combined integral and fe analysis of broad-band random vibration in structural members

E L S E V I E R 0 2 6 6 - 8 9 2 0 ( 9 5 ) 0 0 0 1 9 - 4

Probabilistic Engineering Mechanics 10 (1995) 241-250 Copyright © 1995 Elsevier Science Limited

Printed in Great Britain. All rights reserved 0266-8920/95/$09.50

Combined integral and FE analysis of broad-band random vibration in structural members

Peter Fischer Institute of Engineering Mechanics, University of lnnsbruck, A-6020, Innsbruck, Austria

Alexander K. Belyaev* Institute of Technical Mechanics, Johannes Kepler University of Linz, A-4040, Linz, Austria

&

Helmnt J. Pradlwarter Institute of Engineering Mechanics, University of Innsbruck, A-6020, Innsbruck, Austria

The paper addresses a new approach for the description of broad-band random vibrations of components in complex structures. Random vibrations at low frequencies are determined by methods of computational structural dynamics. Vibrations at high frequencies are described in an integral form, i.e. by means of combination of the Bolotin method of integral estimates and the methods of high-frequency dynamics. The fuzzy intermediate range between the upper frequency bound of efficient numerical computations and the lower bound of the high frequency approach is examined by means of an example. The methodology essentially reduces the computing costs without noticeable loss of information in the high frequency domain. Further, the approach enables one to indicate a frequency domain wherein the vibration localisation within structural members occurs. The applicability of the concept is demonstrated by computing the broad band random vibrations of a thin-walled cover of an engine head.

1 INTRODUCTION

A detailed description of random vibration of engineering structures is made difficult, first, by the complexity of the structure's shape, then, by the assemblage of individual substructures and, finally, by the presence of various secondary systems attached to the primary structure. The conventional well-established methods of structural analysis, such as the substructure synthesis method, finite element (FE) method are not always suitable for description of high frequency dynamics. A fine FE-mesh is required to model short wavelength deformations of higher modes. Apart from the

*On leave from: Department of Mechanics and Control Processes, State Technical University of St Petersburg, 195251, St Petersburg, Russia.

241

computing costs, an excessive number of degrees of freedom produces difficulties in analysis of the obtained results.

Under these circumstances it seems reasonable to use conventional methods of structural mechanics to model the low frequency part of broad-band vibrations 0 < w < O1 and an integral description at high frequencies, 02 < w < 0~. The parameter O1 may be determined as an upper frequency of efficient numerical computations and 02 is the lower theoretical bound of the high frequency approach. Generally, 02 is higher than O1, so that there is an uncertain intermediate range. While O1 is fixed by the individual limitation of numerical expense, the domain of the high frequency approaches can be shifted to lower frequencies by disregarding the theoretical bound 02. So we split the frequency domain into two parts, the frequency range of FE-computations

242 P. Fischer, A. K. Belyaev, H. J. Pradlwarter

0 < 03 < O 1 and the extended range of integral approaches O 1 < 03 < OO and make use of the following spectral representation

u(r, t) = UL (r, w) exp(i cot)dco

(1)

F + ue(r, co) exp(i cot)dw el

Here uL(r,w), 0 < co < O 1 and uH(r, co), O1 < co < o~ are the frequency transforms of the displacements at low and high frequencies, respectively. The properties of the response in the intermediate range O1 < co < O2 are demonstrated exemplarily by a numerical computation with overlapping frequency domains.

The response in the low frequency domain is strongly influenced by structural details, so the discretisation by finite elements is used for the analysis at low frequencies. The FE model consists of the substructure of interest and those parts of the system that influence the modal properties of this substructure. The difficulty of describing the interaction of the modelled subassembly with the neglected one is overcome by means of prescribing stochastic multi-support displacements at the subassembly boundary. To account for the local energy dissipation effects (e.g. gaskets in internal combustion engines) on the modal damping, nonclassical modes are used throughout the entire analysis in the low frequency domain. Efficient computation algorithms for combined force and multi-support excitations are developed.

