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Damped lateral vibration in an axially creeping beam with random material parameters
Inr. J. Solids Snuctures,
1973. Vol. 9, pp. 765 to 788. Pergamon Press. Printed in Great
Britain
DAMPED LATERAL VIBRATION IN AN AX1ALL.Y CREEPING BEAM WITH RANDOM MATERIAL PARAMETERS? W. N. HUANG Argonne
National
Laboratory,
Argonne,
Illinois
and F. A. COZZARELLI State University
of New York,
Buffalo,
New York
Abstract-Damped lateral vibration in an axially creeping beam with random material parameters is considered. The temperature and imperfection density. and as a result two inelastic material parameters. are random functions of the distance along the neutral axis. It is assumed that the material of the beam is governed by a nonlinear Maxwell model, and that the initial tensile stress is much greater than the increment of stress caused by the oscillation so that a perturbation technique may be employed. Two special cases are fully analyzed : (a) the random parameters are random processes which are only slightly random, and (b) the random parameters are random variables which are largely random. Statistical results are obtained for the lateral velocity, bending moment, logarithmic decrement and circular frequency. It is found that in a specific example, where the beam is made of an aluminum alloy, the lateral velocity, bending moment and logarithmic decrement are sensitive to randomness in temperature and imperfection density, whereas the circular frequency is almost deterministic.
1. INTRODUCTION RECENTLY, the
study of the dynamic characteristics of elastic beams having random stiffness parameters has received some attention (e.g., see Bliven and Soong [l]). However, randomness is considerably more pronounced in structure and temperature sensitive nonlinear inelastic material parameters, and this effect has been studied by Soong and Cozzarelli [2], Parkus [3], Parkus and Bargmann [4] and Cozzarelli and Huang [5]. As an extension of the work given in [5], the static problem of steady creep bending in a beam with random inelastic material parameters was considered in [6]. The study of the dynamic characteristics of nonlinear inelastic beams having random inelastic material parameters has received virtually no attention to date, and thus we shall now turn our attention to this problem. The present analysis is concerned specifically with damped lateral vibration in an axially creeping beam, where the inelastic material parameters are described in a stochastic sense. It is assumed that the material of the beam is governed by a nonlinear Maxwell model, composed of a linear elastic element with a deterministic material stiffness and a nonlinear viscous element with two random parameters-a viscosity parameter i, and a creep power n. t This research was supported and in part by the Office of Naval
in part by the National Research under Contract 765
Science Foundation under No. NO001471-Cal08,
Grant
No. GK-1834X,
166
W.
N. HUANG
and F. A. COZZARELLI
The method of approach employed here is analogous to that used in Cozzarelli, Wu and Tang [7], i.e., the initial tensile stress is assumed to be much greater than the increment of stress caused by the oscillation and as a result a perturbation technique may be employed. We then find that the nonlinear viscoelastic problem posed above is replaced by an equivalent linear viscoelastic problem. Also, we separate the problem into two uncoupled problems-the “random temperature problem” and the “random imperfection problem”. In the random temperature problem only k, is random, whereas in the random imperfection problem only n is random. It was shown in [5] that the random inelastic material parameters E, and n have a similar statistical behavior, i.e., they are both lognormal. After the application of the perturbation technique. these two problems reduce to mathematically equivalent problems. Thus, we shall only present the details for the random temperature problem but will give the final results for both problems. In this study it is assumed that the temperature and imperfection density and consequently the inelastic material parameters are random functions of the distance along the neutral axis of the beam. We are then faced with the problem of solving two simultaneous differential equations in velocity and bending moment containing a random process coefficient. The analysis of this problem is difficult, and thus we shall confine ourselves to two special cases: (a) the random parameters are random processes which are only slightly random, and (b) the random parameters are random variables which are largely random. In case (a) the perturbation method suggested by Keller [8] will be used to find the mean velocity and bending moment. In case (b) we employ standard procedures to determine the mean velocity and bending moment; the density function and statistical moments of the logarithmic decrement and circular frequency are also obtained. The formulation of the governing equations with lognormal statistics for the inelastic material parameters is presented in Section 2. This is followed in Sections 3 and 4 by a discussion of the two separate cases mentioned in the preceding paragraph. In both cases a specific numerical example is considered, where the beam is made of aluminum alloy 1100. The results are summarized in the final section.
2. FORMULATION 2.1. Constitutife
OF GOVERNING
EQUATIONS
equation with random parameters
For creeping metals the strain depends not only on time but also exhibits a nonlinear dependence on stress. In order to account for these effects, we shall employ the mathematically convenient as well as physically plausible nonlinear Maxwell model. This model consists of a linear elastic element with a material stiffness E in series with a nonlinear viscous element with two parameters-a creep parameter 1 and a creep power n. Parameters 2 and n are highly sensitive to moderate fluctuations in temperature and imperfection density, whereapE is almost insensitive to such fluctuations. Thus, it is assumed that these two inelastic material parameters are random functions of some space coordinate X whereas E is deterministic. Using a carat over a symbol to indicate a random quantity, the corresponding constitutive equation in the one-dimensional case will then be written as
Damped lateral vibration in an axially creeping beam with random material parameters
767
Here, the stress 8,(X, T) and strain rate &(X, T) are random functions of X, the time T and possibly some other space coordinates, and sgn(&,) is the Signum function defined as 1 a,>0 sgn@,) =
i -1
6, < 0.
Adopting the notation used in [6], we can rewrite equation (1) in a more convenient form as fi+,(X, T) a(x) 1 &3,(X, T) E,(X, T) = --+-8,(X) ~ w(kJ. (3) E
I 1
aT
0,
Here, k,(X) =
@C I(x) i I
B(X) (4)
is a random viscosity parameter equal to the random strain rate obtained when 6,(X, T) = gC, an arbitrary deterministic constant reference stress. In Section 2(b) we will consider a Maxwell material with an initial constant deterministic tensile stress crxi,subjected to a small lateral disturbance at iii = 0. This disturbance is translated into a small additional random increment of stress, i.e. 8,(X, T) = Gxi+ E&(X, 7)
(5)
where 8,(X, T) is a random function and E is a small quantity. Also, as a result of this disturbance the strain rate 2,(X, 7) assumes the form 2,(X, T) = i,,(X) + e&(X, T) + . . . .
(6)
Substituting equations (5) and (6) into equation (3), subtracting the relation of the initial state, and grouping the terms of order E, we obtain a simplified linear constitutive relation for this case as g
F)
(x
*
3
=
L
a~x(xn +~&,(x)~(x)b,(x. 1* T).
-----zz-
E
dT
Here, use has been made of the relation G, = cT,i. For convenience, we summarize some of the statistical results pertaining to $(X) and A(X) obtained in [5]. It is assumed that EC(X)is a function of the random temperature i’(X) only, and that F(X) is a homogeneous normal random process in X with mean T, and variance G$. We define a nondimensional viscosity parameter b(X) and a nondimensional temperature f(X) as E (X) d(X) = +&CO
@a)
where li,, is the nominal value of EC(X)(value at p = To) and B is a deterministic creep constant. The parameter b(X) is related to i(X) by the expression 8(X) = eRX’.
(9)
W. N. HUANG and F. A. COZZARELLI
768
The first order density function of d(X) is then given in terms of the variance of r by
-!!g exp I i
I
J-v’)= J(27+&
I
fJ(&?)
(10)
where U(B) is the unit step function. Equation (10) is the lognormal probability density function [9]. The mean E{8}. variance ai and autocovariance C,(X, -X,) of b(X) are given by E(g) = e”fi2 > 1
(ila)
02 = e”:(e”:- 1) > crf C,(X, -X,)
zz e~:(ec@-x2)-
(lib)
1)
(llc)
where C,(X, -X2) is the autocovariance of t(X). Similarly, ii(X) is assumed to be a function of the random imperfection density A(X) only, where R(X) is a homogeneous normal random process in X with mean N, and variance Specifically, we have
cr;.
t?(X)
=
1 +(n,-
6J~w) - NOI - ~-.--
1)exp i
no--l
(124 i
where
(12b) and thus
f@)=
J(2n;N;;nl)exp~-~(~)2~n~~)]2~~(~-l).
(13)
This is a lognormal density function with two parameters (no and G,F$,) [9], and B(X) has the following statistical properties : E(A) = l+(n,-1)exp G,’ = (nO-l)zexp{($)z~
~1~~)2~ z n _1 [exp{(s)‘)-1]
> n,
UW > (f~~nb)~
(14b)
2.2. Lateral vibration of axially creeping beam As a specific problem consider the free lateral vibration of a nonlinear Maxwell beam, which is subjected initially to a prescribed deterministic axial tensile force followed by a small lateral disturbance at T = 0. The space coordinate X in this problem will be chosen as the neutral axis of the beam which vibrates in the XZ plane. Thus, 6,(X, 2 ; T) and d,(X, Z ; T) are random functions of X, Z and T We make the physically plausible assumption that the deformation of the beam is small. Using the perturbation technique employed
Damped
lateral vibration
in an axially creeping
beam with random
material
parameters
in obtaining equation (7). the classical strain-displacement relation and equation motion for a beam yield the following relations in the first order terms :
a2kifx, T) ax2 f,cx. 2; T) = z aifqx, T) a2&x, q +Q. = -P
8x2
aT
769
of
(15) (16)
In equations (15) and (16) the lateral velocity @X, T) and bending moment &(X, T) = j,- 6,(X, Z ; T)Z dA are random functions of X and T for a beam with cross-section 2; p is the linear density of the beam and Q is an impulsively applied deterministic distributed lateral load which induces a free vibration. Note that, in conformity with the small deformation assumption, the moment due to the axial tensile force PO has been neglected. For convenience, we introduce the following nondimensional quantities :
(17) In these equations, W, is the maximum lateral velocity along the beam at the instant 7; = 0; L is the length of the beam; and 6(x, t), r?r(x,t), q, t and x are the nondimensional lateral velocity, bending moment, lateral load, time and axial coordinate respectively. Two simultaneous differential equations in velocity and bending moment, containing a random process coefficient, may now be obtained from equations (7) (15), (16) and (17) as
a%(X, t)
aG(x, t)
at
i a3qx, t) --= b ax2
Here, b is a nondimensional
ax2 acqx, 7
t)
+q
(18)
i _
+
“o@%MX)lfl(X,0
(19)
deterministic constant given by
(20) with I indicating the second moment of area. We observe that b and A appear in equation (19) side by side. Furthermore, b and ii have a similar statistical behavior, i.e., they are both lognormal [equations (10) and (13)]. Thus, the random temperature problem and the random imperfection problem are mathematically equivalent. We shall therefore only present the details for the random temperature problem but will give the final results for both problems. In Section 3 we shall consider the analysis of equations (18) and (19) for the case where b and A are random processes which are only slightly random [case (a)], and then in Section 4 the case where b and Aare random variables which are largely random [case (b)].
W. N. HUANGand F. A. COZZARELLI
710
3. CASE (a)---r: AND fi RANDOM
PROCESSES
(SLIGHTLY
ENDOW)
3.1. Random temperature problem Let us first consider the random temperature problem in which as we have noted only b(x) is random. Letting T?(x)= n,, equation (19) becomes 1 d%(x, t)
a&(x. t) t+8(x)h(x.
