Non-stationary response of a beam to a moving random force

Journal of Sound and Vibration (1976) 46(3), 323-338

NON-STATIONARY

RESPONSE OF A BEAM TO A MOVING RANDOM FORCE L.

FR+BA

Railway Research Institute, Prague, Czechoslovakia (Received 25 September 1975) The non-stationary random vibration of a beam is investigated. The beam is subjected to a random force with constant mean value which is moving with constant speed along the beam. The statistical characteristics of the first and second order for the deflection and bending moment of the beam are computed by using the correlation method. The numerical results of the coefficient of variation of the deflection at beam span mid-point are given for five basic types of convariances of the force (white noise, constant, exponential cosine, exponential, and cosine wave). The effect of the speed of the movement of the force along the beam as well as the effect of the beam damping is investigated in detail. It is concluded that the resulting beam vibration turns out to be a non-stationary process even though the motion considered is that of a stationary random force. 1. INTRODUCTION Some engineering

structures

are subjected

to moving loads and in many cases the loads obey

laws that are of random character. In moving loads and structures serving in transport, it is particularly the effects of track irregularities, of random motion of vehicles, of composition of trains or traffic flow, etc., that sometimes have a very appreciable random character. Generally, such a moving load is a random function not only of time but also of space co-ordinates. Random, predominantly stationary vibrations of mechanical systems have been studied by Bolotin [l], Crandall [2] and Robson [3], whereas problems involving the effects of moving random loads have not yet been investigated so far. Only Bolotin ([l], section 54) and Robson ([3], section 5.6) have given rough outlines of a procedure for solving the effect of a moving continuous load on a beam. The effect of a moving random force was first tackled by the author [4] and the statistical approach was applied by Zilch and Weissgerber [5] to the vibration of bridges. Here reference [4] and section 26.2 of reference [6] again are assumed to present the solution of the vibration of a beam along which a concentrated force, randomly variable in time, moves with a constant speed. The problem is investigated for several basic types of covariance of the moving force and the effect of its speed and of the beam damping is also taken into account. The problem just mentioned may be applied to the vibration of highway and railway bridges where it represents the basic idealization both for the deterministic and stochastic case (see reference [6]). 2. BASIC THEORY

2.1.

NON-STATIONARY

The

following

RANDOM PROCESS-NOTATION

notation

that is non-stationary

AND NOTATION

is used for the random

function

f(x, t) (or g(x, t), respectively)

in time t and in the space co-ordinate x : 323

L. FRLBA

324

mean value (E is the linear operator of the mean value)

t)lT

E[f(x,

(1)

centred value Sk

f)l,

(2)

= C&,x,t,t),

(3)

t) =.0x, t) - Hf(x,

variance c&J)

=E[&x,t)]

standard deviation (4)

0,(x, t), coefficient of variation

t)l,

V,(x, t) = a&, t)lHf(x,

(5)

covariance Cff(G%,

t1, tz)

=

ah,,

tl)

p

(x,,

&)I,

(6)

cross-covariance C_fe(.%X2,tl,tZ) =

G-~xIJlMwZ)1~

(7)

power spectral density and its relation to the covariance S&w,, wz) = J’ 1 Cff(tl,

tz) e-i(wzz~-o~r~)dtl dt,,

-m -cc

ss m

cfr(tl9 22)=

&

m

Srf(wl,

c~~)e~(“+‘z-~l’~)dw~ do,.

(9)

--m-m If the functionf(t)

is a stationary random process in t then the following relations are valid: Cff(t1 ,tz) = C&z

sr&%

(10)

- r1) = CAr),

(11)

%) = 2~~,(~,) 6(% - 01) = 27cSf(%) b(w, - c-G,

where 6(x) is the Dirac delta function and z = t2 - t,. The Wiener-Khinchine relations for the spectral density and covariance of a stationary random functionf(t) are Sf(o) =

7 Cf(r)e-iordr, -m

(12)

m

C,(r) = &

S,(w) eiro do.

(13)

s -m 2.2.