Section 4 deals with an asymptotic analysis of the high frequency part of the spectrum. Among the integral approaches, the Bolotin method of integral estimates 1 is most prominent. The method implies an absolutely isolated substructure. It was originally applied to estimate a certain mean value of vibration of thin- walled elements under broad-band excitation. Its application results in rather simple formulae in closed form. Instead of eigenfrequencies and normal modes one may use their asymptotic expressions in the high- frequency domain. The main shortcoming of the Bolotin approach is the absolute isolation of substructures considered. Some integral methods have been offered in Ref. 2 to predict the field of high-frequency vibration in complex structures. The main distinctions between low and high frequency vibrations as well as some essential features of high frequency vibration are discussed in Ref. 3. As shown in Refs 2 and 3, high frequency vibrations in engineering structures can be described by means of a relatively simple boundary- value problem. The properties of an actual structure are reflected in this integral theory in the form of certain overall rigidity, mass and absorption characteristics. Hence, some generalized characteristics of the vibration field are obtained. However, the vibration of a particular structural member cannot be computed by

an integral method only, since this member is not represented in the dynamical model. Thus, to analyse the behaviour of a particular member one has to take into account both its individual mechanical characteristics and the nature of its interaction with other structural members. Therefore, the structural member is to be described precisely while the rest of the structure is described in an integral form.

2 LOCAL PRINCIPLE IN STRUCTURAL DYNAMICS

Let us consider a substructure Vn. Because of the localisation of high frequency vibration within the structural members 4'5 the vector of the absolute displacement Un(r, t) in Vn may be represented in form of an expansion in terms of the substructure's normal modes Unk (r),

r G Vn; Un(r , t) = Z Unk(r)qnk(t) + u(r, t) (2) k=l

where qnk(t) is the k-th generalized coordinate of the substructure Vn, r is the reference, t is time. The normal modes are specified such, that they vanish on the substructural boundaries Sn. In this case the function u(r, t) coincides with the actual displacement of the substructural boundaries Sn and it may be referred to as the displacement of the primary structure. To obtain an engineering theory for the description of the high- frequency vibration field in an integral form, it is reasonable to require the maximal spatial smoothness of the function u(r, t) within the whole structure.

Since the normal modes are orthogonal, the expansion (2) allows the following representation of the kinetic energy of the entire structure.

1 N_~ll I J p lin.lind V = p d./ld V T = ~ v~ 2 v (3)

1 N +~ ~=l~=l [OZnk + Onk Jvn punk'ddV]

In view of the spatial smoothness of u(r, t) and essential heterogenity of the structure (p(r) and unk(r) are rapidly changing functions of r) the next estimation appears to be correct

1 I (p)u-ddg T = ~ v (4) N oo

1 -2 + x E Z [qnk + (p)(unk)'fiVnqnk]

A n = l k = l

where

1 I pdV; (5)

1 [ P UnkdV (Unk) = (p)nVn Jv.

Combined integral and FE analysis of broad-band random vibration in structural members 243

are the average density of the structure and the average displacement of the centre of mass of the substructure Vn when it vibrates according to the normal mode Unk(r), respectively.

The equation for the potential energy is obtained analogously:

1 L ( V u ) l ~ l k ~ l 2 2 H = ~ : (C) : (Vu)dV + ~ ~nkqnk

(6) Here (C) is an overall tensor of elastic moduli averaged over the structure, f~,k is the k-th eigenfrequency of substructure Vn, V is the Hamiltonian operator and : denotes the double scalar product. The work of the external surface load F is as follows:

W = .Is F.u dS (7)

Applying the Hamilton variational principle yields the following boundary value problem

l ' e Vn; V ' [ ( C ) : (Vii)l-- li-~t-y~(Unk)0nk ~---0 (8) k=l

2 r 6 Vn; /i/nk + 2¢nk~2~,qnk + f~nkq,k

= --( p)(iink)-iiVn; k = 1,2, . . . oo (9)

r e S; N.[(C) : (Vu)] = F (10)

where N is the unit vector of the outward normal to the surface S. The modal damping is introduced into eqn (9) by the critical damping ratio era,. Equation (8) governs the dynamics of the primary structure while eqns (9) are the equations for the generalized coordinates of the structural members.

Since the problem is split into low and high frequency domains, the boundary value problem is transfered to the frequency domain:

r e Vn; V . [ ( C ) : (VII)]

+(P)W 2 II+ Unk)qnk = 0 (11)

022(p) (Unk) °U Vn = (12)

r E Vn; q.k _022 + 2iCnk~nkW + f~n 2

r e S; N.[(C) : (Vu)] = F (13)

From now on the same designations for the frequency transforms are retained. Substituting qnk, eqn (12), into eqn (11) we obtain the following differential equation:

V.[(C) : (Vu)] + w2A(w).ii = 0 (14)

Here

I _(ii~) (link) Vn .] A(02) = (P/ + ( p ) X~,~_w 2 + 2iCnkf~nkW + f~2kJ k=l

(15)

where I is the identity tensor. The tensor A(w) occupies the place of the mass density in the conventional vibration equation, hence it may be referred to as the generalised mass of the complex structure. This parameter reflects the inertial and spectral properties of the complex structure and may be considered isotrop, 3 i.e.