-----------_ b ~3s’
(21)
t).
The simultaneous partial differential equations (18) and (21) can be combined to yield the following two consecutive nonhomogeneous partial differential equations l3%i(x, t) p+ at2
1 _=-d4A(X,t)
aFq(x, t)
&x,----a+,
+b
at
a2
1 824
(22)
b (ix’
a%i(x, t) ax2 +q’
t)
at
(23)
We will assume that the initial conditions at t = O- are homogeneous and take the load 4 as k = 1,2,3,. . . q = d(t) sin knx (24) where d(t) is the Dirac delta function and k is the vibration mode. The physical meaning of these initial conditions and the lateral load will become apparent in the next paragraph. We may easily convert equations (22) and (23) into two consecutive partial differential equations with zero forcing and nonhomogeneous initial conditions at t = O+. In so doing we obtain the resulting equations
a%ii(x,t) +
dtxl
am,
t)+ 1
at
at2
b
a%w, 0
o
(25)
-=
axa
a2fi(X, t) aqx, t) ~ = -~ ax2 at
(26)
with the initial conditions &(x, 0”) = 0,
ati(x, o+) at
=
-- (W2 sin knx b
k
1,2,3,...
(27)
and t;(x, O+) = sin knx
k = 1,2,3, . . . ,
atik o+) = at
o
*
(28)
Equations (27) and (28) hold with probability one. Thus, we observe that the problem posed in the previous paragraph is equivalent to a free vibration problem with a nonzero deterministic velocity prescribed at t = Of. Let us consider a simply-supported beam. Using equations (25) and (26). we obtain the following boundary conditions on A(x, t) and &(x, t) respectively :
a%(o, t) &(O, t) = &(I, t) = -g-&-
a%(i, t) = ____ = 0 ax2
(29)
Damped lateral vibration in an axially creeping beam with random material parameters
771
and
a2qo,t)
~(O,t)=~(l,t)=dxZ=~=
a2ti(l, t)
ax2
o
.
(30)
Again the above relations hold with probability one. The perturbation method suggested by Keller [8] will now be applied to differential equation (25) with initial and boundary conditions (27) and (29) respectively. Bending moment. In this section it is assumed that b(x) is homogeneous random process which is slightly random, and we may then write d(x) = 1 +&(x).
(31)
Here, E is a small quantity, and C&(X)is a homogeneous equation (31) into equation (25) we obtain (L,+&@r(X,
random process. Substituting
t) = 0
(32)
where
2
L, = g+;+;
(33)
t, =8,(x);. The solution to equation (32) is not separable in x and t, and is extremely difficult to obtain. However, utilizing the perturbation method given in [8], we may obtain an approximate expression for the mean bending moment as E{A} = (1-&L~1E{~1}+&2L~lE{i;lL~l~l))m,.
(35)
Here, me is the solution to the unperturbed deterministic linear partial differential equation
a2mo am, 1 a4mo
-+at+;x at2
=0
(36)
with the prescribed deterministic initial and boundary conditions [equations (27) and (29)]. Also, in equation (35) L; 1 is the inverse operator associated with L,, and is thus an integral operator whose kernel is the Green’s function G(x, 5 ; t) satisfying (37)
with homogeneous initial conditions at t = O-. Note that in equation (35) those terms of order E and .s2satisfy homogeneous initial and boundary conditions. The solution to equation (36) can be assumed in the form m, = T,(t) sin klrx
k = 1,2,3,. . .
(38)
where T1(t) is a function of time t only. Note that the boundary conditions (29) are automatically satisfied by equation (38). Substituting equation (38) into equation (36) yields an ordinary differential equation in Ti. We now make the additional assumption that the
W. N. HUANG and F. A. COZZARELLI
712
vibration obtain
is underdamped.
Thus, setting (knf4/b--$ > 0 and using equations
(knYe _t!2 sin r,t sin km mo= - -
k = 1,2.3,.
br,
where the nondimensional
(27), we
..
frequency is given by (39b)
Next, we must obtain the Green’s function satisfying equation (37). A comparison of equations (36) and (37) indicates that the Green’s function is equal to the deterministic bending moment which results from a unit impulse applied to the beam at the location x = < at time t = 0. This Green’s function vanishes for either t < 0 or x, 5 $ [0, I]. The normal modes obtained from the homogeneous part of equation (37) with the prescribed homogeneous boundary conditions are given by the orthogonal functions Irj(x) = sin j7rx.
(40)
We now try a solution to equation (37) in the form 6(x, C:; t) = f
(41)
sj(tl t)hj(x)
j=l
whereby (42) The orthogonaIity of the normal modes ~j(x) can be used to reduce equation (42) to the system of uncoupled equations
d’qj dqj (jx)’ Tqj -p+dt+
= 26(t) sin j7r<
j=
1,2,3 ,....
Solving equation (43) for qj, we obtain with the use of equation (41)
,jlte-
‘I2sin rjt sin jnc sin jnx,
G(x, 5; t) =
0,
t2O,Olx,<<
1
otherwise
where rj is defined by equation (39b). Finally, with this Green’s function the inverse operator L; ’ operating on some function F(x, t) can be given by L, ‘F(x, t) =
Sf t
* G(x, 5 ; t - z)F( 5, t) dt dr.
0 0
(45)
Let us now return to equation (35). In order to evaluate the terms of order E and E*, we must first determine the mean 8’, and mean square e of 8,(.x) for t(x) a homogeneous random process with mean zero and variance cf. These follow directly from equations (1 I)
Damped
lateral vibration
in an axially creeping
beam with random
material
773
parameters
and (31) as d/2 $1
=
L
-1 E
@=
eW - 2 e+
+ 1
E2
(46)
’
For the term of order s2 we must also specify the autocovariance of ?(x)[C,(v]) where q = x1 -x2]. The analysis would become extremely complicated if we were to choose an exponential expression for C,(q) as in [5] and [6]. On the other hand, the analysis is much simpler if we choose an exponential function for the autocorrelation of &(x), i.e. E{f$(x,)&(x,))
= 8’: e-tVtid
(47)
where d is the nondimensional correlation distance. The autocovariance obtained in inverse fashion from equations (1 lc), (3 l), (46) and (47) as
of t(x) is then
C,(q) = ln[2 e”:12- 1 +(e2”3 - 2 ebfi2 + 1) e-tSt/d] -0:.
(48)
The following limiting values follow from this result : f;iy C,(q) = 0,’
(49)
Lim C,(q) = ln(2 e”:‘* - 1)- cr,‘. V+c=
(50)
Equation (50) gives a very small negative value for 9 + co, since Q has been assumed to be slightly random (i.e., err is small). Equation (48) has been plotted in Fig. 1 for typical values or = 0.1 and d = 0.2, and we note that we still get a physically plausible exponentiallike function for C,. Using the deterministic bending moment m, [equations (39)], the Green’s function G(x, 5 ; t) [equation (44)], and equations (45H47), we may then proceed to evaluate equation (35) and thereby obtain the rather complicated result E(rPr} = -~e~r~2sinr,rsinknx+s~e~ri2[( k
-&+rk’)
sinr,t
k
k
l[(A,
+A2t+A,t2)
k.rrx+ dy’-q6jc3k, ( d4
sin r,t
)
smjnx
]
[(Akj’ + A$) sin r,t + (A$‘+ A$) cos r,t + A$ sin rjt + Ay{ cos rjt] + C (2dY’-d~‘6C,j,k)[AY; j,= 1
sin r,t + (A?: + Ayir) sin rjt + (A?&+ A$$,t) cos rjt
Ifk
+A’& cos r,t] sin&x+
f f (2d’,‘j)-d~‘6i(k_,j,-d~)6i(k+,j,)(Alijg) i ; 2+‘j=z+
sin r,t
+ A?$)sin rit + A(2ijdsin rjt + ,44jj cos r,t + A!$? cos r,t + AyQ cos rjt) sin inx where aij is the Kronecker delta.
(51)
W. N. HUANG and F. A. COZZARELLI
774
d=a,
FIG. 1. Autocovariance
of Q--case
(a)
(5 1)
In equation
4r,2+3 A, =--16rt
A*=-$
4r: - 1 A, = ~ 16r:
A, = -rkA,
k
A”, = (4rjf - l)(rjf - rj2)- 2r:( 1 - 4rj2)
A, = -r,A,
A$i’=
2
6
4r:(r:
’
A!’
“if - ’ 4r,(r,2 - r;)
A:‘& =
=
rk - rj
Ag’ = _
Ayi =
l)+(l-4rf)(r,2
-
rk[2r?(4ri
8ri(r:
-rf)]
At& =
- rj”)’
A”’ A’:‘& = 2
Ayf zz
2 A””
A’& = - Ay&
A?;
18
r,(l-4r,Z)
=
4ri(rE - rF)(rj2 - rf) A(iA 21
=
AU.i) = 23
2rk (r: - rj’)’
r,(l-4rj2) 2rj(r: - rj2)2
A:‘; = _
A’i’i = -Af’
- rj’)’
4r,” - 1 4(ri - rj2)2
-‘k 2rj(r: - rj’)
-rk(l-4rj2) 8rj2(r: - r;) 4r: - 1
=
-
4(r,2 - r;)(r:
- rf)
r,(l-4rf) A(‘” 2o = -4r,(r,‘-rf)(rjf-rj”)
-
‘k
(rjf -rj’)(rj’-rf)
(52)
Damped
lateral vibration
in an axially creeping
beam with random
material
parameters
115
and d, =
3+8d2k2rr2+8d2k4n4 MCI
_ (e lid-l)
C:
l/d= + (k= +j=).rr=+ 8k2j+4[( _ I)“-j e- lid- 11 ~Cti’C”’ 2
d!$ = d, =
(dCy’C”‘)= 3
3
4k3jn4[1 +(-
l)k-j](e-“dc,c$VIi”
1)
1 dC,
dy’ = d;’ = d’:‘” =
4kj3z4[1+(-l)k-j](e-1id-1) C$Wy’( l/d2 +4j2n2) & 4ij2kE4[((_l)i-j+(_l)k-j)e-l/d-(1+(_l)i+k-2j)] d2C:j’C~‘(l/d2+(i-j)2z2)(l/d2+(i+j)2z2)
d(B) =
1 2dC:”
(53)
where 1 C 1 = z+4k2X2
CY' = $+(kkj)“r’.
(54)
CY’
As a particular example we will consider the beam treated deterministically in [7], namely a uniform rectangular cross-section beam made of aluminum alloy 1100. The cross-sectional dimensions are taken to be 2 in in width and 3 in in depth, and the length of the beam is taken as 36 in. The mean temperature of the beam is chosen as 260°C and the axial stress cXi = a, is taken equal to 3000 psi. The material properties are given in Table 1 (see [7X. TABLE 1. MATERIAL PROPERTIESOF ALUMINUMALLOY 1100(260°C) p = E = PI0 = I, =
1.2 x 10-10 8.0x lo6 4.55 364 X 103
lb-h2-in-’ psi
psi-h’/“0
Using the data given in the previous paragraph and in Table 1 and taking the vibration mode k to be one, we obtain the numerical values given in Table 2.
