random vibrations of the frame foundation

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Page 1: Random vibrations of the frame foundation

Section 1-7 S273

BRYJA, D. , MIRONOWICZ, W.

Random vibrations of the frame foundation

The foundation for the machine in the form of the flat Qame made of thick bars is examined in the paper. The Qame is modelled by FEM with the application of the original Timoshenko beam element. The machine is treated as the nondeformable block viscoelastically supported on the pame. The dynamic load is random and has the form of the periodic series of short duration shocks. The solution of the problem is presented in the correlation theory sphere, using the influence response function of the multi-degree-of-fleedom qstem.

1. Statement and solution of the problem

The foundation has the form of the flat frame made of thick bars. The frame is modelled by FEM with the application of the Timoshenko beam element formulated by Langer, Bryja [l]. The machine is treated as the nondeformable block, viscoelastically supported on the frame, as it is shown in Fig. 1 (the shape and dimensions of the fiame and the block as well

as the block and force P(t) localization, are given exemplary, as assumed in the quoted numerical example). The random load P(t), caused by the machine, has the form of the stream of short-duration shocks with the time of duration Tin the period d, where T I A , [2]. It can be described by the discontinuous stochastic process

4.0

0 for t <t i and t >ti +T Y (1) I S( t - t i ) for ti I t I t i + T

A&(t -ti) , s, =

4,8 where ti = id , vd 5 t < (v + 1)d. Ai are the random variables and S(t - t,) is the random h c t i o n designating a single load portion.

The assumed Timoshenko beam finite element (see Fig. 2a) is defined by seven local coordinates gathered in the vector

(2) The shear slope qj = KTIGA = const is regarded to be constant along

the element length and independent of the deflection w, so p i = w,! - 8- pi = W) - 8ii . Displacements of the element approximated by Hermite's polynomials have the form w(x) = (pi + 8,)lHl + ( p j + Jo)lH2 + wiH3 + wj H4 ,

p(x) = (qi + &$)Hi + (qj + LJij)H; - Sg + wil-'H; + wjl-'H; and u(x) = uiH, + u j H, . The potential and kinetic energies are given by the relations

T qe = [Pi 3 Pj 9 gij 9 wi 3 wj ,ui > uj 1 *

rl'

Fig. 1

I * 1 2 O 2

1 2 = -q;fK,q, , Ek = + ti2)& +- Prng2& = -q;fB,qe, (3)

where m denotes mass of the bar per unit length and J, is the axial moment of inertia of the mass m. The matrix equation of motion of the system has the form

Bg + Cg + Kg = FP(t) , (4) \N where g = [q,bIT is the vector of

generalized coordinates, q concerns w''dw'dx the frame and b =[bl,b2,b3IT

MIXI - - N c

0)

ui 0 EJ, GA, EA, @ uj x

dij y\ I.;

I iz

Fig. 2

defines the motion of the block. The matrix coefficients of equation (4) can be obtained after formulas

e e i e i

Page 2: Random vibrations of the frame foundation

S 274 ZAMM Z. Annew. Math. Mech. 80 (2000) S2

where p, x are nonnegative constants, A , , A, , A, , A, - transformation matrices, C, - damping matrix of the constraints,

K, - stiffness matrix of elastic constraints, B,, B, - the inertia matrices of the block and the frame’s rigid node j ,

respectively. Expressions (4) have only a symbolic meaning because matrices A transform local coordinates into corresponding subsets of the generalized coordinates g. Consequently the summation should be done by decomposition.

Let the symbol w(t) denote the structure response which has the form of any frame or block displacement. Such response may be expressed by the general relationship w(t) = uTq(t) in which u is the time-invariant transformation vector. The response of the structure loaded by the random process P(t) may be obtained by summing up the solutions due to the differential impulses P( zjdz The influence response function H , (t - z) = uTq(t - z) is determined by the numerical integration of the equation of motion, as it is formulated in [2]. Coordinates q(t - z) satisfy the equation of motion in which the excitation force has the form of the delta Dirac function P(t) = l.&t - z) . The structure response for zero initial

v-1 tl+T t

conditions takes the form w(t) = vd s t < vd + T and A, I H , (t - z)S(z - t , ) d z + A, I H , (t - z)S(z - t v ) d z for ti I” ,=o

v t l+T

w ( f ) = CAI IH,(t - r)S(z - t,)dz for vd + T I t c (v + l)d . The expected value and covariance of the structure

response may be determined by the assumption that S(t) is the sum of expected value &t) = E[S(t)] and the random

fluctuation g(t) . For example, for vd + T I t c (v + l)A , one obtains E[w(t)] = g E [ A , I IZY,(t - T)&Z - t,)d z ,

c o v w ( t l , t 2 ) = ~ ~ E [ A , A J l I jH12Kssdqdz2 ,where HI2 =Hw(tl -z1)Hw(t2 -z2), Kss = E[$(z, -t,)$(z2 - I , ) ] . The

response process gets cyclostationary (with the cycle A) when E[A,]=E[A], E[A, A, ] =E[A2] and K,(tl , t 2 ) = Kss(t2 - t , ) .

4 r=O

t i +T

ti r=O

I , +T +T v1 vz

ti 1) 1=0 J = o

2. Numerical results and conclusions

It is assumed: m1 = 2457.6 kg/m, m2 = 1474.6 kgm, E = 2.7 - 10” N I m 2 , v = 0.2, K = 1.2, mb = 18000 kg (mass of the

block), k = 2.0.106 Nlm and c = 8.0 .lo3 Nslm (stiffkess and damping of constraints), f i = 6.656 Hz , f 2 = 40.380 Hz (first and second natural frequencies of the frame), 2a1 = 2n;d; + p I 2 4 , 2a2 = 2n;dj + pf 2$2 , al = a2 = 0.025 , i ( t - t i ) = l ,K, =1-S( t i - t j+r2 -zl) -whitenoise, E[A]=2.0.104 N,E[A2]=4.05.108 N 2 , d = 0 . 6 s , T=OSA.

t 6) 0,O 0,6 1,2 1,8 2,4 3,O 3,6 4,2 4,8 5,4 6,O 6,6 7,2 7,8 8,4 9,O 9,6

1,8E-04 1,5E-04

n E 1,2E-04

9,OE-05 v

6,OE-05 3,0E-05

O,OE+OO

Fig. 3

The standard deviation of the vertical displacement of the point L (see Fig.]) is presented in Fig.3.

The proposed Timoshenko FEM model may be used for different bar structures in solving static and dynamic problems. It is described by relatively little number of coordinates, so it is efficient in analysis of stochastic problems.

- 3. References

1. LANGER, J., BRYJA, D.: Timoshenko beam element in FEM analysis of bar systems; Proceedings of XI Polish Conf. on Computer Methods in Mechanics, Kielce-Cedzyna (1 993), 493-500.

2. BRYJA, D., CHROBOK, R., MIRONOWICZ, W : Numerical study of randomly excited response of non-classically damped systems; Proceedings of 2nd European Conference EURODY”93, Trondheim (1 993), 761-767.

Address: DR DANUTA BRYJA, DR WLADYSLAW MIRONOWICZ - Institute of Civil Engineering, Wroclaw University of Technology, 50-370 Wroclaw, Wyb. Wyspianskiego 27, Poland.