A(w) = A(02)I (16)

where

A(02) = ~A(w) : I

1 o~ (__Unk). (link)Vn _] = (p) 1 "~ ~ ( P)032~--~k=l --022 + 2i0nkf~nkW + f~n2kJ

(17)

Due to eqn (17), A(w) is a superposition of single- degree-of-freedom resonance curves. The width of each resonance curve is 2~bnkf~nk at the "half-power" level. If the resonant width is large compared to the eigenfrequency separation A~nk (high modal overlap),

A~nk = ~nk+l -- ~'~nk ~ ff)nk~-~nk -{- ~3nk+l~"~nk+l nnk (18)

or --< ~nk ~nk

the resonance curves in eqn (17) merge to form a smooth frequency function. In this case the sum in eqn (17) can be replaced by an integral over the high-frequency domain (92 < 02 < oo

A(02) = ( p ) [ 1 + 022 I +°° ¢(f~)df] ] 02 _022 ~- 2t-~w~+ ~2j (19)

where ~(f~) is a smooth function of the eigenfrequency distribution. The modal density of any mechanical system is known to increase with the growth of the ordinal number of the eigenfrequencies.1 Hence, for the frequency range f~ > O2, the condition (18) and the replacement of the sum by the integral is valid. Obviously ~2 is specific for each mechanical structure and it depends primarily on the relative density of eigenfrequencies and damping value (cf. eqn (1 8)).

The following representation 2'3

A(02) = (p)[1 - in(w)] 2,

~(02) = 023 1 ~ CfhI)(f~)df~ (20)

introduces a non-dimensional frequency-dependent absorption t¢(02) of high-frequency vibration by the structure. In the case of light damping ¢ << 1 and local smooth function ~(f~) one can carry out an approximate calculation of integral in eqn (20). Integrals of this kind are encountered in calculating the dispersion of steady-state random vibration fields in systems, l i.e.

7f ~(w) = ~ 02 ¢(w) (2 l)


From the latter equation we can see that t~(~v) and consequently the value of the absorption of high- frequency vibration does not depend on the damping ~b at all. The absorption is determined by the distribution function <I,(w). It means that the structural members act as dynamical absorbers with respect to the primary structure. Since the resonance curves corresponding to the internal degrees of freedom merge, a considerable spatial absorption of vibration for the whole high-frequency region is observed.

This integral description can be applied for all substructures except a particular substructure of interest• It allows one to formulate the local principle: 6 for each structure with a relative modal density and damping there is a critical frequency 02. If it is exceeded (w > 02, high frequencies), then any particular substructure can be described precisely, while the others are described integrally by means of the boundary-value problem, eqns (13) and (14). In other words, at high frequencies the vibration of an individual substructure in the structure depends mainly upon (i) the mechanical characteristics of this substructure itself, (ii) its particular coupling to the entire structure, and (iii) certain generalised mechanical properties of the structure. It is worth mentioning that the vibration of any individual substructure does not depend upon details of remote substructures.

3 VIBRATIONS OF LARGE NONCLASSICALLY DAMPED STRUCTURES IN THE LOW FREQUENCY DOMAIN

The local principle cannot be applied for the low frequencies. The dynamical properties of the substructure under consideration are significantly influenced by details of the entire structure. Therefore not only the substructure, but also near-by parts of the whole structure have to be modelled in detail• The mechanical description of such systems is conveniently provided by FE discretization.

The more "remote" in the dynamical sense are the regions of the total structure, the less is their influence on the dynamical properties of the substructure. This principal physical property permits to omit the remote parts having a vanishing influence on the modal properties of the substructure of interest• On the other hand, the domain modelling must not prevent the vibration exchange with the cut-off environment. Prescribing the displacements at the boundary of the isolated system determines also the energy flow. There- fore, combined force and multi-support base excitation has to be applied to compute the low frequency response of members in complex structures.

If the excitation of a linear system is stationary, the frequency representation of the total response may be obtained by superposition of the response due to force

excitation xf(o)) and multi-support excitation xU(w). The partial responses are expressed by the excitation forces f and the boundary displacements u, utilizing the corresponding transfer matrices V f and V u, respectively. The procedure is identical to that for velocities and accelerations in the frequency domain:

= xf( ) + xU( ) = vf( )f(w) +

(22)

The covariance matrix of the displacement response reads:

0 d ~ + OO

E[x(t)x(t) *w ] = ([vf(w)Sff(w)vf(w) *T ] k ) = - - OO

+ *w]

+ 2Re[Vf(w)Sfu(O.;)VV(w)*T])dw.