W. N. HUANG and F. A. COZZARELLI
716
TABLE 2. NUMERICALVALUES
I = 4.5
EC0 = bxi i-i2.0
inJ
“” = 0.415
hr-’
= 1.419 x lo-” = 828.68 for/i
= 1
[equation
(4)]
[equation
(20 11
[equation
(39b)]
It may be seen from Table 2 that, as a consequence of the small damping in this example (b << l), ri >> 1 and jr; - $1 >> 1 for j # i. We then find that, for any values of err and d, the expression of order E in equation (51) is completely dominated by the r,t sin r,t term and the expression of order t.’ is completely dominated by the d, A,t2 sin r,t term. Thus, we may now considerably simplify the expression for E{Gi!, with negligible loss in accuracy, to the following form which is separated in x and t : ; e-“‘sin r,t sin nx. 1 -cf$+c22d,A,~t2 EjG), = -f: 1
With the use of equations (17) (20) (39b) and (52) equation (55) may be rewritten terms of the physical parameters and dimensional time T (in seconds) as
in
(56) where
and where m* is a normalizing
factor defined as
In the present example rx = 0.7 and /I = 1158.7, and thus we see that as a consequence of small damping b >> CI. For practical purposes it is sufficient to confine our attention to the time interval over which the exponential function in equation (56) decays to one per cent of its initial value. i.e., 0 I T I 4,6/a. Within this interval, we will consider that the “slightly random” assumption implies that &22d,&:~2T2 c 1 in equation (56). This requires that i g2F 1 = e20s_ 2 e”z’2 + 1 << 42.32
c
3 + 8d2n2
4d(l/dz+4s2)+
8d2;rr4(e-lid_ I) ( l/d2 +47r2)’
11 1
(58)
Equation (58) will be used as a condition on 6, for a given d. In Fig. 2(a), a single curve showing the decay of E{h/m*),,, (LMP = locus of maximum points) for the mean bending moment [equation (56)]] at the midspan of the beam (x = )) is plotted vs time 7 in seconds for typical values g‘r = 0.1 and d = 0,0.2, co-all of which
Damped
lateral vibration
in an axially creeping
beam with random
material
Ill
parameters
.__ 0
.-_-mm__.
ki
0
J
T I -
L_y-ooo2
-
crT=O.I (Random
temp. plob.1
-0-002
sh -
t lb)
FIG. 2. Decay curves--case
----o--n,,
=0.4(Rondom
imp. prob.)
(a). (a) LMP of mean bending moment and velocity. LMP of deterministic bending moment and velocity.
(b) deviation
from
satisfy condition (58). The deviation of the LMP of the mean bending moment from the LMP of the deterministic bending moment, [E(r?t/m*} --n~e/rn*]~~~, has been plotted in Fig. 2(b) (solid curves). Note that for d > 0 this deviation vanishes at a root T = T, and that the LMP of the mean bending moment is less than the LMP of the deterministic bending moment in the interval 0 < T < 7;,, and is greater for T > ‘i;,. We also see that there exist two critical times Tmi, and T,,, where the random effect is an extremum. We obtain T, and these two critical times from equations (56) and (57) as
With CJ~= 0.1 and d = 0.2 in this example equations (59) yield 7;, = 3.147 set, Tmin= 0.877 set and T,,, = 5.131 set, which approximately equal 580, 161 and 946 cycles of vibration respectively. Also, for these values equations (lla) yield E{c?} = 1.005, and Fig. 2 yields E{A/m,},,, = 0.9977 at T = Tin = 0.877 sec. Thus, the bending moment is sensitive to small variations in temperature. Velocity. Turning now to the velocity, we obtain from equation (26)
for fi(x, t) and A(x, t) mean square differentiable with respect to time t and axial coordinate x respectively. Substituting equation (51) into equation (60) integrating the result with
W. N. HUANG and F. A. COZZARELLI
778
respect to t, and then using initial conditions
(28). we obtain
Isinr
k
t-tcosr
k
t sinkm
rk
d,(kz)2[(B,+B2t+BBjt2)sin
-cz~6?emt~z
br, + 2 j= j+k
1
d’j,‘(kn)2 sin knx+
sin kxx
rkt+(B4t+B,t2)cosrkt]
&)-Ab.
+ (B$j’+ B$‘t) cos r,t + B’$, sin rjt + By’, cos rjt] + i
1
j= jfk
(2dy’-d~‘~S~,~,,
. (jn)2[B’fi sin r,t +(B$$ + Byit) sin rjt + (Byi + B’&t) cos rjt + B’/: cos rkt] .sin jzx+
(2d:‘“-d~‘6i(k_zj,-d~‘Sick+2j,)(i~)2(B(;’js) sin r,t
i f i=l j=l
i#kfj#i
+
In equation
B$$) sin rit + Byd sin rjt + Bg) cos r,t + B$$ cos rit + Bit cos rjt) sin (61) B, =$
k
B, =$
=$
k
B, = -r,B,
B, = -rkB2
By’ =
B,
-1 r: - rj’
By: = -
r,(r,’ - 3rT) 4rj(ri - rj’)’
B$ = Zr,Byi
@’ =
@’ =
Byi
=
B$
=
(3r: - r,“) 2rjf(r: -rj2)2 - 2r, (r,‘-rj2)’
-’ -
2(r,Z r;)* rk
2rj(rjf - r;)
Byi = g J
@/&I= _
1 2(ri - rj’)(r: - rf)
.
(61)
Damped lateral vibration in an axially creeping beam with random material parameters
719
(62) Again as a specific example we will consider the beam with small damping discussed as in our analysis of the bending moment, we find that the expression of order 1 in equation (61) is completely dominated by the rk cos r,t term, the expression of order E is completely dominated by the t cos rkt term, and the expression of order c2 is completely dominated by the d,B,r2 cos r,t term. Thus, we may in this case considerably simplify equation (61) for Ef8), with negligible loss in accuracy, to the following form : in [i’]. Proceeding
E{ ti!) = (1 -s81rzT+s2UI~~2iiii2)
eWaTcos BTsin 7r.x.
(63)
Here, T is the time in seconds, and TVand /.I are as defined in equations (57). The expression for E{$} given by equation (63) is identical with the expression for E{fi/m*~ given by equation (56) except that sin fir is replaced by cos /?T Since /I >> 1, these two expressions give essentially the same locus of the maximum points. Thus, Fig. 2 also applies in this case. The velocity is clearly sensitive to small fluctuations in temperature. 3.2. reborn imperfection problem As we have previously stated the random imperfection problem and the random temperature problem are mathematicaliy equivalent. Instead of using equation (31) we start with the relation ii(x) = n,[l -+-&fir(X)]
(64)
where fir(x) is a homogeneous random process. Its mean ti, and mean square z are obtained from equations (14) and (64) as
(~~-l)2[exp~2(~)z~-2exp~~(~)2~+l] +
n2c2 0’ The autocorrelation
(66)
of iii(x) is assumed to be E{A,(x,)A,(x2)) = Te-Iqt’d
(67)
based upon the same argument given in Section 3.1. Proceeding as in the random temperature problem, we obtain the same results [i.e., equations (51), (56), (58), (59), (61) and (63)] with the substitutions
W. N. HUANG and F. A. COZZARELLI
780
The curve shown in Fig. 2(a) also applies in this problem for values a,& = 0.4 and d = 0, 0.2, m-all of which satisfy condition (58) with the substitutions (68). The deviation of the LMP of the mean bending moment and of the mean velocity from the LMP of the deterministic bending moment and velocity have also been plotted in Fig. 2(b) (dashed curves). For the values a,,& = 0.4 and d = 0.2, equations (59) with the substitutions (68) yields E{A/n,) = 1.005, and Fig. 2 yields E{A/m,),,, = E{G/w,},,, = 0.9974 at T = Tmin 740, 179 and 1088 cycles of vibration respectively. Also, for these values equation (14a) yields E{fi/n,f = 1.005, and Fig. 2 yields E(&i/m,J,,, = E{$/w,j,,, = 0.9974 at T = Tmin = 0.973 sec. Thus, the bending moment and the velocity are also sensitive to small variations in imperfection density. We conclude that both bending moment and velocity are sensitive to small fluctuations in temperature and in imperfection density. This differs from the results obtained for the 3-bar truss in [5] and statically loaded beam in [6], where the stress was essentially deterministic and only the velocity was sensitive to fluctuations in temperature and in imperfection density. 4. CASE (b)--c?
AND fi RANDOM
4.1. Random temperature
VARIABLES
(LARGELY
RANDOM)
problem
Again let us first consider the random temperature problem in which only C?is random. In this section d is taken to be a random variable which is largely random. Thus, whereas in equation (31) E may no longer be used as a perturbation parameter, ~?i(x) is simply a random variable c$. It follows that the consecutive differential equations (25) and (26) may be simplified to two uncoupled linear differential equations in A(x, t) and 6(x, t). each with a random constant coefficient, i.e., &ii(x, at2
t) +cfi.dh(x, t)+l at
d4rFi(x. t) b -=
a2qx, t) +@2=++1
at2
at
(69)
aXa
w b
o
a.2
= o,
(70)
The deterministic initial conditions are given by equations (27) and (28) and the deterministic homogeneous boundary conditions are again given by equations (29) and (30). Note that whereas equations (69) and (70) are identical, the initial conditions on #I and G are different. We shall only present the details of the velocity solution but will give the final results for both velocity and bending moment. Ve/elocity. The solution to equation (70) is separable, and may be expressed as the product of a random function of time and a deterministic function of space in the form G(x, t) = T*(t) sin kzx
k = 1,2, 3;.
..
(71)
Substituting equation (71) into equation (70), solving the resulting ordinary differential equation in Yf2(t) with the initial conditions (28), and using equations (17), (20) and (57), we obtain the velocity for the underdamped case (b < fik*/u) as
qx, t)
=
Bk* q/[@k2/a)*
a -aiT sin(a,,/[(/Ik2/a)’ - g21e
- g2]T+
0) sin kzx
k = 1,2,3,.
(72)
781
Damped lateral vibration in an axially creeping beam with random material parameters
where the phase angle is given by S = tan-’
&W/a)’
- $21 F
(73)
I.
Since there is zero probability that b in this case is greater than /?k2/a, the lognormal density function for 8 [equation (lo)] must be replaced by a truncated lognormal density function. Thus, we let 1 (744 f(&‘) = n,J(2rr)a,B exp and R, = ++erf
VW
where R, is such that equation (74a) satisfies the normalization erf x is the error function defined as J;2X) erf x = __
condition.
Note that
1 e - Y’lzdy.
(75)
s
The mean E{@ and variance gi based on the truncated lognormal density function [equation (74a)] are given by
(764 and
W-W where H, = i+erf
ln(flk’/a) - 0:
(774
cr H, = ++erf
ln(flk2/a) - 2af or
VW
I
Expressions for the first order density function and the statistical moments of G(x, t) may be found from equations (72) and (74). For example, the mean velocity is given by
sin(aJ[(bk2/a)2
- &‘]T+
f3)db.