RANDOM

VIBRATION

OF A BEAM

Vibration of a Bernoulli-Euler equation :

beam with viscous damping is described by the following

L[u(x, t)) = EJ a4 u(x, t)/i?x” + p 8 u(x, t)/iB 2 + 2pcob h(x,

t)/iB = Ax,

t >,

(14)

where L is a linear operator, u(x, t) is the vertical deflection of the beam at point x and time t, EJ is the constant bending stiffness of the beam, /* is the constant mass per unit length of the beam, ob is the coefficient of the viscous damping of the beam, and the transient load per unit length on the beam, p(x, t), is given by p(x,t)=(P(:‘))

for

t(:J.

BEAM UNDER MOVING RANDOM FORCE

325

Equation (14) is conveniently solved by the normal-mode analysis :

P(X,t) = 5 w,(x) Q,(r),

(17)

J-1

where O,(X) are the normal modes of free vibration depending on boundary conditions, qj(t) is the generalized deflection obtained with respect to the initial conditions from the equation d%,(O/dt2 + 2a&,(Oldt

+ c&,(t)

=

Q,(t),

(18)

and the generalized force Q,(t) is

(19)

The following notation is further introduced : I = length (span) of the beam, o, = 27rfi = circular natural frequency of the undamped beam, 07 = e.$ - e.$ = circular natural frequency of the damped beam, 1

IPJ(X)U~(X)~= [7)

for

(;;E),

(20)

0

(1lo;) e-“b’sin 0; t ) ()

for *[:i). (21) ( The impulse function h,(t), defined in equation (21), represents a response of the system (18) to an impulse 6(t) for zero initial conditions. Then, the solution of equation (18) may be written in the following form (by using equations (21) and (19), respectively) : MO =

q,(t) = j h,(t - 3 Q,W dr = 1 h,W Q,O- 4 dz 0

0 =

r

h,(7)

-m

Q,(t - z) dr = r h,(’ . -02

7)

Q,(z) dr.

(22)

Let the external load p(x, t) be a non-stationary process with mean (deterministic) value E[p(x, t)] and with a centred (random) value j(x, t) : Pee f> = E[Pb,

01 +dk 0.

(23)

The response of the linear equation (14) is assumed in a similar form : u(x, t) = E[u(x, t)] + 8(x, t).

(24)

Putting expressions (23) and (24) in equation (14) gives L{E[u(x,

01+ w, a=

E[Pk

r)l +iG, 2).

(25)

Upon taking into account the linear operators L and E equation (25) may be rewritten as two equations : L{E Mx,

011= E[P& 01,

LP(x, a= a, t).

(26) (27)

326

L. FR+RA

The first, equation (26), is valid for mean deterministic values of random functions 15(x,t ). p(x, I ), qj(t) and Qj( f ) while the second, equation (27 J, is valid for their centred random components. It is pre-assumed that the solution of the deterministic case (76) can easily be calculated. Thus, the statistical characteristics of the first order (mean values) ma! be obtained from equation (26) and the statistical characteristics of the second order are described by the covariances. By applying the definition (6) the covariances of the random functions can be calculated. The covariance of the generalized deflection is obtained from equations (6) and (22): CYlq*(f*, f2) =

The covariance

W,(~I)4J~Jl

of the deflection

is, from equations

(6) and (161,

(29) The covariance

(6) and (I 7),

of the load is, from equations

2 pVj(X,) d,(tl)

C~~(XI*XZ, f~,r~) = E

=

Finally,

the covariance

5 k=l

pL uj(xl)

of the generalized

Mk

of the deflection =

pL’k(xZ)

dk(f2)

uk(x.2)

I

Ca,ak(t,,

(30)

fz).

force is, from equations

(6) and ( 19),

I uj(X1)

Mj

d(ht)

1

1

ZY-

The variance

$ j-1

y k:l

[ j=l

uk(x2)

CppCxl,

XZ,

t19

TV) dx,

(31)

dx,.

ss

00

may be calculated

C”“(&x,t,t)

=

from equations

i

-?

j=l

kyl

L’J(X)Uk(X)C4j4k(t,~).

(3) and (29)

:

(32)

As the cross-covariances C4,ak(t,, f2) are for j # k sufficiently small in comparison to the components C4,4,(fl,t2) for j= k (for a proof, see reference [l]), they may be neglected, CsJsk(tI, tJ = 0, and equation (32) is then simplified to the form a:(xY r) = $

U:(X) C4j4jt1, f )’

j=l

For the bending

moment

(33)

of the beam, M(x, t) = -EJ~Zv(x,t)/i3x2,

(34)

BEAMUNDER MOVING RANDOM the

321

FORCE

covariance is obtained in a similar way as for the deflection : G&l9

x2, I,, a = 3

2 wy

qxd

4(x*) C4,4&,

tz).