(23)

The assumption of Rayleigh damping does not hold due to the damping localization in bolted and welted joints. The modal damping of the lowest eigensolutions depends strongly on the modeshapes, 7 that means nonclassical damping is observed. The eigensolutions become complex and they are determined by the following quadratic eigenproblem 8'9

[K~ + C ~ A + M ~ A 2] = [0], (24)

where K, C and M are the stiffness, damping and mass matrix of the structure, • and A are the complex eigenvectors and eigenvalues, respectively. Utilizing the property of pairwise conjugate complex eigensolutions, a numerically efficient summation rule for the components of the transfer matrix of excitation forces vf@) is utilized) ° The index " j " is the modecounter of those solutions of a conjugate complex pair, where the imaginary part of the eigenvalue is positive.

nmod Vfrs(~) ---- Z [(hj(oJ) + ]~j(og))rlr~ + / ( h i ( w )

j = l

- i m - h/w))Orsj] (25)

1 'rsj = (b~j(I)sj; hi(w) = 2Aj(i~o - Aj)'

/~(W) hj(wl Aim im = = - A j ). (26)

Compared to the summation rule of the classically damped system, i.e.

nmod ,o[~class(o3) = ,~"~ r,,re, class/ti,/,/j [,O3)Tlrsj, class,)

j = l

.1_ .z/im, classz x class\~ t(nj ~w)r/rsj )1

hjClaSs, , = 1 . kw) fZj 2 - w 2 + i2~jf~jw'

(27)

The? ss = q~/rj II/sj,

(28)


nonclassical damping requires the same number of summation loops. The only difference is that the functions of the classical modal summation depend on the real modeshapes ~j, eigenfrequencies f~j and modal damping ratios ~j, whereas the nonclassical problem deals with the corresponding complex quantities.

To obtain the transfer matrix of the multi-support base excitation, the common procedure of splitting the equation of motion into support excited DOFs x and nonsupport DOFs u is applied

M C Cc Mc + Cu] { ti(t ) [M~ M u ] { ~(t) x(t)

E cl{x t, ) + K T Ku u(t) = f"(t) "

(29)

After transformation into the frequency domain, eqn (29) can be reordered so that the support excitation acts as additional equivalent forcing vector on the non- supported DOFs

[-Mw 2 + iwC + K]x(w) (30)

= f(t) + [,;2M c - iaJCc - Kc]u(~o)

feq(w) = U(~)u(a;) (31)

= [~2M C - iwC C - Kc]U(W ).

The advantage of this formulation is that in the frequency domain the derivatives with respect to time are expressed in powers of i~o. This allows one to avoid the definition of dynamic and pseudostatic parts of the displacements. Moreover, the damping coupling matrix Cc, neglected by formulating multi-support base excitation in the time domain, 11'12 is fully taken into account.

Correspondingly, the support excitation in eqn (22) can be expressed by equivalent forces. The DOFs of equivalent force applications are determined by the non- zero rows of the coupling matrices Mc, Cc and K c

X(W) = vf(w)f(w) + vf(w)feq(w). (32)

Combining eqns (32) and (31) with (22), the transfer matrix of support excitation can be easily obtained as a product of the force transfer matrix by the support transformation matrix U(~)

vU(~) = vf(o3)U(w). (33)

Using eqn (33) the covariance matrix of the response evaluates to the following expression

~ = + O O

E[x(t)x(t) *T] = vr(w)[Sff(w) .d = - - O O

+ *T +

+ Sfu (o2)U(w)*T]v f (w)*Tdw. (34)

The equation has the same structure as in the case of force excitation. The contributions of the support excitation are taken into account by transformation into spectra of equivalent forces that are added to form the total forcing spectra. Since the excitation data are generally incomplete, the excitations u(t) and f(t) are to be reconstructed by means of linear combination of available data U(t) and f(t), cf. 13

u(t) = Eufi(t); f(t) = Err(t). (35)

Analogously the response to ~(t) is generalised to any linear combination of the DOF-displacements x(t) by means of matrix R

~(t) = Rx(t). (36)

Therefore, the generalised formulation for the covariance matrix of the response for combined force and multi-support excitation is given by

E[x(t)x(t) *T ] R (Jf__=_+~ = vf(o3) [Sff(0d) (37) \

+ S~fU (~) -~- Sffuf(oj)]V f(03)*Td0)) R T ,

with

sff( ) = EoS

s fu( ) = U( ,)EuS u " ( )EuVt ) ; (38)

S~rf(~v) = U(w)EuS~? (a;)E T + x- - ,,T EfS~ (a))EuU(W)

where equivalent force spectra (38) are used to account exactly for multi-support excitation of nonclassically damped systems.

