an analogy with the viscousdamping case when the system is excited by means of a harmonic forcingfunction Nevertheless, experimental tests are often performed in a forcedvibration condition and it is undoubtedly useful to obtain a response modelfor these systems in terms of eigenvalues and mode shapes, howeverquestionable this free-vibration solution may be Therefore, provided thatthe results are used judiciously, we justify the considerations that follow onthe basis of physical sense

In general, hysteretic damping is taken into account by expressing theequations of motion in the form (6.150), where the damping matrix is written

as iγK In the homogeneous case, assuming a solution in the form

(7.90)

leads to

which admits a nontrivial solution if It is not

difficult to see that we obtain now a set of n complex eigenvalues (because

the coefficients of the characteristic polynomial are complex) and that the

set of n real eigenvectors z j are the same as for the undamped case.The eigenvalues contain information on both frequency and dampingcharacteristics and they can be written as

(7.91)where

(7.92)

Note that, as in the previous cases, the ωjs have well-defined values but the

values of the K jj s and M jjs depend on the normalization that we choose.Once again, it is common practice to fix the indeterminacy on the eigenvectors

by choosing, out of the many possibilities, the vectors pj (j=1, 2,…, n) which

satisfy the relations

Incidentally, it may be worth noting that a more general case ofproportional hysteretic damping can be considered by writing the equations

of motion as

(7.93a)

and assuming that the hysteretic damping matrix H (not to be confused

with a FRF matrix) is given by

(7.93b)

where a and b are two constants We will not deal specifically with this case

because, as the reader can verify, the nature of the eigensolution is the same

as before and nothing is added to the essence of the problem

With the above considerations in mind, it is now only a small effort toarrive at a response model in the case of the harmonic excitation

We start from the equations of motion (6.150) and perform the change ofcoordinates

where P is the matrix of mass orthonormal eigenvectors Next, we premultiply

the resulting equation by P and arrive at

(7.94a)

which represents a set of n uncoupled equations The jth equation reads

explicitly

(7.94b)Assuming a harmonically varying response it is not difficult toretrieve the solution in physical coordinates as

(7.97)

where now we have

(7.98)

Despite the discussion at the beginning of this section, there seems to be

no difficulty in the derivation of the response model of eqs (7.96a, b) and(7.97) There is, however, a subtle conceptual problem Our FRFs must be

the Fourier transform of real impulse response functions, and this implies(Section 7.4) that the conditions (7.47) apply This is not the case for theFRFs of eqs (7.96b) and (7.97); furthermore, if these latter are modified toagree with eqs (7.47) it follows that our FRFs do not satisfy the requirement

of causality We will not go proceed further in this discussion which is beyondthe scope of this book, but it seems that these conceptual problems—althoughthey can be ignored in many practical situations—are the price that we mustpay for the inadequacy of a free vibration solution in the hysteretic case

The interested reader can refer, for example, to Nashif et al [3] and

Newland [4]

7.8 A few remarks on other solution strategies: Laplace

transform and direct integration

This chapter has dealt in some detail with the response properties of varioustypes of MDOF system However, special attention has been intentionallygiven to the so-called modal approach (or modal superposition, modalexpansion techniques), where the dynamic response is expressed as a seriesexpansion of eigenmodes The reason is twofold: first of all, this text is mainlyconcerned with linear vibrations of structural and mechanical systems and,second, the modal approach has considerable importance in many aspects

of experimental vibration measurements Nevertheless, the reader would beright in assuming that other approaches are available in order to solve theforced vibration problem of MDOF systems

7.8.1 Laplace transform method

At least in principle, the Laplace transform method can be directly applied

to eq (7.1) to obtain

(7.99)

where U=U(s) and F(s) are, respectively, the Laplace transforms of u(t) and

f(t), s is the Laplace operator and are the vectors of initial displacementsand velocities For zero initial conditions eq (7.99) can be rewritten as

(7.100)and consequently

(7.101)

where the last expression on the right-hand side for G–1 can be found in any

book on matrix algebra: adj(G) is called classical adjoint (or adjugate, to avoid confusion with the Hermitian adjoint) of G and is the transposed matrix

of cofactors of G From eq (7.101) we recognize G–1 as the matrix of receptance

transfer functions R ij (s) and we note that, in the inverse transformation of eq

(7.101), the poles from the transfer functions are the eigenvalues of our systembecause they are obtained from the characteristic equation

This method also applies only for linear systems and may be useful for systemswith nonproportional damping, where eqs (7.1) cannot be uncoupled by

means of the classical modal matrix P Note also that the terms ‘poles’ and

‘residues’ come directly from the Laplace transform approach

In addition to what has been said above and in Section 5.3.3, we canbriefly review the case of a SDOF system—thus keeping the mathematicsextremely simple—and get an idea of how this technique works as far astransfer and frequency response functions are concerned

If we take the Laplace transform of both sides of the equation

and assume zero initial conditions (which amounts toneglecting the solution of the homogeneous equation), we arrive at the SDOFcounterpart of eq (7.100) which can be written as

(7.102)where the meaning of the symbols is obvious and

(7.103)

is the (complex-valued) receptance transfer function The denominator of eq(7.103) is the characteristic equation, whose roots (the poles) can be writtenfor an underdamped system as

(7.104)

Now we note that H(s) can be rewritten as

(7.105a)and expanded in partial fractions as

(7.105b)

where the coefficients (residues) A j can be obtained from

(7.106)

It is straightforward, in this case, to determine that and obtain

(7.107)

If now we consider that the frequency response function is simply the

transfer function evaluated along the iω axis, we obtain from eq (7.105b)

by an observer who looks down on a plane which cuts through the surfaceand whose normal is parallel to the σ-axis (Fig 7.1)

A different approach to the solution of the forced-vibration problem forboth SDOF and MDOF systems consists of a direct numerical integration ofthe equation(s) of motion in the time domain The details of this approachbelong rightfully to the subject of numerical techniques and are beyond thescope of this book; however, some general comments on the advantages andlimitations of these methods are not out of place

In particular, the reader is warned against the temptation to use directintegration as a ‘black box’, where you input the right equations and obtainthe correct response time history

The major advantage of direct integration is that it applies both to linearand nonlinear problems and, as a matter of fact, it is the only generallyapplicable method for the analysis of nonlinear systems Nonetheless, as far

as linear vibrations are concerned, direct integration methods may also be aneffective alternative to the modal approach For example, in the case of a