(78)
Decay curves for the mean velocity [equation (7811at the midspan of the beam discussed in the previous section with k = 1, and cr = 0.2 and 0.5 are presented in Fig. 3. In this example p >>a (small damping), and as a result the normalization factor R, is very close to unity [equation (74b)] and the density function of the phase angle 6’ is sharply peaked around 7r/2 [equation (73)]. Results have also been obtained for or = 0.1 in the present case (not plotted in Fig. 3) and these results checked very closely with the curves for cr = 0.1 and d = cc in Fig. 2 for case (a). For these values of o, and d, b in both cases is essentially a random variable which is only slightly random.
782
W. N. HUANG and F. A. COZZARELLI
x=0.5 k=
0
I
2
3
I
4
5
6
65
4
5
6
65
7
0.03 - ( b )
0
FIG. 3. Decay
curves*ase
I
2
3
(b). (a) LMP of mean velocity and bending moment. LMP of deterministic velocity and bending moment.
(b) Deviation
from
The logarithmic decrement S, a quantity widely used as a measure of damping, is given by the expression (79) where WI and W, are the velocities at two successive maxima. It then follows from equation (72) that
(80) Also, the circular frequency in cycles per second follows as (;,=
fxJ[(p/2/ay -8’1
(81)
271 For convenience, we rewrite equations (SO)and (81) in nondimensional
s^
bJ[(pk2/a)*-
I]
& = J[(pc/a)2 - 8’1 ci,
J[(p#v/or)2-2’1
00 = J[(/?k2/lx)‘-
l]
form as
(824
@W
where 6, and o,, are the nominal values (at p = T,) given by (83)
Damped lateral vibration in an axially creeping beam with random material parameters
783
Given that d is truncated lognormal [equation (74)], the density functions of 6/S, and I%/w, follow from equations (82) as
[ln(r,(Dk2/a)(6/6,)~~' u 2 10 do
1844 Wb)
Equation (84a) has been plotted in Fig. 4 for the previously discussed example (solid curves) with k = 1 and typical values for u,. The lower limit on the logarithmic decrement
FIG. 4. Density function of &&,--case (b).
(S/6, = 0) corresponds with 6 = 0, while the upper limit (6/h, + co) corresponds with the cutoff d = p/a. Equation (85a) has also been plotted in Fig. 5 (solid curves) for the same values of k and or. Here, the lower limit on the circular frequency (o/w0 = 0) corresponds with the cutoff cfp= /3/a, while the upper cutoff (o,AB~= ~/a)/~[(~/a)‘I]) corresponds with ~5= 0. The means and the standard deviations of s^jS, and Q/o, are readily obtained by numerical integration. For G‘,equal to 0.2 and 0.5, values of these statistical properties and those for $ [equations (I I)] are given in Table 3.
784
W. N. HUANG and F. A. COZZARELLJ
- -Random
temp
----Random
prob.
imp. prob.
50-
IO-
0-999999
09999995
w/w, FIG.5. Density function
of &j/w,+ase
(b)
TABLET. MEANSANDSTANDARDDEVIATIONSOF~/&,AND~&,
or 0.2 0.5
E{C$
0s
E{&S,~
I .0202 1.1331
0.2061 0.6039
1.1331
1.0202
ud/dn 0.2061 0.6039
E{W%l 0.999999 0.999590
uwwo om12 0.0203
It may be seen from Table 3 that in this example the statistics of the logarithmic decrement are numerically equal to the statistics of the material parameter 2. This follows directly from equation (82a) since the density function of b peaks near value 1.0 and /I/al >> 1 (small damping). Applying the same argument to equation (82b), we find that the density function of G/w, is very sharply concentrated near 1.0, and this is apparent in Fig. 5 and Table 3. Thus, we find in this example that whereas the logarithmic decrement is sensitive to fluctuations in temperature, the circular frequency is almost insensitive to such fluctuations. Bending moment. Since equations (69) and (70) with their associated initial and boundary conditions are mathematically similar, we may proceed in an analogous manner to obtain the bending moment and its statistics. We obtain for the bending moment
@x.t)
=
- (PM2 2n2ci2&pk2/ct)2 -
g2]
emNiTsin(aJ[(pk2/oc)2 -J2]T)
and for the mean bending moment
where m* is defined in equation (57b).
sin knx
k = 1,2, 3,. . . (86)
Damped lateral vibration in an axially creeping beam with random material parameters
785
The expression for E{&/m*} given by equation (87) is identical with the expression for E{$j given by equation (78) except that here the phase angle 6 is equal to zero. Due to the high frequency in the previously discussed example we may expect that the locus of maximum points in that example is the same for bending moment and velocity. Thus, Fig. 3 also applies in this case. Likewise, equations (82), (84) and (85) for 6^/6, and h/o,, and their density functions also hold for the bending moment solution. Thus the bending moment and its logarithmic decrement also show significant random fluctuations as the temperature fluctuates. 4.2. Random imperfection problem Velocity. We now turn to the random imperfection problem in which only li is random. In this problem iQ,J//s, and c?+B, can be easily obtained from equations (72), (73), (82) and (83) in the random temperature problem with the substitution B+fi/n,.
(88)
Now, 6, and oO are nominal values evaluated at fl = NO. The underdamped in this problem requires that
restriction
where
(89b) Again, since there is zero probability that A in this case is greater than /?k2/d, the lognormal density function for A [equation (13)] must be replaced by a truncated lognormal density function. Thus, we write f(n) =
no -l
Q2JGWa&(n-
e~P{-~(~)2~n(~)]2~[U(n-l)-U(n-~)]
1)
(90a)
PW The mean E(A) and variance CJ~are then given by E{li} = 1 +n(no-
l)exp
2
-
(914
{:($q’)
{($)‘)r3exP{($!j2}-I]
(9Ib)
where H, = ~+erf{~[ln((“~~~)[‘)
H, = i+erf{s[ln(
(‘YE);
-(%)‘I}
(924
‘) -2( $)l).
(92b)
786
W. N. HUANG and F. A. COZZARELLI
Given that ii is truncated lognormal [equation (9Oa)], the density functions of &S, and &)/CL+, are obtained from equations (82) with substitution (88) as
Y, =
PW
_ti’i 00
= (~~-~)(~)[(~)~-~:lexp~ Z,(Z,-
l)Q,J(27c)@,nb
-- l(“1)‘[ln~z2-l)~‘) ___-._ 2
lT&
no-i
(94a)
W-4 Equations (93a) and (94a) have been plotted in Figs. 4 and 5 respectively (dashed curves) for the previously discussed example with k = 1 and typical values for a,&. The means and the standard deviations of (F/S, and cS,/W, are also readily obtained. For o,& = 0.8 and 2.0, values of these statistical properties and those for ii [equations (14)] are given in Table 4. TABLE4. MEANS AND STANDARDDEVIATIONS OF $6,
0.8 2.0
1.0201 1.1342
0.1827 0.5589
I.0201 1.1342
0.1827 0.5589
AND Q/w,
0.999938 0.994273
0@079 0.0755
As in the random temperature problem we find that in this example the statistics of the logarithmic decrement are numerically equal to those of e/n,. and that the density function of Q/w0 is very sharply concentrated near one. Thus, we again find that whereas the logarithmic decrement shows significant random fluctuation as the imperfection density fluctuates, the circular frequency is almost deterministic. Bending moment. The bending moment for the random imperfection problem may be obtained from equation (86) with the substitution (88). We then find that the statistics of
Damped
the bending discussion.
lateral vibration
moment
in an axially creeping
are identical
beam with random
with those of the velocity,
5. SUMMARY
material
parameters
and thus require
181
no further
OF RESULTS
The present study is an investigation of the effect of randomness in the material parameters .$ and ii on damped lateral vibration in an axially creeping beam. The randomness in these parameters results from randomness in the temperature and imperfection density, which have been assumed to be independent homogeneous normal random processes. Two separate cases have been considered, i.e., (a) b and A random processes which are slightly random, and (b) 8 and A random variables which are largely random. In each case the problem has been separated into two uncoupled parts-the “random temperature problem” and the “random imperfection problem”. In the random temperature problem only E, is random, whereas in the random imperfection problem only n is random. For both cases a simply-supported nonlinear Maxwell beam made of aluminum alloy 1100 has been taken as an example. In this example the damping is small, and the vibration is underdamped. In case (a) we have obtained the mean bending moment and velocity by using exponential-like autocovariances for the temperature and the imperfection density. As a consequence of the small damping, very simple and useful expressions for the mean bending moment and the mean velocity have been obtained. Two critical times, where the effect of material parameter randomness on the bending moment and velocity is an extremum. have been found. It has been observed that both the bending moment and velocity are sensitive to randomness in temperature and in imperfection density. This differs from the results obtained for the 3-bar truss in [5] and statically loaded beam in [6], where the stress was essentially deterministic and only the velocity was sensitive to such randomness. In case (b) we have found the means of the velocity and the bending moment. Comparing these means with those in case (a), we have observed that the two cases give equivalent results when the correlation distance in case (a) is large and the standard deviation in case (b) is small. The statistics of the logarithmic decrements and circular frequencies for the velocity and bending moment have also been presented. It has been found that as a consequence of the small damping the velocity, bending moment and their associated logarithmic decrements are sensitive to randomness in temperature and in imperfection density. whereas the associated circular frequencies are almost deterministic.
REFERENCES [l] D. 0. BLIVEN and T. T. SOONC. On frequencies of elastic beams with random imperfections. J. Frunklin Institute 287, 291 (1969). [2] T. T. S~ONG and F. A. COZZARELLI, Effect of random temperature distributions on creep in circular plates. Int. J. Non-Linear Mech. 2, 21 (1967). [3] H. PARKUS, On the Lifetime of Viscoelastic Structures in a Random Temperature Field. Recent Progress in Applied Mechanics-Folke Odqoisr Volume, p. 391. John Wiley (1967). [4] H. PARKUS and H. BARGMANN, Note on the behavior of thermorheologically simple materials in random temperature fields. Acia Mech. 9, 152 (1970). [5] F. A. COZZARELLI and W. N. HUANG, Effect of random material parameters on nonlinear steady creep solutions. Int. J. Solids Swucr. 7. 1477 (1971). [6] W. N. HUANG and F. A. COZZARELLI, Steady creep bending in a beam with random material parameters, J. Franklin Institute 294, 323 (1972).
788
W. N. HUANG
and F. A.
COZZARELLI
[7] F. A. CUZZARELLI, J. Wu and S.
TANG, Lateral vibration of a nonlinear viscoelastic beam under initial axial tension. J. Sound Vibr. 13, 147 (1970). [S] J. B. KELLER,Stochastic Equations and Wave Propagation in Random Media. Proc. Symp. on A& Math. 16, 145 (1964). [9] J. AITCHISONand J. A. C. BROWN, The Lognormai ~jsfrjbuf~on. Cambridge
(Received 16 Augur
Akrparr-Mccnenyerccn
University
Press (1957).
