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Stability of Viscoelastic Structural Members under Periodic and Random Loads
217
CHAPTER 3 STABILITY OF VISCOELASTIC STRUCTURAL MEMBERS UNDER PERIODIC AND RANDOM LOADS
In this chapter dynamic stability problems are studied for viscoelastic structural members subjected to ageing under periodic and stochastic compressive loads. The mechanical behavior of structures is described by linear governing equations. In order to derive stability conditions we employ the direct Lyapunov method and construct stability functionals. This method is well-known in the theory of stability for functional differential equations, see e.g. Burton (1982,1983,1985), Corduneanu (1991), Hale (1977), Kolmanovskii & Nosov (1986). We suggest new stability functionals which utilize the specific properties of relaxation measures formulated in Chapter 1. By using these functionals, explicit restrictions to compressive loads are developed. The influence of rheological and geometrical parameters on the critical load is studied both analytically and numerically. Section 1 deals with the stability problem for a viscoelastic cylindrical shell under the action of a time-varying axial compressive force and a distributed radial traction. Section 2 is concerned with stability for the zero solution of a linear integro-differential equation with a periodic coefficient. As an example, stability conditions are derived for a viscoelastic bar compressed by a periodic longitudinal force. Section 3 is devoted to the analysis of the mean square stability for a viscoelastic cylindrical shell compressed by axial and radial stochastic loads of the white noise type. In Section 4 we derive some stability conditions for a viscoelastic bar driven by random compressive loads. We confine ourselves to non-ageing viscoelastic materials and develop some explicit restrictions on deterministic and stochastic components of external forces. Section 5 is concerned with an extension of the results derived in Section 4 to integro-differential equations with operator coefficients with nonconvolutive integral kernels.
Chapter 3
218 1. STABILITY OF A VISCOELASTIC SHELL UNDER TIME-VARYING LOADS
In this section the stability problem is studied for a viscoelastic cylindrical shell under the action of a time-varying axial compressive force p = p(t) and a uniformly distributed radial compressive load q = q(t). Our purpose is to derive restrictions on these functions which guarantee the dynamic stability of a viscoelastic shell. The stability conditions for a viscoelastic shell under a time-independent load were derived by Drozdov et al. (1991). Stability of viscoelastic structural members under time-dependent compressive forces was analysed e.g. by Akbarov et al. (1992), Belen'kaya (1987), Belen'kaya & Yudovich (1978), Bogdanovich (1973), Bolotin (1985), Cederbaum & Mond (1992), Eshmatov & Kurbanov (1975), Matyash (1967), Moskvin et al. (1984), Rahn & Mote (1993). 1.1. Formulation of the problem and governing equations Let us consider a thin-walled, circular, cylindrical shell with length 1, radius R and thickness h. Introduce orthogonal coordinates (x1 ,x 2 ) in the middle surface of the shell, such that r1-axis coincides with the longitudinal axis. Denote by y(t, x i , x 2 ) the shell deflection. We assume that: (i) deflection y is so small that the nonlinear terms with respect toy and its derivatives can be neglected in the formulas for the strains and curvatures of the middle surface; (ii) Kirchhoff's hypotheses hold; (iii) the forces in the middle surface are essentially larger that the transverse and inertia forces, and we can neglect the transverse and inertia forces in the equilibrium equations for the stresses in the middle surface; (iv) the displacements in the middle surface are sufficiently less that deflection y, and we can neglect them in the formulas for the curvatures of the middle surface; (v) the material behavior obeys the constitutive equations of a linear viscoelastic solid (1.5.46) with the relaxation operator (1.5.22), (1.5.26). The relaxation measure Q0 (t) satisfies conditions (1.6.35) — (1.6.37) and (1.6.51). Then y satisfies the motion equations, see e.g. Volmir (1967), Phy(t, ci, x2) + E* [D 2 y(t, xi, x2) — z +ph
—
[
2
. (t i
ac2i
C1, c
t
Jo
2 Qo(t — s) D y(s, c i
a2y (t, x i, x2) + g Ra x2 (t, x i , x2) =
2
—
h a2
i
f i Qo(t — s) ac? (s, C1 , x 2 )ds] = DZ Y(t,
Cylindrical shell under time-varying loads
219
with the initial data y
y It=o = yl(xi, x2)•
It =o = y0(x1,x2),
Here p is the material density, Y(t, xi , x2) is the Airy function, yo is the initial deflection, y1 is the initial speed of the deflection, E* = Eh3[12(1 — n2 )]-1 is the bending rigidity, E is the Young modulus, v is Poisson's ratio, D is the Laplace operator. We will not fix the boundary conditions, and suppose only that these conditions permit the following expansions of the solutions 00
m,n=1 00 0
=
S
m,n=1
ir mx1
Ymn (t ) sin
1
¢,,,h (t) sin
sin R ,
pmcl ttx2 sin . 1 R
(1.2)
t
Substitution of (1.2) into (1.1) yields rhymn(t) + Amn (P, 4)ymn (t) + Bmn
~ Q0(t
ymn ( 0) = y 0, mit
— S)ymn (S)ds = 0, ymn ( 0) = y 1, m p
(1.3)
Here Amn (P, 4) = E*Amn +
Eh ( irr 4D-i 1 ) mit
R2
2 h 2 , [rh( pm 1 ) + 4R( R ) ]
B,hp = E* Dpap +
Eh pm 4 D -1 > 0, ( 1 ) mn R2
Dmh
= [( pI )2 + ( R)2]2,
where yo, mn , yi, mi are the Fourier coefficients for the initial conditions and Omn
—R
n
. (~ ~ )2 = « m~
Introduce the following Definition. A shell is stable if for any inequality f
S ( U, mn + y 1, mp )< d
m,n=1
implies
S 00
sup y t>0 m,n=1
( )<
S.
e > 0 there is a d >
0 such that the
Chapter 3
220
The purpose of this section is to derive some restrictions on functions p(t) and q(t) which would ensure the stability of a viscoelastic shell. 1.2. Stability conditions In order to derive stability conditions we introduce the functionals
n$) = rhymn(t) + [Amn (t) + BmnQO(t)1ymn(t) f Q 0 (t t
—Bmn
— s)(ymn(t
) 2
t n$, mn = IRrhy m h ( t) + Bmn J~ (Qo(t — s) —
Qo(f))ymn(s)ds]
+ph[Amn (0 + BmnQo( o's)]y2mn (t),
(1.4)
(1.5) Vmn = N + &mn1 lm, where Amn (t) = Amn (r(t), q(t)) and parameters amn > 0 will be determined below. It follows from (1.3) — (1.5) that Vmn = [Bmn (Amn (t) + Bmn Qi(00)) ]'t (QP( s) — QO(oo))ds 0 — amp BmnQO (t) — (Ph + amh )Ah,p (t)]ymn (t) —
Bmn f t IR~mnO(t ~ s) C( `i0(t —
(Amn(t) + Bmn QO ( f))
s) — Qi(°c))(ymn(i) — ymn(s))
BmpIAm) + BmnQO()] Let us assume that
_
j [Qo(t —
s)
— QO() ] y
amn = mio IRAmn(t) + BmnQO(oo)] > 0.
2 ds
(s)ds.
(1.6)
(1.7)
Set amn = T2 amn, where constant T = ~2 is determined by formula (1.6.51), and suppose that
G
mion IR
t
(Ph + amn ) m~~ I Amp (t) I
f (Qo(s) — Qo( oo ))ds 0 According to (1.6.55), the function in the brackets decreases monotonously. Taking into account that
j
Bmn amn
[Qo(s) — Q0(oo)Jds =fIQO(s)1sds 0
221
Cylindrical shell under time-varying loads we can rewrite this inequality as
amn Bmn
IQo(s) I sds.
m ó 1 Amn(t) 1< _ ph +Taamn to
It follows from (1.6.55) and (1.6) — (1.8) that [~ t >0:
(1.8) mn
< 0. Therefore, for any
2
T2 am n ym n (t) < [T amnfimn (0) + ph(Amn (0) + BmnQs(ss))] yÓ,
+ph(ph + T2 amn )yl
m~
mn
This inequality implies the following Theorem 1.1. Suppose that inequalities (1.7) and (1.8) hold. Then a viscoelastic shell is stable. q
1.3. Example Let us consider a viscoelastic cylindrical shell under the periodic axial compressive force p(t) = Po lR- R l sin wt, where R o , Pl and w are positive constants. Introduce the dimensionless variables P: R
Rt'
*
Eh
_ ~T*, T * _v
/
pR12 p2Eh
According to Theorem 1.1, sufficient conditions of shell stability are
po + r < min F(C), m2F(C)[F(V) — r —14] Pw *~ * < N min m ,' 1 + (T/T* ) 2 m2 [F(C) — RPó where _ n2 h nmR
n!
R [ h!
F(C) = [1
+ fR
v2 )
2
+ ( ]'
1 ~Qo(s)~sds. T* [l + Qo ( f)] j
The minimal value of function F(V) is F0 =
+ Qo(oo) ‚/3(1 — n2) 1
i]
Chapter 3
222 For a fixed m value, the function
m2 F(F — r — Pi )[ I + (T/T*)2 m 2 ( F — Pó —
r* )]
-1
increases monotonously with the growth of F. Therefore this function reaches its minimal value with respect to when F = F0 . For F = F0 , this function reaches its minimal value with respect to m for m = 1. Therefore, a viscoelastic shell is stable if 1+(T/T*)2(
iw* < NFo(Fo — R~~— R~ )[
P
r + R~~
Fo — R~~— R1)]_ i •
(1.9)
~~
3.2
* ~max
0 0
0
R~~~
0
0.4
Figure 1.1: Dimensionless critical frequency w~,, of the sinusoidal load vs dimensionless amplitude ri of the periodic part of the compressive load. The calculations are carried out for v = 0.3 and ró = 0.3. Light points correspond to c = 0.4, and black points correspond to c = 0.8. Let us consider the standard viscoelastic material with relaxation measure (1.5.40) Qo(t) =
—(1 —
Eo)[1 — e
cr( —)j,
Linear equation with periodic coefficients
223
where E and E0 are the current and limiting elastic moduli, respectively, and T is the characteristic time of relaxation. The second condition (1.9) can be presented as follows: * P i w < F1 k(1 —
c)(F0
—
Ró
—
2 Pi){c[1 -1- k (F0
—
Ró —
-1
ri)]}
,
(1.10)
where c = E0 /E and k = T/T*. The right-hand side of (1.10) reaches its maximal value with respect to k when k = ( F0 _r0 — R~ )-112°
(1.11)
This means that a viscoelastic shell has maximal reserve of the stability when the characteristic times for material T and for structure T* are related by formula (1.11). The critical frequency of the periodic compressive load is = F0( 1 — c)(F0 — ró —
r~ )1/2 /( 2CR*1)•
The dependence of w~,, on amplitude ri is plotted in Fig. 1.1. The numerical results show that the dimensionless critical frequency is not very large except for a narrow domain of the pi values in the neighbourhood of zero where w,*,. tends to infinity as ri —> 0. When parameter c grows, the critical frequency increases for "large" pi values and decreases for "small" ones.
2. STABILITY OF A LINEAR INTEGRO-DIFFERENTIAL EQUATION WITH PERIODIC COEFFICIENTS This section is concerned with the stability analysis for a linear integrodifferential equation with periodic coefficients. A similar equation arises in the dynamics of thin-walled viscoelastic elements of structures under periodic compressive loading. The equation under consideration has a specific peculiarity which makes its analysis difficult: in the absence of the integral term it is only stable, but not asymptotically stable. Therefore, in order to derive stability conditions we are to introduce some specific restrictions on kernels of the integral operator, which are taken from the study of relaxation measures for linear viscoelastic media. Employing the direct Lyapunov method and constructing new stability functionals, we derive some new sufficient stability conditions which are close enough to the necessary ones. In particular, when the integral term vanishes, our stability conditions turn into the well-known stability criterion for a linear differential equation with periodic coefficients, see e.g. Yakubovich
224
Chapter 3
& Starzhinskii (1975). In the general case, the proposed stability conditions have the following mechanical meaning: a viscoelastic structure under periodic excitations is asymptotically stable if the corresponding elastic structure is stable and the material viscosity is sufficiently large. As an example, the stability problem is considered for a linear viscoelastic bar compressed by time-periodic loads. Explicit limitations on the material parameters are developed which guarantee the bar stability, and the dependence of the critical relaxation rate on the material viscosity is analysed numerically for different frequencies of the periodic compressive load. Stability of ordinary differential equations with periodic coefficients was discussed in detail by Erugin (1966), Malkin (1956), McLachlan (1964) and Yakubovich & Starzhinskii (1975). It was shown that time-periodic loads can induce parametric oscillations with unbounded growth of small initial perturbations. This analysis was based on the Floquet presentation of the fundamental matrix of a linear differential equation with periodic coefficients. An extension of this theory to functional-differential equations was suggested by Stokes (1962) and Hale (1977). Parametric instabilities in elastic systems were studied e.g. by Bolotin (1964), Evan-Iwanowski (1976) and Herrmann (1967). Some results in the stability theory for viscoelastic thin-walled elements of structures were obtained by Belen'kaya (1987), Belen'kaya & Yudovich (1978), Bolotin (1985), Cederbaum & Mond (1992), Eshmatov & Kurbanov (1975), Matyash (1967), Moskvin et al. (1984), Rahn & Mote 91993) and Stevens (1966). It is worth noting that the stability conditions were obtained in these works either by using simplified constitutive equations reducing the integro-differential equation to the differential one, or by applying approximate methods (averaging techniques, multiple scales analysis, etc). To the best of our knowledge, in this sphere of applications the stability conditions have been studied in detail only for integro-differential equations with constant coefficients, but up to date there is no general theory for the case of time-varying, and, in particular, periodic coefficients. In this section we develop the direct Lyapunov method for such problems and derive some sufficient stability conditions. 2.1. Formulation of the problem and basic assumptions Let us consider the following integro-differential equation: ii(t) -} a(t)u(t) + b J t Qe (t — 0
s)u(s)ds = 0.
(2.1)
Here t(i): [0, cc) [0, cc) is an unknown function, a(t) is a positive, continuously differentiable, periodic function with a period T, b is a positive constant, Qo (t) is the integral kernel. Denote by ao = T-1 a(t)dt the mean value fo of the function a(t). We suppose that ao > 0. Function Q0 (t) is assumed to
Linear equation with periodic coef~cients
225
be twice continuously differentiable and to satisfy conditions (1.6.35)-(1.6.37) and (1.6.51). Let us consider the differential equation ü a(t) u = 0,
(2.2)
which corresponds to the case when the integral term in (2.1) vanishes. In the new variables u1 = u(t), u2 = II(t), Eqn. (2.2) can be re-written in the matrix form
= A(t)U,
(2.3)
where U
r = G ~Z
1,
A(t)=
r
~ a~t)
~~.
L
Introduce vector-function V(t) which satisfies the adjoint differential equation V = -A7'(t)V,
(2.4)
where the superscript T denotes transpose. Let F(i) —
F12( t)
F22( t) J
be the fundamental matrix for Eqn. (2.4), i.e. the solution of (2.4) with the initial condition V(0) = I, where I is the unit matrix. Here we employ nonstandard notation for elements of matrix F(t) in order to emphasize that functions F (t) located in any column of this matrix satisfy independent scalar equations: 'ii = a(t)4'12, 4
'21 = a(t)F22,
4)
12 =
—
'22 =
Ol1, —
(2.5)
O21.
For a constant coefficient a = a0 , matrix F(t) can be presented as follows: G
—
s c l Q sin ~a o t
Qos ln aot ct
, '
(2.6)
Similarly, let
*(t)
[ 4'12(t) ~22(t) J
be the fundamental matrix for Eqn. (2.3). It can be shown that *11(0 = F22(t), *21(1) = —4 '12(1),
012(t)
=
022(t) = F11(i).
(2.7)
226
Chapter 3
It follows from the Liouville theorem that for any t > 0 det ' I'()= det F(t) = 1.
(2.8)
Using this fact we can write the characteristic equation for matrix 0(T) as follows: A2 — 1i (0(T))l + 1 = 0,
(2.9)
where I is the first invariant of matrix. It is well known, see e.g. Yakubovich & Starzhinskii (1975), that the zero solution of (2.2) is stable if and only if all eigenvalues of matrix 0(T) are single and lie on the unit circle. Hence, according to (2.9), the zero solution of (2.2) is stable if and only if I 11 (0(T)) l< 2.
(2.10)
Our objective is to extend this result to integro-differential equation (2.1). Namely, we will derive conditions of asymptotic stability for the zero solution of Eqn. (2.1), which turn into criterion (2.10) when the integral term in (2.1) tends to zero. 2.2. Stability conditions In this subsection we derive some sufficient conditions of asymptotic stability for the zero solution of Eqn. (2.1). In order to formulate these conditions let us introduce the following notation. Denote by f() the functions: f 1(t) = ao fi i (t)f12 (t) + F21(t)F22 (t), f2(t) = aoF i2(t) + Fi2(t),
fa(t) = aof?1(t) + Fi1(1).
(2.11)
It follows from (2.5) and the above formulas that functions f( t) satisfy the differential equations = a(t)f2 — f3,
/2 =
—2
fi(0,
(2.12)
fa = 2a(t)ff
with the initial conditions 0
fi ( ) = 0 ,
f2( 0 ) = 1,
f3(0) = a o .
It is easy to check that for a constant a = a 0 , f 1 (t) = 0,
f2(t) = 1,
f3( 1 ) = a0.
Let á(t)= a(t) + bQo (oo). Introduce the functions _ F(t)
max{~(t),0} (t)b á
F1(t) = :L(t)
a(t) '
F2 (t) = f2(t) ~(t)
Linear equation with periodic coefficients
227
Denote by Y, U1 and U2 their maximal and minimal values
U = sup F(t),
U1 = sllR IFi(t) I,
U2 =
t>o
t>0
inf F2(t),
t >o
We assume that = min á(t)> 0, t>0
min[a s + bQo (oo) f2 (t)] > O. t>0
(2.13)
Note that for a constant coefficient a(t) = ao , the second ine q uality (2.13) follows from the first one. Let H(t) =
Jo
[Qo(s) —
Qo (oo)jds.
It follows from (1.6.52) that function H(t) is bounded.
Theorem 2.1. Suppose that conditions (1.6.35)-(1.6.37), (1.6.51) and (2.13) are fulfilled, U1 <
min{4
'H(oo) + T 1Q0(00) ~~
}~
_l U2
(2.14)
and Y<
1(oo)U2 — a [H(oo ) +Ti ~ Qo (oo)~}U1 Y2 — ~Yl
(2.15)
where a = 3TT 1T2 . Then the zero solution of E qn. (2.1) is stable. q In order to prove Theorem 2.1 we construct the Lyapunov functionals L 1 (1), ..., L6 () such that L 6 (t) is positive and non-increasing in time. Let us introduce new variables z1 = u1f11 + u2 4112,
z2 = u1f21 + u2 f22.
Differentiation of these expressions with the use of (2.1) and (2.5) yields z l = —b(12(t)
J
0
(t — s) u 1 (s)ds,
/ Qo (t — s)ul (s)ds. o Let us consider the function z2 = —b4
(2.16)
22 (t)
L 1 (t) = 2 [ao zi (t) + z(t)J.
(2.17)
For a constant a = a , employing (2.6) we find that L1 = i [a o u2(t) + 2 (t)j, i.e. L1 eq uals the total energy of the conservative system without the integral term. Differentiation of (2.17) with the use of (2.16) leads to the e quality L i = —b[f
i (t)u(t)
+ f2(t)u(t))
Jo
Qo(t —
s)u(s)ds.
(2.18)
228
Chapter 3
In order to transform the second term in (2.18) we introduce the functional L2(t) =
jr0
2
Qo(t — s)[ u(t) — u(s)]
ds — Q
o
(t)u 2 (t).
(2.19)
Differentiating (2.19) and using (2.1) and conditions (1.6.35) we arrive at t
L2 = f
o
t — 2~(t) l Q0 (t —
(t — s)[ u(t) —
u(8)]2 ds
s)u(s)ds — Q 0 (t)u 2 (t).
0
(2.20)
In follows from (2.12), (2.18) and (2.20) that the derivative of the functional L3 =
2L1 (t) — bf
2 (t)L 2 (t)
(2.21)
can be written as follows: L3 (t) = 2bf1 ()[L 2 (i) — u(t) jr o o
Qo(t — s)( u(t) —
0 (t
u(s)) 2 ds —
— s)u(s)dsj Qo(t)u2 (t)]
Since —
u(t)u(s) =
[(ti(t) — u(s))2 — u2 (t) — u
2
(s)],
(2.22)
the latter formula has the form L3(t) = bf1(t)[3L2(t) — jr — bf 2 (t)[
rt
Q0 (t — s u )(t (— )
o
u(s))g ds —
Jo Let us consider now the functional L4(t) = 2{~(t) + b rt [Qo(t — s) — Q
o
Qo(t —
s)u2 (s)ds] 2 Qo(t ) u (t ) j •
o(f)] u(s)ds}2.
(2.23)
(2.24)
Differentiation of (2.24) with the use of (2.1) yields L4 (t) = [t~(t) — bQ
0 (oo)u(t)
+ b l t Q0 (t — s) u(s)ds] N2L4 (t) o =
This relation together with (2.24) implies that the derivative of the functional L5(t) = L4(t) + ~(t)u2(i)
(2.25)
229
Linear equation with periodic coefIcients
can be calculated as follows: L5 (t) = 24(t)112 (t) — á(t)b u(t) j [Qo (t — s) — Q
o()] u(s)ds .
We transform this equation with the use of (2.22) and finally obtain 2L 5 (t) = ~(t)u 2 (t) — á(t)b{H(t)u 2 (t)
— j
+j
[Q o (t
)]u2(s)ds
[Q o(t — s) — Qo(
— s) — Q o()] ((t) —
u(s)) 2 ds}.
(2.26)
Let us introduce the functional L 6 (t) = L3(t) + 2ßL 5 (1),
(2.27)
where b is a positive constant which will be determined below. It follows from (2.23), (2.26) and (2.27) that L6 (t) = — ~(t)b[G(t)u 2 (t) + /
+ Jo
lt o
G1 (1, 1 — s) u2 (s)ds
G2 (t, t — s)(i() —
u(s)) 2 ds],
(2.28)
where G(t) = 1[ß(t) —
-b ] + 3F1(t)Q8(t) —
Gi (t, t) = Q[Qo (t) — Qo (oo)] + Fi (t)Qo(t), G2(t, t) = F2(t)Q o(t) — b[Q o(t) — Qo( oo)] — 3F i(t)Q0(0).