Remarks on the numerical efficiency of the procedure

The organisation of the equations in matrix form is significantly more efficient than the commonly applied plain evaluation of a quadruple sum 14 for computing the variances of the response. The size of the excitation spectra and the response vector is, in general, considerably smaller than the actual number of DOFs of the structure. Correspondingly, only those components are to be considered for assembling the transfer matrix vf(w) that are related to excitation or response DOFs. The matrices Eu, Ef and R as well as the support transformation matrix U(~o) are commonly sparse matrices. Using compact matrix algebra, the computation time for evaluating the response depends rather on the number of excitation and response DOFs than on the size of the structure. Compared to classical damping (without taking into account the expense for solving the eigenproblem) the additional numerical expense is negligible.

Numerical example for FE-structure

Let the substructure of interest be a five-walled


~Y t]OO

Fig. 1. Spatial distribution of variances of surface normal velocities for "rain-on-the-roof' excitation.

rectangle box of the size a × b × c = 0.08 x 0.2 × 0.45 m, h = 0 . 001m , which is connected to the supporting structure by means of a standard-linear-solid joint model, whereas the structural parts are considered to be undamped. The assembly may, e.g. represent an internal combustion engine's rocker cover attached to the cylinder head by means of a gasket. The FE modelling is restricted to the cover and the cylinder head, assuming that the effects of the cut-off motor block on the local modes of the cover are vanishing. Figure 1 shows the spatial distribution of the variances of surface normal velocities for spatially uncorrelated white noise excitation (Sff(x, ~) -- 1) at the cover ("rain- on- the-roof ' excitation). Figure 2 considers the effect of

the omitted parts of the structure by subjecting the base of the cylinder head to fully correlated white accelerations in the three spatial directions. To obtain the total response for arbitrary levels of excitation, the responses of the two cases have to be linearly combined by using the respective excitation ratios. The structure has 2700 DOFs, while the response was evaluated at 200 DOFs of the cover. Force excitation was applied to 200 cover nodes and 90 DOFs were base-excited. The frequency range for both examples covers the terzband from 425 to 531Hz. The computation time for evaluating the response to force excitation was about 10CPUmin, while the case of base excitation required 25 CPU min, both on a HP-workstation Apollo, series 700.

E-2

Fig. 2. Spatial distribution of variances of surface normal velocities for multi-support base excitation.

Combined integral and FE analysis o f broad-band random vibration in structural members 247

4 INTEGRAL DESCRIPTION OF A THIN- WALLED ELEMENT AT HIGH FREQUENCIES

Intregral description is especially effective for thin-walled structural members having a constant mass per unit area. A mean value of their vibration could be evaluated by means of the Bolotin method of integral estimates I that uses the asymptotic expressions for eigenfrequencies and normal modes. The absolute isolation of the substructures was mentioned to be the main shortcoming of this method. A generalisation is suggested 15 which requires no information about modes and takes into account the interaction of structural members. We consider a thin- walled box that may be interpreted, e.g. as an abstraction of a cover of an engine head, which is important for the prediction of the engine noise. Assume the cover is modelled by five walls, where all walls are plates of thickness h. In this case the box is a structural member having constant mass per unit area # = ph = const, where p is the mass density. Let u(r, t) and v~(r) denote the absolute displacement of the cover and its normal modes, respectively. Provided that the modes Vn(r) are normalised, the generalised coordinates q, (t) are given as

damping i , are given as 16

KvXn 1 -~ Clf~n(1 + ~2)j fin = + 2

t~alK ] 2 [ al K [1 + Clf~n( 1 +t~2) j + [-Clf~n--0 ~_ ~2) ]

(42)

a l K Zv 2

C1~2(1 + / ~ 2 )

in : ~ n "+-

Clf~n(1 +~2) +[Cla~i-~_~2~

(43)

Here C1 is an averaged longitudinal rigidity of the engineering structure and al = V/--~1/M1 is the group velocity in the structure. The damping factor and the eigenfrequencies of the isolated substructure are denoted by ~n and f~n, respectively. The contribution of the generalised parameters of the entire structure is positive valued, therefore the response of the coupled structure is reduced compared to the isolated case. Substituting eqn (40) into eqn (41) yields