1972; rccised 26 November 1972)
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1973. Vol. 9, pp. 765 to 788. Pergamon Press. Printed in Great
Britain
DAMPED LATERAL VIBRATION IN AN AX1ALL.Y CREEPING BEAM WITH RANDOM MATERIAL PARAMETERS? W. N. HUANG Argonne
National
Laboratory,
Argonne,
Illinois
and F. A. COZZARELLI State University
of New York,
Buffalo,
New York
Abstract-Damped lateral vibration in an axially creeping beam with random material parameters is considered. The temperature and imperfection density. and as a result two inelastic material parameters. are random functions of the distance along the neutral axis. It is assumed that the material of the beam is governed by a nonlinear Maxwell model, and that the initial tensile stress is much greater than the increment of stress caused by the oscillation so that a perturbation technique may be employed. Two special cases are fully analyzed : (a) the random parameters are random processes which are only slightly random, and (b) the random parameters are random variables which are largely random. Statistical results are obtained for the lateral velocity, bending moment, logarithmic decrement and circular frequency. It is found that in a specific example, where the beam is made of an aluminum alloy, the lateral velocity, bending moment and logarithmic decrement are sensitive to randomness in temperature and imperfection density, whereas the circular frequency is almost deterministic.
1. INTRODUCTION RECENTLY, the
study of the dynamic characteristics of elastic beams having random stiffness parameters has received some attention (e.g., see Bliven and Soong [l]). However, randomness is considerably more pronounced in structure and temperature sensitive nonlinear inelastic material parameters, and this effect has been studied by Soong and Cozzarelli [2], Parkus [3], Parkus and Bargmann [4] and Cozzarelli and Huang [5]. As an extension of the work given in [5], the static problem of steady creep bending in a beam with random inelastic material parameters was considered in [6]. The study of the dynamic characteristics of nonlinear inelastic beams having random inelastic material parameters has received virtually no attention to date, and thus we shall now turn our attention to this problem. The present analysis is concerned specifically with damped lateral vibration in an axially creeping beam, where the inelastic material parameters are described in a stochastic sense. It is assumed that the material of the beam is governed by a nonlinear Maxwell model, composed of a linear elastic element with a deterministic material stiffness and a nonlinear viscous element with two random parameters-a viscosity parameter i, and a creep power n. t This research was supported and in part by the Office of Naval
in part by the National Research under Contract 765
Science Foundation under No. NO001471-Cal08,
Grant
No. GK-1834X,
166
W.
N. HUANG
and F. A. COZZARELLI
The method of approach employed here is analogous to that used in Cozzarelli, Wu and Tang [7], i.e., the initial tensile stress is assumed to be much greater than the increment of stress caused by the oscillation and as a result a perturbation technique may be employed. We then find that the nonlinear viscoelastic problem posed above is replaced by an equivalent linear viscoelastic problem. Also, we separate the problem into two uncoupled problems-the “random temperature problem” and the “random imperfection problem”. In the random temperature problem only k, is random, whereas in the random imperfection problem only n is random. It was shown in [5] that the random inelastic material parameters E, and n have a similar statistical behavior, i.e., they are both lognormal. After the application of the perturbation technique. these two problems reduce to mathematically equivalent problems. Thus, we shall only present the details for the random temperature problem but will give the final results for both problems. In this study it is assumed that the temperature and imperfection density and consequently the inelastic material parameters are random functions of the distance along the neutral axis of the beam. We are then faced with the problem of solving two simultaneous differential equations in velocity and bending moment containing a random process coefficient. The analysis of this problem is difficult, and thus we shall confine ourselves to two special cases: (a) the random parameters are random processes which are only slightly random, and (b) the random parameters are random variables which are largely random. In case (a) the perturbation method suggested by Keller [8] will be used to find the mean velocity and bending moment. In case (b) we employ standard procedures to determine the mean velocity and bending moment; the density function and statistical moments of the logarithmic decrement and circular frequency are also obtained. The formulation of the governing equations with lognormal statistics for the inelastic material parameters is presented in Section 2. This is followed in Sections 3 and 4 by a discussion of the two separate cases mentioned in the preceding paragraph. In both cases a specific numerical example is considered, where the beam is made of aluminum alloy 1100. The results are summarized in the final section.
2. FORMULATION 2.1. Constitutife
OF GOVERNING
EQUATIONS
equation with random parameters
For creeping metals the strain depends not only on time but also exhibits a nonlinear dependence on stress. In order to account for these effects, we shall employ the mathematically convenient as well as physically plausible nonlinear Maxwell model. This model consists of a linear elastic element with a material stiffness E in series with a nonlinear viscous element with two parameters-a creep parameter 1 and a creep power n. Parameters 2 and n are highly sensitive to moderate fluctuations in temperature and imperfection density, whereapE is almost insensitive to such fluctuations. Thus, it is assumed that these two inelastic material parameters are random functions of some space coordinate X whereas E is deterministic. Using a carat over a symbol to indicate a random quantity, the corresponding constitutive equation in the one-dimensional case will then be written as
Damped lateral vibration in an axially creeping beam with random material parameters
767
Here, the stress 8,(X, T) and strain rate &(X, T) are random functions of X, the time T and possibly some other space coordinates, and sgn(&,) is the Signum function defined as 1 a,>0 sgn@,) =
i -1
6, < 0.
Adopting the notation used in [6], we can rewrite equation (1) in a more convenient form as fi+,(X, T) a(x) 1 &3,(X, T) E,(X, T) = --+-8,(X) ~ w(kJ. (3) E
I 1
aT
0,
Here, k,(X) =
@C I(x) i I
B(X) (4)
is a random viscosity parameter equal to the random strain rate obtained when 6,(X, T) = gC, an arbitrary deterministic constant reference stress. In Section 2(b) we will consider a Maxwell material with an initial constant deterministic tensile stress crxi,subjected to a small lateral disturbance at iii = 0. This disturbance is translated into a small additional random increment of stress, i.e. 8,(X, T) = Gxi+ E&(X, 7)
(5)
where 8,(X, T) is a random function and E is a small quantity. Also, as a result of this disturbance the strain rate 2,(X, 7) assumes the form 2,(X, T) = i,,(X) + e&(X, T) + . . . .
(6)
Substituting equations (5) and (6) into equation (3), subtracting the relation of the initial state, and grouping the terms of order E, we obtain a simplified linear constitutive relation for this case as g
F)
(x
*
3
=
L
a~x(xn +~&,(x)~(x)b,(x. 1* T).
-----zz-
E
dT
Here, use has been made of the relation G, = cT,i. For convenience, we summarize some of the statistical results pertaining to $(X) and A(X) obtained in [5]. It is assumed that EC(X)is a function of the random temperature i’(X) only, and that F(X) is a homogeneous normal random process in X with mean T, and variance G$. We define a nondimensional viscosity parameter b(X) and a nondimensional temperature f(X) as E (X) d(X) = +&CO
@a)
where li,, is the nominal value of EC(X)(value at p = To) and B is a deterministic creep constant. The parameter b(X) is related to i(X) by the expression 8(X) = eRX’.
(9)
W. N. HUANG and F. A. COZZARELLI
768
The first order density function of d(X) is then given in terms of the variance of r by
-!!g exp I i
I
J-v’)= J(27+&
I
fJ(&?)
(10)
where U(B) is the unit step function. Equation (10) is the lognormal probability density function [9]. The mean E{8}. variance ai and autocovariance C,(X, -X,) of b(X) are given by E(g) = e”fi2 > 1
(ila)
02 = e”:(e”:- 1) > crf C,(X, -X,)
zz e~:(ec@-x2)-
(lib)
1)
(llc)
where C,(X, -X2) is the autocovariance of t(X). Similarly, ii(X) is assumed to be a function of the random imperfection density A(X) only, where R(X) is a homogeneous normal random process in X with mean N, and variance Specifically, we have
cr;.
t?(X)
=
1 +(n,-
6J~w) - NOI - ~-.--
1)exp i
no--l
(124 i
where
(12b) and thus
f@)=
J(2n;N;;nl)exp~-~(~)2~n~~)]2~~(~-l).
(13)
This is a lognormal density function with two parameters (no and G,F$,) [9], and B(X) has the following statistical properties : E(A) = l+(n,-1)exp G,’ = (nO-l)zexp{($)z~
~1~~)2~ z n _1 [exp{(s)‘)-1]
> n,
UW > (f~~nb)~
(14b)
2.2. Lateral vibration of axially creeping beam As a specific problem consider the free lateral vibration of a nonlinear Maxwell beam, which is subjected initially to a prescribed deterministic axial tensile force followed by a small lateral disturbance at T = 0. The space coordinate X in this problem will be chosen as the neutral axis of the beam which vibrates in the XZ plane. Thus, 6,(X, 2 ; T) and d,(X, Z ; T) are random functions of X, Z and T We make the physically plausible assumption that the deformation of the beam is small. Using the perturbation technique employed
Damped
lateral vibration
in an axially creeping
beam with random
material
parameters
in obtaining equation (7). the classical strain-displacement relation and equation motion for a beam yield the following relations in the first order terms :
a2kifx, T) ax2 f,cx. 2; T) = z aifqx, T) a2&x, q +Q. = -P
8x2
aT
769
of
(15) (16)
In equations (15) and (16) the lateral velocity @X, T) and bending moment &(X, T) = j,- 6,(X, Z ; T)Z dA are random functions of X and T for a beam with cross-section 2; p is the linear density of the beam and Q is an impulsively applied deterministic distributed lateral load which induces a free vibration. Note that, in conformity with the small deformation assumption, the moment due to the axial tensile force PO has been neglected. For convenience, we introduce the following nondimensional quantities :
(17) In these equations, W, is the maximum lateral velocity along the beam at the instant 7; = 0; L is the length of the beam; and 6(x, t), r?r(x,t), q, t and x are the nondimensional lateral velocity, bending moment, lateral load, time and axial coordinate respectively. Two simultaneous differential equations in velocity and bending moment, containing a random process coefficient, may now be obtained from equations (7) (15), (16) and (17) as
a%(X, t)
aG(x, t)
at
i a3qx, t) --= b ax2
Here, b is a nondimensional
ax2 acqx, 7
t)
+q
(18)
i _
+
“o@%MX)lfl(X,0
(19)
deterministic constant given by
(20) with I indicating the second moment of area. We observe that b and A appear in equation (19) side by side. Furthermore, b and ii have a similar statistical behavior, i.e., they are both lognormal [equations (10) and (13)]. Thus, the random temperature problem and the random imperfection problem are mathematically equivalent. We shall therefore only present the details for the random temperature problem but will give the final results for both problems. In Section 3 we shall consider the analysis of equations (18) and (19) for the case where b and A are random processes which are only slightly random [case (a)], and then in Section 4 the case where b and Aare random variables which are largely random [case (b)].
W. N. HUANGand F. A. COZZARELLI
710
3. CASE (a)---r: AND fi RANDOM
PROCESSES
(SLIGHTLY
ENDOW)
3.1. Random temperature problem Let us first consider the random temperature problem in which as we have noted only b(x) is random. Letting T?(x)= n,, equation (19) becomes 1 d%(x, t)
a&(x. t) t+8(x)h(x.