We estimate functions G1 and G2 using Egns. (1.6.54) and (1.6.55) as follows: Gi(t, r) > [b — T 1 ~ Fi(t)~~ ][Qo(r) — Qo(oo)J , G2 (i, t) > [~2 2 F2 (t) — 37'1 1F i(t)1 — b][Qo( t) — Qo(oo)i. It follows from these inequalities that functions G1 (t, r) and G2 (t, i- ) are nonnegative if U1 < bTi < 3a — ß(Y 2 — o Ui ).
(2.29)
Eqn. (2.14) implies that (2.29) is valid, and we put -1 b = 3(aT1 ) (U2 — & U1 ).
In this case, Egns. (2.13) and (2.28) yield L 6 (t) < —á(t)bG(t) u 2 (t).
(2.30)
Chapter 3
230
Substituting expressions (2.21), (2.25) into (2.27) and employing (2.17), (2.19) and (2.24) we get t
— bfz(t)[J
o
L6(t) = a o c~~(t) + z(t)
Qo(t — s)( u(t) — u(s))
+b{[~(t) + b 1 ( Q o (t — s) — Q
z
ds — Qo(t)u2(t)]
o(oo))u(s)ds]
2
+ a(t)u 2 (t)}.
(2.31)
It follows from this formula and conditions (1.6.36) and (1.6.42) that for any t >0 2 L6(t) > b~(t)u2 (t) +aoc~ (t) + ci(t) +bQo(f)fz(t)u (t).
Substitution of expressions for z1 and z2 into this inequality with the use of (2.11) yields 2 (t) + r(t, u(t), ii(t)), L6 (t) ? 0á(011
(2.32)
where r(t, u4, uz) = [f3(t) + bQo(f)f2(t)]u~~+ 2f1(t)u4uz + fz(t)u2. The quadratic form r(t, u~, u2 ) is positive definite if and only if for any t > 0 [fs(t) + bW o(f)f2(t)]fz(t) — f () > 0.
(2.33)
It follows from (2.12) that the expression f 2 (i)f 3 (i) — f() is independent of t and equals a 0 . Substitution of this expression into (2.33) implies that ao + bQo( oo) fz (t) > 0. This inequality together with (2.32) and condition (2.13) yields L 6 (t) > bku 2 (t).
(2.34)
Putting t = 0 in (2.31) and utilizing the initial conditions for matrix function F(t) we find 2
L6(0) = [a o + bá(0)]u(0) + (1 + 19)~iZ(0).
(2.35)
Integrating (2.30) from 0 to t and using (2.34) and (2.35) we obtain bku2 (t) <_ [a 0 + b~(o)]u2 (0) + (i + b)~2 (o) + br(t) f t 112(s)ds, o
where r(t) = max á(s)~ G(s)~ . 0(3 Ki
Linear equation with periodic coefficients
231
This formula together with the Gronwall inequality implies that 112 1
( )5
Q
2 [(~o + b~(0))?1 (0) + (1 + b) ~ 2 ( 0)]
t
x [1 + — ~ G(t) jt exp(b
G(t)dt)ds].
J
(2.36)
Substitution of (2.36) into (2.30) yields Ls(t) <
[(~o + b~(0))u2(0) + (1 + b) it2 ( 0 )] t
1'(t)
x G(t)[1 + 6
J
G(t)dt)ds].
t
exp( —~
J
Integration of this inequality with the use of (2.35) implies that there is a continuous, monotonously increasing, positive function M(t) such that for any t > 0 Ls(t) < M(t)[(a o + b~(0))u2(0) + (1 + a)~2 ( 0)].
(2.37)
Since functions F1 (t), F2 (t) are bounded, conditions (1.6.35) and (1.6.36) imply that for any € > 0 there is t o (e) > 0 such that for any t > t o (e) I G(t) — G(t) l< €,
(2.38)
where Goo (t) = iRb[H(oo)
(t~b ]
3
+ F1(t)Qo(c )•
Estimating this function we find Goo(t) > 1[1(oo) —Y —
3 3-1
/
IQo(oo)~ U1] •
It follows from this inequality, (2.14) and (2.15) that there is an € > 0 such that G(t) > 2e1 for any t > 0. Choosing c = € and applying (2.38) we obtain that G(t) > q for t > i (e l ). This inequality together with (2.30) implies L 6 (t) < — k 1 bu 2 (t) for t > t 0 ( 1 ). It follows from this formula and (2.34) that u 2 (s)ds] < L6(t o(e~ )), k[Qu 2 (t) + e l b 1 t o (~ I )
t> _ t o( 6 0.
Finally, by estimating L6 (t 0 ( 1 )) with the use of (2.37) we find that for any t > to(ci): k[/3u 2 (t) + ei b
ft
J o( E,) u2(s)ds] t
< M(io (ci ))[(ao – b~(0))ti 2 (0) + (1 + r)~2(0)]•
Chapter 3
232
This inequality implies Theorem 2.1. q Remark. For the standard viscoelastic material with relaxation measure (1.6.7), inequalities (2.13) — (2.15) can be presented as follows: min a(t) > cb, t>0
Y1
6
"s > bc max f2 (t), t>0
U1c U< gRY2-6 7U2 — 3 UI U
(2.39)
2.3. Stability of the integro-differential equation and the corresponding ordinary differential equation The above stability conditions for Eqn. (2.1) are formulated in such a way that, at first sight, they have no connection with the well-known stability condition (2.10) for Eqn. (2.2). In this subsection it is demonstrated that our conditions of asymptotic stability for Eqn. (2.1) are fulfilled if and only if the zero solution of (2.2) is stable and functions a(t) and Q0 (t) satisfy some additional conditions. We confine ourselves to the non-critical case when all the eigenvalues of matrix F(T) are single. Denote by X(i) a matrix function with components X;3 (t), and by 0(n) a matrix function with components Y13 (n). Function C(t) is determined in [0, Ti, and coincides with F(t) in this interval. Function 0(n) is determined for nonnegative integers n and equals F(nT). It can be shown that F(t) = X(t — nT)O(n),
nT < t < (n + 1)T.
(2.40)
Putting t = (n + 1)T we obtain from (2.40) that Y(n + 1) = F(T)Y(n). Since Y(0) = 4)(0) = I, this equality implies that 0(n) = F'a (T).
(2.41)
Substitution of (2.40) into (2.11) yields
± (C11(
f1 (t) = ao[C11 (t — n T)X12(t — nT)Y 1 (n) nT)X (t — nT) + C12 ( — nT)X21 (t — nT))Y i i (n) Y12(n) 22
hT)C22(t — nT)Y i2(n)] + [Cii(t — hT)Ci2(t — hT)fz i(h) nT) + C12 ( — hT)C21 (t — hT))f2i( n)022(n) + C2i (t — n T) X22 (t — n T)q4(n)j +C2i(t —
± ( C11(t — nT)X22(t —
Let us consider the quadratic form 2
= X11(t)~ 2 + bi2 (t)~h + X22 (t) h2 , where b 12 () = Cii(t —
b11 (i) = Cii (t — hT)Ci2(t — nT), hT)C22 (t — nT) + C12 ( — hT)C2i (t — nT), d22(t) = C2i(t — n T) X22 (t — nT).
233
Linear equation with periodic coefficients
It follows from (2.3) and (2.8) that det X (t) = det X (0) = 1 for any t E [0, Ti. Eqn. (2.5) implies that there is a sufficiently small t 1 > 0 such that X11(t1) > 0 and C12 (t 1 ) < 0. Therefore, for t = t 1 -I- nT, the form L(t, x, h) is negative definite, and there is a d > 0 such that L(t 1 + nT, x, h) < — d(x 2 + h2 ). This means that f 1 (i 1 + nT) K — d[ao(Yi i(n) + 0i2(n)) + (Y (n) + 0i2(n))]•
It follows from this inequality that there is a d l > 0 such that (2.42)
I fi (ti + nT) I ? d1 11 0(n) ~~ 2 .
Equalities (2.41) and (2.42) imply that 11 Fn (T) 112 . sup ~~ f1(t) > d 1 max n
(2.43)
t>0
For any positive integer n, we have Fn(T) ~~ > rn( f(T)), where p is the spectral radius of matrix, cf. e.g. Horn & Johnson (1985). This inequality together with (2.43) leads to the estimate sup 1 f1(t) I> d 1 max n
r2n
t>0
(F(T)) = d1 max
r2n
(y(T)).
(2.44)
Obviously, (2.45)
f2( 0) = 1. t>ó f2(t) ~
According to (2.44) and (2.45), stability condition (2.14) is valid only for r(y(T)) < 1. On the other hand, it is known, see e.g. Yakubovich & Starzhinskii (1975), that r(y(T)) > 1. Therefore, inequality (2.14) implies r(tY(T)) = 1. For the non-critical case, the necessary and sufficient stability condition (2.10) for Eqn. (2.2) follows from this equality and (2.9). Thus, we have proved that for the non-critical case, the stability conditions for integrodifferential equation (2.1) imply the stability conditions for the corresponding differential equation (2.2). Now we derive an inverse result, namely, we show that the stability of ordinary differential equation (2.2) ensures the stability of integro-differential equation (2.1) under some assumptions regarding kernel Q0 (t) of the integral operator and the periodic coefficient a(t). Stability of equation (2.2) implies boundedness of fundamental matrix function F(t). This means that there exists a positive constant c1 such that for any t > 0: F~,i (t) ~
(i, J = 1, 2).
It follows from this inequality and (2.11) that there is a positive constant c2 such that for any t > O f() I < C2,
0 < f2(t) < C2,
0 < f3(t) <
C2.
(2.46)
234
Chap ter 3
Let us prove that there is a positive constant c3 such that for any t > 0 J2(Í) > c3 .
(2.47)
Suppose that this hypothesis is not true. Therefore, there is a sequence {t m } such that f 2 (t m ) < m-2 . It follows from this inequality and (2.11) that 4 I F12(tm) I< cm ,
4 I F22(tm) I< c 4 m' ,
where c4 = max(1, a o 2 ). These estimates together with condition det F(t m ) = F n.(tm) f22(tm) —
F12(tm)F2 i (tm ) = 1,
see (2.8), imply that 1 < c m -1[I Fii(tm) I+ I F12(tm) I] < 2c4 rn -1 NF~i (tm) + Fi2(tm) < 2c4m-1 ,fs(tm)• Therefore, f 3 (t m ) > [m/(2c4)]2 . Since this inequality contradicts (2.46), our assumption is not true and inequality (2.47) holds. It follows from formulas (2.46) and (2.47) that stability condition (2.10) for ordinary differential equation (2.2) yields U1 < c2 , U2 > c3 . In this case, inequality (2.14) can be treated only as a restriction on function Qo (t) which guarantees asymptotic stability of the zero solution of (2.1). For a given kernel Qo (t) satisfying (2.14), inequality (2.15) can be considered as a restriction on the coefficient a(t) which ensures the asymptotic stability. 2.4. Stability of a viscoelastic bar under periodic compressive. load In this subsection we derive stability conditions for a rectilinear viscoelastic bar under the action of a compressive load. Let us consider the plane bending of a viscoelastic bar with length 1, cross-section area S and moment of inertia of the cross-section J. At moment t = 0, compressive forces R = P(t) are applied to the bar ends. Under the action of external forces, the bar deforms. Denote by y(t, x) the bar deflection at the point with longitudinal coordinate x at moment t > 0. We suppose that (a) function y and its derivative are so small that we can neglect the nonlinear terms in the expression for the curvature of the longitudinal axis; (b) the hypothesis regarding plane sections in the bending is fulfilled; (c) the stress s(t) is connected with the strain e(s), (0 < s < t), by the constitutive equation of a linear viscoelastic solid (1.6.1). Under the above assumptions function y(t, x) satisfies the equation pSy(t, x) + E J[D4 y(t, x) + J t Q0(t — s)D4 y(s, x)ds] 0
+R(t)D2 y(t, x)
=0
(2.48)
Linear equation with periodic coefficients
235
with the initial conditions y(O, cR) = n2(cR)•
y(O,cR) = n1(x),
Here p is mass density, n~ (x) is the initial deflection, n2 (x) is the initial speed of deflection, D is the operator of differentiation with respect to x, Dy = áy/áx. 10
o
O
o
7
G
•
O
. **
*
0
O
.
O
¤
•~ * * + *
*
O
*
.
*
O
*
,
0
~ O
~
~~
o
C~
1
Figure 2.1: Stability domain for a beam driven by periodic excitations. The calculations are carried out for Po = 0.2/3e and P1 = O.1/3e , i.e. for m = 0.125. Light points correspond to w = 0.2, black points correspond to w = 0.5, and asterisks correspond to w = 0.8 .
We confine ourselves to a simply supported bar with the boundary conditions D2 y(t, 0) = D2 y(t, l) = 0.
y(t, 0) = y(i, l) = 0,
In order to satisfy these conditions we seek a solution of Eqn. (2.48) in the form f
= S un (t) sin -i
n
phc l
(2.49)
Chap ter 3
236 Substitution of (2.49) into (2.48) yields 4 pSii (t) +
4
!)// Pe\n2 ) un (t)
14 E J [(1 —
+j Qo(t — s) t
up (s)dsj = 0,
(2.50)
where Pe = i2 EJ1-2 is the Euler critical force. Suppose that the load /3(t) has the form P(t) = PO + P1 sin W!,
where R0 , P1 and W are positive constants. In the following, the main stability region is analysed, and only the term with n = 1 is considered. Eqn. (2.50) implies that
~(t) + [i + m sin wt
I
(t) +
j
.(t. — s)u(s)ds, = 0,
(2.51)
where
t, = t/T, w = w7, 4
T
_ / r$1
P1
R Qo (t)
Let us restrict ourselves to the stability analysis for a bar made of the standard viscoelastic material with the relaxation measure (1.6.7). Applying Theorem 2.1 to Eqn. (2.51) we obtain the following stability conditions: 1—
M> Pe C(Re
where
and y1
72
—
Po) -~,
11<7<72,
are the roots of the quadratic equation
Ug2 72 — (3 gg1 + cg2 )7 + 6cU1 = 0. The boundaries of the stability region in the (c, 7)-plane are plotted in Fig. 2.1. The numerical analysis shows that the lower boundary of the stability region is very close to the abscissa axis practically for the whole range of w values. The upper boundary of this region has an essential maximum for c .: 0.35 which corresponds to half the maximal admissible c value. With the growth of frequency w, the stability region significantly decreases and disappears for w > 1. This means that for a fixed intensity of the compressive load a bar driven by periodic forces is stable only for sufficiently small frequencies of periodic excitations.
Cylindrical shell under random loads
237
3. STABILITY OF A VISCOELASTIC SHELL DRIVEN BY RANDOM LOADS In this section we derive stability conditions for a viscoelastic cylindrical shell considered in Section 1 under the action of stochastic compressive loads P(t) = ro + Pi wi(t),
4(t) = qo + gi ~2 (t),
(3.1)
where w(t) are standard Wiener processes. Theory of stochastic processes and stochastic differential and integrodifferential equations has been in the focus of attention during the past three decades, see e.g. Gikhman & Skorokhod (1969), Schuss (1980). Here we confine ourselves to the stability problems for stochastic integrodifferential equations. The stability theory for stochastic differential equations was discussed e.g. by Hasminskii (1980). The problems of existence and uniqueness for stochastic integro - differential equations were studied e.g. by Mao (1989), Mizel & Trutzer (1984), see also Andreeva et al. (1992) and the bibliography therein. Gikhman (1980) analysed these problems for stochastic partial differential equation of hyperbolic type. Some stability conditions for stochastic integro - differential equations with delay were developed by Hausmann (1978), Ichikawa (1982), Kadiev & Ponosov (1992), Kozin & Prodromou (1971), Mao (1990), Mizel & Trutzer (1984), Potapov (1993), Zelentsovskii (1991). In these works some different concepts of stochastic stability were considered: almost sure stability, stability in mean square, stability with probability 1. Below we restrict ourselves to the analysis of the mean square stability for the zero solution. Stochastic differential and integro-differential equations describe the dynamic behavior of physical systems driven by random noises. A few examples of these systems were presented by Hotsthemke & Lefever (1984). Stability of elastic systems under the action of random excitations was analysed by Bolotin (1979) and Dimentberg (1989). Stability and instability of elastic bars and shells were studied e.g. by Asokanthan & Ariaratnam (1992) and Lepore & Stoltz (1972). Some stability conditions for viscoelastic structural members under the action of random perturbations were derived by Drozdov (1993, 1994), Drozdov & Kolmanovskii (1991, 1992), Potapov (1984, 1989), Potapov & Marasanov (1992), Tylikowski (1991). In this chapter we confine ourselves to the study of structures driven by "white noise" type excitations, see Egns. (3.1). This model describes random forces with a rapidly decreasing correlation, see Horsthemke & Lefever (1984). A more adequate model of random loads is a colored noise. Stability of nonlinear dynamic systems under the action of colored noises was analysed by Klosek-Dygas et al. (1988). The objective of this section is to derive restrictions on forces pi* and qi in Eqn. (3.1) which would ensure the mean-square stability of a viscoelastic
Chapter 3
238 shell. 3.1. Stability conditions
Substituting (3.1) into (1.3) we obtain the following Ito integro - differential equations:
phdu 2, mn =— IAmpu1, mn() t) + Bmn
rt
du1, mn = u2, mn(t)dt9
QO(t
— s)ui, mn (s)ds ]dt
Jo0 +C~i)u1 mn (t)*ui (t) + C,i,2)
1,
m.n(t)dw2(t )•
(3.2)
Here 111, mn = ymn (t)+
U2, mn =
irr
A
mn _ Amn(Po +90),
m
2
C~2) — h
=Pih(_ )2+
Definition. A shell is stable in the mean-square sense if for any e > 0 there is a d > 0 such that the inequality ~ / (yo, mn + yl, mn ) < d S m,n-1
implies sup e t>0
00
E
m,n-1
1l2, mn()
< e,
where symbol E denotes mathematical expectation. Calculate the differential of functional (1.5) with Amn = A. Using Ito's formula and (3.2) we obtain t
dVmn = f — Bmn
—
[ Bmn (—amnQO(t)
l[ mnQO(t — s) — a
JrO
( Am n + Bmn Qi(oO))
x (Qo(t — s) — Qo(f))I(ui, mn(t) —
2 ui, mn(8)) ds
° + (Amp + BmnQ0( f)) j t (Qo(t — s)
— QO(f))ds)
—(1 + am.n )(C( bi)2 + C,V 2)2 )Iu~ , mn (t) (QO(t — —( Amh + BmnQO(f))Bmp
J
ph
s) — QO( f))u1, (s)ds}dt mn
+2u1, mn(t)[( 1 + Ph
)Rh u2, mn(t)
239
Cylindrical shell under random loads
2) 1 ))ui, mn.(s)ds][C~ >dwi(t) + C dw2(t)]. (3.3)
±Bmn j (Qo(/ — s) — Qo ( Suppose that rriih[A m,n
an
+ Bmn Qi(Oi)] > 0.
(3.4)
and choose a mn = T2 [Am n +Bmn Qo (oo)] > 0, where T = T2 is determined according to (1.6.51). Integrating (3.3) from zero to t, taking the mathematical expectation and using (1.6.55) and (3.4) we find
SVmn (l) — eVmn ( 0 ) <
Hm p ( S)eu1, mit('
(3.5)
where
Hmh (t)- Bmn [`4mn + Bmn Q0 (f)] C
[ — T2 Q0(t) +
I
(Q0 (1 — s) — Q0 (f))ds]
2 2)2 {1 + P [Amn + BmnQO(00)]}[C2 + c~ ].
h It follows from (1.6.55) and (3.4) that fImn (t) < 0.
Suppose that H,nn (oo) > 0 for any positive integers m and n. This condition can be written in the form C(1)2 + C12)2 < —
Bmn [Am,n + BmnQs(oo)]
1 + T2/(rh)[A, nn + Bmn Q0 (°°)] Jo
I~~(s) sds.
(3.6)
Then Eqn. (3.5) implies eNmn(t) < e Vmn(0)•
Substitution of (1.5) into this inequality yields Theorem 3.1. Suppose that conditions (3.4) and (3.6) hold. Then a u scoelastic shell is stable in the mean-s q uare sense. q
3.2. Example Let us consider a cylindrical shell under a random radial compressive load q . In the new notation 4~~
4;. R
q1 R
7
—
/ rR 1/
2
,
240
Chapter 3
stability conditions (3.4) and (3.6) can be written as 2.10-5
*
ql
er 0•
0
k
0
—>
10 3
Figure 3.1: Dimensionless critical intensity of the random load qi cr vs the
ratio k of the characteristic time for material T to the characteristic time for structure T* . Light points correspond to c = 0.4, and black points correspond to c = 0.8. qó < 4'ó cr
qi
~ 4~~er'
where qó *z 9i er
= min I
cr =
min F( p), n
NF(n)[F(n) — qó] + (T/T
)2 2 [F(n)
(3.7)
— go],
and F(n) = [1 + Qo( O)]{ 12(1 2 n2 ) R)z[1 + ( jR )2]2
4 ± -14---
R z -z [ I ± ( ~) ] }.
Let us consider a viscoelastic shell with R/1 = 0.2 and h/R = 0.001. The material obeys the constitutive equation (1.5.45) with the relaxation measure (1.6.7) and Poisson's ratio v = 0.3. Calculation of the first minimum in (3.7) yields qó er = 5.863 10-6 c.
Viscoelastic bar under random loads
241
For qo = 0, the dependence of the critical random load qi ,. on k = T/T* is plotted in Fig. 3.1. The numerical analysis shows that the critical random load reaches its maximal value for k 50 — 70 and has a relatively narrow peak of the maximum. Therefore, by choosing an appropriate characteristic time of material we provide a significant reserve of the shell stability. All the above considerations correspond to the case of isothermal loading. Let us now consider the shell stability when the temperature can change. We suppose that admissible variations of temperature are not very large, and we can neglect their influence on elastic moduli. The relaxation measure is assumed to be highly sensitive to the temperature changes. The material behavior is governed by the standard time-temperature shift principle, see e.g. Christensen (1982), Pipkin (1972). This phenomenon takes place, for example, in the neighbourhood of the glass-rubber transition point in polymers. In this case, the deterministic critical load qó is independent of temperature, whereas small variations of temperature imply significant changes in the stochastic critical load qi , as we pass through the peak on the curves plotted in Fig. 3.1.