1 ~ 1 oo oc cx~ t t I i(~o-w')t t (1~!2) = ~ E{02(t)} = ~ - A ~ L L l-oc [ Vn(r)'E{f(r'03)f(r"03')}'Vn(r')e~--- 0303

= = J-co (-03 2 + iin03 + (22)(_03t 2 _ iinO 3, + fi2) dAdA'd03dJ (44)

qn(t) : IA #Vn(r) 'u(r ' t )dA; JA #vk(r)*vn(r)dA = tSkn'

(39)

where A = bib2 + 2(bib3 + b2b3) is the area of the box faces. An averaged value of the square of the velocity may be expressed in the form

1 L E{fi2(r' t)}dA

1 } [ = ~-A~j~ I E { O j O n : .: JA #Vn(r)-vj(r)dA (40)

1 oo E{O2(t)}.

~ A n = l

The local principle, Section 2 of the present paper, allows one to consider any structural element alone. As shown in Ref. 15 the influence of the rest structure and the coupling of rigidity K is taken into account by the effective eigenfrequencies ~ , and the effective damping factors in, i.e.

f

qn(03) = JA f(r, 03).Vn(r)dZ

_032 + iin03 + ~ 2n ' (41)

where an effective eigenfrequency (~, and an effective

The engine head and its cover form an "acoustical chamber". Since there is a little attenuation due to damping and the reflection of propagating waves is strong, the wave is reflected many times before it is absorbed. It results in a highly reverberant field, which is more or less uniformly distributed throughout the cover. 16 Such a "rain-on-the-roof" loading (after Maidanik) is a typical spatial white noise, which can be expressed in the following form

E{f(r,w)f*(r ' ,w')} = Sp(03)t~(03 - 03t)t~(t - r ')I (45)

where Sp(03) is the spectral density of the acoustic wave pressure, I is a unit tensor and * denotes the complex conjugate of a quantity. Substitution of eqn (45) into eqn (44) yields

1 , ~ [oo 032Sp(03)d03 (46) (112) = , , 2 ~ Z...a I 032 iia03 + 02[ 2. - ~ , = l a - ~ [ - +

For a lightly damped box and wide-band excitation, the integral in eqn (46) can be evaluated,1 to give

71" ~ Sp(fin) (/~2) = __~A n=~ 1 ~ " (47)

where in = in((~n) is given parametrically by eqns (42) and (43). For the case of a high density of eigenfrequencies the sum in eqn (47) may be replaced by an


integral 1

7r f+~Sp(f~) ON (/j2) = ~---~J0 ~ ~ d a , (48)

where the density of eigenfrequencies dN/df~ is introduced. An asymptotic density of eigenfrequencies of the flexural vibrations of a plate bl x b E is known to be independent of the boundary conditions and it is equal to 1

dN blb2 ~ ~ - ~ , (49)

where D is the flexural rigidity of the plate. Non- degeneration of the dynamic fringe effect for plates admits an approximate matching of solutions for adjacent subregions. Consequently, the density of the cover eigenfrequencies is equal to the sum of the densities of these plates, i.e.

d N _ A p~ (50)

df~ 47r VD"

Inserting this result into eqn (48) we have

_ l sg )d (51) (92) 4ph p v / ~ j 0 ~f~) "

The spectral density of spatially averaged velocities of the cover is obtained from eqn (51)

S<,~> (w) = IH(w)12Sp(w)

in( )l 2 = 7r (52)

4phpx/'~(w ) '

Here, H(w) is the transfer function for the local vibration that depends on mechanical parameters of the substructure, i.e. h, D and p as well as the effective damping ~n. The latter is strongly influenced by the overall parameters of the whole structure, coupling rigidity and the local material damping in, cf. eqn (43). Note that this result does not contradict the assertion of

Section 2 that the overall absorption of the entire structure at high frequencies is actually independent of the local material damping in substructures, since the absorption in complex structures is mainly of resonant character, cf. eqn (21).

For numerical calculations the following parameters were taken: b 1 = 0.2m, b 2 = 0-45m, b E = 0"08m, h = 1 × 10-3m, u = 0 . 3 , E = 2.1 × 101]Nm-E,p = 7.8 × 103kgm -3, ~=0 .2 , K = 1.51 × 107Nm -1, CI = 2 . 5 x 109 N, M 1 = 160kgm -1 . The critical ratio of damping is chosen to be 5 x 10 -3 at all frequencies. Performing the computations, the lower bound of the high frequency approaches was shifted to the first eigenfrequency, whereas the FE-computation comprises the large frequency range of the lowest 100 modes. Comparing the results in the overlapping frequency domain allows to draw some conclusions about the accuracy and applicability of the discussed approaches. In particular, the consequences of setting the individual frequency limit O1 of the FE-computations can be pointed out.