-----------_ b ~3s’
(21)
t).
The simultaneous partial differential equations (18) and (21) can be combined to yield the following two consecutive nonhomogeneous partial differential equations l3%i(x, t) p+ at2
1 _=-d4A(X,t)
aFq(x, t)
&x,----a+,
+b
at
a2
1 824
(22)
b (ix’
a%i(x, t) ax2 +q’
t)
at
(23)
We will assume that the initial conditions at t = O- are homogeneous and take the load 4 as k = 1,2,3,. . . q = d(t) sin knx (24) where d(t) is the Dirac delta function and k is the vibration mode. The physical meaning of these initial conditions and the lateral load will become apparent in the next paragraph. We may easily convert equations (22) and (23) into two consecutive partial differential equations with zero forcing and nonhomogeneous initial conditions at t = O+. In so doing we obtain the resulting equations
a%ii(x,t) +
dtxl
am,
t)+ 1
at
at2
b
a%w, 0
o
(25)
-=
axa
a2fi(X, t) aqx, t) ~ = -~ ax2 at
(26)
with the initial conditions &(x, 0”) = 0,
ati(x, o+) at
=
-- (W2 sin knx b
k
1,2,3,...
(27)
and t;(x, O+) = sin knx
k = 1,2,3, . . . ,
atik o+) = at
o
*
(28)
Equations (27) and (28) hold with probability one. Thus, we observe that the problem posed in the previous paragraph is equivalent to a free vibration problem with a nonzero deterministic velocity prescribed at t = Of. Let us consider a simply-supported beam. Using equations (25) and (26). we obtain the following boundary conditions on A(x, t) and &(x, t) respectively :
a%(o, t) &(O, t) = &(I, t) = -g-&-
a%(i, t) = ____ = 0 ax2
(29)
Damped lateral vibration in an axially creeping beam with random material parameters
771
and
a2qo,t)
~(O,t)=~(l,t)=dxZ=~=
a2ti(l, t)
ax2
o
.
(30)
Again the above relations hold with probability one. The perturbation method suggested by Keller [8] will now be applied to differential equation (25) with initial and boundary conditions (27) and (29) respectively. Bending moment. In this section it is assumed that b(x) is homogeneous random process which is slightly random, and we may then write d(x) = 1 +&(x).
(31)
Here, E is a small quantity, and C&(X)is a homogeneous equation (31) into equation (25) we obtain (L,+&@r(X,
random process. Substituting
t) = 0
(32)
where
2
L, = g+;+;
(33)
t, =8,(x);. The solution to equation (32) is not separable in x and t, and is extremely difficult to obtain. However, utilizing the perturbation method given in [8], we may obtain an approximate expression for the mean bending moment as E{A} = (1-&L~1E{~1}+&2L~lE{i;lL~l~l))m,.
(35)
Here, me is the solution to the unperturbed deterministic linear partial differential equation
a2mo am, 1 a4mo
-+at+;x at2
=0
(36)
with the prescribed deterministic initial and boundary conditions [equations (27) and (29)]. Also, in equation (35) L; 1 is the inverse operator associated with L,, and is thus an integral operator whose kernel is the Green’s function G(x, 5 ; t) satisfying (37)
with homogeneous initial conditions at t = O-. Note that in equation (35) those terms of order E and .s2satisfy homogeneous initial and boundary conditions. The solution to equation (36) can be assumed in the form m, = T,(t) sin klrx
k = 1,2,3,. . .
(38)
where T1(t) is a function of time t only. Note that the boundary conditions (29) are automatically satisfied by equation (38). Substituting equation (38) into equation (36) yields an ordinary differential equation in Ti. We now make the additional assumption that the
W. N. HUANG and F. A. COZZARELLI
712
vibration obtain
is underdamped.
Thus, setting (knf4/b--$ > 0 and using equations
(knYe _t!2 sin r,t sin km mo= - -
k = 1,2.3,.
br,
where the nondimensional
(27), we
..
frequency is given by (39b)
Next, we must obtain the Green’s function satisfying equation (37). A comparison of equations (36) and (37) indicates that the Green’s function is equal to the deterministic bending moment which results from a unit impulse applied to the beam at the location x = < at time t = 0. This Green’s function vanishes for either t < 0 or x, 5 $ [0, I]. The normal modes obtained from the homogeneous part of equation (37) with the prescribed homogeneous boundary conditions are given by the orthogonal functions Irj(x) = sin j7rx.
(40)
We now try a solution to equation (37) in the form 6(x, C:; t) = f
(41)
sj(tl t)hj(x)
j=l
whereby (42) The orthogonaIity of the normal modes ~j(x) can be used to reduce equation (42) to the system of uncoupled equations
d’qj dqj (jx)’ Tqj -p+dt+
= 26(t) sin j7r<
j=
1,2,3 ,....
Solving equation (43) for qj, we obtain with the use of equation (41)
,jlte-
‘I2sin rjt sin jnc sin jnx,
G(x, 5; t) =
0,
t2O,Olx,<<
1
otherwise
where rj is defined by equation (39b). Finally, with this Green’s function the inverse operator L; ’ operating on some function F(x, t) can be given by L, ‘F(x, t) =
Sf t
* G(x, 5 ; t - z)F( 5, t) dt dr.
0 0
(45)
Let us now return to equation (35). In order to evaluate the terms of order E and E*, we must first determine the mean 8’, and mean square e of 8,(.x) for t(x) a homogeneous random process with mean zero and variance cf. These follow directly from equations (1 I)
Damped
lateral vibration
in an axially creeping
beam with random
material
773
parameters
and (31) as d/2 $1
=
L
-1 E
@=
eW - 2 e+
+ 1
E2
(46)
’
For the term of order s2 we must also specify the autocovariance of ?(x)[C,(v]) where q = x1 -x2]. The analysis would become extremely complicated if we were to choose an exponential expression for C,(q) as in [5] and [6]. On the other hand, the analysis is much simpler if we choose an exponential function for the autocorrelation of &(x), i.e. E{f$(x,)&(x,))
= 8’: e-tVtid
(47)
where d is the nondimensional correlation distance. The autocovariance obtained in inverse fashion from equations (1 lc), (3 l), (46) and (47) as
of t(x) is then
C,(q) = ln[2 e”:12- 1 +(e2”3 - 2 ebfi2 + 1) e-tSt/d] -0:.
(48)
The following limiting values follow from this result : f;iy C,(q) = 0,’
(49)
Lim C,(q) = ln(2 e”:‘* - 1)- cr,‘. V+c=
(50)
Equation (50) gives a very small negative value for 9 + co, since Q has been assumed to be slightly random (i.e., err is small). Equation (48) has been plotted in Fig. 1 for typical values or = 0.1 and d = 0.2, and we note that we still get a physically plausible exponentiallike function for C,. Using the deterministic bending moment m, [equations (39)], the Green’s function G(x, 5 ; t) [equation (44)], and equations (45H47), we may then proceed to evaluate equation (35) and thereby obtain the rather complicated result E(rPr} = -~e~r~2sinr,rsinknx+s~e~ri2[( k
-&+rk’)
sinr,t
k
k
l[(A,
+A2t+A,t2)
k.rrx+ dy’-q6jc3k, ( d4
sin r,t
)
smjnx
]
[(Akj’ + A$) sin r,t + (A$‘+ A$) cos r,t + A$ sin rjt + Ay{ cos rjt] + C (2dY’-d~‘6C,j,k)[AY; j,= 1
sin r,t + (A?: + Ayir) sin rjt + (A?&+ A$$,t) cos rjt
Ifk
+A’& cos r,t] sin&x+
f f (2d’,‘j)-d~‘6i(k_,j,-d~)6i(k+,j,)(Alijg) i ; 2+‘j=z+
sin r,t
+ A?$)sin rit + A(2ijdsin rjt + ,44jj cos r,t + A!$? cos r,t + AyQ cos rjt) sin inx where aij is the Kronecker delta.
(51)
W. N. HUANG and F. A. COZZARELLI
774
d=a,
FIG. 1. Autocovariance
of Q--case
(a)
(5 1)
In equation
4r,2+3 A, =--16rt
A*=-$
4r: - 1 A, = ~ 16r:
A, = -rkA,
k
A”, = (4rjf - l)(rjf - rj2)- 2r:( 1 - 4rj2)
A, = -r,A,
A$i’=
2
6
4r:(r:
’
A!’
“if - ’ 4r,(r,2 - r;)
A:‘& =
=
rk - rj
Ag’ = _
Ayi =
l)+(l-4rf)(r,2
-
rk[2r?(4ri
8ri(r:
-rf)]
At& =
- rj”)’
A”’ A’:‘& = 2
Ayf zz
2 A””
A’& = - Ay&
A?;
18
r,(l-4r,Z)
=
4ri(rE - rF)(rj2 - rf) A(iA 21
=
AU.i) = 23
2rk (r: - rj’)’
r,(l-4rj2) 2rj(r: - rj2)2
A:‘; = _
A’i’i = -Af’
- rj’)’
4r,” - 1 4(ri - rj2)2
-‘k 2rj(r: - rj’)
-rk(l-4rj2) 8rj2(r: - r;) 4r: - 1
=
-
4(r,2 - r;)(r:
- rf)
r,(l-4rf) A(‘” 2o = -4r,(r,‘-rf)(rjf-rj”)
-
‘k
(rjf -rj’)(rj’-rf)
(52)
Damped
lateral vibration
in an axially creeping
beam with random
material
parameters
115
and d, =
3+8d2k2rr2+8d2k4n4 MCI
_ (e lid-l)
C:
l/d= + (k= +j=).rr=+ 8k2j+4[( _ I)“-j e- lid- 11 ~Cti’C”’ 2
d!$ = d, =
(dCy’C”‘)= 3
3
4k3jn4[1 +(-
l)k-j](e-“dc,c$VIi”
1)
1 dC,
dy’ = d;’ = d’:‘” =
4kj3z4[1+(-l)k-j](e-1id-1) C$Wy’( l/d2 +4j2n2) & 4ij2kE4[((_l)i-j+(_l)k-j)e-l/d-(1+(_l)i+k-2j)] d2C:j’C~‘(l/d2+(i-j)2z2)(l/d2+(i+j)2z2)
d(B) =
1 2dC:”
(53)
where 1 C 1 = z+4k2X2
CY' = $+(kkj)“r’.
(54)
CY’
As a particular example we will consider the beam treated deterministically in [7], namely a uniform rectangular cross-section beam made of aluminum alloy 1100. The cross-sectional dimensions are taken to be 2 in in width and 3 in in depth, and the length of the beam is taken as 36 in. The mean temperature of the beam is chosen as 260°C and the axial stress cXi = a, is taken equal to 3000 psi. The material properties are given in Table 1 (see [7X. TABLE 1. MATERIAL PROPERTIESOF ALUMINUMALLOY 1100(260°C) p = E = PI0 = I, =
1.2 x 10-10 8.0x lo6 4.55 364 X 103
lb-h2-in-’ psi
psi-h’/“0
Using the data given in the previous paragraph and in Table 1 and taking the vibration mode k to be one, we obtain the numerical values given in Table 2.