4. STABILITY OF A VISCOELASTIC BAR DRIVEN BY RANDOM COMPRESSIVE LOADS In this section the stability problem is studied for a non-ageing viscoelastic bar driven by random perturbations of "white noise" type. Using the direct Lyapunov method some sufficient stability conditions are derived. Stability of elastic bars and shells under the action of a random compressive load was studied by Asokanthan & Ariaratnam (1992), Bolotin (1979), Lepore & Stoltz (1972), Potapov (1985). It was shown that purely elastic structural members are unstable under the action of "white noise" excitations. In order to derive stability conditions the materials were assumed to exhibit the viscoelastic behavior (Voigt's model). It was proved that the critical load is proportional to the viscosity coefficient. For simple integral models of viscoelasticity, stability of viscoelastic bars was analysed by Potapov (1984, 1989), Potapov & Marasanov (1992) by using explicit solutions for the corresponding Kolmogorov equations. 4.1. Formulation of the problem and basic assumptions Let us consider plane bending of a rectilinear bar with length 1, crosssection area S and moment of inertia of the cross-section. J. The bending
242
Chapter 3
occurs in the plane which passes through the longitudinal axis of the bar and its axis of symmetry. The bar is made of a non-ageing linear viscoelastic material with the Young modulus E and the relaxation measure Qo (t — s), which satisfies conditions (1.6.35) — (1.6.37) and (1.6.51). At moment t = 0, external forces P = P(t) are applied to the bar ends. Denote by y(t, x) the bar deflection at the point with longitudinal coordinate x E [0,1] at moment t E [0, oo). Function y and its derivatives are assumed to be so small that all the nonlinear terms in the expression for the curvature of the longitudinal axis can be neglected. Function y(t , x) satisfies the equation t pSy(t, c) + EJ[D4 y(t, c) -} l Qo (t — s)D 4 y(s, x)ds] o +R(t)D2 y(t,x) = 0,
(4.1)
initial conditions (2.1.33) and one of boundary conditions (2.1.6) — (2.1.8). Here p is mass density, and the superscript dot denotes differentiation with respect to time t. Suppose that 3 R(t) = Po + / 1,14 ),
(4.2)
where w(t) is the standard Wiener process. Egns. (4.1), (4.2) together with initial conditions (2.1.33) and boundary conditions (2.1.6) — (2.1.8) describe plane bending of a viscoelastic bar driven by random compressive load. According to Gikhman (1980), this initialboundary problem has a unique generalized solution provided initial data yo (x) and y1 (x) belong to the Sobolev space WZ with the norm ~k ' II 2 = j [D2yi(r)]2dC. Definition. The bar is stable in the mean-square sense if for any e > 0 there is a d = d(e) > 0 such that the inequality II yoII+IIyi I< implies the estimate sup Ey2 (t, x) <6, t,s
x E [0,1],
t E [0, oo).
Our objective is to derive some restrictions on Po and R1 , which would ensure the bar stability. 4.2. Transformation of the governing equations
Denote by y„ the maximal value of the bar deflection. Introduce the
Viscoelastic bar under random loads
243
following dimensionless variables and parameters: c*
_
c I
t t*
,
,
yo (c) ni(c *) =
To
y(t, c ) / u1(t* i x* ) = y*
y*
y(t c)To
u2 (t* i c* ) —
y*
n2 (x) = yi(c)To y*
, i
w* (t* ) =
1'~l2
E JTo o = 2, a = pSl 4 7
EJ
w(t)Tr
i
3
,
r1~o -1 * = PSI2 ,
1
where constant T2 is determined by inequality (1.6.52). According to Gikhman & Skorokhod (1969), w* (t * ) is a Wiener process. In the new notation, Eqn. (4.1), (4.2) and (2.1.6) — (2.1.8), (2.1.33) can be presented in the following form (for simplicity asterisks and argument x are omitted): du1 = u2 (t)di, du2 = — a[D4 u i () + j Qo(t — s)D4 u1(s)ds] — Ri D2 ui(t)dw(t), o ui( 0 ) = ni, u2( 0 ) = n2, 2 2 D u1(t, 0) = D u1(t,1) = 0, u 1 (t, 0) = u i (t, 1) = 0, Du l (t, 0) = Du i (t,1) = 0, ul(t, 0) = u1 (,1) = 0, Du1 (t, 0) = D2 u1(t, 1) = 0. u 1 (t, 0) = u1 (t, l) = 0,
(4.3) (4.4)
(4.5)
Denote by Yk (x) the eigenfunctions and by l k the eigenvalues for the differential equation D 4 Y + l D2 Y = 0
(4.6)
with one of the boundary conditions (4.5). It is well known that there is a sequence of eigenfunctions {Y„(x)} such that
0
),:
I D Yh (c)D9/Im (2)dx = dhm,
Jo ~ D 0 n(x)D 2
2
0 m(x)d C = lndnm,
(4.7)
where 6flm is the Kronecker delta. Sequence {Y„(x)} is complete in the subspace of Wz whose elements satisfy boundary conditions (4.5). Therefore, functions u~(t, x) and n( x) can be expressed in the form:
z S m=l f
u~ (i x) =
~ m( t )Ym( c),
S m1 00
n~ (t ic) =
S~m (t )Ym (x )•
(4.8)
244
Chapter 3
We substitute (4.8) into (4.3) and (4.4), multiply each e q uality by D2Y (x) and integrate from 0 to 1. Integrating by parts and utilizing (4.5) and (4.7) we find dz1 = z 2 (t)dt,
dz 2n = — al~[(1 — Ro l,~ 1 )z i,•,(t) +
jo Qo (t —
s)z in (s)ds]dt
+Rilh zi n(t)dw(t) i (n = 1, 2,...).
zin(0) = z1n,
z2n(0) = V2n,
(4.9)
It follows from (4.5), (4.7) and (4.8) and the Cauchy ine quality that for any r E [0,1] and > 0:
ui(t, x) = [
Jo
r
G
c
Jo
< ~[Du i (t , x)] 2 dx =
c
c
Du i (t, x)dx]2 ~ f [Du i (t, ~)]2 dx f dx o
S=i m,h
o
zim (t) cih(t) f ~ D~m (x)D~n ( c)dz
o
f
= Scih(t)i /
11 n~ 11 2 =
i
I
Jo
c
[D2v~ ( x )]Zd =
f
S V~mV~ h J m n=1 /
/r
n=i
i
/'
/,
D2 Ym(x)D2 Yn( c)d2
,
~
=
lhz
.
(4.10)
4.3. Stability conditions In this subsection we prove the followin g Theorem 4.1. Suppose that Po < )4 [1 + Qo(oo)]
(4.11)
and Oo
<
~
1± [al?(1 + Qo(oo) — Ro lT 1)]-1 Jo
~ Qo(s)1sds.
(4.12)
Then a viscoelastic bar driven by a random compressive load is stable in the mean-s q uare sense. q 4.3.1. Remarks 1. Condition (4.11) coincides with the necessary and sufficient stability
Viscoelastic bar under random loads
245
condition for a viscoelastic bar under the action of a deterministic compressive load R0 , see Chapter 2, Section 1. 2. Let us consider lateral oscillations of an elastic bar with the Young modulus E0 = E[1 + Qo(oo)]. It can be shown that the maximal period of natural oscillations is T
= 2t
p514 E0 J li
Denote by Pe = E° Jl 11-2 the Euler critical force for an elastic bar with the Young modulus E0 . Then stability conditions (4.11) and (4.12) can be written as follows:
P° < P1 2
1 2
—
f- 1Qo(s)1sds
Po/Pe
1 + 4p (Ts /T)2 (1 —
1,
R0/Pe)
1
+ Qo( oo)
( 4.13 )
3. Let us consider the standard viscoelastic solid with the relaxation measure (1.5.40) Qo(t) = —(1 —
1
É )[
— exp(-7`- )],
where E and E0 are the current and the limiting moduli of elasticity. In the absence of the deterministic component of the compressive load, PO = 0, stability conditions (4.13) take the form
I
R1
I< Re[( E o
-
1)To]' 1
2 ° 2 -1 [ + 4p (7) ] .
4.3.2. Proof Let us suggest some preliminary estimates. According to (4.11) there is a positive constant integer n > 1 a„ = 1 + Qo(oo) —
R° l~, 1 > 1 + Qo (oo) —
Ro li 1 > a.
a such that for any (4.14)
Function «5(x) = a x 2 [1 + Qo(oo) —
Ro x -1]
increases monotonously for x > R0 [2(1 + Qo (oo)]-1. This assertion together with (4.11) implies that for any integer n > 1 (4.15)
Chapter 3
246 where Y„ = f(l,2 ). Calculate the derivative with respect to time of function L(t) =
—
Q0(t) + j [Qo (t — s) — Qo(c )lds.
Employing (1.6.55) we obtain L(t) =
~~0(t) + [Qo(O — Q
-
o (oo)1
< 0.
Therefore, for any t > 0 L(t) > L(oo). With the use of (1.6.36) this inequality can be presented as follows: fR
L(t) > f 0
[Qo(s) - QO(f))ds.
Integration by parts with the use of (1.6.36) and (1.6.52) yields L(t)
f
> s[Qo(s) - Qo(f)]J-~~- f
Qo (s)sds =
j
~Qo (s)~ sds.
(4.16)
It follows from (4.12) that f
~Qo (s)~ sds - Pi a -1 (1 + 0A1 ) > 0.
This inequality together with (4.16) implies that for any t > 0 and any integer n>1
L(I) - R a-1(l+ Y )>0.
(4.17)
Now let us construct the Lyapunov functionals. First, calculate the differential of the functional W1 (t) = z (i) + al{[1 + Q 0 (t) — -
j
Ro l
1
jz (i)
z i (s)J 2 ds}.
0(t — s)[z1() —
(4.18)
Ito's formula and Egns. (4.9) imply that
Wih( t) = -al~ {-QO(t) c~h(t) + f -Ria
1
o
t Qo(t —
s)[z1 (t) — cih( s)] 2 ds
zim (t)}dt+2R1 ln cln (t)c2n (t)dw(t).
Now calculate the differential of the functional t 2 h (t — s) W2 (i) = Z 2 (t) + al,, ~ [Q o — Q0( f) j cih(s)ds.
(4.19)
(4.20)
Viscoelastic bar under random loads
247
It follows from this equality and (4.9) that dW2i (t) = -al~ o„ z 1,a (t)dt + R1 l„ z~„ (t)dw(t).
(4.21)
Egns. (4.9), (4.20) and (4.21) imply that the differential of the functional
W3 (i) = w2„ (t) + Y„ [cip (t) + W1(t)J can be calculated as follows:
(4.22)
dW3n (t) = -al~~Yp {[-Qo(1) - R~~a -1( 1 + Yp 1 )]z~p (~ ) +J
- s)[ci n (t) - z (s)J 2 ds
o(
o
[Qo(t - s) - Qo(f)]zin(s)ds}dt Jo + 2 R1lmz1»(t)[W2~ (t) + Y z 2 ()]dw().
+2z(I)
(4.23)
Transform the expression in the right-hand side of (4.23) 2z(1)
j
[Qo(t
=-
Jo
[Qo(t - s) - Qo(f)]z i n(s)ds (
-
s)
-
Qo l f)][ Z i(t ) - zi(s)]2 ds t
+z~z(t) j [Qo(i- s) - Qo()]ds t + [Qo(t — s ) — Qo()]z(s)ds.
j
Utilizing this equality we find
+ j[
dW3„(t) = -al~ fp {[L(t) - Ri a -1(1 + F„ 1 )]c~„(t) o (i - s) -(Qo(t
-
s)
+ ro t
-
Qo())][ zi(t) - zi(s)] 2 ds
[Qo (t —
s ) — Qo (oo)]ci „ (s)ds }dt
+2R1l„z1„(t)[W2n(t) + Ynz2n(t)]dw(t).
(4.24)
Integration of (4.24) from 0 to t and calculation of the mathematical expectation with the use of (1.6.52), (1.6.55) and (4.17) yields e W3„(t) < £W 3m (0). Substitution of Eqn. (4.18), (4.20) and (4.22) into this inequality implies that
S{c2p(t) + al n
f
[Q o(t —
s) —
QO(f)]zip( s)ds}2
Chapter 3
248 +YhR £{ c~h(t) + Z(t) + al~ [(1 +Qo(t) —
-
j
Rol
)z~hR(t)
0(t — s)( ci(t) — z i(ls))2ds]}
< — Y[1 + al,2, (1 + Qo (0) — Ro lp 1)]V~p + (1 + Fp )V2h It follows from this inequality, (1.6.35), (1.6.36), (4.14) and (4.15) that
aal(1 + i al~ )£z?hR (t) < (1 + al~ )( al,zihR +
(4.25)
Eqn. (4.25) implies that there is a positive constant c such that for any integer n >1 £ zihR (~) < c(V~hR + z2hR) •
(4.26)
Sum up inequalities (4.26) with respect to n from 1 to infinity. Using (4.20) we obtain oo
£ SR z (t) < c( h=1
11
11 2 ±11 12 112 ).
The assertion of Theorem 4.1 follows from this inequality and (4.10). 0. 4.4. Instability of an elastic bar under random compressive load
Let us consider plane bending of an elastic bar with the Young modulus E. The bar is compressed by the forces P(t) = Ro + P1 ~(t) applied to its ends. It is assumed that P0 < R .
Let us choose the dimensionless initial perturbation in the form
0 ni = ,
n2 = di~i (x),
where d is an arbitrary positive constant. The dimensionless variables ti,(t,x) are as follows: UI(t,x) = d zi(t)Yi( x),
u2(t, x) = dz2(t) Y1( x),
where functions z1 (t) and z2 (1) satisfy the equations dz1 = z 2 dt, dz 2 = —a li(1 — P oli 1 )z1dt+ Pi l1z1dw(t)
(4.27)
with the initial conditions zß(0) = 0,
2.2(0) = 1.
It follows from (4.27) and Ito's formula that the deterministic functions X~ (t) = £ci (t),
C2(t) = £zi z2(t),
C3(t) = £z2(t)
Viscoeiastic bar under random loads
249
satisfy the differential equations
·
l=
C3 = qC1 — 2FRC2
X2 = -FXi + C3,
2c2,
(4.28)
with the initial conditions C2 (0) = 0,
x1 (0) = 0,
C3 (0) = 1,
where N = ~IAT.
FR = aA7(1 - i 0 A1l),
The characteristic equation for system (4.28) is written as follows:
f(k) = k 3 + 40k — 28 = 0.
(4.29)
For P1 Y 0, function f(k) grows monotonously, f(0) = —28 < 0 and f (oo) = oo. Therefore, Eqn. (4.29) has only one positive root k1 = k. Two other complex conjugate roots can be found as k 2, 3 = 2 where w 2 = 3k 2 + 160> 0. It can be shown that solutions of Egns. (4.28) have the form C1 (t) = —[ k(Ci cos — +w(Ci sin 2
C2 sin 1)
+ C2 cos 2 )] exp(— 2) + 2C3 k exp(kt), 2
C2(t) = l [( k — w2)( C1 cos 2 — C2 sin 2 ) ki 2 +2kw(Cl sin + C2 cos 2 )] exp(— 2) + C3 k exp(Kt), 2 wt C3(t) = [(28 + k0)(Cl cos 2 —C 2 sin ) ±Ow(Cj sin
2
+ C2 cos 2 )] ecr(-
where 42 1
=
8(9
2
+ w2) '
2
)+
2C3 (8 — kFR) e cr(kt),
(4.30)
2 2 3 K 2 + CV 3k 2 ) C3 — qw(9k 2 + w2) ' 28(9? + w 2 ) . 2
C2
=
It follows from (4.30) that for any d > 0 lim sup £u1 (t, x) = oo. oo xE[o,1] Therefore, an elastic bar driven by random load is unstable in the mean-square sense for any nonzero intensity of "white noise" excitations. This result was obtained by using another technique by Ariarathnam & Xie (1990).
250
Chapter 3
According to this result and Theorem 4.1 the material viscosity plays a two-faced role in the stability problems. On one hand, viscosity leads to a decrease of the critical force for deterministic loading, i.e. to a reduced resource of stability. On the other hand, it implies a growth of the critical random loads and an increased resource of stability for structures driven by "white noise" perturbations.
5. STABILITY OF A CLASS OF STOCHASTIC INTEGRO-DIFFERENTIAL EQUATIONS In Section 4 the stability problem was analysed for a system of stochastic Volterra integro-differential equations with convolution kernels of integral operators. In this section we extend the above results to the case of nonconvolution kernels. The corresponding equations describe the mechanical behaviour of viscoelastic structural members subjected to ageing under the action of random forces. We derive explicit conditions of stability in the mean square sense. These conditions are obtained by using the Lyapunov direct method and constructing stability functionals. As examples, we consider the stability problem for an ageing viscoelastic bar under stochastic compressive loading and formulate some conditions on the load, which ensure the bar stability for arbitrary relaxation measures and for various types of end supports. 5.1. Formulation of the problem In this subsection the stability problem is formulates for a class of stochastic integro-differential equations. To explain why this class of equations is considered, we begin with a typical problem of the bar stability under the action of stochastic loading. Then, we generalize this problem introducing some additional assumptions about external loads. Finally, we replace these mechanical problems by the problem of the mean square stability for stochastic operator equations. Let us consider a rectilinear viscoelastic bar with length 1, cross-section area S and moment of inertia J. At moment t = 0, compressive forces P are applied to the bar ends. Under the action of external forces, the bar deforms. Denote by y(t, x) the bar deflection at point x at moment t > 0. We suppose that (1) function y and its derivatives are so small that we can neglect the nonlinear terms in the formula for the curvature of the longitudinal axis, (2) the hypothesis regarding plane sections in the bending is fulfilled.
251
Stability of stochastic equations
For the linear viscoelastic material (1.6.3), function y(t, s) satisfies the equation, cf, (4.1), pSy(t, x) = —EJ[D
4
y(t, x) — ( -- (t, s)D4 y(s, x)ds] — R D2 y(t, t) Jo
(5.1)
with the initial data ji( 0,x) =
y(0,x) = y0(x),
and one of boundary conditions (2.1.6) — (2.1.8) D2 y(t, 0) = D2 y(t, l) = 0, Dy(t, 0) = Dy(t,1) = 0, Dy(t, 0) = D2 y(t,1) = 0.
y(t, 0) = y(t, I) = 0, y(t, 0) = y(t,1) = 0, 11(1,0) = y(1, l) = 0,
(5.2)
Here P is mass density, yo (x) is the initial deflection, yl(x) is the initial speed of deflection, D is the operator of differentiation with respect to x, (Du = áu/óx). Relaxation measure Q(t, s) is assumed to satisfy the constitutive restrictions (1.6.61) and (1.6.62). Suppose that (5.3)
P = PO + R1~(t),
where Po , P1 are constants, w(t) is a standard Wiener process and 11)(t) is a white noise. Expression (5.3) allows the deflection of a viscoelastic bar to be studied under the action of a random load with a rapidly decreasing correlation. Substitution of (3.3) into (5.1) yields (Ju l = u 2 (t,x)dt, du 2 = —[Au
i (i,x)
—
l
(t, s)Bu l (s, x)ds]dt — C ul (t, t)dw(t)
(5.4)
where ui = y(1, r),
u2 =
~ D2 . A = p(EJD4 + P1 D2 ), B = PS D4 , C = P
(5.5)
The specific form of operators A, B and C is determined by external load and types of support. Egns. (5.5) correspond to a bar under the action of compressive forces applied to its ends. For a bar on an elastic Vinkler foundation, operator A has the form, see e.g. Volmir (1967),
A=S P
(EJD4
2
+ RoD + kI),
(5.6)
252
Chapter 3
where I is the unit operator and k is the rigidity coefficient. Some other examples of operators A are given by Drozdov et al. (1991). As a generalization of the above mentioned problems, we will consider the system (5.4) under the following assumptions: HI: A, B, C are selfadjoint commuting operators; H2: B is a positive definite operator, its inverse operator B-1 is bounded. It follows from HI that there is a sequence of the eigenfunctions {Yi (x)} such that (yi, y j) = iii and Ayi = lA y
,
B0i = lB Yi i
CYi
where VA, lB and l are the eigenvalues of A, B and C, respectively, d the Kronecker delta, and (j,.> denotes the inner product in L2. Function ui (t, x) can be presented in the form ui(t,x) = S Ui(t) y~ (c),
is
(5.7)
d =~
where Ui (t) = (u i (t, x), Yi (x)). Introduce the following Definition. The zero solution of system (5.4) is stable in the mean square sense if for any e > 0 there exists a d > 0 such that the inequality (v1, Av4) + (12,12) < d implies sup £(u i (t), u 1 (t)) < e, t>o
where £ denotes the mathematical expectation. This definition means that if the total (kinematic & potential) initial energy is small, then the displacements are small in the mean square sense for any moment of time. 5.2. Stability conditions In this subsection we derive some sufficient stability conditions for the zero solution of system (5.4). For this purpose we construct Lyapunov's functionals under the assumption that min (lA — I B0 I lB) > 0,
(5.8)
Stability of stochastic eq uations
253
where 1 Ro 1= s p Ro(t), R0 (t) = R(t, t) = 1 u
( ( s, t)ds.
First, we employ the functional W~ (t) = (12 (t),112(1)) + u1(t), Au1(t))
According to I to's formula and (5.4), we find dW1 (t) = [2
/ . 1
Q (t, s)(u2 (t), Bu1 (s))ds + (Cu' (t), Cz1 (t))]dt a
—2(u
2 (t),
Cu i (t))dw(t).
(5.9)
Introduce the functional W2(t) = W1(~) ±
J
s)(u1(t) — uI(s), B(ui(i) — u i (s)))ds.
Eqn. (5.9) implies dW2 (t) = [-2Q(t, 0)(u2 (t), Bu i (t)) + (Cu1 (i), Cu i ()) +
02Q
-(i, s)(u
J o atas
— ui(s), B(ui(t) — ui(s)))ds]dt —2( u 2 (t), Cu 1 (t))dw(t).