(35)

J=lk=l

The dashes denote the derivatives with respect to x.

3. MOTION OF A RANDOM FORCE ALONG A BEAM Assume that one has a concentrated force, P(t), randomly variable in time, which is moving with a constant speed c along a beam from its left to its right-hand side (see Figure 1). Therefore, the load p(x, t) in equation (14) is p(x, t) =6(x - ct)P(t).

(36)

The force P(t) has a constant mean value P and a centred random value p(t) : P(t) = P + P(t), E[P(t)]

(37)

= P.

(38)

The covariance of the force is CPP(f1,tz) =

mw-T~,)l.

(39)

Upon putting expressions (37) and (38) into equation (36) the load p(x, t) resolves into the mean and centred values, respectively : E[p(x, t)] = 6(x - ct) P,

(W

i(x, t) = 6(x - ct)~(t).

(41) By using the definition (6) the covariance of the load is calculated from expressions (41) and (39) : c&l,XZ,fl,~Z)

=a% = &b

- cwyt&(x2 - Cfl)@,

- ctJP(t,)]

- %)

(42)

CPP(fl,f2).

The procedure prescribed by expressions (28) and (3 1) gives the covariance of the generalized force, 1I CQ,&l9

b) = L MjMk

uj(xl) ok(x2)

&6

-

Ch)

6(x,

-

~4)

Cdl,

22) dx,

dx,

ss 00

=

(l/W,

Mk))

~&I)

Uk('%)~I'P(h,

tz),

and of the generalized deflection, m m

Figure 1. The motion of a random force P(t) along the beam.

(43)

328

L. FR$RA

while the covariance of the deflection and its variance remain the same as in equations (29) and (32), respectively. In computing expressions (43) and (44) the properties of the Dirac delta function were used (see reference [6]). With respect to equations (26) and (40), the mean value of the deflection of the beam may be calculated as a response of the beam to a moving constant single force P. The solution may be written in the following form (see reference [6], equation (1.24)):

2 j=l

E[u(x, t)] = u. g

-

1

1

jaU2(j2 - a2) - 2B21 (j"

Here the following

j2

j’(

a’)

_

sin

j'[j'( j'- a2)2+ 4a2 j’] ’

_

e-W

sin wi

I-

!+Y!_

2ja~(Cos~_

e-mbr

cos

0;

f)

j32)l/z

notation

sin?.

1

has been introduced: v0 = P13/(48EJ)

is the deflection

(45)

(46)

at the middle of the beam span under a load P at the same point, a = cl(2f, I)

is the dimensionless

speed parameter,

(47)

and jI = e&/WI = 9/( 277)

(48)

is the dimensionless damping parameter (9 being the logarithmic decrement of damping). The calculation of the mean value and of the covariance of the deflection and of the bending moment gives all the statistical characteristics of the first and second order. One of the most important characteristics among them is the variance and/or the standard deviation. For their calculation the approximate relation (33) is used and C4,4,(f, t) will have, with respect to the intervals in expressions (15), (18) and (21) and with respect to the equations (22), (28) and (31), the following form: t f cl&t) The normal

= &

SI ’ 00

hj(t - T,)hj(l-

modes of a freely supported

72) U,(c71) Uj(C7z)Cpp(71,72) d7, d72. beam are

nj(x) = sin (j7cx/Z), so that the variance

(33) is, in accordance

X sin 7

sin 7

with equation

(50) (21),

CPP(7r, r2) dr, dr,.

The following dimensionless independent variables and parameters numerical calculations, together with the notation (47) and (48) : z = et/l,

(49)

x1 = cz,/l,

x2 = c72/,,

(51) are introduced

for the

(52)

a = (rr/a) (j” - /-12)‘j2,

(53)

b=jrt,

(54)

d = (n/a) /I.

(55)

329

BEAM UNDER MOVING RANDOM FORCE

After putting expressions (55) to (58) into equation (51) the variance may be written as 0,2(x,2) =

~[~l’jje

-d(z-Xl)sin a(z - x1) x

x e-r(*-x2) sin a(z - x2) sin bx, sin bxz C,,

dx, dx,.