Figure 3 shows the square of the absolute value of transfer function IH(w)l 2 of eqn (52) for the isolated cover and highlights the effect of the frequency averaging by integral description compared to FE- computation. Setting the frequency limit Ol at low frequencies results in the loss of individual peaks for w > O1, whereas the average response of any sufficient wide frequency band is obtained correctly. The significance of the mode peaks as well as the accuracy of the eigenfrequencies decrease with increasing frequency w. Therefore, the integral method reveals to be a powerful tool for vibration analysis, even at rather low frequencies. Please note, that for prediction of the engine born noise the computations are usually performed for the frequency range up to 3000 Hz, cf. Ref. 17.

The coupling of the cover to the entire structure is considered in Fig. 4. The upper smooth curve corresponds to the averaged value of IH(w)l 2 for the

isolated cover; FE analysis o isolated cover; Bolotin method

log IH(ra)l 2 -1.51

-2.0

-2.5

-3,0

-3.5

-4.0

-4.5

-5.0

0.0 I I

0.5 1.0

Fig. 3. Effect of frequency averaging of integral description.

I 1.5E3

frequency [Hz]


Finite Element analysis -..e- isolated cover; Bolotin method --e.- present approach

log IH(oj)I 2 -2.0

-2.5

-3.0

-3.5

-4.0

-4.5

-5.0

-5.5

-6.0 I I

0.0 0.5 1.0 1.5 2.0

I I 2.5

I 3.0E3

frequency [Hz]

Fig. 4. Combination of FE computation at low frequencies and integral descriptions at high frequencies.

absolutely isolated cover (4 = (, Bolotin's method). The lower one represents the integral description of the present study, i.e. it is calculated by means of eqn (52). We neglect the frequency drift, eqn (42), since it does not affect the asymptotic density of eigenfrequencies. The Bolotin method and the present approach are seen to be the upper and lower integral bounds for the actual vibration. In the Bolotin method any structural member is considered as absolutely isolated and hence the vibrational flow to the other structural members is prevented. Contrary to this an integral description of the present approach corresponds to a perfect bond between the cover and the rest structure. They are assumed to be fully coupled which results in overrating of the vibrational flow into the structure. The actual resonance curve is at some frequencies higher than the Bolotin curve. This fact does not contradict the assertion that the Bolotin method represents the upper bound, because it delivers a frequency-averaged estimate of the vibration field.

The FE-calculation presented in Fig. 4 was performed for the frequencies up to 1100Hz. The reason for this frequency truncation is the excessive computing cost and the observation that the individual resonances are hardly observable (compared to the resonance peaks at low frequencies). When limiting the FE-computations to a set of lowest modes, the frequency average of the modal response can be extrapolated asymptotically between the upper and lower bound given by the Bolotin method and the present approach. This procedure allows to estimate the response in the frequency domain w > O1 as well as taking into account the influence of the primary structure and the substructure coupling. The difference between the Bolotin method and the present approach becomes insignificant for high frequencies. The current example shows this behaviour in the domain larger than about 1500 Hz. Thus, for f > 1500 Hz the vibrations localise

within the cover and the cover may be considered separately.

5 CONCLUSIONS

Three methods: FE analysis of nonclassically damped systems, Bolotin's method of integral estimates and the theory of high-frequency vibration are combined in order to compute the broad-band random vibrations of structural members in complex systems. It allows one to overcome certain shortcominigs of these methods which are as follows. A fine FE mesh is required to model higher normal modes which causes extraordinary computing costs. Therefore, generally an intermediate frequency range exists between the upper bound of the FE-computations Ol and the lower theoretical bound of the high frequency approaches O2, where none of the approaches seems to be applicable. The application of the theory of high-frequency vibration results only in certain integral fields of vibration, however the structural members are not represented in this theory. Structural elements are considered in the method of integral estimates, but they are assumed to be absolutely isolated.

Based on the observation that the dynamic properties are different at low and high frequencies, the following strategy is proposed. The FE-computation is suggested to be performed in the low frequency domain 0 < w < 01 where the frequency Ol may be determined as an upper frequency of efficient numerical computation. For the higher frequencies, i.e. w > O1, the upper and lower integral bounds of the vibrations are calculated by means of the present approach. The frequency average of the numerical calculation may be extrapolated asymptotically between the upper and lower bound. This leads to an estimate of the substructure response, including the influence of the


entire complex structure. The result is an essential reduction of the computing cost without considerable loss of information.