It follows from this relation that the functional W3(t) = W2(t) + Q(t, 0)(ui (t), Bui (t)) has the differential dW3(1) = [ ±1
dtQ
(t, 0)( i (t), 1uß (t)) + (Cul(t), Cui(t))
` á á (t, s)(u1(t) — ui (s),B(ui (i) — u i (s)))ds]dt —2( u2 (t), Cu i (t))dw(t).
(5.10)
We now introduce the functional W4 ( ) = 112 (t) + / R(t, s)1 u i (s)ds , 3o
where function R(t, s) is the relaxation kernel satisfying conditions (1.6.67) (1.6.69). Using Ito's formula we obtain from (5.4) d1474 (t) = —( A — Ro (t)B)ui (t)dt — Cui(t)dw(t).
Chapter 3
254 It follows from this equality that the differential of the functional / W5(t) = (W4(t), B
-1
W4(t))
has the form dW5(t) = [-2(u2(t), ( AB 1 - Ro(t)I)u1(t)) + (Cu1(t),
2 J]
'
0
B-~C
ul(t))
R(t, s)((A - Rp (2)B)u i (t), /11 (s))ds
-2(Cu l (t) , B-1 u2 (t) +
o
R(t, s)ul (s)ds)dw(t).
(5.11)
Let W6(t) = W5(t) + (u1(t), ( AB-1 - Rfl(t)I)ui(1))• This equation together with (5.11) implies that 1 1 dW6(t) = [ k (t)(u1(0, 4(t)) + (Cu1(i), B Cu1~t)) -2 t R(t, s)((A - Ro (t)B)u i (t), u i (s))ds]dt J0
-2(B-1Cu1 (t),u2 (t) + J R(t, s)Bu i (s)ds)dw(t). (5.12) 0 Transform the third term in the right-hand side of (5.12). Using (5.7) we have
-2
J
0
R(t, 5)((A - Ro(t)B)u i(t), ui(s))ds
= -2 s(l4 - Ro (t)lB) J R(t, s)U: (t)U1(s)ds ~
= S(
Ro(t)l8)
J
R(t, s)(U=(t) - UU(s)) 2 ds
- °
R (t) S(kA i=1
00
- S(lA -R0(t)l) i=1
where R°(t) =
~ o
R(t, s)ds.
Introduce the functional W7(t) = S6 (t) + aW3 (t),
i
( R(t, s)Us (s)ds,
Jo
(5.13)
Stability of stochastic equations
255
where a is a constant to be determined below. It follows from (5.12) and (5.13) that (t, 0 )(uß (t), Bui (t)) + 00 1(t), (aI + 13-1)C2 u i (t))
dW,(t) = { —
+
S g
l t [al6
~
R° (t) (lA — Ro(t)lB)½?(t)
R4(t)(u1(t), u1(t)) —
at a (t,S)
+ (l - Ro (t)lB)R(t, S)](U,(t) - U (s)) 2 ds —
S(l 4 — R o (t)l6) I t R(t, s)U,?(s)ds}dt i
—2(Cu
~=1
i (t), (aI
0
=i
+ B-1)u2 (t) + l R(t, s)u i (s)ds)dw(i). o
Integrate this equation from 0 to t and calculate the mathematical expectation. We obtain £W7() - ew,(o) / Ti o)(ui(T)i Bu i (T)) — ~ a ( Jo f + (tYi (T), (oJ + B-1)C2 ui(T)) — R° (T) S (lA — R o(T)lB)U ?(T) d=1 =e
— S Jr [— alB a(t, s) — ( lA — Ro( T)lB)R( T, s)] ~ =i o 2 C(U (T) — U,(s)) ds
—
Ro (t)l6) jo
S( E=1
R(t, s)U?(s)ds}dt.
(5.14)
Choose a — R4I l 2 lA I a=T max ~~ A6 where constant T = T2 is determined by Eqn. (1.6.62). Condition (5.8) implies that a > 0. It follows from this inequality, (1.6.61), (1.6.64), (5.14), and the properties of Ito's integrals that — eW,(0) <—
l GH~ (t) — S ~ 30 t
=1
~
- S / t eUs (s)ds i =i
33 0
(a + e)(lC)
l~~
I
2
]eU?(t)dt
lt (lA _ I R0 lB) R(t, s)dt, a
(5.15)
Chapter 3
256 where H(I) =
— a 5Q (t,0)l~~ + 1 0 (i) + R0(I)(l
I Ro I
lr)
Suppose that for any i = 1, 2, .. .
(a+ e)(07)z < inf 112 (1).
(5.16)
It follows from Egns. (5.15) and (5.16) that e W7 (t) < 8W7(0). Substitution of the expressions for Wk (t) into this inequality yields a(u2(t), uz(t)) + (W4(t ), B W4(t )) 11 +(41(1), [cv(A + Q(t, 0 )1) + (A — Ro (t) B)B-1] 1(1)) A-a r ~Q (t, s)( 74(t) — u i(s), B(ui(t) — ui(s)))ds J
< (ui (0), [aA + (A — Ro(
0
)B)B-1lu i ( 0 ))
(5.17)
+(uß(0), ( aI+ B-1)uz( 0)). Suppose that min [lA + Q(fR, 0)l ~~ > 0.
(5.18)
Eqn. (5.17) implies the following
Theorem 5.1. Suppose that inequalities (5.8), (5.16) and (5.18) hold. Then the zero solution of system (5..') is stable in the mean square sense. q For a non-ageing viscoelastic material with Q(t, s) = Qo(t —
s),
we have
R(, s) —
~
JQ (t, s)dt = — as
Qo(t — s)dr =
J
Qo (t — s) — Q o(f), = R(t t) =
R°(t) = / R(t, s)ds = o
J o
Jo
t
—
Qo(f),
[Q (s) — Qo(oo)]ds.
Condition (5.8) can be written as min [4A + Qo (oo)l j > 0.
(5.19)
Stability of stochastic equations
257
This inequality implies (5.18) . Functions H;(t ) have the form H; (t ) = — il BQ0(t) +
[la
+ Qo(c )lr] j [Qo(s) — Qo( f)jds.
We calculate the derivatives of H; (t ) with respect to time and estimat e them get
with the use of (1.6.35) — (1.6.27), (1.6.55) and (5.19). As a result we
~ = T2 l B max A
+ QoBoo)lB Qo (t)
s
s
—[ ~~ + Qo(oo)lB][Qo(t) — Qo(oo)] > [l; + Qo (oo ) l Therefore, h
)
j{T2~
o (t ) — [Q o(t) — Q0(oo)]} > 0.
0, and
H ( t ) > H (oo) = [lrR + Qo( co)lr]j: [Qo(s) — Qo(oo)]ds. Integration by parts with the use of (1.6.35 ) yields Jo f
Qo ( oo )j ds
[Qo(s) —
—/
3o
= s[Qo(s) — Qo( f)]~-~~
f o( s )sds =
Jo
I
o( s) I sds.
Thus, condition (5.16 ) can be written as follows:
(l)2
< ( + B) — i [ll + Qo(oo)lB]
~ I Qo(s) 1 sds.
jro
( 5.20 )
We arrive at the following Theorem 5.2. Suppose that inequalities (5.19) and (5.20) are vali d. Then the zero solution of system (5.4) is stable in the mean square sense. q
5.3. Examples 1. Let us develop stability conditions for a simply supported viscoelastic bar compressed by a random load. In this case, Y;(c) = sin piZ/I, and the eigenvalues of operators (5.5) have the form lA
—
n4i4EJ pS14
1 —
~
Ro
Pe2 2 )i
lB a
— z4i4
EJ
pS14
i
lC a
—
p 2 i 2 R1 m S12
where Pe = it2E J1-2 denotes the Euler critical force.
(5.21)
Chapter 3
258
First, we analyse the stability for a bar made of an ageing viscoelastic material. For simplicity, we assume that R'(t) > 0 for anyt > 0. Substitution of (5.21) into (5.8) and (5.18) yields
R° <1—
—
max[~~ R° ,
(5.22)
Q(oo, O)]•
2 It follows from (5.21) that a = 7' (1— (5.21) imply
I
RO ). This expression, (5.16) and
~
( R1 )2 < H0 [1 -{- 4 p2(1- I R o )( T ) 2 ] -1 Pe T*
(5.23)
Here
— RO ~ ~[(1 I i>
H0
° — R )R°(t) — (1—
I
Ro I) 8 (t, 0 )],
and
T —2
pS14
p2 E J
is the maximal period of natural oscillations. Stability conditions for an ageing viscoelastic bar consist of restriction (5.22) for the deterministic compressive force PO and restriction (5.23) for the random component R1 . Now let us consider a non-ageing viscoelastic material. In this case, stability conditions have the form (5.19) and (5.20). These inequalities imply Ro < Pocr = R [1 + Qo(oo)]•
(5.24)
Substitution of (5.21) into (5.20) yields 4 •4 PO2 )[1+T2(1+ Qo(~)) ~p EJ]-1, SI Roc r 2
1
(R )2
(5.25)
where
- f°O1 -l-I
Qo ( s )
I sds
Q'(oo)
The right-hand side of (5.25) increases monotonously in i . Therefore, inequality (5.25) holds for any positive integer i if and only if ( ócr )2
<1(1 —
°
r
2 )[1 + 4p (1+ Qo( f))(l ) 2 ]
1
-
(5.26)
Stability of stochastic equations
259
Egns. (5.24) and (5.26) guarantee the stability of a non-ageing viscoelastic bar under stochastic loading. For the standard viscoelastic material with the relaxation measure (1.5.40), these conditions take the form 1
c
Pe
P Eo
E
< Roer = Re ~i
2 —E E~ (I —T,* )2] -1. )[1+4i
The dependence of the dimensionless critical load P1 , on the parameter c = E0 /E is plotted in Fig. 5.1. The critical load decreases with the growth of c and vanishes when c = 1 (purely elastic material). When c tends to zero (i.e. when the material viscosity grows), the critical load increases and tends to infinity. With the growth of T/T, value, the critical force diminishes. 10
Rlcr
0 0
c —i•
1
Figure 5.1: The dimensionless critical load Ri ~,. vs parameter c. The calculations are carried out for Po = 0. Black points correspond to T/T„ = 0.1, light points correspond to T/T, = 1.0, and asterisks correspond to T/T* = 10.0. 2. Let us consider the stability problem for a simply supported bar on an elastic foundation. We restrict our consideration to the case of a non-ageing viscoelastic material.
260
Chapter 3
The eigenvalues of operator (5.6) can be written as _ ~4 i 4 EJ
where k (5.21).
=
k
PO
(5.27)
rei2
Pcl4
kl 4 /(P4 EJ). The eigenvalues of operators B and
C have the form
7
.,~. • •~ 'N. •·•%..... *
N~ NNNN..~NN~NNNNN~NNNmN
.... ~
***
*** **
yy TT #* * * **
C ** ** yy } }*
1
i
**~
0 Figure 5.2: The dimensionless critical loads
and
Pier vs the dimensionless rigidity of an elastic foundation k1. The calculations are carried out for Qo (oo) = —0.2. Black points correspond to T/T* = 0.1, and light points correspond to T/T* = 1.0 . 1
r
First, let us consider a viscoelastic bar under the action of a deterministic load (P1 = 0). Substitution of (5.21) and (5.27) into (5.19) implies the only stability condition
j cr
< PO
cr
,
where 2 4 R cr = min i (1 + k 0 i )
(5.28)
Concluding remarks
261
and k o = k/[1 + Qo (oo)]. It follows from (5.29) that the critical load grows when the rigidity of an elastic foundation increases. The dependence of the dimensionless critical load Poor on the dimensionless rigidity k o is plotted in Fig. 5.2 by asterisks. Let us now consider a viscoelastic bar under the action of a random load ( Po = 0). In this case, stability condition (5.20) can be presented as follows: Pl V `V ROer
Plcr
where // P~Cr = min /(1 +
i4
)[4 + 4p2 ( —7T,* )2 (1 + Qo( oo))(1 ~- Ko)]-1.
(5.29)
The dimensionless critical load R1~ r vs the dimensionless rigidity k o is plotted in Fig. 5.2. The numerical results show that the growth of the rigidity of an elastic foundation leads to an increase of the deterministic critical load 1 cr and to a decrease of the random critical load R ,.. When T/T, value increases, the critical random load Pier diminishes.
6. CONCLUDING REMARKS In this chapter the stability of viscoelastic elements of structures is studied under the action of time-varying and random loads. For this purpose we employ the Lyapunov direct method and have constructed new stability functionals. These functionals utilize the specific properties of relaxation measures for viscoelastic materials suggested in Chapter 1. In Section 1 the stability problem is considered for a cylindrical shell made of a non-ageing viscoelastic material under compressive longitudinal and radial loads which can vary in time, and stability conditions (1.7) and (1.8) are derived. The well-known stability condition (1.7) imposes restrictions only on current values of compressive loads. This inequality is determined by the "zero" moment of the relaxation measure J ~o
IQo(oo)I =
0
lQo(s) ~ ds.
(6.1)
Condition (1.8) is new. It imposes limitations on the rate of change of external forces. These constrains are determined by the first moment of the relaxation measure f
J
oo
~ Qo(s)(sds.
(6.2)
262
Chapter 3
Since the zero moment of the relaxation measure does not depend practically on the temperature, while the first moment depends on it drastically, these stability conditions allow the critical load to be increased significantly by changing the temperature conditions in an appropriate way. Numerical analysis has been carried out for a viscoelastic shell under the action of a periodic longitudinal load. The results of calculations show that the critical frequency of periodic excitations depends significantly on the intensity of the sinusoidal compressive load. The frequency does not vanish only in the narrow vicinity of zero, in which it tends to infinity when the intensity of the periodic load tends to zero. With the growth of the material viscosity, the behaviour of critical frequency has a non-monotonous character. It increases for large intensities of periodic compressive load, and it decreases for small intensities. Results of Section 1 were derived by Drozdov (1993). In Section 2 we analyse in detail the stability problem for a linear integrodifferential equation of the second order with a periodic coefficient. By employing a similar approach for a narrow class of stability problems (only one equation with an additional assumption about the periodic character of a time-varying coefficient) we have derived more sophisticated stability conditions (2.13) — (2.15). It has been shown that these sufficient conditions are very close to the necessary stability conditions for ordinary differential equations. Namely, the stability conditions obtained require the stability of the corresponding differential equations and, additionally, impose some limitations on the relaxation measures. These restrictions are plotted in Fig. 2.1 for the standard viscoelastic solid with the exponential relaxation kernel. This kernel is determined by two positive constants: the characteristic relaxation time -y -1 = T and the material viscosity c. The numerical results show that for a fixed frequency of the compressive load, the domain of stability is a bounded region in the (c, ·c ) plane, which collapses with the growth of the frequency of the sinusoidal compressive load. Results of Section 2 were developed by Drozdov & Gil (1994). Sections 3 — 5 are concerned with the study of the mean square stability for viscoelastic structural members driven by random excitations. It is assumed that random load has so rapidly decreasing correlations that it can be described by the "white noise" model. This leads to the governing integro-differential equations in the Ito form. We have proposed new Lyapunov functionals to study the stability of these equations and have derived some sufficient stability conditions. First, we analyse the mean square stability for a non-ageing viscoelastic cylindrical shell and develop stability conditions (3.4) and (3.6). It is worth noting that these conditions are determined by the same moments of relaxation measure (6.1) and (6.2) as in the case of time-varying loading. For the standard viscoelastic material it has been shown that the main physical parameter determining resource of stability is the ratio of the characteristic time of a viscoelastic material T to the characteristic time (maximal
References
263 T*.
period of natural oscillations) of a structure Dependence of the critical intensity of random load on this parameter is plotted in Fig. 3.1. The curves demonstrate the drastically non-monotonous character of this dependence. This means that by choosing an appropriate viscoelastic material one can increase significantly the resource of stability for a given structure driven by random excitations. Results of Section 3 were derived by Drozdov (1993). Section 4 deals with the stability problem for a viscoelastic bar driven by a random compressive load. It has been shown that for a sufficiently small intensity of the stochastic load a viscoelastic bar is stable in the mean square sense, while the corresponding elastic bar is unstable. At first sight, this fact contradicts the well-known assertion that the material viscosity leads only to the decrease of the stability resource. The explanation is rather simple: the material viscosity leads to the decrease of the critical load only for deterministic loading, whereas for stochastic loading we have the inverse picture. Results of Section 4 were derived by Drozdov & Kolmanovskii (1991, 1992). The method employed in Section 4 can be extended to arbitrary integrodifferential equations with operator coefficients both for convolutive and nonconvolutive kernels of integral operators. In Section 5, general stability conditions (5.8), (5.16) and (5.18) have been derived for integro-differential equations with operator coefficients and non-convolutive kernels. As an example, the stability problem has been analysed for a viscoelastic bar lying on an elastic foundation under the action of a random compressive load. The numerical analysis demonstrates that an additional rigidity (or "elasticity") of a system leads to the increase of the deterministic critical load and to the decrease of the stochastic one, see Fig. 5.2. In particular, the presence of an elastic foundation implies a growth of the Euler critical force and a decrease of the random critical load. On the contrary, the growth of the material viscosity diminishes the deterministic critical force, and increases the stochastic one. Results of Section 5 were developed by Drozdov (1994).
REFEREN CES 1. Akbarov, U., Badalov, F.B., & Eshmatov, Kh. Stability of a viscoelastic rod under dynamic loading. J. Appl. Mech. Techn. Phys. 33(1992) 613-616. 2. Andreeva, E., Kolmanovskii, V., & Shaikhet, L. Control of Hereidtary Systems. Nauka, Moscow, 1992. 3. Ariaratnam, S.T. & Xie, W.C. Liapunov exponent and rotation number
264
Chapter 3
of a two-dimensional nilpotent stochastic system. Dynamic Stab. Syst. 5(1990) 1-9. 4. Asokanthan, S.F. & Ariaratnam, S.T. Dynamic stability of axially moving bands under random tension fluctuations. In: "The 18th International Congress of Theoretical and Applied Mechanics. Abstracts." 1992, 15. 5. Arutyunyan, N.Kh., Drozdov, A.D., & Naumov, V.E. Mechanics of Growing Viscoelastoplastic Bodies. Nauka, Moscow, 1987 (in Russian). 6. Belen'kaya, L.Kh. A study of the parametric oscillations of a viscoelastic cylindrical shell. Mech. Solids. 22(1987) 142-148. 7. Belen'kaya, L.Kh., & Vudovich, V.I. Stability of a viscoelastic rod under a periodic load. Mech. Solids. 13(1978) 128-133. 8. Bogdanovich, A.E. Dynamic stability of elastoviscous orthotropic cylindrical shell. Mech. Polimers. 4(1973) 714-721 (in Russian). 9. Bolotin, V.V. Dynamic Stability of Elastic Systems. Holden Day, San Francisco, 1964. 10. Bolotin, V.V. Stochastic Oscillations of Elastic Systems. Nauka, Moscow, 1979 (in Russian). 11. Bolotin, V.V. Parametrically excited oscillations in systems with delay. Mech. Solids. 20(1985) 12-19. 12. Burton, T.A. Construction of Liapunov functionals for Volterra equations. J. Math. Anal. Appl. 85(1982) 90-105. 13. Burton,T.A. Volterra Integral and Differential Equations. Academic Press, N.-Y., 1983. 14. Burton, T.A. Stability and Periodic Solutions of Ordinary and Functional Differential Equations. Academic Press, N.-Y., 1985. 15. Cederbaum, G., & Mond, M. Stability properties of a viscoelastic column under a periodic force. Trans. ASME. J. App!. Mech. 59(1992) 16-19. 16. Christensen, R.M. Theory of Viscoelasticity. An Introduction. Academic Press, N.-Y., 1982. 17. Corduneanu, C. Integral Equations and Applications. Cambridge Univ. Press, Cambridge, 1991. 18. Dimentberg, M.F. Random Processes in Dynamic Systems with Variable Parameters. Nauka, Moscow, 1989 (in Russian).
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19. Dobrovolskii, S.M., Kotiurgina, A.S. & Romanovskii, R.K. On stability of solutions to linear systems with almost-periodic matrix. Math. Notes. 52(1992) 10-14 (in Russian). 20. Drozdov, A. Stability of viscoelastic shells under periodic and stochastic loading. Mech. Res. Comm. 20(1993) 481-486. 21. Drozdov, A. Stability of a class of stochastic integro-differential equations. Stochastic Anal. App!. (1994) to appear. 22. Drozdov, A.D. & Gil, M.I. Stability of linear integro-differential equations with periodic coefficients. Quart. App!. Math. (1994) to appear. 23. Drozdov, A.D., & Kolmanovskii, V.B. Stability of viscoelastic rods under a random longitudinal load. J. App!. Mech. Techn. Phys. 32(1991) 772-779. 24. Drozdov, A.D., & Kolmanovskii, V.B. Stochastic stability of viscoelastic bars. Stochastic Anal. App!. 10(1992) 265-276. 25. Drozdov, A.D., Kolmanovskii, V.B., & Velmisov, P.A. Stability of Viscoelastic Systems. Saratov Univ. Press, Saratov, 1991 (in Russian). 26. Erugin, N.P. Linear Systems of Ordinary Differential Equations with Periodic and Quasi-Periodic Coefficients. Academic Press, N.-Y., 1966. 27. Eshmatov, Kh. & Kurbanov, P. Parametric oscillations in a viscoelastic rod with a nonlinear inheritance characteristic. J. App!. Math. Mech. 39(1975). 28. Evan-Iwanowski, R.M. Resonance Oscillations in Mechanical Systems. Elsevier, Amsterdam, 1976. 29. Gikhman, I.I. On the first initial-boundary problem for a stochastic hyperbolic equation. Theory of Stochastic Processes 8(1980) 20-31 (in Russian). 30. Gikhman, I.I. & Skorokhod, A.V. Introduction to the Theory of Random Processes. Saunders, Philadelphia, 1969. 31. Hale, J.K. Theory of Functional Differential Equations. Springer - Verlag, N.-Y., 1977. 32. Hasminskii, R.Z. Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Groningen, 1980. 33. Hausmann, U.G. Asymptotic stability of the linear Ito equations in infinite - dimensions. J. Math. Anal. Appl. 65(1978) 219-235.