W. N. HUANG and F. A. COZZARELLI
716
TABLE 2. NUMERICALVALUES
I = 4.5
EC0 = bxi i-i2.0
inJ
“” = 0.415
hr-’
= 1.419 x lo-” = 828.68 for/i
= 1
[equation
(4)]
[equation
(20 11
[equation
(39b)]
It may be seen from Table 2 that, as a consequence of the small damping in this example (b << l), ri >> 1 and jr; - $1 >> 1 for j # i. We then find that, for any values of err and d, the expression of order E in equation (51) is completely dominated by the r,t sin r,t term and the expression of order t.’ is completely dominated by the d, A,t2 sin r,t term. Thus, we may now considerably simplify the expression for E{Gi!, with negligible loss in accuracy, to the following form which is separated in x and t : ; e-“‘sin r,t sin nx. 1 -cf$+c22d,A,~t2 EjG), = -f: 1
With the use of equations (17) (20) (39b) and (52) equation (55) may be rewritten terms of the physical parameters and dimensional time T (in seconds) as
in
(56) where
and where m* is a normalizing
factor defined as
In the present example rx = 0.7 and /I = 1158.7, and thus we see that as a consequence of small damping b >> CI. For practical purposes it is sufficient to confine our attention to the time interval over which the exponential function in equation (56) decays to one per cent of its initial value. i.e., 0 I T I 4,6/a. Within this interval, we will consider that the “slightly random” assumption implies that &22d,&:~2T2 c 1 in equation (56). This requires that i g2F 1 = e20s_ 2 e”z’2 + 1 << 42.32
c
3 + 8d2n2
4d(l/dz+4s2)+
8d2;rr4(e-lid_ I) ( l/d2 +47r2)’
11 1
(58)
Equation (58) will be used as a condition on 6, for a given d. In Fig. 2(a), a single curve showing the decay of E{h/m*),,, (LMP = locus of maximum points) for the mean bending moment [equation (56)]] at the midspan of the beam (x = )) is plotted vs time 7 in seconds for typical values g‘r = 0.1 and d = 0,0.2, co-all of which
Damped
lateral vibration
in an axially creeping
beam with random
material
Ill
parameters
.__ 0
.-_-mm__.
ki
0
J
T I -
L_y-ooo2
-
crT=O.I (Random
temp. plob.1
-0-002
sh -
t lb)
FIG. 2. Decay curves--case
----o--n,,
=0.4(Rondom
imp. prob.)
(a). (a) LMP of mean bending moment and velocity. LMP of deterministic bending moment and velocity.
(b) deviation
from
satisfy condition (58). The deviation of the LMP of the mean bending moment from the LMP of the deterministic bending moment, [E(r?t/m*} --n~e/rn*]~~~, has been plotted in Fig. 2(b) (solid curves). Note that for d > 0 this deviation vanishes at a root T = T, and that the LMP of the mean bending moment is less than the LMP of the deterministic bending moment in the interval 0 < T < 7;,, and is greater for T > ‘i;,. We also see that there exist two critical times Tmi, and T,,, where the random effect is an extremum. We obtain T, and these two critical times from equations (56) and (57) as
With CJ~= 0.1 and d = 0.2 in this example equations (59) yield 7;, = 3.147 set, Tmin= 0.877 set and T,,, = 5.131 set, which approximately equal 580, 161 and 946 cycles of vibration respectively. Also, for these values equations (lla) yield E{c?} = 1.005, and Fig. 2 yields E{A/m,},,, = 0.9977 at T = Tin = 0.877 sec. Thus, the bending moment is sensitive to small variations in temperature. Velocity. Turning now to the velocity, we obtain from equation (26)
for fi(x, t) and A(x, t) mean square differentiable with respect to time t and axial coordinate x respectively. Substituting equation (51) into equation (60) integrating the result with
W. N. HUANG and F. A. COZZARELLI
778
respect to t, and then using initial conditions
(28). we obtain
Isinr
k
t-tcosr
k
t sinkm
rk
d,(kz)2[(B,+B2t+BBjt2)sin
-cz~6?emt~z
br, + 2 j= j+k
1
d’j,‘(kn)2 sin knx+
sin kxx
rkt+(B4t+B,t2)cosrkt]
&)-Ab.
+ (B$j’+ B$‘t) cos r,t + B’$, sin rjt + By’, cos rjt] + i
1
j= jfk
(2dy’-d~‘~S~,~,,
. (jn)2[B’fi sin r,t +(B$$ + Byit) sin rjt + (Byi + B’&t) cos rjt + B’/: cos rkt] .sin jzx+
(2d:‘“-d~‘6i(k_zj,-d~‘Sick+2j,)(i~)2(B(;’js) sin r,t
i f i=l j=l
i#kfj#i
+
In equation
B$$) sin rit + Byd sin rjt + Bg) cos r,t + B$$ cos rit + Bit cos rjt) sin (61) B, =$
k
B, =$
=$
k
B, = -r,B,
B, = -rkB2
By’ =
B,
-1 r: - rj’
By: = -
r,(r,’ - 3rT) 4rj(ri - rj’)’
B$ = Zr,Byi
@’ =
@’ =
Byi
=
B$
=
(3r: - r,“) 2rjf(r: -rj2)2 - 2r, (r,‘-rj2)’
-’ -
2(r,Z r;)* rk
2rj(rjf - r;)
Byi = g J
@/&I= _
1 2(ri - rj’)(r: - rf)
.
(61)
Damped lateral vibration in an axially creeping beam with random material parameters
719
(62) Again as a specific example we will consider the beam with small damping discussed as in our analysis of the bending moment, we find that the expression of order 1 in equation (61) is completely dominated by the rk cos r,t term, the expression of order E is completely dominated by the t cos rkt term, and the expression of order c2 is completely dominated by the d,B,r2 cos r,t term. Thus, we may in this case considerably simplify equation (61) for Ef8), with negligible loss in accuracy, to the following form : in [i’]. Proceeding
E{ ti!) = (1 -s81rzT+s2UI~~2iiii2)
eWaTcos BTsin 7r.x.
(63)
Here, T is the time in seconds, and TVand /.I are as defined in equations (57). The expression for E{$} given by equation (63) is identical with the expression for E{fi/m*~ given by equation (56) except that sin fir is replaced by cos /?T Since /I >> 1, these two expressions give essentially the same locus of the maximum points. Thus, Fig. 2 also applies in this case. The velocity is clearly sensitive to small fluctuations in temperature. 3.2. reborn imperfection problem As we have previously stated the random imperfection problem and the random temperature problem are mathematicaliy equivalent. Instead of using equation (31) we start with the relation ii(x) = n,[l -+-&fir(X)]
(64)
where fir(x) is a homogeneous random process. Its mean ti, and mean square z are obtained from equations (14) and (64) as
(~~-l)2[exp~2(~)z~-2exp~~(~)2~+l] +
n2c2 0’ The autocorrelation
(66)
of iii(x) is assumed to be E{A,(x,)A,(x2)) = Te-Iqt’d
(67)
based upon the same argument given in Section 3.1. Proceeding as in the random temperature problem, we obtain the same results [i.e., equations (51), (56), (58), (59), (61) and (63)] with the substitutions
W. N. HUANG and F. A. COZZARELLI
780
The curve shown in Fig. 2(a) also applies in this problem for values a,& = 0.4 and d = 0, 0.2, m-all of which satisfy condition (58) with the substitutions (68). The deviation of the LMP of the mean bending moment and of the mean velocity from the LMP of the deterministic bending moment and velocity have also been plotted in Fig. 2(b) (dashed curves). For the values a,,& = 0.4 and d = 0.2, equations (59) with the substitutions (68) yields E{A/n,) = 1.005, and Fig. 2 yields E{A/m,),,, = E{G/w,},,, = 0.9974 at T = Tmin 740, 179 and 1088 cycles of vibration respectively. Also, for these values equation (14a) yields E{fi/n,f = 1.005, and Fig. 2 yields E(&i/m,J,,, = E{$/w,j,,, = 0.9974 at T = Tmin = 0.973 sec. Thus, the bending moment and the velocity are also sensitive to small variations in imperfection density. We conclude that both bending moment and velocity are sensitive to small fluctuations in temperature and in imperfection density. This differs from the results obtained for the 3-bar truss in [5] and statically loaded beam in [6], where the stress was essentially deterministic and only the velocity was sensitive to fluctuations in temperature and in imperfection density. 4. CASE (b)--c?
AND fi RANDOM
4.1. Random temperature
VARIABLES
(LARGELY
RANDOM)
problem
Again let us first consider the random temperature problem in which only C?is random. In this section d is taken to be a random variable which is largely random. Thus, whereas in equation (31) E may no longer be used as a perturbation parameter, ~?i(x) is simply a random variable c$. It follows that the consecutive differential equations (25) and (26) may be simplified to two uncoupled linear differential equations in A(x, t) and 6(x, t). each with a random constant coefficient, i.e., &ii(x, at2
t) +cfi.dh(x, t)+l at
d4rFi(x. t) b -=
a2qx, t) +@2=++1
at2
at
(69)
aXa
w b
o
a.2
= o,
(70)
The deterministic initial conditions are given by equations (27) and (28) and the deterministic homogeneous boundary conditions are again given by equations (29) and (30). Note that whereas equations (69) and (70) are identical, the initial conditions on #I and G are different. We shall only present the details of the velocity solution but will give the final results for both velocity and bending moment. Ve/elocity. The solution to equation (70) is separable, and may be expressed as the product of a random function of time and a deterministic function of space in the form G(x, t) = T*(t) sin kzx
k = 1,2, 3;.
..
(71)
Substituting equation (71) into equation (70), solving the resulting ordinary differential equation in Yf2(t) with the initial conditions (28), and using equations (17), (20) and (57), we obtain the velocity for the underdamped case (b < fik*/u) as
qx, t)
=
Bk* q/[@k2/a)*
a -aiT sin(a,,/[(/Ik2/a)’ - g21e
- g2]T+
0) sin kzx
k = 1,2,3,.
(72)
781
Damped lateral vibration in an axially creeping beam with random material parameters
where the phase angle is given by S = tan-’
&W/a)’
- $21 F
(73)
I.
Since there is zero probability that b in this case is greater than /?k2/a, the lognormal density function for 8 [equation (lo)] must be replaced by a truncated lognormal density function. Thus, we let 1 (744 f(&‘) = n,J(2rr)a,B exp and R, = ++erf
VW
where R, is such that equation (74a) satisfies the normalization erf x is the error function defined as J;2X) erf x = __
condition.
Note that
1 e - Y’lzdy.
(75)
s
The mean E{@ and variance gi based on the truncated lognormal density function [equation (74a)] are given by
(764 and
W-W where H, = i+erf
ln(flk’/a) - 0:
(774
cr H, = ++erf
ln(flk2/a) - 2af or
VW
I
Expressions for the first order density function and the statistical moments of G(x, t) may be found from equations (72) and (74). For example, the mean velocity is given by
sin(aJ[(bk2/a)2
- &‘]T+
f3)db.