Broad-band random vibrations of a thin-walled cover were calculated using the concept. Analysis of Fig. 4 allows us to make some conclusions about the localisation of vibrations also known as strong vibration localisation or normal mode localisation. 16 The differences of the curves corresponding to the isolated cover and the cover attached to the structure become insignificant for the frequencies f > 1500Hz. This means that the vibration localisation takes place at high frequencies and the tendency to localisation becomes stronger with the growth of frequency. This conclusion fully agrees with the central result of the papers. 4'18 Moreover, f o r f > 1500 Hz one may consider the cover separately and ignore the rest of the structure which considerably simplifies the computations.

A C K N O W L E D G E M E N T

The research was supported by the Austrian Industrial Research Promotion Fund under contract No. 6/546, which is gratefully acknowledged by the authors.

REFERENCES

1. Bolotin, V. V., Random Vibrations of Elastic Systems. Nijhoff, The Hague, 1984.

2. Belyaev, A. K. & Palmov, V. A., Integral theories of random vibration of complex structures. In Random Vibration - - Status and Recent Developments, ed. I. Elishakoff & R. Lyon. Elsevier, Amsterdam, 1986, pp. 19-38.

3. Belyaev, A. K., Vibrational state of complex mechanical structures under broad-band excitation. Int. J. of Solids and Structures, 27 (1991) 811-823.

4. Pierre, C., Weak and strong vibration localisation in disordered structures: a statistical investigation. J. Sound Vibration, 139 (1990) 111-132.

5. Li, D. & Benaroya, H., Dynamics of periodic and near- periodic structures. Appl. Mech. Rev., 45 (1992) 447-459.

6. Belyaev, A. K., On the application of the locality principle in structural dynamics. Acta Mechanica, 83 (1990) 213- 222.

7. Schober, U., Untersuchung der Krrperschalld/impfung durch Fiigestellen im Moter. Abschluflbericht: Gehiiuse- diimpfung, Vorhaben Nr. 391, Institut f~ir Technische Akustik, Technische Universit~it Berlin, 1989.

8. Chen, H. C., Partial eigensolution of damped structural systems by Arnoldi's method. J. Earthq. Engng. Struct. Dyn., 22 (1993) 63-74.

9. Lang, G. F., Demystifying complex modes. Sound & Vibr., January 1989, 36-40.

10. Fischer, P., Pradlwarter, H. J. & Priebsch, H. H., Prediction of noise emission of engines using random vibration analysis of non-classically damped structures. In Structural Safaety and Reliability, Proc. of ICOSSAR'93, ed. G. I. Schu~ller, M. Shinozuka & J. P. Yao. Balkema, Rotterdam, 1993, Vol. 1, pp. 239-242.

11. Berrah, M. & Kausel, M., Response spectrum analysis of structures subjected to spatially varying motions. J. Earthq. Engng. Struct. Dyn., 21 (1992) 461-470.

12. Der Kiureghian, A. & Neuenhofer, A., Response spectrum method for multi-support seismic excitations. J. Earthq. Engng. Struct. Dyn., 21 (1992) 713-740.

13. Zerva, A., Seismic ground motion simulations from a class of spatial variability models. J. Earthq. Engng. Struct. Dyn., 21 (1992) 351-361.

14. Harichandran, R. S., An efficient adaptive algorithm for large-scale random vibration analysis. J. Earthq. Engng. Struct. Dyn., 22 (1993) 151-165.

15. Belyaev, A. K. & Pradlwarter, H. J., Broad-band vibration of driven components of complex structures. In Structural Safety and Reliability, Proc. of ICOSSAR'93, ed. G. I. Schu~ller, M. Shinozuka & J. P. Yao. Balkema, Rotter- dam, 1993, Vol. 1, pp. 85-91.

16. Hodges, C. H. & Woodhouse, J., Theories of noise and vibration transmission in complex structures. Reports in Progress in Physics, 49 (1986) 107-170.

17. Herster, P, Gschweitl, E. & Rainer, G. Ph., Use of air borne noise calculation to develop low noise engines. In Structural Dynamics: Recent Advances, ed. N. S. Ferguson, H. W. Wolfe & C. Mei. University of Southampton, England, 1994, Vol. 2, pp. 1033-1044.

18. Cha, P. D. & Pierre, C., Vibration localisation by disorder in assemblies of monocoupled, multimode component systems. ASME J. Applied Mechanics, 58 (1991) 1072- 1081.

combined integral and fe analysis of broad-band random vibration in structural members

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