(56)

The numerical results will be arranged in the form of a ratio of the standard deviation (equation (4)) of the deflection at mid-span point of the beam to the value u,, (equation (46)) and this ratio will be expressed as a product of the coefficient of variation VP of the force P(t) and of the function v(z): V”(Z)= O”(V2,WIJ = VPY(Z). (57) Cp(T)

S,(w)

C,(T)

.sil(w)

Ji_

:-..j

Cp(T)

S,(w)

+&ge) A-p!?!. Figure 2. Basic types of covariances CAT) and corresponding power spectral densities S&J) of the force P(t): (a) white noise, (b) constant covariance, (c) exponential cosine covariance, (d) exponential covariance, (e) cosine wave covariance.

L. FR+BA

330

Therefore, V,,(z) is a certain coefficient of variation of the deflection at the beam span midpoint and it has a similar form as the impact factor (dynamic coefficient) in the deterministic approach of the present problem. Both the mean value (45) and the variance (5 1) of the deflection of the beam are functions of the co-ordinate s and of the time f and, therefore. the response of the beam is nonstationary for any (stationary or non-stationary) random behaviour of the moving force P(r) because the response of a finite beam to a moving force has always been a transient process. In what follows the function v(z), dependent on time t or z (see equation (52)), is calculated for several types of covariances of the force P(t) to obtain the coefficient of variation of the deflection of the beam (equation (57)). The force P(t) is assumed to be a stationary random function of t and the following basic types of covariances (equation (12)) or power spectral densities (equation (I 3)) are taken into account (see Figure 2) : (a) white noise CP(T) = sp 6(r), S,(w) = s,; (58) (b) constant

covariance C,(r) = up’,

(c) exponential

Sp(w) = 2r&6(0);

(59)

cosine C,(z) = a: e- WZ’cos 00 z, Sp(w) = a;o,D/{w29 + (w + oO)z) + l/+0,2 + (0 - cfQ)J;

(d) exponential,

(60)

o. = 0 CP(z) = a: e-“g’r’,

Sp(0) = a; 2W,/(02 + 0;) ;

(61)

SP(W) = rNJ;[G(w - wg) + S(0 + oo)].

(62)

(e) cosine wave, 0, = 0 C,(T) = a; cos 00 z, 3.1.

WHITE

NOISE

The constant spectral density SP with the dimension covariance in accordance with expression (58) :

[force2 x time] corresponds

to the

CPP(fl, tJ = S,&f2 - fl), CPP(lXl/C, I&/C) = (c/l) S,S(x, - x,). After inserting the last expression in equation (56) and after tedious integrations and rearrangements the coefficient of variation (57) appears with the following expressions for VP and y(z) : VP = (spw1p2/P, (63) ,I1 dz)

=

g6vj(x)

2

714(j4

[ u~~416pl)l’2(~a~~b+d2x _

B’)

x [sin 2bz + eezdzsin 2uz + (&a

+ b)} (cos 262 - e-zd~cos 2uz)] +

+ {(a - b)/[(u - 6)2 + dZJ} [-sin 262 + e-2dzsin 2uz -t + {d/(a - 6)}( cos 262 - eezdz cos2az)]

- {2d/(b2 + d2)} [(b/d) sin 262 t

+ cos 262 - e-2dz] - {2d/(u* + d’)} [I - e-2dr{cos 2az Ii2 -

(a/d) sin 2az}] + (2/d) (1 - e-Mz>

.

(64)

BEAM UNDER MOVING RANDOM I -

,

I

!

I

,

I

(a)

I

,

I

0=0.25

331

FORCE /

I

(

I

I

I

I

I

I

/

1

I

(

I

I

(b)

I.5 (r=o.5 o’il

I 0

,

I

,

I

0.5

I

I

/

I

0

0.5

I

.? = Cl/i

Figure 3. Ekhaviourof the function v(z) in time for various speed parameters a, (a) /I = 0, white noise; (b) B = O-1, white noise.

Here and in the following in accordance with equations (50) and (57), use has been made of the expressions uJ(x) = u,(1/2) = sinjx/2.

(65)

Equation (64), or a direct integration, gives several special cases : (a) Quasi-static, a = 0, /? > 0 The limiting case when the random force is moving with zero speed along the beam is analyzed in detail in section 4. Here only the result is quoted (x0 is the distance of the instantaneous action of the force measured from the left-hand side of the beam) : sin bx,,/l.