266
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34. Herrmann, G. Dynamic Stability of Structures. Pergamon Press, Oxford, 1967. 35. Horn, R.A. & Johnson, C.R. Matrix Analysis. Cambridge Univ. Press, Cambridge, 1985. 36. Horsthemke, W. & Lefever, R. Noise-Induced Transitions. Theory and Applications in Physics, Chemistry and Biology. Springer-Verlag, N.-Y., 1984. 37. Ichikawa, A. Stability of semi-linear stochastic evolution equations. J. Math. Anal. App!. 90(1982) 12-44. 38. Kadiev, R.I. & Ponosov, A.V. Stability of linear stochastic functional di ferential equations under constantly acting perturbations. Diff. Egns. 28(1992) 173-179. 39. Klosek-Dygas, M.M., Matkowsky, B.J., & Schuss, Z. Colored noise in dynamic systems. SIAM J. Appl. Math. 48(1988) 425-441. 40. Kolmanovskii, V.B. & Nosov, V.R. Stability of Functional Differential Equations, Academic Press, N.-Y., 1986. 41. Kozin, F. & Prodromou, S. Necessary and sufficient conditions for almost sure sample stability of linear Ito equations. SIAM J. App!. Math. 21(1971) 413-424. 42. Lepore, J.A. & Stoltz, R.A. Stability of linear cylindrical shells subjected to stochastic excitations. In: Curtain, R.F. (ed.) "Stability of Stochastic Dynamic Systems". Lecture Notes in Mathematics 294(1972) 239-251. Springer-Verlag, Berlin. 43. Malkin, I.G. Some Problems of the Theory of Nonlinear Oscillations. Gostechizdat, Moscow, 1956 (in Russian). 44. Mao, X. Existence and uniqueness of solution of delay stochastic integral equations. Stochastic Anal. Appl. 7(1989) 59-75. 45. Mao, X. Exponential stability in mean square for stochastic differential equations. Stochastic Anal. Appl. 8(1990) 91-113. 46. Matyash, V.I. Dynamic stability of a hinged viscoelastic bar. Mech. Polymers. 2(1967) 293-300. 47. McLachlan, N.W. Theory and Applications of Mathieu Functions, Dover Publications, N.-Y., 1964. 48. Mize!, V. & Trutzer, V. Stochastic hereditary equations: existence and asymptotic stability. J. Integral Egns. 7(1984) 1-72.
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49. Moskvin, V.G., Semenov, V.A. & Smirnov, A.I. On the investigation of parametric vibrations of systems with hereditary friction. Mech. Solids. 19(1984) 31-36. 50. Pipkin, A.C. Lectures on Viscoelasticity Theory. Springer-Verlag, N.-Y., 1972. 51. Potapov, V.D. Stability of viscoelastic constructions under stationary stochastic compressive loads. Mech. Solids. 19(1984) 153-159. 52. Potapov, V.D. Stability of Viscoelastic Structural Members. Strojizdat, Moscow, 1985 (in Russian). 53. Potapov, V.D. On the stability of rods for stochastic excitation. J. App!. Math. Mech. 53(1989) 797-803. 54. Potapov, V.D. The stability of motion of a stochastic viscoelastic system. J. App!. Math. Mech. 57(1993) 537-545. 55. Potapov, V.D. & Marasanov, A.I. Buckling and stability of polymeric composite beams under stochastic excitation. J. App!. Math. Mech. (ZAMM) 72(1992) T97-T100. 56. Rahn, C.D. & Mote, C.D. On the stability of the damped Hill's equation wzth arbitrary, bounded parametric excitation, Trans. ASME. J. App!. Mech. 60(1993) 366-370. 57. Schuss, Z. Theory and Applications of Stochastic Differential Equations. Wiley, N.-Y., 1980. 58. Stevens, K.K. On the parametric excitation of a viscoelastic column. LIEA J. 12(1966) 2111-2116. 59. Stokes, A. A Floquet theory for functional differential equations. Proc. Nat. Acad. Sci. USA. 48(1962) 1330-1334. 60. Tylikowski, A. Stability and bounds on a motion of viscoelastic columns with imperfections and time-dependent forces. In: "Creep in Structures". IUTAM Symposium, Cracow, Poland, 1990. Springer - Verlag, Berlin, 1991 653-658. 61. Volmir, A.S. Stability of Elastic Systems. Nauka, Moscow, 1967 (in Russian). 62. Yakubovich, V.A., & Starzhinskii, V.M. Linear Differential Equations with Periodic CoefJicients. Vol. 1, 2. Wiley, N.-Y., 1975. 63. Zelentsovskii, A.L. Stability with probability 1 of solutions of systems of linear stochastic Ito differential-difference equations. Ukrainian Math. J. 2(1991) 123-126.
CHAPTER 3 STABILITY OF VISCOELASTIC STRUCTURAL MEMBERS UNDER PERIODIC AND RANDOM LOADS
In this chapter dynamic stability problems are studied for viscoelastic structural members subjected to ageing under periodic and stochastic compressive loads. The mechanical behavior of structures is described by linear governing equations. In order to derive stability conditions we employ the direct Lyapunov method and construct stability functionals. This method is well-known in the theory of stability for functional differential equations, see e.g. Burton (1982,1983,1985), Corduneanu (1991), Hale (1977), Kolmanovskii & Nosov (1986). We suggest new stability functionals which utilize the specific properties of relaxation measures formulated in Chapter 1. By using these functionals, explicit restrictions to compressive loads are developed. The influence of rheological and geometrical parameters on the critical load is studied both analytically and numerically. Section 1 deals with the stability problem for a viscoelastic cylindrical shell under the action of a time-varying axial compressive force and a distributed radial traction. Section 2 is concerned with stability for the zero solution of a linear integro-differential equation with a periodic coefficient. As an example, stability conditions are derived for a viscoelastic bar compressed by a periodic longitudinal force. Section 3 is devoted to the analysis of the mean square stability for a viscoelastic cylindrical shell compressed by axial and radial stochastic loads of the white noise type. In Section 4 we derive some stability conditions for a viscoelastic bar driven by random compressive loads. We confine ourselves to non-ageing viscoelastic materials and develop some explicit restrictions on deterministic and stochastic components of external forces. Section 5 is concerned with an extension of the results derived in Section 4 to integro-differential equations with operator coefficients with nonconvolutive integral kernels.
Chapter 3
218 1. STABILITY OF A VISCOELASTIC SHELL UNDER TIME-VARYING LOADS
In this section the stability problem is studied for a viscoelastic cylindrical shell under the action of a time-varying axial compressive force p = p(t) and a uniformly distributed radial compressive load q = q(t). Our purpose is to derive restrictions on these functions which guarantee the dynamic stability of a viscoelastic shell. The stability conditions for a viscoelastic shell under a time-independent load were derived by Drozdov et al. (1991). Stability of viscoelastic structural members under time-dependent compressive forces was analysed e.g. by Akbarov et al. (1992), Belen'kaya (1987), Belen'kaya & Yudovich (1978), Bogdanovich (1973), Bolotin (1985), Cederbaum & Mond (1992), Eshmatov & Kurbanov (1975), Matyash (1967), Moskvin et al. (1984), Rahn & Mote (1993). 1.1. Formulation of the problem and governing equations Let us consider a thin-walled, circular, cylindrical shell with length 1, radius R and thickness h. Introduce orthogonal coordinates (x1 ,x 2 ) in the middle surface of the shell, such that r1-axis coincides with the longitudinal axis. Denote by y(t, x i , x 2 ) the shell deflection. We assume that: (i) deflection y is so small that the nonlinear terms with respect toy and its derivatives can be neglected in the formulas for the strains and curvatures of the middle surface; (ii) Kirchhoff's hypotheses hold; (iii) the forces in the middle surface are essentially larger that the transverse and inertia forces, and we can neglect the transverse and inertia forces in the equilibrium equations for the stresses in the middle surface; (iv) the displacements in the middle surface are sufficiently less that deflection y, and we can neglect them in the formulas for the curvatures of the middle surface; (v) the material behavior obeys the constitutive equations of a linear viscoelastic solid (1.5.46) with the relaxation operator (1.5.22), (1.5.26). The relaxation measure Q0 (t) satisfies conditions (1.6.35) — (1.6.37) and (1.6.51). Then y satisfies the motion equations, see e.g. Volmir (1967), Phy(t, ci, x2) + E* [D 2 y(t, xi, x2) — z +ph
—
[
2
. (t i
ac2i
C1, c
t
Jo
2 Qo(t — s) D y(s, c i
a2y (t, x i, x2) + g Ra x2 (t, x i , x2) =
2
—
h a2
i
f i Qo(t — s) ac? (s, C1 , x 2 )ds] = DZ Y(t,
Cylindrical shell under time-varying loads
219
with the initial data y
y It=o = yl(xi, x2)•
It =o = y0(x1,x2),
Here p is the material density, Y(t, xi , x2) is the Airy function, yo is the initial deflection, y1 is the initial speed of the deflection, E* = Eh3[12(1 — n2 )]-1 is the bending rigidity, E is the Young modulus, v is Poisson's ratio, D is the Laplace operator. We will not fix the boundary conditions, and suppose only that these conditions permit the following expansions of the solutions 00
m,n=1 00 0
=
S
m,n=1
ir mx1
Ymn (t ) sin
1
¢,,,h (t) sin
sin R ,
pmcl ttx2 sin . 1 R
(1.2)
t
Substitution of (1.2) into (1.1) yields rhymn(t) + Amn (P, 4)ymn (t) + Bmn
~ Q0(t
ymn ( 0) = y 0, mit
— S)ymn (S)ds = 0, ymn ( 0) = y 1, m p
(1.3)
Here Amn (P, 4) = E*Amn +
Eh ( irr 4D-i 1 ) mit
R2
2 h 2 , [rh( pm 1 ) + 4R( R ) ]
B,hp = E* Dpap +
Eh pm 4 D -1 > 0, ( 1 ) mn R2
Dmh
= [( pI )2 + ( R)2]2,
where yo, mn , yi, mi are the Fourier coefficients for the initial conditions and Omn
—R
n
. (~ ~ )2 = « m~
Introduce the following Definition. A shell is stable if for any inequality f
S ( U, mn + y 1, mp )< d
m,n=1
implies
S 00
sup y t>0 m,n=1
( )<
S.
e > 0 there is a d >
0 such that the
Chapter 3
220
The purpose of this section is to derive some restrictions on functions p(t) and q(t) which would ensure the stability of a viscoelastic shell. 1.2. Stability conditions In order to derive stability conditions we introduce the functionals
n$) = rhymn(t) + [Amn (t) + BmnQO(t)1ymn(t) f Q 0 (t t
—Bmn
— s)(ymn(t
) 2
t n$, mn = IRrhy m h ( t) + Bmn J~ (Qo(t — s) —
Qo(f))ymn(s)ds]
+ph[Amn (0 + BmnQo( o's)]y2mn (t),
(1.4)
(1.5) Vmn = N + &mn1 lm, where Amn (t) = Amn (r(t), q(t)) and parameters amn > 0 will be determined below. It follows from (1.3) — (1.5) that Vmn = [Bmn (Amn (t) + Bmn Qi(00)) ]'t (QP( s) — QO(oo))ds 0 — amp BmnQO (t) — (Ph + amh )Ah,p (t)]ymn (t) —
Bmn f t IR~mnO(t ~ s) C( `i0(t —
(Amn(t) + Bmn QO ( f))
s) — Qi(°c))(ymn(i) — ymn(s))
BmpIAm) + BmnQO()] Let us assume that
_
j [Qo(t —
s)
— QO() ] y
amn = mio IRAmn(t) + BmnQO(oo)] > 0.
2 ds
(s)ds.
(1.6)
(1.7)
Set amn = T2 amn, where constant T = ~2 is determined by formula (1.6.51), and suppose that
G
mion IR
t
(Ph + amn ) m~~ I Amp (t) I
f (Qo(s) — Qo( oo ))ds 0 According to (1.6.55), the function in the brackets decreases monotonously. Taking into account that
j
Bmn amn
[Qo(s) — Q0(oo)Jds =fIQO(s)1sds 0
221
Cylindrical shell under time-varying loads we can rewrite this inequality as
amn Bmn
IQo(s) I sds.
m ó 1 Amn(t) 1< _ ph +Taamn to
It follows from (1.6.55) and (1.6) — (1.8) that [~ t >0:
(1.8) mn
< 0. Therefore, for any
2
T2 am n ym n (t) < [T amnfimn (0) + ph(Amn (0) + BmnQs(ss))] yÓ,
+ph(ph + T2 amn )yl
m~
mn
This inequality implies the following Theorem 1.1. Suppose that inequalities (1.7) and (1.8) hold. Then a viscoelastic shell is stable. q
1.3. Example Let us consider a viscoelastic cylindrical shell under the periodic axial compressive force p(t) = Po lR- R l sin wt, where R o , Pl and w are positive constants. Introduce the dimensionless variables P: R
Rt'
*
Eh
_ ~T*, T * _v
/
pR12 p2Eh
According to Theorem 1.1, sufficient conditions of shell stability are
po + r < min F(C), m2F(C)[F(V) — r —14] Pw *~ * < N min m ,' 1 + (T/T* ) 2 m2 [F(C) — RPó where _ n2 h nmR
n!
R [ h!
F(C) = [1
+ fR
v2 )
2
+ ( ]'
1 ~Qo(s)~sds. T* [l + Qo ( f)] j
The minimal value of function F(V) is F0 =
+ Qo(oo) ‚/3(1 — n2) 1
i]
Chapter 3
222 For a fixed m value, the function
m2 F(F — r — Pi )[ I + (T/T*)2 m 2 ( F — Pó —
r* )]
-1
increases monotonously with the growth of F. Therefore this function reaches its minimal value with respect to when F = F0 . For F = F0 , this function reaches its minimal value with respect to m for m = 1. Therefore, a viscoelastic shell is stable if 1+(T/T*)2(
iw* < NFo(Fo — R~~— R~ )[
P
r + R~~
Fo — R~~— R1)]_ i •
(1.9)
~~
3.2
* ~max
0 0
0
R~~~
0
0.4
Figure 1.1: Dimensionless critical frequency w~,, of the sinusoidal load vs dimensionless amplitude ri of the periodic part of the compressive load. The calculations are carried out for v = 0.3 and ró = 0.3. Light points correspond to c = 0.4, and black points correspond to c = 0.8. Let us consider the standard viscoelastic material with relaxation measure (1.5.40) Qo(t) =
—(1 —
Eo)[1 — e
cr( —)j,
Linear equation with periodic coefficients
223
where E and E0 are the current and limiting elastic moduli, respectively, and T is the characteristic time of relaxation. The second condition (1.9) can be presented as follows: * P i w < F1 k(1 —
c)(F0
—
Ró
—
2 Pi){c[1 -1- k (F0
—
Ró —
-1
ri)]}
,
(1.10)
where c = E0 /E and k = T/T*. The right-hand side of (1.10) reaches its maximal value with respect to k when k = ( F0 _r0 — R~ )-112°
(1.11)
This means that a viscoelastic shell has maximal reserve of the stability when the characteristic times for material T and for structure T* are related by formula (1.11). The critical frequency of the periodic compressive load is = F0( 1 — c)(F0 — ró —
r~ )1/2 /( 2CR*1)•
The dependence of w~,, on amplitude ri is plotted in Fig. 1.1. The numerical results show that the dimensionless critical frequency is not very large except for a narrow domain of the pi values in the neighbourhood of zero where w,*,. tends to infinity as ri —> 0. When parameter c grows, the critical frequency increases for "large" pi values and decreases for "small" ones.
2. STABILITY OF A LINEAR INTEGRO-DIFFERENTIAL EQUATION WITH PERIODIC COEFFICIENTS This section is concerned with the stability analysis for a linear integrodifferential equation with periodic coefficients. A similar equation arises in the dynamics of thin-walled viscoelastic elements of structures under periodic compressive loading. The equation under consideration has a specific peculiarity which makes its analysis difficult: in the absence of the integral term it is only stable, but not asymptotically stable. Therefore, in order to derive stability conditions we are to introduce some specific restrictions on kernels of the integral operator, which are taken from the study of relaxation measures for linear viscoelastic media. Employing the direct Lyapunov method and constructing new stability functionals, we derive some new sufficient stability conditions which are close enough to the necessary ones. In particular, when the integral term vanishes, our stability conditions turn into the well-known stability criterion for a linear differential equation with periodic coefficients, see e.g. Yakubovich
224
Chapter 3
& Starzhinskii (1975). In the general case, the proposed stability conditions have the following mechanical meaning: a viscoelastic structure under periodic excitations is asymptotically stable if the corresponding elastic structure is stable and the material viscosity is sufficiently large. As an example, the stability problem is considered for a linear viscoelastic bar compressed by time-periodic loads. Explicit limitations on the material parameters are developed which guarantee the bar stability, and the dependence of the critical relaxation rate on the material viscosity is analysed numerically for different frequencies of the periodic compressive load. Stability of ordinary differential equations with periodic coefficients was discussed in detail by Erugin (1966), Malkin (1956), McLachlan (1964) and Yakubovich & Starzhinskii (1975). It was shown that time-periodic loads can induce parametric oscillations with unbounded growth of small initial perturbations. This analysis was based on the Floquet presentation of the fundamental matrix of a linear differential equation with periodic coefficients. An extension of this theory to functional-differential equations was suggested by Stokes (1962) and Hale (1977). Parametric instabilities in elastic systems were studied e.g. by Bolotin (1964), Evan-Iwanowski (1976) and Herrmann (1967). Some results in the stability theory for viscoelastic thin-walled elements of structures were obtained by Belen'kaya (1987), Belen'kaya & Yudovich (1978), Bolotin (1985), Cederbaum & Mond (1992), Eshmatov & Kurbanov (1975), Matyash (1967), Moskvin et al. (1984), Rahn & Mote 91993) and Stevens (1966). It is worth noting that the stability conditions were obtained in these works either by using simplified constitutive equations reducing the integro-differential equation to the differential one, or by applying approximate methods (averaging techniques, multiple scales analysis, etc). To the best of our knowledge, in this sphere of applications the stability conditions have been studied in detail only for integro-differential equations with constant coefficients, but up to date there is no general theory for the case of time-varying, and, in particular, periodic coefficients. In this section we develop the direct Lyapunov method for such problems and derive some sufficient stability conditions. 2.1. Formulation of the problem and basic assumptions Let us consider the following integro-differential equation: ii(t) -} a(t)u(t) + b J t Qe (t — 0
s)u(s)ds = 0.
(2.1)
Here t(i): [0, cc) [0, cc) is an unknown function, a(t) is a positive, continuously differentiable, periodic function with a period T, b is a positive constant, Qo (t) is the integral kernel. Denote by ao = T-1 a(t)dt the mean value fo of the function a(t). We suppose that ao > 0. Function Q0 (t) is assumed to
Linear equation with periodic coef~cients
225
be twice continuously differentiable and to satisfy conditions (1.6.35)-(1.6.37) and (1.6.51). Let us consider the differential equation ü a(t) u = 0,
(2.2)
which corresponds to the case when the integral term in (2.1) vanishes. In the new variables u1 = u(t), u2 = II(t), Eqn. (2.2) can be re-written in the matrix form
= A(t)U,
(2.3)
where U
r = G ~Z
1,
A(t)=
r
~ a~t)
~~.
L
Introduce vector-function V(t) which satisfies the adjoint differential equation V = -A7'(t)V,
(2.4)
where the superscript T denotes transpose. Let F(i) —
F12( t)
F22( t) J
be the fundamental matrix for Eqn. (2.4), i.e. the solution of (2.4) with the initial condition V(0) = I, where I is the unit matrix. Here we employ nonstandard notation for elements of matrix F(t) in order to emphasize that functions F (t) located in any column of this matrix satisfy independent scalar equations: 'ii = a(t)4'12, 4
'21 = a(t)F22,
4)
12 =
—
'22 =
Ol1, —
(2.5)
O21.
For a constant coefficient a = a0 , matrix F(t) can be presented as follows: G
—
s c l Q sin ~a o t
Qos ln aot ct
, '
(2.6)
Similarly, let
*(t)
[ 4'12(t) ~22(t) J
be the fundamental matrix for Eqn. (2.3). It can be shown that *11(0 = F22(t), *21(1) = —4 '12(1),
012(t)
=
022(t) = F11(i).
(2.7)
226
Chapter 3
It follows from the Liouville theorem that for any t > 0 det ' I'()= det F(t) = 1.
(2.8)
Using this fact we can write the characteristic equation for matrix 0(T) as follows: A2 — 1i (0(T))l + 1 = 0,
(2.9)
where I is the first invariant of matrix. It is well known, see e.g. Yakubovich & Starzhinskii (1975), that the zero solution of (2.2) is stable if and only if all eigenvalues of matrix 0(T) are single and lie on the unit circle. Hence, according to (2.9), the zero solution of (2.2) is stable if and only if I 11 (0(T)) l< 2.
(2.10)
Our objective is to extend this result to integro-differential equation (2.1). Namely, we will derive conditions of asymptotic stability for the zero solution of Eqn. (2.1), which turn into criterion (2.10) when the integral term in (2.1) tends to zero. 2.2. Stability conditions In this subsection we derive some sufficient conditions of asymptotic stability for the zero solution of Eqn. (2.1). In order to formulate these conditions let us introduce the following notation. Denote by f() the functions: f 1(t) = ao fi i (t)f12 (t) + F21(t)F22 (t), f2(t) = aoF i2(t) + Fi2(t),
fa(t) = aof?1(t) + Fi1(1).
(2.11)
It follows from (2.5) and the above formulas that functions f( t) satisfy the differential equations = a(t)f2 — f3,
/2 =
—2
fi(0,
(2.12)
fa = 2a(t)ff
with the initial conditions 0
fi ( ) = 0 ,
f2( 0 ) = 1,
f3(0) = a o .
It is easy to check that for a constant a = a 0 , f 1 (t) = 0,
f2(t) = 1,
f3( 1 ) = a0.