(78)
Decay curves for the mean velocity [equation (7811at the midspan of the beam discussed in the previous section with k = 1, and cr = 0.2 and 0.5 are presented in Fig. 3. In this example p >>a (small damping), and as a result the normalization factor R, is very close to unity [equation (74b)] and the density function of the phase angle 6’ is sharply peaked around 7r/2 [equation (73)]. Results have also been obtained for or = 0.1 in the present case (not plotted in Fig. 3) and these results checked very closely with the curves for cr = 0.1 and d = cc in Fig. 2 for case (a). For these values of o, and d, b in both cases is essentially a random variable which is only slightly random.
782
W. N. HUANG and F. A. COZZARELLI
x=0.5 k=
0
I
2
3
I
4
5
6
65
4
5
6
65
7
0.03 - ( b )
0
FIG. 3. Decay
curves*ase
I
2
3
(b). (a) LMP of mean velocity and bending moment. LMP of deterministic velocity and bending moment.
(b) Deviation
from
The logarithmic decrement S, a quantity widely used as a measure of damping, is given by the expression (79) where WI and W, are the velocities at two successive maxima. It then follows from equation (72) that
(80) Also, the circular frequency in cycles per second follows as (;,=
fxJ[(p/2/ay -8’1
(81)
271 For convenience, we rewrite equations (SO)and (81) in nondimensional
s^
bJ[(pk2/a)*-
I]
& = J[(pc/a)2 - 8’1 ci,
J[(p#v/or)2-2’1
00 = J[(/?k2/lx)‘-
l]
form as
(824
@W
where 6, and o,, are the nominal values (at p = T,) given by (83)
Damped lateral vibration in an axially creeping beam with random material parameters
783
Given that d is truncated lognormal [equation (74)], the density functions of 6/S, and I%/w, follow from equations (82) as
[ln(r,(Dk2/a)(6/6,)~~' u 2 10 do
1844 Wb)
Equation (84a) has been plotted in Fig. 4 for the previously discussed example (solid curves) with k = 1 and typical values for u,. The lower limit on the logarithmic decrement
FIG. 4. Density function of &&,--case (b).
(S/6, = 0) corresponds with 6 = 0, while the upper limit (6/h, + co) corresponds with the cutoff d = p/a. Equation (85a) has also been plotted in Fig. 5 (solid curves) for the same values of k and or. Here, the lower limit on the circular frequency (o/w0 = 0) corresponds with the cutoff cfp= /3/a, while the upper cutoff (o,AB~= ~/a)/~[(~/a)‘I]) corresponds with ~5= 0. The means and the standard deviations of s^jS, and Q/o, are readily obtained by numerical integration. For G‘,equal to 0.2 and 0.5, values of these statistical properties and those for $ [equations (I I)] are given in Table 3.
784
W. N. HUANG and F. A. COZZARELLJ
- -Random
temp
----Random
prob.
imp. prob.
50-
IO-
0-999999
09999995
w/w, FIG.5. Density function
of &j/w,+ase
(b)
TABLET. MEANSANDSTANDARDDEVIATIONSOF~/&,AND~&,
or 0.2 0.5
E{C$
0s
E{&S,~
I .0202 1.1331
0.2061 0.6039
1.1331
1.0202
ud/dn 0.2061 0.6039
E{W%l 0.999999 0.999590
uwwo om12 0.0203
It may be seen from Table 3 that in this example the statistics of the logarithmic decrement are numerically equal to the statistics of the material parameter 2. This follows directly from equation (82a) since the density function of b peaks near value 1.0 and /I/al >> 1 (small damping). Applying the same argument to equation (82b), we find that the density function of G/w, is very sharply concentrated near 1.0, and this is apparent in Fig. 5 and Table 3. Thus, we find in this example that whereas the logarithmic decrement is sensitive to fluctuations in temperature, the circular frequency is almost insensitive to such fluctuations. Bending moment. Since equations (69) and (70) with their associated initial and boundary conditions are mathematically similar, we may proceed in an analogous manner to obtain the bending moment and its statistics. We obtain for the bending moment
@x.t)
=
- (PM2 2n2ci2&pk2/ct)2 -
g2]
emNiTsin(aJ[(pk2/oc)2 -J2]T)
and for the mean bending moment
where m* is defined in equation (57b).
sin knx
k = 1,2, 3,. . . (86)
Damped lateral vibration in an axially creeping beam with random material parameters
785
The expression for E{&/m*} given by equation (87) is identical with the expression for E{$j given by equation (78) except that here the phase angle 6 is equal to zero. Due to the high frequency in the previously discussed example we may expect that the locus of maximum points in that example is the same for bending moment and velocity. Thus, Fig. 3 also applies in this case. Likewise, equations (82), (84) and (85) for 6^/6, and h/o,, and their density functions also hold for the bending moment solution. Thus the bending moment and its logarithmic decrement also show significant random fluctuations as the temperature fluctuates. 4.2. Random imperfection problem Velocity. We now turn to the random imperfection problem in which only li is random. In this problem iQ,J//s, and c?+B, can be easily obtained from equations (72), (73), (82) and (83) in the random temperature problem with the substitution B+fi/n,.
(88)
Now, 6, and oO are nominal values evaluated at fl = NO. The underdamped in this problem requires that
restriction
where
(89b) Again, since there is zero probability that A in this case is greater than /?k2/d, the lognormal density function for A [equation (13)] must be replaced by a truncated lognormal density function. Thus, we write f(n) =
no -l
Q2JGWa&(n-
e~P{-~(~)2~n(~)]2~[U(n-l)-U(n-~)]
1)
(90a)
PW The mean E(A) and variance CJ~are then given by E{li} = 1 +n(no-
l)exp
2
-
(914
{:($q’)
{($)‘)r3exP{($!j2}-I]
(9Ib)
where H, = ~+erf{~[ln((“~~~)[‘)
H, = i+erf{s[ln(
(‘YE);
-(%)‘I}
(924
‘) -2( $)l).
(92b)
786
W. N. HUANG and F. A. COZZARELLI
Given that ii is truncated lognormal [equation (9Oa)], the density functions of &S, and &)/CL+, are obtained from equations (82) with substitution (88) as
Y, =
PW
_ti’i 00
= (~~-~)(~)[(~)~-~:lexp~ Z,(Z,-
l)Q,J(27c)@,nb
-- l(“1)‘[ln~z2-l)~‘) ___-._ 2
lT&
no-i
(94a)
W-4 Equations (93a) and (94a) have been plotted in Figs. 4 and 5 respectively (dashed curves) for the previously discussed example with k = 1 and typical values for a,&. The means and the standard deviations of (F/S, and cS,/W, are also readily obtained. For o,& = 0.8 and 2.0, values of these statistical properties and those for ii [equations (14)] are given in Table 4. TABLE4. MEANS AND STANDARDDEVIATIONS OF $6,
0.8 2.0
1.0201 1.1342
0.1827 0.5589
I.0201 1.1342
0.1827 0.5589
AND Q/w,
0.999938 0.994273
0@079 0.0755
As in the random temperature problem we find that in this example the statistics of the logarithmic decrement are numerically equal to those of e/n,. and that the density function of Q/w0 is very sharply concentrated near one. Thus, we again find that whereas the logarithmic decrement shows significant random fluctuation as the imperfection density fluctuates, the circular frequency is almost deterministic. Bending moment. The bending moment for the random imperfection problem may be obtained from equation (86) with the substitution (88). We then find that the statistics of
Damped
the bending discussion.
lateral vibration
moment
in an axially creeping
are identical
beam with random
with those of the velocity,
5. SUMMARY
material
parameters
and thus require
181
no further
OF RESULTS
The present study is an investigation of the effect of randomness in the material parameters .$ and ii on damped lateral vibration in an axially creeping beam. The randomness in these parameters results from randomness in the temperature and imperfection density, which have been assumed to be independent homogeneous normal random processes. Two separate cases have been considered, i.e., (a) b and A random processes which are slightly random, and (b) 8 and A random variables which are largely random. In each case the problem has been separated into two uncoupled parts-the “random temperature problem” and the “random imperfection problem”. In the random temperature problem only E, is random, whereas in the random imperfection problem only n is random. For both cases a simply-supported nonlinear Maxwell beam made of aluminum alloy 1100 has been taken as an example. In this example the damping is small, and the vibration is underdamped. In case (a) we have obtained the mean bending moment and velocity by using exponential-like autocovariances for the temperature and the imperfection density. As a consequence of the small damping, very simple and useful expressions for the mean bending moment and the mean velocity have been obtained. Two critical times, where the effect of material parameter randomness on the bending moment and velocity is an extremum. have been found. It has been observed that both the bending moment and velocity are sensitive to randomness in temperature and in imperfection density. This differs from the results obtained for the 3-bar truss in [5] and statically loaded beam in [6], where the stress was essentially deterministic and only the velocity was sensitive to such randomness. In case (b) we have found the means of the velocity and the bending moment. Comparing these means with those in case (a), we have observed that the two cases give equivalent results when the correlation distance in case (a) is large and the standard deviation in case (b) is small. The statistics of the logarithmic decrements and circular frequencies for the velocity and bending moment have also been presented. It has been found that as a consequence of the small damping the velocity, bending moment and their associated logarithmic decrements are sensitive to randomness in temperature and in imperfection density. whereas the associated circular frequencies are almost deterministic.
REFERENCES [l] D. 0. BLIVEN and T. T. SOONC. On frequencies of elastic beams with random imperfections. J. Frunklin Institute 287, 291 (1969). [2] T. T. S~ONG and F. A. COZZARELLI, Effect of random temperature distributions on creep in circular plates. Int. J. Non-Linear Mech. 2, 21 (1967). [3] H. PARKUS, On the Lifetime of Viscoelastic Structures in a Random Temperature Field. Recent Progress in Applied Mechanics-Folke Odqoisr Volume, p. 391. John Wiley (1967). [4] H. PARKUS and H. BARGMANN, Note on the behavior of thermorheologically simple materials in random temperature fields. Acia Mech. 9, 152 (1970). [5] F. A. COZZARELLI and W. N. HUANG, Effect of random material parameters on nonlinear steady creep solutions. Int. J. Solids Swucr. 7. 1477 (1971). [6] W. N. HUANG and F. A. COZZARELLI, Steady creep bending in a beam with random material parameters, J. Franklin Institute 294, 323 (1972).
788
W. N. HUANG
and F. A.
COZZARELLI
[7] F. A. CUZZARELLI, J. Wu and S.
TANG, Lateral vibration of a nonlinear viscoelastic beam under initial axial tension. J. Sound Vibr. 13, 147 (1970). [S] J. B. KELLER,Stochastic Equations and Wave Propagation in Random Media. Proc. Symp. on A& Math. 16, 145 (1964). [9] J. AITCHISONand J. A. C. BROWN, The Lognormai ~jsfrjbuf~on. Cambridge
(Received 16 Augur
Akrparr-Mccnenyerccn
University
Press (1957).
1972; rccised 26 November 1972)
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