(66)

The coefficient of variation has infinite values for /I = 0 and a = 0. (b) Undamped beam, fl = 0, a > 0, a # b

8jh(ay_ b2) [sin 2az - (a/b)' sin 2bz +(2a/b2)(a2 - b2) z] ] li2.

(67)

(c) Undamped beam, fi = 0, a = j, a = b

y(z)=2.396U,(X)(taz + *az cos 2az ,=I

J 71

&sin 2az)“‘.

The expressions (64) to (68) were calculated for j = 1 and for the following values of the parameters : z=O(O*l) l;a=O(O-1)

l;/?=O,O-05,O.1.

(69)

The time behaviour of the functions y(z) during the motion of the force along the beam is drawn in Figures 3(a) and 3(b) for various dimensionless values of the speed, a, and of the damping, /?. The maximum values, max y(z), of these functions are summarized in Figure 6(a) of section 3.3 and here they are represented as functions of the speed parameter a for various damping parameters /I.

332

L. FR+BA

3.2. CONSTANT COVARIANCE In this case, the coefficient

of variation

(57) is calculated

from equations

VP = fJPlP> Y(Z) =

c

(70)

a2

m 96 7

(56) and (59):

uj(x) (j4 _ jjz) [(a2 + d2 _ /jZ)Z + 4d2 b2]'

j=l

x {(a2 + d2 - b2)sinbz

- 2dbcosbz

+ emd’[(b/a)(d2 - a2 + b2)sinaz + 2dbcosazJ]. (71)

Special cases : (a) Quasi-static, c1= 0, /? arbitrary (72) (b) Undamped beam, /? = 0, CI> 0, a # b sinbz-

b -smaz a

.

(73)

(c) Undamped beam, /? = 0, a = j, a = b y(z)

=S”

I=1 j4n4

Vj(X) (sin az - azcos az).

(74)

The expressions (71) to (74) are illustrated in Figures 4(a) and (b) and the maxima, max y(z), are summarized in Figure 6(b) of section 3.3 as functions of a and /I.

Figure 4. Behaviour of the function y(z) in time for various speed parameters a, (a) b = 0, constant covariante; (b) /3= 0.1, constant covariance.

3.3. EXPONENTIAL COSINE Additional dimensionless of exponential cosine :

parameters

are introduced

for the covariance

(60) in the form

g = % UC,

(75)

y = 00 l/c.

(76)

BEAM UNDER MOVING RANDOM FORCE

333

The coefficient of variation (57) is obtained from expression (56) with equations (60), (53) to (55), (75) and (76) substituted therein: VP = CPIP,

1 1, l/Z

+ j f(z,

x1,

x2) e-g(x2-xl)dxz dx,

Xl where

(77)

f(z, x1, x2) = e -d(Z--xl)sin a(z - x1) e- d(r--x2)sin a(z - x2) sin bx, sin bx, cos y(xZ - x1). Equation (77) has been programmed for an EC 1030 computer and calculated for j = 1 and for the following values of the parameters : a = 0*05,0*1 (O-1) 1; z = 0 (0.05) 1; /I = 0,0~05,0*1;

y = 0, n/(2a), K/a.

g = 0,O.l;

(78)

If y = n/a, the frequencies w0 and o1 are equal, c_+,= wl, because y = n/a = o1 l/c = wa l/c. For y = 0 the covariance is the exponential function (61) while for g = 0 it is the cosine wave (62). If y = 0 and g = 0 the constant covariance (59) results. Some samples of calculations of y(z) are represented in Figures 5(a) to (j) for a = 0,0*25, 0*5,1;~=0,0~1,g=0,0~1;y=0,n/(2a),1r/a. The maxima of y(z) are summarized in Figures 6(a) to (g) for all calculated cases. 4. QUASI-STATIC CASE A special case is the motion of a random force P(t) with an infinitely low speed along the beam. Thus, the force remains for infinitely long time at the distance x,, from the left-hand side of the beam; however, x0 is assumed to be a variable parameter. The load of the beam is in this case p(x, t) = 6(x - xg) P(t). (79) The generalized force (19) corresponding to the load (79) is

and the generalized deflection (22) becomes

1

q,(t) = (l/M,) u,(xo) h,(t - 2)P(r) dr.