Let á(t)= a(t) + bQo (oo). Introduce the functions _ F(t)
max{~(t),0} (t)b á
F1(t) = :L(t)
a(t) '
F2 (t) = f2(t) ~(t)
Linear equation with periodic coefficients
227
Denote by Y, U1 and U2 their maximal and minimal values
U = sup F(t),
U1 = sllR IFi(t) I,
U2 =
t>o
t>0
inf F2(t),
t >o
We assume that = min á(t)> 0, t>0
min[a s + bQo (oo) f2 (t)] > O. t>0
(2.13)
Note that for a constant coefficient a(t) = ao , the second ine q uality (2.13) follows from the first one. Let H(t) =
Jo
[Qo(s) —
Qo (oo)jds.
It follows from (1.6.52) that function H(t) is bounded.
Theorem 2.1. Suppose that conditions (1.6.35)-(1.6.37), (1.6.51) and (2.13) are fulfilled, U1 <
min{4
'H(oo) + T 1Q0(00) ~~
}~
_l U2
(2.14)
and Y<
1(oo)U2 — a [H(oo ) +Ti ~ Qo (oo)~}U1 Y2 — ~Yl
(2.15)
where a = 3TT 1T2 . Then the zero solution of E qn. (2.1) is stable. q In order to prove Theorem 2.1 we construct the Lyapunov functionals L 1 (1), ..., L6 () such that L 6 (t) is positive and non-increasing in time. Let us introduce new variables z1 = u1f11 + u2 4112,
z2 = u1f21 + u2 f22.
Differentiation of these expressions with the use of (2.1) and (2.5) yields z l = —b(12(t)
J
0
(t — s) u 1 (s)ds,
/ Qo (t — s)ul (s)ds. o Let us consider the function z2 = —b4
(2.16)
22 (t)
L 1 (t) = 2 [ao zi (t) + z(t)J.
(2.17)
For a constant a = a , employing (2.6) we find that L1 = i [a o u2(t) + 2 (t)j, i.e. L1 eq uals the total energy of the conservative system without the integral term. Differentiation of (2.17) with the use of (2.16) leads to the e quality L i = —b[f
i (t)u(t)
+ f2(t)u(t))
Jo
Qo(t —
s)u(s)ds.
(2.18)
228
Chapter 3
In order to transform the second term in (2.18) we introduce the functional L2(t) =
jr0
2
Qo(t — s)[ u(t) — u(s)]
ds — Q
o
(t)u 2 (t).
(2.19)
Differentiating (2.19) and using (2.1) and conditions (1.6.35) we arrive at t
L2 = f
o
t — 2~(t) l Q0 (t —
(t — s)[ u(t) —
u(8)]2 ds
s)u(s)ds — Q 0 (t)u 2 (t).
0
(2.20)
In follows from (2.12), (2.18) and (2.20) that the derivative of the functional L3 =
2L1 (t) — bf
2 (t)L 2 (t)
(2.21)
can be written as follows: L3 (t) = 2bf1 ()[L 2 (i) — u(t) jr o o
Qo(t — s)( u(t) —
0 (t
u(s)) 2 ds —
— s)u(s)dsj Qo(t)u2 (t)]
Since —
u(t)u(s) =
[(ti(t) — u(s))2 — u2 (t) — u
2
(s)],
(2.22)
the latter formula has the form L3(t) = bf1(t)[3L2(t) — jr — bf 2 (t)[
rt
Q0 (t — s u )(t (— )
o
u(s))g ds —
Jo Let us consider now the functional L4(t) = 2{~(t) + b rt [Qo(t — s) — Q
o
Qo(t —
s)u2 (s)ds] 2 Qo(t ) u (t ) j •
o(f)] u(s)ds}2.
(2.23)
(2.24)
Differentiation of (2.24) with the use of (2.1) yields L4 (t) = [t~(t) — bQ
0 (oo)u(t)
+ b l t Q0 (t — s) u(s)ds] N2L4 (t) o =
This relation together with (2.24) implies that the derivative of the functional L5(t) = L4(t) + ~(t)u2(i)
(2.25)
229
Linear equation with periodic coefIcients
can be calculated as follows: L5 (t) = 24(t)112 (t) — á(t)b u(t) j [Qo (t — s) — Q
o()] u(s)ds .
We transform this equation with the use of (2.22) and finally obtain 2L 5 (t) = ~(t)u 2 (t) — á(t)b{H(t)u 2 (t)
— j
+j
[Q o (t
)]u2(s)ds
[Q o(t — s) — Qo(
— s) — Q o()] ((t) —
u(s)) 2 ds}.
(2.26)
Let us introduce the functional L 6 (t) = L3(t) + 2ßL 5 (1),
(2.27)
where b is a positive constant which will be determined below. It follows from (2.23), (2.26) and (2.27) that L6 (t) = — ~(t)b[G(t)u 2 (t) + /
+ Jo
lt o
G1 (1, 1 — s) u2 (s)ds
G2 (t, t — s)(i() —
u(s)) 2 ds],
(2.28)
where G(t) = 1[ß(t) —
-b ] + 3F1(t)Q8(t) —
Gi (t, t) = Q[Qo (t) — Qo (oo)] + Fi (t)Qo(t), G2(t, t) = F2(t)Q o(t) — b[Q o(t) — Qo( oo)] — 3F i(t)Q0(0).
We estimate functions G1 and G2 using Egns. (1.6.54) and (1.6.55) as follows: Gi(t, r) > [b — T 1 ~ Fi(t)~~ ][Qo(r) — Qo(oo)J , G2 (i, t) > [~2 2 F2 (t) — 37'1 1F i(t)1 — b][Qo( t) — Qo(oo)i. It follows from these inequalities that functions G1 (t, r) and G2 (t, i- ) are nonnegative if U1 < bTi < 3a — ß(Y 2 — o Ui ).
(2.29)
Eqn. (2.14) implies that (2.29) is valid, and we put -1 b = 3(aT1 ) (U2 — & U1 ).
In this case, Egns. (2.13) and (2.28) yield L 6 (t) < —á(t)bG(t) u 2 (t).
(2.30)
Chapter 3
230
Substituting expressions (2.21), (2.25) into (2.27) and employing (2.17), (2.19) and (2.24) we get t
— bfz(t)[J
o
L6(t) = a o c~~(t) + z(t)
Qo(t — s)( u(t) — u(s))
+b{[~(t) + b 1 ( Q o (t — s) — Q
z
ds — Qo(t)u2(t)]
o(oo))u(s)ds]
2
+ a(t)u 2 (t)}.
(2.31)
It follows from this formula and conditions (1.6.36) and (1.6.42) that for any t >0 2 L6(t) > b~(t)u2 (t) +aoc~ (t) + ci(t) +bQo(f)fz(t)u (t).
Substitution of expressions for z1 and z2 into this inequality with the use of (2.11) yields 2 (t) + r(t, u(t), ii(t)), L6 (t) ? 0á(011
(2.32)
where r(t, u4, uz) = [f3(t) + bQo(f)f2(t)]u~~+ 2f1(t)u4uz + fz(t)u2. The quadratic form r(t, u~, u2 ) is positive definite if and only if for any t > 0 [fs(t) + bW o(f)f2(t)]fz(t) — f () > 0.
(2.33)
It follows from (2.12) that the expression f 2 (i)f 3 (i) — f() is independent of t and equals a 0 . Substitution of this expression into (2.33) implies that ao + bQo( oo) fz (t) > 0. This inequality together with (2.32) and condition (2.13) yields L 6 (t) > bku 2 (t).
(2.34)
Putting t = 0 in (2.31) and utilizing the initial conditions for matrix function F(t) we find 2
L6(0) = [a o + bá(0)]u(0) + (1 + 19)~iZ(0).
(2.35)
Integrating (2.30) from 0 to t and using (2.34) and (2.35) we obtain bku2 (t) <_ [a 0 + b~(o)]u2 (0) + (i + b)~2 (o) + br(t) f t 112(s)ds, o
where r(t) = max á(s)~ G(s)~ . 0(3 Ki
Linear equation with periodic coefficients
231
This formula together with the Gronwall inequality implies that 112 1
( )5
Q
2 [(~o + b~(0))?1 (0) + (1 + b) ~ 2 ( 0)]
t
x [1 + — ~ G(t) jt exp(b
G(t)dt)ds].
J
(2.36)
Substitution of (2.36) into (2.30) yields Ls(t) <
[(~o + b~(0))u2(0) + (1 + b) it2 ( 0 )] t
1'(t)
x G(t)[1 + 6
J
G(t)dt)ds].
t
exp( —~
J
Integration of this inequality with the use of (2.35) implies that there is a continuous, monotonously increasing, positive function M(t) such that for any t > 0 Ls(t) < M(t)[(a o + b~(0))u2(0) + (1 + a)~2 ( 0)].
(2.37)
Since functions F1 (t), F2 (t) are bounded, conditions (1.6.35) and (1.6.36) imply that for any € > 0 there is t o (e) > 0 such that for any t > t o (e) I G(t) — G(t) l< €,
(2.38)
where Goo (t) = iRb[H(oo)
(t~b ]
3
+ F1(t)Qo(c )•
Estimating this function we find Goo(t) > 1[1(oo) —Y —
3 3-1
/
IQo(oo)~ U1] •
It follows from this inequality, (2.14) and (2.15) that there is an € > 0 such that G(t) > 2e1 for any t > 0. Choosing c = € and applying (2.38) we obtain that G(t) > q for t > i (e l ). This inequality together with (2.30) implies L 6 (t) < — k 1 bu 2 (t) for t > t 0 ( 1 ). It follows from this formula and (2.34) that u 2 (s)ds] < L6(t o(e~ )), k[Qu 2 (t) + e l b 1 t o (~ I )
t> _ t o( 6 0.
Finally, by estimating L6 (t 0 ( 1 )) with the use of (2.37) we find that for any t > to(ci): k[/3u 2 (t) + ei b
ft
J o( E,) u2(s)ds] t
< M(io (ci ))[(ao – b~(0))ti 2 (0) + (1 + r)~2(0)]•
Chapter 3
232
This inequality implies Theorem 2.1. q Remark. For the standard viscoelastic material with relaxation measure (1.6.7), inequalities (2.13) — (2.15) can be presented as follows: min a(t) > cb, t>0
Y1
6
"s > bc max f2 (t), t>0
U1c U< gRY2-6 7U2 — 3 UI U
(2.39)
2.3. Stability of the integro-differential equation and the corresponding ordinary differential equation The above stability conditions for Eqn. (2.1) are formulated in such a way that, at first sight, they have no connection with the well-known stability condition (2.10) for Eqn. (2.2). In this subsection it is demonstrated that our conditions of asymptotic stability for Eqn. (2.1) are fulfilled if and only if the zero solution of (2.2) is stable and functions a(t) and Q0 (t) satisfy some additional conditions. We confine ourselves to the non-critical case when all the eigenvalues of matrix F(T) are single. Denote by X(i) a matrix function with components X;3 (t), and by 0(n) a matrix function with components Y13 (n). Function C(t) is determined in [0, Ti, and coincides with F(t) in this interval. Function 0(n) is determined for nonnegative integers n and equals F(nT). It can be shown that F(t) = X(t — nT)O(n),
nT < t < (n + 1)T.
(2.40)
Putting t = (n + 1)T we obtain from (2.40) that Y(n + 1) = F(T)Y(n). Since Y(0) = 4)(0) = I, this equality implies that 0(n) = F'a (T).
(2.41)
Substitution of (2.40) into (2.11) yields
± (C11(
f1 (t) = ao[C11 (t — n T)X12(t — nT)Y 1 (n) nT)X (t — nT) + C12 ( — nT)X21 (t — nT))Y i i (n) Y12(n) 22
hT)C22(t — nT)Y i2(n)] + [Cii(t — hT)Ci2(t — hT)fz i(h) nT) + C12 ( — hT)C21 (t — hT))f2i( n)022(n) + C2i (t — n T) X22 (t — n T)q4(n)j +C2i(t —
± ( C11(t — nT)X22(t —
Let us consider the quadratic form 2
= X11(t)~ 2 + bi2 (t)~h + X22 (t) h2 , where b 12 () = Cii(t —
b11 (i) = Cii (t — hT)Ci2(t — nT), hT)C22 (t — nT) + C12 ( — hT)C2i (t — nT), d22(t) = C2i(t — n T) X22 (t — nT).
233
Linear equation with periodic coefficients
It follows from (2.3) and (2.8) that det X (t) = det X (0) = 1 for any t E [0, Ti. Eqn. (2.5) implies that there is a sufficiently small t 1 > 0 such that X11(t1) > 0 and C12 (t 1 ) < 0. Therefore, for t = t 1 -I- nT, the form L(t, x, h) is negative definite, and there is a d > 0 such that L(t 1 + nT, x, h) < — d(x 2 + h2 ). This means that f 1 (i 1 + nT) K — d[ao(Yi i(n) + 0i2(n)) + (Y (n) + 0i2(n))]•
It follows from this inequality that there is a d l > 0 such that (2.42)
I fi (ti + nT) I ? d1 11 0(n) ~~ 2 .
Equalities (2.41) and (2.42) imply that 11 Fn (T) 112 . sup ~~ f1(t) > d 1 max n
(2.43)
t>0
For any positive integer n, we have Fn(T) ~~ > rn( f(T)), where p is the spectral radius of matrix, cf. e.g. Horn & Johnson (1985). This inequality together with (2.43) leads to the estimate sup 1 f1(t) I> d 1 max n
r2n
t>0
(F(T)) = d1 max
r2n
(y(T)).
(2.44)
Obviously, (2.45)
f2( 0) = 1. t>ó f2(t) ~
According to (2.44) and (2.45), stability condition (2.14) is valid only for r(y(T)) < 1. On the other hand, it is known, see e.g. Yakubovich & Starzhinskii (1975), that r(y(T)) > 1. Therefore, inequality (2.14) implies r(tY(T)) = 1. For the non-critical case, the necessary and sufficient stability condition (2.10) for Eqn. (2.2) follows from this equality and (2.9). Thus, we have proved that for the non-critical case, the stability conditions for integrodifferential equation (2.1) imply the stability conditions for the corresponding differential equation (2.2). Now we derive an inverse result, namely, we show that the stability of ordinary differential equation (2.2) ensures the stability of integro-differential equation (2.1) under some assumptions regarding kernel Q0 (t) of the integral operator and the periodic coefficient a(t). Stability of equation (2.2) implies boundedness of fundamental matrix function F(t). This means that there exists a positive constant c1 such that for any t > 0: F~,i (t) ~
(i, J = 1, 2).
It follows from this inequality and (2.11) that there is a positive constant c2 such that for any t > O f() I < C2,
0 < f2(t) < C2,
0 < f3(t) <
C2.
(2.46)
234
Chap ter 3
Let us prove that there is a positive constant c3 such that for any t > 0 J2(Í) > c3 .
(2.47)
Suppose that this hypothesis is not true. Therefore, there is a sequence {t m } such that f 2 (t m ) < m-2 . It follows from this inequality and (2.11) that 4 I F12(tm) I< cm ,
4 I F22(tm) I< c 4 m' ,
where c4 = max(1, a o 2 ). These estimates together with condition det F(t m ) = F n.(tm) f22(tm) —
F12(tm)F2 i (tm ) = 1,
see (2.8), imply that 1 < c m -1[I Fii(tm) I+ I F12(tm) I] < 2c4 rn -1 NF~i (tm) + Fi2(tm) < 2c4m-1 ,fs(tm)• Therefore, f 3 (t m ) > [m/(2c4)]2 . Since this inequality contradicts (2.46), our assumption is not true and inequality (2.47) holds. It follows from formulas (2.46) and (2.47) that stability condition (2.10) for ordinary differential equation (2.2) yields U1 < c2 , U2 > c3 . In this case, inequality (2.14) can be treated only as a restriction on function Qo (t) which guarantees asymptotic stability of the zero solution of (2.1). For a given kernel Qo (t) satisfying (2.14), inequality (2.15) can be considered as a restriction on the coefficient a(t) which ensures the asymptotic stability. 2.4. Stability of a viscoelastic bar under periodic compressive. load In this subsection we derive stability conditions for a rectilinear viscoelastic bar under the action of a compressive load. Let us consider the plane bending of a viscoelastic bar with length 1, cross-section area S and moment of inertia of the cross-section J. At moment t = 0, compressive forces R = P(t) are applied to the bar ends. Under the action of external forces, the bar deforms. Denote by y(t, x) the bar deflection at the point with longitudinal coordinate x at moment t > 0. We suppose that (a) function y and its derivative are so small that we can neglect the nonlinear terms in the expression for the curvature of the longitudinal axis; (b) the hypothesis regarding plane sections in the bending is fulfilled; (c) the stress s(t) is connected with the strain e(s), (0 < s < t), by the constitutive equation of a linear viscoelastic solid (1.6.1). Under the above assumptions function y(t, x) satisfies the equation pSy(t, x) + E J[D4 y(t, x) + J t Q0(t — s)D4 y(s, x)ds] 0
+R(t)D2 y(t, x)
=0
(2.48)
Linear equation with periodic coefficients
235
with the initial conditions y(O, cR) = n2(cR)•
y(O,cR) = n1(x),
Here p is mass density, n~ (x) is the initial deflection, n2 (x) is the initial speed of deflection, D is the operator of differentiation with respect to x, Dy = áy/áx. 10
o
O
o
7
G
•
O
. **
*
0
O
.
O
¤
•~ * * + *
*
O
*
.
*
O
*
,
0
~ O
~
~~
o
C~
1
Figure 2.1: Stability domain for a beam driven by periodic excitations. The calculations are carried out for Po = 0.2/3e and P1 = O.1/3e , i.e. for m = 0.125. Light points correspond to w = 0.2, black points correspond to w = 0.5, and asterisks correspond to w = 0.8 .
We confine ourselves to a simply supported bar with the boundary conditions D2 y(t, 0) = D2 y(t, l) = 0.
y(t, 0) = y(i, l) = 0,
In order to satisfy these conditions we seek a solution of Eqn. (2.48) in the form f
= S un (t) sin -i
n
phc l
(2.49)
Chap ter 3
236 Substitution of (2.49) into (2.48) yields 4 pSii (t) +
4
!)// Pe\n2 ) un (t)
14 E J [(1 —
+j Qo(t — s) t
up (s)dsj = 0,
(2.50)
where Pe = i2 EJ1-2 is the Euler critical force. Suppose that the load /3(t) has the form P(t) = PO + P1 sin W!,
where R0 , P1 and W are positive constants. In the following, the main stability region is analysed, and only the term with n = 1 is considered. Eqn. (2.50) implies that
~(t) + [i + m sin wt
I
(t) +
j
.(t. — s)u(s)ds, = 0,
(2.51)
where
t, = t/T, w = w7, 4
T
_ / r$1
P1
R Qo (t)
Let us restrict ourselves to the stability analysis for a bar made of the standard viscoelastic material with the relaxation measure (1.6.7). Applying Theorem 2.1 to Eqn. (2.51) we obtain the following stability conditions: 1—
M> Pe C(Re
where
and y1
72
—
Po) -~,
11<7<72,
are the roots of the quadratic equation
Ug2 72 — (3 gg1 + cg2 )7 + 6cU1 = 0. The boundaries of the stability region in the (c, 7)-plane are plotted in Fig. 2.1. The numerical analysis shows that the lower boundary of the stability region is very close to the abscissa axis practically for the whole range of w values. The upper boundary of this region has an essential maximum for c .: 0.35 which corresponds to half the maximal admissible c value. With the growth of frequency w, the stability region significantly decreases and disappears for w > 1. This means that for a fixed intensity of the compressive load a bar driven by periodic forces is stable only for sufficiently small frequencies of periodic excitations.
Cylindrical shell under random loads
237
3. STABILITY OF A VISCOELASTIC SHELL DRIVEN BY RANDOM LOADS In this section we derive stability conditions for a viscoelastic cylindrical shell considered in Section 1 under the action of stochastic compressive loads P(t) = ro + Pi wi(t),
4(t) = qo + gi ~2 (t),
(3.1)
where w(t) are standard Wiener processes. Theory of stochastic processes and stochastic differential and integrodifferential equations has been in the focus of attention during the past three decades, see e.g. Gikhman & Skorokhod (1969), Schuss (1980). Here we confine ourselves to the stability problems for stochastic integrodifferential equations. The stability theory for stochastic differential equations was discussed e.g. by Hasminskii (1980). The problems of existence and uniqueness for stochastic integro - differential equations were studied e.g. by Mao (1989), Mizel & Trutzer (1984), see also Andreeva et al. (1992) and the bibliography therein. Gikhman (1980) analysed these problems for stochastic partial differential equation of hyperbolic type. Some stability conditions for stochastic integro - differential equations with delay were developed by Hausmann (1978), Ichikawa (1982), Kadiev & Ponosov (1992), Kozin & Prodromou (1971), Mao (1990), Mizel & Trutzer (1984), Potapov (1993), Zelentsovskii (1991). In these works some different concepts of stochastic stability were considered: almost sure stability, stability in mean square, stability with probability 1. Below we restrict ourselves to the analysis of the mean square stability for the zero solution. Stochastic differential and integro-differential equations describe the dynamic behavior of physical systems driven by random noises. A few examples of these systems were presented by Hotsthemke & Lefever (1984). Stability of elastic systems under the action of random excitations was analysed by Bolotin (1979) and Dimentberg (1989). Stability and instability of elastic bars and shells were studied e.g. by Asokanthan & Ariaratnam (1992) and Lepore & Stoltz (1972). Some stability conditions for viscoelastic structural members under the action of random perturbations were derived by Drozdov (1993, 1994), Drozdov & Kolmanovskii (1991, 1992), Potapov (1984, 1989), Potapov & Marasanov (1992), Tylikowski (1991). In this chapter we confine ourselves to the study of structures driven by "white noise" type excitations, see Egns. (3.1). This model describes random forces with a rapidly decreasing correlation, see Horsthemke & Lefever (1984). A more adequate model of random loads is a colored noise. Stability of nonlinear dynamic systems under the action of colored noises was analysed by Klosek-Dygas et al. (1988). The objective of this section is to derive restrictions on forces pi* and qi in Eqn. (3.1) which would ensure the mean-square stability of a viscoelastic
Chapter 3
238 shell. 3.1. Stability conditions
Substituting (3.1) into (1.3) we obtain the following Ito integro - differential equations:
phdu 2, mn =— IAmpu1, mn() t) + Bmn
rt
du1, mn = u2, mn(t)dt9
QO(t
— s)ui, mn (s)ds ]dt
Jo0 +C~i)u1 mn (t)*ui (t) + C,i,2)
1,
m.n(t)dw2(t )•
(3.2)
Here 111, mn = ymn (t)+
U2, mn =
irr
A
mn _ Amn(Po +90),
m
2
C~2) — h
=Pih(_ )2+
Definition. A shell is stable in the mean-square sense if for any e > 0 there is a d > 0 such that the inequality ~ / (yo, mn + yl, mn ) < d S m,n-1
implies sup e t>0
00
E
m,n-1
1l2, mn()
< e,
where symbol E denotes mathematical expectation. Calculate the differential of functional (1.5) with Amn = A. Using Ito's formula and (3.2) we obtain t
dVmn = f — Bmn
—
[ Bmn (—amnQO(t)
l[ mnQO(t — s) — a
JrO
( Am n + Bmn Qi(oO))
x (Qo(t — s) — Qo(f))I(ui, mn(t) —
2 ui, mn(8)) ds
° + (Amp + BmnQ0( f)) j t (Qo(t — s)
— QO(f))ds)
—(1 + am.n )(C( bi)2 + C,V 2)2 )Iu~ , mn (t) (QO(t — —( Amh + BmnQO(f))Bmp
J
ph
s) — QO( f))u1, (s)ds}dt mn
+2u1, mn(t)[( 1 + Ph
)Rh u2, mn(t)
239
Cylindrical shell under random loads
2) 1 ))ui, mn.(s)ds][C~ >dwi(t) + C dw2(t)]. (3.3)
±Bmn j (Qo(/ — s) — Qo ( Suppose that rriih[A m,n
an
+ Bmn Qi(Oi)] > 0.