(81)

0

Equation (81) gives the deterministic solution (16) of the deflection of the beam. With respect to the definition (6) the covariance of the load (79) is given by &(x1, x2, t,, tz) = 6(x, - x0) 6(x, - x0) CPdfl,

t*),

(82)

the covariance of the generalized force (31) by

ss 1 1

CQ,Qk(fl, tz) = =

-IM,"ko (l/CM,

~_dxl)

uk (x2)

&xl

-

x0)

%x2

-

xO)CPP(h,

h)

h

h

o

Mk))

u,(xO)

xk 60)

CPP(ti

3 h),

(83)

334

3

0.5

Figure 5. Behaviour of the function y(z) in time for various speed parameters a, (a) /I = 0, exponential covariance,g = 0.1, y = 0; (b) /3 = 0.1, exponential covariance,g = 0.1, y = 0; (c)b = 0, cosine wave covariance, g = 0, y = a/(2a); (d) B = 0.1, cosine wave covariance, g = 0, y = rc/(2a); (e) /I = 0, cosine wave covariance, g = 0, y = n/a; (f) /I = 0.1, cosine wave covariance, g = 0, y = n/a; (g) B = 0, exponential cosine covariance, g = 0.1, y = n/(2a); (h) jI = 0.1, exponential cosine covariance, g = 0.1, y = n/(2a); (i) B = 0, exponential cosine covariance, g = 0.1, y = n/a; (j) b = 0.1, exponential cosine covariance, g = 0.1, y = n/a.

335

BEAM UNDER MOVING RANDOM FORCE 3c’,,,,,,,,,

,

,j

2.5

0.5

m

-

p=0 p = 0.05 p = 04

p=0 p = 0.05 p=o4

Figure 6. Maxima of y(z) as functions of speed parameter a for various damping /?. (a) White noise; (b) constant covariance; (c)exponential covariance; (d) cosine wave covariance, g = 0, y = n/(2$; (e) cosine wave covariance, g= 0, y= z/a; (f) exponential cosine covariance, g=O*l, y= n/(2a); (g) exponential cosine covariance, g = 0.1, y = z/u.

336

L. FRhA

and the covariance

of the generalized

(28) by

21 12 L.j(sO)L’k(xCl) hj(f, - rl)h,(t, IS k 0 0

CQ,+(tI, t2) = & .I The variance

deflection

of the deflection

(33) is approximated

by

a:(~, t) = 2 & L’:(X)~J;(x,) 1 fhj(r - 71) h,(t - r,)C,,(rr, j=1 J 0 0 As the force P(t) is assumed (IO), that

to be a stationary

(84)

-7r)C,,(s,,7,)dT,,drz.

function

CPP(fl, f2) = C&f?

72) dr,dr,.

it follows, according

(85) to equation

- [I).

(86)

Furthermore, the force P(t) acts at every point x0 for an infinitely long time and, therefore, it is convenient to introduce new independent variables in expression (85) : $1 = t - T1,

s2 = I -

dr, = -ds,, Upon inserting

72 -

71 =

f -s,

-

t +

s1 =

s,

-

s2.

dr, = -ds,.

expressions

c

z

j=l

(87)

(86) and (87), the variance

m 1

0,2(x,a) =

72,

(85) becomes,

for t +

W,

rr

u;(X)$(X0)

hj(s,) hj(S2)CPdSl - s2)dsl ds2

’

0

0

Equation (88) has been evaluated for the covariance of the force P(t) in the form of white noise and of a constant and these results have been given previously in equations (66) and (72). 4.1. EXPONENTIALCOSINE If the covariance of the force P(t) has the form of an exponential is calculated from expression (88) as cc

r$(x, W) =

cosine (60) the variance

c

j=l

1 ~ v?(x) $(x0) a; 4MjzWjz J

+ 46, bz co;]

+ (l/D,)

[b: + oy - w: - (bz/wb) (6: + oi2 + co;)] , I

(89)

with the notation b, = co, - cob,

Di = (bf + w’f - w;)’ + 46: wo’,

62 = -(CO, + (+,),

i= 1, 2.

4.2. EXPONENTIAL The covariance of P(t) in the form of an exponential function for VP = up/P and j = 1, the following expression for y(z) :

Y(Xo)=

I

2 %(X0)

1

i *cl

I - 2flW,/Wl + w;/w: 1

+

1 + 2~0,/0,

i %+2 + w;/w; ’ j3Wl

)I

--2+ /?w,

(61) gives, in equation

(57)

4 1 - 2fiw,/w 1+ w’g/w: I

+

112

.

(90)

331

BEAM UNDER MOVING RANDOM FORCE

4.3.

COSINE WAVE

If the covariance ofP(t) possesses the form (62) of a cosine wave the coefficient of variation (57) is calculated as y(xo) = (96/n4) ur(xo) [l/{(l - c&B:)~ + 4~20~/o:}]“2.

(91)

With the exception of the case o0 = ol, expression (91) has a finite value even for /I = 0: Y(XO)= (96/x4) r&,)/(1

- w@:).

(92)

The results of calculations according to expressions (89) to (92) are also represented in Figures 5 and 6. It is necessary to assume the following numeric parameters in the quasistatic cases in the limit for 01= 0 : g=O*l,

o,/ol=

wolw1= l/2,

O-1 for a/n + 0, y= n/a,

’

Y= nl(2a),

WO/Wl= 1.

5. CONCLUSIONS

The variance of the deflection at the mid-span point of the beam is variable during the passage of a random force along the beam. Its maximum is reached approximately at the moment when the force crosses the middle of the span, or a little later for the case of low speeds. At higher speeds of the force the maximum variance appears at the moment when the force comes near to the right-hand side of the beam (see Figures 3-5). This event is quite analogous to the classic dynamic effects of vehicles on bridges (see reference [6]). The variance was expressed in the form of the coefficient of variation (57) which corresponds to the dynamical coefficient (impact factor) of bridges in their deterministic investigation. Five basic types of covariances of the moving force were investigated : white noise, constant, exponential cosine, exponential, and cosine wave (Figure 2). The speed of the movement of the force on the beam has the following effect on the maximum of the variance of the beam vibration: the variance diminishes with increasing speed for white noise (Figure 6(a)). For constant covariance of the force, the maximum variance of the beam first increases for values of the dimensionless speed parameter up to about a = 0.5 to 0.7 and then slightly falls (Figure 6(b)). A similar behaviour has appeared for an exponential covariance of the force with a peak at about a = 0.6 (Figure 6(c)). For a cosine and exponential cosine covariance of the force it is necessary to distinguish two basic cases. If w0 # w, the variance always has a finite value and, e.g., it reaches its maxima at about a = 0.3 for wO/wl = l/2, y = ?r/(Za) (see Figures 6(d) and 6(f)). If, however, w,, = o, (e.g., w,, = or, y = n/a) the variance diminishes with increasing speed and it even reaches infinitely high values at a = 0 for an undamped beam (see Figures 6(e) and 6(g)). The last case is similar to the case of white noise (Figure 6(a)), where the mechanical system responds from the broad band of frequencies to the natural frequencies only. The increasing damping of the beam causes a diminishing of the variance of the deflection for all investigated types of covariances. Given knowledge of the statistical characteristics of axle loads (P, VP, type of covariance, a, fl), one can use the methods presented here to calculate the stochastic response of a beam (see equation (57) and Figures 6(a) to (g)). Therefore, these methods can be used for applications to highway and railway bridges and to other structures in civil and mechanical engineering subjected to moving random loads.

338

L. FRhA

REFERENCES 1. V. V. BOLOTIN 1965 Statistical Methods in Engineering Mechanics (in Russian, 2nd edition). Moscow: Stroiizdat. 2. S. H. CRANDALLand W. D. MARK 1963 Random Vibration in Mechanical Systems. New York, London : Academic Press. 3. J. D. ROBSON 1964 An Introduction to Random Vibration. Amsterdam, London, New York: Elsevier Publishing Co. (Edinburgh University Press). 4. L. FR~BA 1968 Eighth Congress of the International Association for Bridge and Structural Engineering (ZABSE), New York, September 9-14, 1968, Final Report, pp. 122331236. Non-stationary vibrations of bridges under random moving load. Zurich: IABSE Secretariat. 5. K. ZILCH and V. WEISSGERBER 1971 Strasse, Briicke. Tunnel 23(10), 257-262. Zur Formulierung eines Schwingbeiwertes. 6. L. FR~BA 1972 Vibration of Solids and Structures Under Moving Loads. Groningen : Noordhoff International Publishing (Prague: Academia).