(3.4)
and choose a mn = T2 [Am n +Bmn Qo (oo)] > 0, where T = T2 is determined according to (1.6.51). Integrating (3.3) from zero to t, taking the mathematical expectation and using (1.6.55) and (3.4) we find
SVmn (l) — eVmn ( 0 ) <
Hm p ( S)eu1, mit('
(3.5)
where
Hmh (t)- Bmn [`4mn + Bmn Q0 (f)] C
[ — T2 Q0(t) +
I
(Q0 (1 — s) — Q0 (f))ds]
2 2)2 {1 + P [Amn + BmnQO(00)]}[C2 + c~ ].
h It follows from (1.6.55) and (3.4) that fImn (t) < 0.
Suppose that H,nn (oo) > 0 for any positive integers m and n. This condition can be written in the form C(1)2 + C12)2 < —
Bmn [Am,n + BmnQs(oo)]
1 + T2/(rh)[A, nn + Bmn Q0 (°°)] Jo
I~~(s) sds.
(3.6)
Then Eqn. (3.5) implies eNmn(t) < e Vmn(0)•
Substitution of (1.5) into this inequality yields Theorem 3.1. Suppose that conditions (3.4) and (3.6) hold. Then a u scoelastic shell is stable in the mean-s q uare sense. q
3.2. Example Let us consider a cylindrical shell under a random radial compressive load q . In the new notation 4~~
4;. R
q1 R
7
—
/ rR 1/
2
,
240
Chapter 3
stability conditions (3.4) and (3.6) can be written as 2.10-5
*
ql
er 0•
0
k
0
—>
10 3
Figure 3.1: Dimensionless critical intensity of the random load qi cr vs the
ratio k of the characteristic time for material T to the characteristic time for structure T* . Light points correspond to c = 0.4, and black points correspond to c = 0.8. qó < 4'ó cr
qi
~ 4~~er'
where qó *z 9i er
= min I
cr =
min F( p), n
NF(n)[F(n) — qó] + (T/T
)2 2 [F(n)
(3.7)
— go],
and F(n) = [1 + Qo( O)]{ 12(1 2 n2 ) R)z[1 + ( jR )2]2
4 ± -14---
R z -z [ I ± ( ~) ] }.
Let us consider a viscoelastic shell with R/1 = 0.2 and h/R = 0.001. The material obeys the constitutive equation (1.5.45) with the relaxation measure (1.6.7) and Poisson's ratio v = 0.3. Calculation of the first minimum in (3.7) yields qó er = 5.863 10-6 c.
Viscoelastic bar under random loads
241
For qo = 0, the dependence of the critical random load qi ,. on k = T/T* is plotted in Fig. 3.1. The numerical analysis shows that the critical random load reaches its maximal value for k 50 — 70 and has a relatively narrow peak of the maximum. Therefore, by choosing an appropriate characteristic time of material we provide a significant reserve of the shell stability. All the above considerations correspond to the case of isothermal loading. Let us now consider the shell stability when the temperature can change. We suppose that admissible variations of temperature are not very large, and we can neglect their influence on elastic moduli. The relaxation measure is assumed to be highly sensitive to the temperature changes. The material behavior is governed by the standard time-temperature shift principle, see e.g. Christensen (1982), Pipkin (1972). This phenomenon takes place, for example, in the neighbourhood of the glass-rubber transition point in polymers. In this case, the deterministic critical load qó is independent of temperature, whereas small variations of temperature imply significant changes in the stochastic critical load qi , as we pass through the peak on the curves plotted in Fig. 3.1.
4. STABILITY OF A VISCOELASTIC BAR DRIVEN BY RANDOM COMPRESSIVE LOADS In this section the stability problem is studied for a non-ageing viscoelastic bar driven by random perturbations of "white noise" type. Using the direct Lyapunov method some sufficient stability conditions are derived. Stability of elastic bars and shells under the action of a random compressive load was studied by Asokanthan & Ariaratnam (1992), Bolotin (1979), Lepore & Stoltz (1972), Potapov (1985). It was shown that purely elastic structural members are unstable under the action of "white noise" excitations. In order to derive stability conditions the materials were assumed to exhibit the viscoelastic behavior (Voigt's model). It was proved that the critical load is proportional to the viscosity coefficient. For simple integral models of viscoelasticity, stability of viscoelastic bars was analysed by Potapov (1984, 1989), Potapov & Marasanov (1992) by using explicit solutions for the corresponding Kolmogorov equations. 4.1. Formulation of the problem and basic assumptions Let us consider plane bending of a rectilinear bar with length 1, crosssection area S and moment of inertia of the cross-section. J. The bending
242
Chapter 3
occurs in the plane which passes through the longitudinal axis of the bar and its axis of symmetry. The bar is made of a non-ageing linear viscoelastic material with the Young modulus E and the relaxation measure Qo (t — s), which satisfies conditions (1.6.35) — (1.6.37) and (1.6.51). At moment t = 0, external forces P = P(t) are applied to the bar ends. Denote by y(t, x) the bar deflection at the point with longitudinal coordinate x E [0,1] at moment t E [0, oo). Function y and its derivatives are assumed to be so small that all the nonlinear terms in the expression for the curvature of the longitudinal axis can be neglected. Function y(t , x) satisfies the equation t pSy(t, c) + EJ[D4 y(t, c) -} l Qo (t — s)D 4 y(s, x)ds] o +R(t)D2 y(t,x) = 0,
(4.1)
initial conditions (2.1.33) and one of boundary conditions (2.1.6) — (2.1.8). Here p is mass density, and the superscript dot denotes differentiation with respect to time t. Suppose that 3 R(t) = Po + / 1,14 ),
(4.2)
where w(t) is the standard Wiener process. Egns. (4.1), (4.2) together with initial conditions (2.1.33) and boundary conditions (2.1.6) — (2.1.8) describe plane bending of a viscoelastic bar driven by random compressive load. According to Gikhman (1980), this initialboundary problem has a unique generalized solution provided initial data yo (x) and y1 (x) belong to the Sobolev space WZ with the norm ~k ' II 2 = j [D2yi(r)]2dC. Definition. The bar is stable in the mean-square sense if for any e > 0 there is a d = d(e) > 0 such that the inequality II yoII+IIyi I< implies the estimate sup Ey2 (t, x) <6, t,s
x E [0,1],
t E [0, oo).
Our objective is to derive some restrictions on Po and R1 , which would ensure the bar stability. 4.2. Transformation of the governing equations
Denote by y„ the maximal value of the bar deflection. Introduce the
Viscoelastic bar under random loads
243
following dimensionless variables and parameters: c*
_
c I
t t*
,
,
yo (c) ni(c *) =
To
y(t, c ) / u1(t* i x* ) = y*
y*
y(t c)To
u2 (t* i c* ) —
y*
n2 (x) = yi(c)To y*
, i
w* (t* ) =
1'~l2
E JTo o = 2, a = pSl 4 7
EJ
w(t)Tr
i
3
,
r1~o -1 * = PSI2 ,
1
where constant T2 is determined by inequality (1.6.52). According to Gikhman & Skorokhod (1969), w* (t * ) is a Wiener process. In the new notation, Eqn. (4.1), (4.2) and (2.1.6) — (2.1.8), (2.1.33) can be presented in the following form (for simplicity asterisks and argument x are omitted): du1 = u2 (t)di, du2 = — a[D4 u i () + j Qo(t — s)D4 u1(s)ds] — Ri D2 ui(t)dw(t), o ui( 0 ) = ni, u2( 0 ) = n2, 2 2 D u1(t, 0) = D u1(t,1) = 0, u 1 (t, 0) = u i (t, 1) = 0, Du l (t, 0) = Du i (t,1) = 0, ul(t, 0) = u1 (,1) = 0, Du1 (t, 0) = D2 u1(t, 1) = 0. u 1 (t, 0) = u1 (t, l) = 0,
(4.3) (4.4)
(4.5)
Denote by Yk (x) the eigenfunctions and by l k the eigenvalues for the differential equation D 4 Y + l D2 Y = 0
(4.6)
with one of the boundary conditions (4.5). It is well known that there is a sequence of eigenfunctions {Y„(x)} such that
0
),:
I D Yh (c)D9/Im (2)dx = dhm,
Jo ~ D 0 n(x)D 2
2
0 m(x)d C = lndnm,
(4.7)
where 6flm is the Kronecker delta. Sequence {Y„(x)} is complete in the subspace of Wz whose elements satisfy boundary conditions (4.5). Therefore, functions u~(t, x) and n( x) can be expressed in the form:
z S m=l f
u~ (i x) =
~ m( t )Ym( c),
S m1 00
n~ (t ic) =
S~m (t )Ym (x )•
(4.8)
244
Chapter 3
We substitute (4.8) into (4.3) and (4.4), multiply each e q uality by D2Y (x) and integrate from 0 to 1. Integrating by parts and utilizing (4.5) and (4.7) we find dz1 = z 2 (t)dt,
dz 2n = — al~[(1 — Ro l,~ 1 )z i,•,(t) +
jo Qo (t —
s)z in (s)ds]dt
+Rilh zi n(t)dw(t) i (n = 1, 2,...).
zin(0) = z1n,
z2n(0) = V2n,
(4.9)
It follows from (4.5), (4.7) and (4.8) and the Cauchy ine quality that for any r E [0,1] and > 0:
ui(t, x) = [
Jo
r
G
c
Jo
< ~[Du i (t , x)] 2 dx =
c
c
Du i (t, x)dx]2 ~ f [Du i (t, ~)]2 dx f dx o
S=i m,h
o
zim (t) cih(t) f ~ D~m (x)D~n ( c)dz
o
f
= Scih(t)i /
11 n~ 11 2 =
i
I
Jo
c
[D2v~ ( x )]Zd =
f
S V~mV~ h J m n=1 /
/r
n=i
i
/'
/,
D2 Ym(x)D2 Yn( c)d2
,
~
=
lhz
.
(4.10)
4.3. Stability conditions In this subsection we prove the followin g Theorem 4.1. Suppose that Po < )4 [1 + Qo(oo)]
(4.11)
and Oo
<
~
1± [al?(1 + Qo(oo) — Ro lT 1)]-1 Jo
~ Qo(s)1sds.
(4.12)
Then a viscoelastic bar driven by a random compressive load is stable in the mean-s q uare sense. q 4.3.1. Remarks 1. Condition (4.11) coincides with the necessary and sufficient stability
Viscoelastic bar under random loads
245
condition for a viscoelastic bar under the action of a deterministic compressive load R0 , see Chapter 2, Section 1. 2. Let us consider lateral oscillations of an elastic bar with the Young modulus E0 = E[1 + Qo(oo)]. It can be shown that the maximal period of natural oscillations is T
= 2t
p514 E0 J li
Denote by Pe = E° Jl 11-2 the Euler critical force for an elastic bar with the Young modulus E0 . Then stability conditions (4.11) and (4.12) can be written as follows:
P° < P1 2
1 2
—
f- 1Qo(s)1sds
Po/Pe
1 + 4p (Ts /T)2 (1 —
1,
R0/Pe)
1
+ Qo( oo)
( 4.13 )
3. Let us consider the standard viscoelastic solid with the relaxation measure (1.5.40) Qo(t) = —(1 —
1
É )[
— exp(-7`- )],
where E and E0 are the current and the limiting moduli of elasticity. In the absence of the deterministic component of the compressive load, PO = 0, stability conditions (4.13) take the form
I
R1
I< Re[( E o
-
1)To]' 1
2 ° 2 -1 [ + 4p (7) ] .
4.3.2. Proof Let us suggest some preliminary estimates. According to (4.11) there is a positive constant integer n > 1 a„ = 1 + Qo(oo) —
R° l~, 1 > 1 + Qo (oo) —
Ro li 1 > a.
a such that for any (4.14)
Function «5(x) = a x 2 [1 + Qo(oo) —
Ro x -1]
increases monotonously for x > R0 [2(1 + Qo (oo)]-1. This assertion together with (4.11) implies that for any integer n > 1 (4.15)
Chapter 3
246 where Y„ = f(l,2 ). Calculate the derivative with respect to time of function L(t) =
—
Q0(t) + j [Qo (t — s) — Qo(c )lds.
Employing (1.6.55) we obtain L(t) =
~~0(t) + [Qo(O — Q
-
o (oo)1
< 0.
Therefore, for any t > 0 L(t) > L(oo). With the use of (1.6.36) this inequality can be presented as follows: fR
L(t) > f 0
[Qo(s) - QO(f))ds.
Integration by parts with the use of (1.6.36) and (1.6.52) yields L(t)
f
> s[Qo(s) - Qo(f)]J-~~- f
Qo (s)sds =
j
~Qo (s)~ sds.
(4.16)
It follows from (4.12) that f
~Qo (s)~ sds - Pi a -1 (1 + 0A1 ) > 0.
This inequality together with (4.16) implies that for any t > 0 and any integer n>1
L(I) - R a-1(l+ Y )>0.
(4.17)
Now let us construct the Lyapunov functionals. First, calculate the differential of the functional W1 (t) = z (i) + al{[1 + Q 0 (t) — -
j
Ro l
1
jz (i)
z i (s)J 2 ds}.
0(t — s)[z1() —
(4.18)
Ito's formula and Egns. (4.9) imply that
Wih( t) = -al~ {-QO(t) c~h(t) + f -Ria
1
o
t Qo(t —
s)[z1 (t) — cih( s)] 2 ds
zim (t)}dt+2R1 ln cln (t)c2n (t)dw(t).
Now calculate the differential of the functional t 2 h (t — s) W2 (i) = Z 2 (t) + al,, ~ [Q o — Q0( f) j cih(s)ds.
(4.19)
(4.20)
Viscoelastic bar under random loads
247
It follows from this equality and (4.9) that dW2i (t) = -al~ o„ z 1,a (t)dt + R1 l„ z~„ (t)dw(t).
(4.21)
Egns. (4.9), (4.20) and (4.21) imply that the differential of the functional
W3 (i) = w2„ (t) + Y„ [cip (t) + W1(t)J can be calculated as follows:
(4.22)
dW3n (t) = -al~~Yp {[-Qo(1) - R~~a -1( 1 + Yp 1 )]z~p (~ ) +J
- s)[ci n (t) - z (s)J 2 ds
o(
o
[Qo(t - s) - Qo(f)]zin(s)ds}dt Jo + 2 R1lmz1»(t)[W2~ (t) + Y z 2 ()]dw().
+2z(I)
(4.23)
Transform the expression in the right-hand side of (4.23) 2z(1)
j
[Qo(t
=-
Jo
[Qo(t - s) - Qo(f)]z i n(s)ds (
-
s)
-
Qo l f)][ Z i(t ) - zi(s)]2 ds t
+z~z(t) j [Qo(i- s) - Qo()]ds t + [Qo(t — s ) — Qo()]z(s)ds.
j
Utilizing this equality we find
+ j[
dW3„(t) = -al~ fp {[L(t) - Ri a -1(1 + F„ 1 )]c~„(t) o (i - s) -(Qo(t
-
s)
+ ro t
-
Qo())][ zi(t) - zi(s)] 2 ds
[Qo (t —
s ) — Qo (oo)]ci „ (s)ds }dt
+2R1l„z1„(t)[W2n(t) + Ynz2n(t)]dw(t).
(4.24)
Integration of (4.24) from 0 to t and calculation of the mathematical expectation with the use of (1.6.52), (1.6.55) and (4.17) yields e W3„(t) < £W 3m (0). Substitution of Eqn. (4.18), (4.20) and (4.22) into this inequality implies that
S{c2p(t) + al n
f
[Q o(t —
s) —
QO(f)]zip( s)ds}2
Chapter 3
248 +YhR £{ c~h(t) + Z(t) + al~ [(1 +Qo(t) —
-
j
Rol
)z~hR(t)
0(t — s)( ci(t) — z i(ls))2ds]}
< — Y[1 + al,2, (1 + Qo (0) — Ro lp 1)]V~p + (1 + Fp )V2h It follows from this inequality, (1.6.35), (1.6.36), (4.14) and (4.15) that
aal(1 + i al~ )£z?hR (t) < (1 + al~ )( al,zihR +
(4.25)
Eqn. (4.25) implies that there is a positive constant c such that for any integer n >1 £ zihR (~) < c(V~hR + z2hR) •
(4.26)
Sum up inequalities (4.26) with respect to n from 1 to infinity. Using (4.20) we obtain oo
£ SR z (t) < c( h=1
11
11 2 ±11 12 112 ).
The assertion of Theorem 4.1 follows from this inequality and (4.10). 0. 4.4. Instability of an elastic bar under random compressive load
Let us consider plane bending of an elastic bar with the Young modulus E. The bar is compressed by the forces P(t) = Ro + P1 ~(t) applied to its ends. It is assumed that P0 < R .
Let us choose the dimensionless initial perturbation in the form
0 ni = ,
n2 = di~i (x),
where d is an arbitrary positive constant. The dimensionless variables ti,(t,x) are as follows: UI(t,x) = d zi(t)Yi( x),
u2(t, x) = dz2(t) Y1( x),
where functions z1 (t) and z2 (1) satisfy the equations dz1 = z 2 dt, dz 2 = —a li(1 — P oli 1 )z1dt+ Pi l1z1dw(t)
(4.27)
with the initial conditions zß(0) = 0,
2.2(0) = 1.
It follows from (4.27) and Ito's formula that the deterministic functions X~ (t) = £ci (t),
C2(t) = £zi z2(t),
C3(t) = £z2(t)
Viscoeiastic bar under random loads
249
satisfy the differential equations
·
l=
C3 = qC1 — 2FRC2
X2 = -FXi + C3,
2c2,
(4.28)
with the initial conditions C2 (0) = 0,
x1 (0) = 0,
C3 (0) = 1,
where N = ~IAT.
FR = aA7(1 - i 0 A1l),
The characteristic equation for system (4.28) is written as follows:
f(k) = k 3 + 40k — 28 = 0.
(4.29)
For P1 Y 0, function f(k) grows monotonously, f(0) = —28 < 0 and f (oo) = oo. Therefore, Eqn. (4.29) has only one positive root k1 = k. Two other complex conjugate roots can be found as k 2, 3 = 2 where w 2 = 3k 2 + 160> 0. It can be shown that solutions of Egns. (4.28) have the form C1 (t) = —[ k(Ci cos — +w(Ci sin 2
C2 sin 1)
+ C2 cos 2 )] exp(— 2) + 2C3 k exp(kt), 2
C2(t) = l [( k — w2)( C1 cos 2 — C2 sin 2 ) ki 2 +2kw(Cl sin + C2 cos 2 )] exp(— 2) + C3 k exp(Kt), 2 wt C3(t) = [(28 + k0)(Cl cos 2 —C 2 sin ) ±Ow(Cj sin
2
+ C2 cos 2 )] ecr(-
where 42 1
=
8(9
2
+ w2) '
2
)+
2C3 (8 — kFR) e cr(kt),
(4.30)
2 2 3 K 2 + CV 3k 2 ) C3 — qw(9k 2 + w2) ' 28(9? + w 2 ) . 2
C2
=
It follows from (4.30) that for any d > 0 lim sup £u1 (t, x) = oo. oo xE[o,1] Therefore, an elastic bar driven by random load is unstable in the mean-square sense for any nonzero intensity of "white noise" excitations. This result was obtained by using another technique by Ariarathnam & Xie (1990).
250
Chapter 3
According to this result and Theorem 4.1 the material viscosity plays a two-faced role in the stability problems. On one hand, viscosity leads to a decrease of the critical force for deterministic loading, i.e. to a reduced resource of stability. On the other hand, it implies a growth of the critical random loads and an increased resource of stability for structures driven by "white noise" perturbations.
5. STABILITY OF A CLASS OF STOCHASTIC INTEGRO-DIFFERENTIAL EQUATIONS In Section 4 the stability problem was analysed for a system of stochastic Volterra integro-differential equations with convolution kernels of integral operators. In this section we extend the above results to the case of nonconvolution kernels. The corresponding equations describe the mechanical behaviour of viscoelastic structural members subjected to ageing under the action of random forces. We derive explicit conditions of stability in the mean square sense. These conditions are obtained by using the Lyapunov direct method and constructing stability functionals. As examples, we consider the stability problem for an ageing viscoelastic bar under stochastic compressive loading and formulate some conditions on the load, which ensure the bar stability for arbitrary relaxation measures and for various types of end supports. 5.1. Formulation of the problem In this subsection the stability problem is formulates for a class of stochastic integro-differential equations. To explain why this class of equations is considered, we begin with a typical problem of the bar stability under the action of stochastic loading. Then, we generalize this problem introducing some additional assumptions about external loads. Finally, we replace these mechanical problems by the problem of the mean square stability for stochastic operator equations. Let us consider a rectilinear viscoelastic bar with length 1, cross-section area S and moment of inertia J. At moment t = 0, compressive forces P are applied to the bar ends. Under the action of external forces, the bar deforms. Denote by y(t, x) the bar deflection at point x at moment t > 0. We suppose that (1) function y and its derivatives are so small that we can neglect the nonlinear terms in the formula for the curvature of the longitudinal axis, (2) the hypothesis regarding plane sections in the bending is fulfilled.
251
Stability of stochastic equations
For the linear viscoelastic material (1.6.3), function y(t, s) satisfies the equation, cf, (4.1), pSy(t, x) = —EJ[D
4
y(t, x) — ( -- (t, s)D4 y(s, x)ds] — R D2 y(t, t) Jo
(5.1)
with the initial data ji( 0,x) =
y(0,x) = y0(x),
and one of boundary conditions (2.1.6) — (2.1.8) D2 y(t, 0) = D2 y(t, l) = 0, Dy(t, 0) = Dy(t,1) = 0, Dy(t, 0) = D2 y(t,1) = 0.
y(t, 0) = y(t, I) = 0, y(t, 0) = y(t,1) = 0, 11(1,0) = y(1, l) = 0,
(5.2)
Here P is mass density, yo (x) is the initial deflection, yl(x) is the initial speed of deflection, D is the operator of differentiation with respect to x, (Du = áu/óx). Relaxation measure Q(t, s) is assumed to satisfy the constitutive restrictions (1.6.61) and (1.6.62). Suppose that (5.3)
P = PO + R1~(t),
where Po , P1 are constants, w(t) is a standard Wiener process and 11)(t) is a white noise. Expression (5.3) allows the deflection of a viscoelastic bar to be studied under the action of a random load with a rapidly decreasing correlation. Substitution of (3.3) into (5.1) yields (Ju l = u 2 (t,x)dt, du 2 = —[Au
i (i,x)
—
l
(t, s)Bu l (s, x)ds]dt — C ul (t, t)dw(t)
(5.4)
where ui = y(1, r),
u2 =
~ D2 . A = p(EJD4 + P1 D2 ), B = PS D4 , C = P
(5.5)
The specific form of operators A, B and C is determined by external load and types of support. Egns. (5.5) correspond to a bar under the action of compressive forces applied to its ends. For a bar on an elastic Vinkler foundation, operator A has the form, see e.g. Volmir (1967),
A=S P
(EJD4
2
+ RoD + kI),
(5.6)
252
Chapter 3
where I is the unit operator and k is the rigidity coefficient. Some other examples of operators A are given by Drozdov et al. (1991). As a generalization of the above mentioned problems, we will consider the system (5.4) under the following assumptions: HI: A, B, C are selfadjoint commuting operators; H2: B is a positive definite operator, its inverse operator B-1 is bounded. It follows from HI that there is a sequence of the eigenfunctions {Yi (x)} such that (yi, y j) = iii and Ayi = lA y
,
B0i = lB Yi i
CYi
where VA, lB and l are the eigenvalues of A, B and C, respectively, d the Kronecker delta, and (j,.> denotes the inner product in L2. Function ui (t, x) can be presented in the form ui(t,x) = S Ui(t) y~ (c),
is
(5.7)
d =~
where Ui (t) = (u i (t, x), Yi (x)). Introduce the following Definition. The zero solution of system (5.4) is stable in the mean square sense if for any e > 0 there exists a d > 0 such that the inequality (v1, Av4) + (12,12) < d implies sup £(u i (t), u 1 (t)) < e, t>o
where £ denotes the mathematical expectation. This definition means that if the total (kinematic & potential) initial energy is small, then the displacements are small in the mean square sense for any moment of time. 5.2. Stability conditions In this subsection we derive some sufficient stability conditions for the zero solution of system (5.4). For this purpose we construct Lyapunov's functionals under the assumption that min (lA — I B0 I lB) > 0,
(5.8)
Stability of stochastic eq uations
253
where 1 Ro 1= s p Ro(t), R0 (t) = R(t, t) = 1 u
( ( s, t)ds.
First, we employ the functional W~ (t) = (12 (t),112(1)) + u1(t), Au1(t))
According to I to's formula and (5.4), we find dW1 (t) = [2
/ . 1
Q (t, s)(u2 (t), Bu1 (s))ds + (Cu' (t), Cz1 (t))]dt a
—2(u
2 (t),
Cu i (t))dw(t).
(5.9)
Introduce the functional W2(t) = W1(~) ±
J
s)(u1(t) — uI(s), B(ui(i) — u i (s)))ds.
Eqn. (5.9) implies dW2 (t) = [-2Q(t, 0)(u2 (t), Bu i (t)) + (Cu1 (i), Cu i ()) +
02Q
-(i, s)(u
J o atas
— ui(s), B(ui(t) — ui(s)))ds]dt —2( u 2 (t), Cu 1 (t))dw(t).
It follows from this relation that the functional W3(t) = W2(t) + Q(t, 0)(ui (t), Bui (t)) has the differential dW3(1) = [ ±1
dtQ
(t, 0)( i (t), 1uß (t)) + (Cul(t), Cui(t))
` á á (t, s)(u1(t) — ui (s),B(ui (i) — u i (s)))ds]dt —2( u2 (t), Cu i (t))dw(t).
(5.10)
We now introduce the functional W4 ( ) = 112 (t) + / R(t, s)1 u i (s)ds , 3o
where function R(t, s) is the relaxation kernel satisfying conditions (1.6.67) (1.6.69). Using Ito's formula we obtain from (5.4) d1474 (t) = —( A — Ro (t)B)ui (t)dt — Cui(t)dw(t).
Chapter 3
254 It follows from this equality that the differential of the functional / W5(t) = (W4(t), B
-1
W4(t))
has the form dW5(t) = [-2(u2(t), ( AB 1 - Ro(t)I)u1(t)) + (Cu1(t),
2 J]
'
0
B-~C
ul(t))
R(t, s)((A - Rp (2)B)u i (t), /11 (s))ds
-2(Cu l (t) , B-1 u2 (t) +
o
R(t, s)ul (s)ds)dw(t).
(5.11)
Let W6(t) = W5(t) + (u1(t), ( AB-1 - Rfl(t)I)ui(1))• This equation together with (5.11) implies that 1 1 dW6(t) = [ k (t)(u1(0, 4(t)) + (Cu1(i), B Cu1~t)) -2 t R(t, s)((A - Ro (t)B)u i (t), u i (s))ds]dt J0
-2(B-1Cu1 (t),u2 (t) + J R(t, s)Bu i (s)ds)dw(t). (5.12) 0 Transform the third term in the right-hand side of (5.12). Using (5.7) we have
-2
J
0
R(t, 5)((A - Ro(t)B)u i(t), ui(s))ds
= -2 s(l4 - Ro (t)lB) J R(t, s)U: (t)U1(s)ds ~
= S(
Ro(t)l8)
J
R(t, s)(U=(t) - UU(s)) 2 ds
- °
R (t) S(kA i=1
00
- S(lA -R0(t)l) i=1
where R°(t) =
~ o
R(t, s)ds.
Introduce the functional W7(t) = S6 (t) + aW3 (t),
i
( R(t, s)Us (s)ds,
Jo
(5.13)
Stability of stochastic equations
255
where a is a constant to be determined below. It follows from (5.12) and (5.13) that (t, 0 )(uß (t), Bui (t)) + 00 1(t), (aI + 13-1)C2 u i (t))
dW,(t) = { —
+
S g
l t [al6
~
R° (t) (lA — Ro(t)lB)½?(t)
R4(t)(u1(t), u1(t)) —
at a (t,S)
+ (l - Ro (t)lB)R(t, S)](U,(t) - U (s)) 2 ds —
S(l 4 — R o (t)l6) I t R(t, s)U,?(s)ds}dt i
—2(Cu
~=1
i (t), (aI
0
=i
+ B-1)u2 (t) + l R(t, s)u i (s)ds)dw(i). o
Integrate this equation from 0 to t and calculate the mathematical expectation. We obtain £W7() - ew,(o) / Ti o)(ui(T)i Bu i (T)) — ~ a ( Jo f + (tYi (T), (oJ + B-1)C2 ui(T)) — R° (T) S (lA — R o(T)lB)U ?(T) d=1 =e
— S Jr [— alB a(t, s) — ( lA — Ro( T)lB)R( T, s)] ~ =i o 2 C(U (T) — U,(s)) ds
—
Ro (t)l6) jo
S( E=1
R(t, s)U?(s)ds}dt.
(5.14)
Choose a — R4I l 2 lA I a=T max ~~ A6 where constant T = T2 is determined by Eqn. (1.6.62). Condition (5.8) implies that a > 0. It follows from this inequality, (1.6.61), (1.6.64), (5.14), and the properties of Ito's integrals that — eW,(0) <—
l GH~ (t) — S ~ 30 t
=1
~
- S / t eUs (s)ds i =i
33 0
(a + e)(lC)
l~~
I
2
]eU?(t)dt
lt (lA _ I R0 lB) R(t, s)dt, a
(5.15)
Chapter 3
256 where H(I) =
— a 5Q (t,0)l~~ + 1 0 (i) + R0(I)(l
I Ro I
lr)
Suppose that for any i = 1, 2, .. .
(a+ e)(07)z < inf 112 (1).
(5.16)
It follows from Egns. (5.15) and (5.16) that e W7 (t) < 8W7(0). Substitution of the expressions for Wk (t) into this inequality yields a(u2(t), uz(t)) + (W4(t ), B W4(t )) 11 +(41(1), [cv(A + Q(t, 0 )1) + (A — Ro (t) B)B-1] 1(1)) A-a r ~Q (t, s)( 74(t) — u i(s), B(ui(t) — ui(s)))ds J
< (ui (0), [aA + (A — Ro(
0
)B)B-1lu i ( 0 ))
(5.17)
+(uß(0), ( aI+ B-1)uz( 0)). Suppose that min [lA + Q(fR, 0)l ~~ > 0.
(5.18)
Eqn. (5.17) implies the following
Theorem 5.1. Suppose that inequalities (5.8), (5.16) and (5.18) hold. Then the zero solution of system (5..') is stable in the mean square sense. q For a non-ageing viscoelastic material with Q(t, s) = Qo(t —
s),
we have
R(, s) —
~
JQ (t, s)dt = — as
Qo(t — s)dr =
J
Qo (t — s) — Q o(f), = R(t t) =
R°(t) = / R(t, s)ds = o
J o
Jo
t
—
Qo(f),
[Q (s) — Qo(oo)]ds.
Condition (5.8) can be written as min [4A + Qo (oo)l j > 0.
(5.19)
Stability of stochastic equations
257
This inequality implies (5.18) . Functions H;(t ) have the form H; (t ) = — il BQ0(t) +
[la
+ Qo(c )lr] j [Qo(s) — Qo( f)jds.
We calculate the derivatives of H; (t ) with respect to time and estimat e them get
with the use of (1.6.35) — (1.6.27), (1.6.55) and (5.19). As a result we
~ = T2 l B max A
+ QoBoo)lB Qo (t)
s
s
—[ ~~ + Qo(oo)lB][Qo(t) — Qo(oo)] > [l; + Qo (oo ) l Therefore, h
)
j{T2~
o (t ) — [Q o(t) — Q0(oo)]} > 0.
H ( t ) > H (oo) = [lrR + Qo( co)lr]j: [Qo(s) — Qo(oo)]ds. Integration by parts with the use of (1.6.35 ) yields Jo f
Qo ( oo )j ds
[Qo(s) —
—/
3o
= s[Qo(s) — Qo( f)]~-~~
f o( s )sds =
Jo
I
o( s) I sds.
Thus, condition (5.16 ) can be written as follows:
(l)2
< ( + B) — i [ll + Qo(oo)lB]
~ I Qo(s) 1 sds.
jro
( 5.20 )
We arrive at the following Theorem 5.2. Suppose that inequalities (5.19) and (5.20) are vali d. Then the zero solution of system (5.4) is stable in the mean square sense. q
5.3. Examples 1. Let us develop stability conditions for a simply supported viscoelastic bar compressed by a random load. In this case, Y;(c) = sin piZ/I, and the eigenvalues of operators (5.5) have the form lA
—
n4i4EJ pS14
1 —
~
Ro
Pe2 2 )i
lB a
— z4i4
EJ
pS14
i
lC a
—
p 2 i 2 R1 m S12
where Pe = it2E J1-2 denotes the Euler critical force.
(5.21)
Chapter 3
258
First, we analyse the stability for a bar made of an ageing viscoelastic material. For simplicity, we assume that R'(t) > 0 for anyt > 0. Substitution of (5.21) into (5.8) and (5.18) yields
R° <1—
—
max[~~ R° ,
(5.22)
Q(oo, O)]•
2 It follows from (5.21) that a = 7' (1— (5.21) imply
I
RO ). This expression, (5.16) and
~
( R1 )2 < H0 [1 -{- 4 p2(1- I R o )( T ) 2 ] -1 Pe T*
(5.23)
Here
— RO ~ ~[(1 I i>
H0
° — R )R°(t) — (1—
I
Ro I) 8 (t, 0 )],
and
T —2
pS14
p2 E J
is the maximal period of natural oscillations. Stability conditions for an ageing viscoelastic bar consist of restriction (5.22) for the deterministic compressive force PO and restriction (5.23) for the random component R1 . Now let us consider a non-ageing viscoelastic material. In this case, stability conditions have the form (5.19) and (5.20). These inequalities imply Ro < Pocr = R [1 + Qo(oo)]•
(5.24)
Substitution of (5.21) into (5.20) yields 4 •4 PO2 )[1+T2(1+ Qo(~)) ~p EJ]-1, SI Roc r 2
1
(R )2
(5.25)
where
- f°O1 -l-I
Qo ( s )
I sds
Q'(oo)
The right-hand side of (5.25) increases monotonously in i . Therefore, inequality (5.25) holds for any positive integer i if and only if ( ócr )2
<1(1 —
°
r
2 )[1 + 4p (1+ Qo( f))(l ) 2 ]
1
-
(5.26)
Stability of stochastic equations
259
Egns. (5.24) and (5.26) guarantee the stability of a non-ageing viscoelastic bar under stochastic loading. For the standard viscoelastic material with the relaxation measure (1.5.40), these conditions take the form 1
c
Pe
P Eo
E
< Roer = Re ~i
2 —E E~ (I —T,* )2] -1. )[1+4i
The dependence of the dimensionless critical load P1 , on the parameter c = E0 /E is plotted in Fig. 5.1. The critical load decreases with the growth of c and vanishes when c = 1 (purely elastic material). When c tends to zero (i.e. when the material viscosity grows), the critical load increases and tends to infinity. With the growth of T/T, value, the critical force diminishes. 10
Rlcr
0 0
c —i•
1
Figure 5.1: The dimensionless critical load Ri ~,. vs parameter c. The calculations are carried out for Po = 0. Black points correspond to T/T„ = 0.1, light points correspond to T/T, = 1.0, and asterisks correspond to T/T* = 10.0. 2. Let us consider the stability problem for a simply supported bar on an elastic foundation. We restrict our consideration to the case of a non-ageing viscoelastic material.
260
Chapter 3
The eigenvalues of operator (5.6) can be written as _ ~4 i 4 EJ
where k (5.21).
=
k
PO
(5.27)
rei2
Pcl4
kl 4 /(P4 EJ). The eigenvalues of operators B and
C have the form
7
.,~. • •~ 'N. •·•%..... *
N~ NNNN..~NN~NNNNN~NNNmN
.... ~
***
*** **
yy TT #* * * **
C ** ** yy } }*
1
i
**~
0 Figure 5.2: The dimensionless critical loads
and
Pier vs the dimensionless rigidity of an elastic foundation k1. The calculations are carried out for Qo (oo) = —0.2. Black points correspond to T/T* = 0.1, and light points correspond to T/T* = 1.0 . 1
r
First, let us consider a viscoelastic bar under the action of a deterministic load (P1 = 0). Substitution of (5.21) and (5.27) into (5.19) implies the only stability condition
j cr
< PO
cr
,
where 2 4 R cr = min i (1 + k 0 i )
(5.28)
Concluding remarks
261
and k o = k/[1 + Qo (oo)]. It follows from (5.29) that the critical load grows when the rigidity of an elastic foundation increases. The dependence of the dimensionless critical load Poor on the dimensionless rigidity k o is plotted in Fig. 5.2 by asterisks. Let us now consider a viscoelastic bar under the action of a random load ( Po = 0). In this case, stability condition (5.20) can be presented as follows: Pl V `V ROer
Plcr
where // P~Cr = min /(1 +
i4
)[4 + 4p2 ( —7T,* )2 (1 + Qo( oo))(1 ~- Ko)]-1.
(5.29)
The dimensionless critical load R1~ r vs the dimensionless rigidity k o is plotted in Fig. 5.2. The numerical results show that the growth of the rigidity of an elastic foundation leads to an increase of the deterministic critical load 1 cr and to a decrease of the random critical load R ,.. When T/T, value increases, the critical random load Pier diminishes.
6. CONCLUDING REMARKS In this chapter the stability of viscoelastic elements of structures is studied under the action of time-varying and random loads. For this purpose we employ the Lyapunov direct method and have constructed new stability functionals. These functionals utilize the specific properties of relaxation measures for viscoelastic materials suggested in Chapter 1. In Section 1 the stability problem is considered for a cylindrical shell made of a non-ageing viscoelastic material under compressive longitudinal and radial loads which can vary in time, and stability conditions (1.7) and (1.8) are derived. The well-known stability condition (1.7) imposes restrictions only on current values of compressive loads. This inequality is determined by the "zero" moment of the relaxation measure J ~o
IQo(oo)I =
0
lQo(s) ~ ds.
(6.1)
Condition (1.8) is new. It imposes limitations on the rate of change of external forces. These constrains are determined by the first moment of the relaxation measure f
J
oo
~ Qo(s)(sds.
(6.2)
262
Chapter 3
Since the zero moment of the relaxation measure does not depend practically on the temperature, while the first moment depends on it drastically, these stability conditions allow the critical load to be increased significantly by changing the temperature conditions in an appropriate way. Numerical analysis has been carried out for a viscoelastic shell under the action of a periodic longitudinal load. The results of calculations show that the critical frequency of periodic excitations depends significantly on the intensity of the sinusoidal compressive load. The frequency does not vanish only in the narrow vicinity of zero, in which it tends to infinity when the intensity of the periodic load tends to zero. With the growth of the material viscosity, the behaviour of critical frequency has a non-monotonous character. It increases for large intensities of periodic compressive load, and it decreases for small intensities. Results of Section 1 were derived by Drozdov (1993). In Section 2 we analyse in detail the stability problem for a linear integrodifferential equation of the second order with a periodic coefficient. By employing a similar approach for a narrow class of stability problems (only one equation with an additional assumption about the periodic character of a time-varying coefficient) we have derived more sophisticated stability conditions (2.13) — (2.15). It has been shown that these sufficient conditions are very close to the necessary stability conditions for ordinary differential equations. Namely, the stability conditions obtained require the stability of the corresponding differential equations and, additionally, impose some limitations on the relaxation measures. These restrictions are plotted in Fig. 2.1 for the standard viscoelastic solid with the exponential relaxation kernel. This kernel is determined by two positive constants: the characteristic relaxation time -y -1 = T and the material viscosity c. The numerical results show that for a fixed frequency of the compressive load, the domain of stability is a bounded region in the (c, ·c ) plane, which collapses with the growth of the frequency of the sinusoidal compressive load. Results of Section 2 were developed by Drozdov & Gil (1994). Sections 3 — 5 are concerned with the study of the mean square stability for viscoelastic structural members driven by random excitations. It is assumed that random load has so rapidly decreasing correlations that it can be described by the "white noise" model. This leads to the governing integro-differential equations in the Ito form. We have proposed new Lyapunov functionals to study the stability of these equations and have derived some sufficient stability conditions. First, we analyse the mean square stability for a non-ageing viscoelastic cylindrical shell and develop stability conditions (3.4) and (3.6). It is worth noting that these conditions are determined by the same moments of relaxation measure (6.1) and (6.2) as in the case of time-varying loading. For the standard viscoelastic material it has been shown that the main physical parameter determining resource of stability is the ratio of the characteristic time of a viscoelastic material T to the characteristic time (maximal
References
263 T*.
period of natural oscillations) of a structure Dependence of the critical intensity of random load on this parameter is plotted in Fig. 3.1. The curves demonstrate the drastically non-monotonous character of this dependence. This means that by choosing an appropriate viscoelastic material one can increase significantly the resource of stability for a given structure driven by random excitations. Results of Section 3 were derived by Drozdov (1993). Section 4 deals with the stability problem for a viscoelastic bar driven by a random compressive load. It has been shown that for a sufficiently small intensity of the stochastic load a viscoelastic bar is stable in the mean square sense, while the corresponding elastic bar is unstable. At first sight, this fact contradicts the well-known assertion that the material viscosity leads only to the decrease of the stability resource. The explanation is rather simple: the material viscosity leads to the decrease of the critical load only for deterministic loading, whereas for stochastic loading we have the inverse picture. Results of Section 4 were derived by Drozdov & Kolmanovskii (1991, 1992). The method employed in Section 4 can be extended to arbitrary integrodifferential equations with operator coefficients both for convolutive and nonconvolutive kernels of integral operators. In Section 5, general stability conditions (5.8), (5.16) and (5.18) have been derived for integro-differential equations with operator coefficients and non-convolutive kernels. As an example, the stability problem has been analysed for a viscoelastic bar lying on an elastic foundation under the action of a random compressive load. The numerical analysis demonstrates that an additional rigidity (or "elasticity") of a system leads to the increase of the deterministic critical load and to the decrease of the stochastic one, see Fig. 5.2. In particular, the presence of an elastic foundation implies a growth of the Euler critical force and a decrease of the random critical load. On the contrary, the growth of the material viscosity diminishes the deterministic critical force, and increases the stochastic one. Results of Section 5 were developed by Drozdov (1994).
REFEREN CES 1. Akbarov, U., Badalov, F.B., & Eshmatov, Kh. Stability of a viscoelastic rod under dynamic loading. J. Appl. Mech. Techn. Phys. 33(1992) 613-616. 2. Andreeva, E., Kolmanovskii, V., & Shaikhet, L. Control of Hereidtary Systems. Nauka, Moscow, 1992. 3. Ariaratnam, S.T. & Xie, W.C. Liapunov exponent and rotation number
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