Ghostscript wrapper for D:\Digitalizacja\MTS86_t24z1_4_PDF_artyku³y\mts86_t24z1_2.pdf M E C H A N I K A TEORETYCZNA I STOSOWANA 1/2, 24, (1986) DYNAMIC STABILITY OF VISCOELASTIC CONTINUOUS SYSTEMS UNDER TIME-DEPENDENT LOADINGS ANDRZEJ TYLIKOWSKI Warsaw Technical University 1. Introduction The problem of static buckling of viscoelastic columns under constant axial forces has been solved by DE LEEUW [1]. Applying the correspondence principle and analysing the properties of elasticity moduli the critical loadings for several viscoelastic models have been obtained. One of the first analyses of the dynamic stability of viscoelastic conti- nuous systems has been made by GENIN and MAYBEE [2]. In this paper the stability of a beam made up of a linear Voigt-Kelvin material with viscoelastic boundary conditions, has been investigated. In the next significant study PLAUT [3] has used the Liapunov method to determine the stability criteria of viscoelastic columns subjected to compresive axial loadings. Using the same method WALKER and DIXON [4] have examined the effect of a linear structural damping on the stability of plane membranes adjacent to a supersonic airstream. The dynamic stability of continuous systems under time-dependent deterministic or stochastic loadings has also received much attention, e.g. (KOZIN [5], ARIARATNAM and TAMM [6], TYLIKOWSKI [7]). The problem was solved not only for a simple elastic column subjected to an axial time-dependent force but also for arches, panels, plates and shells. In most papers the dissipation of energy was described by an external viscous model of damping. In the present article the applicability of the Liapunov method is extended to linear Voigt-Kelvin systems subjected to time-dependent deterministic or stochastic parametric excitations. Using appropriate functionals general sufficient conditions for the asymptotic stability, the almost sure asymptotic stability as well as the uniform stochastic stability are derived. The paper describes the two general approaches to the stability analysis and present some illustrative examples. 2. Problem Formulation Consider a Hilbert space Jf of all summable functions having all generalized deriva- tives of order ^ 2« on the open set Q, summable to the power 2, independent of time, 128 A. TYLIKOWSKI possessing a suitable inner product <.,.> and a dynamic system, which is assumed to be -well defined by the equation u + Du+Ku+ yj (Si+qdLiii = 0, xeQ, (i) 1 = 1 under the condition that at every fixed t e [0, oo) the state of the system («, it) belongs t o the product F x Y, where Y (Y a jf) is a subset of functions belonging to jf, which satisfy given linear time-independent boundary conditions on the boundary 3Q of Q. Operators K, D, Lt are linear differential with respect to spatial variables. AT is self-adjoint of order 2n, D is of order «$2«, L; are self-adjoint of order < « , qt and £; are constant and time-dependent loading components, respectively. The question of interest is the stability of the eqilibrium (it, it) = (0, 0) for a general • system of the form (1). To estimate deviations of solutions from the equilibrium state we introduce formal stability definitions using a scalar measure || • ||, which is the distance between a solution of equation (1) with nontrivial initial conditions and the trivial solu- tion. The study of stability of equilibrium state splits into three branches. First, under the assumption that trie time-dependent components of forces are deterministic functions of time, conditions of the asymptotic stability of the trivial solution, i.e. conditions that imply Um||«H = 0 are derived. Our second purpose is to discuss the almost sure asymptotic stability of the trivial solution, i.e. that corresponding to the equality P { l i m |M | = 0 } = 1, t~> 4< s > 0 d > O r > o (t ^ U I We are going to study the foregoing kinds of stability via the Liapunov functional approach. In order to employ the direct Liapunov method we construct the class of func- tionals as follows / V= a{u,u} + (l—a)^u + Du,u + Duy + {itiKu}+Cu,Y'qiLlw), (2) 1 = 1 where 0 < a ^ — . For a = — we have the functional similar to „the best" functional applied by KOZIN [5]. The mentioned functionals are the same only if the dissipation operator corresponds ST ABI LI T Y O F VISC OELASTIC SYSTEM S 129 to the viscous model of damping. F or a arbitrarily small but positive we obtain the func- tional  similar  to  that  introduced  by  PLAU T  and  IN F AN TE  [8]. The functional V satisfies  the desired  positive  definite  property  if + y< M  +  2JM!M  +  I>M> +  < M , X u > + ( M , ^ ? A«) -   (4) If condition  (3) is  satisfied  functional  (4) is  positive  definite  and its  time- derivative  along equation  (1)  is 4 T T  - - \ (K+  2 W u,Du)- (lu+Du,  ^  £ ( 0A«) .  (5) a i  (= i  i= i Our  object  is  to  obtain  bounds  on  F t  that will  guarantee  the  asymptotic  stability  or  the almost sure asymptotic stability.  In order to do this we transform  (5) into the form ^   (6) • where U ±  is the known functional  and A is a parameter describing  the intensity of  damping. We  now  attempt  to  construct  a  bound - where the function %  is  to  be  determined.  Substituting  (7) into  (6)  and solving  the  obtai- ned  differential  inequality  we  have ^ ( 0  < V 10 exp{- Ut+2J X (s)ds\ .  (8) o Thus,  it  immediately  follows  that  the  sufficient  stability  condition  for  the  asymptotic stability  with  respect  to  the  measure  11  •   11  = y V x   is \ im±- j X (.s)ds Ś X,  (9) r- >- oo  *  o • 9  M ech .  T eo r et .  i  Stos.  1—2/ 86 130 A. TYLIKOWSKI or for the almost sure asymptotic stability, if the processes £( are ergodic and stationary, is E% < A, (10) where E denotes the operator of the mathematical expectation. 4. Uniform Stochastic Stability If the excitations are the Gaussian white noises equation (1) should be rewritten in the Itó differential form du = vdt, i i dv = ~[KU+DV+ YqtLtu)dt- Y^Ltudwt, (11) 1 = 1 /=! where wt are the standard uncorrelated Wiener processes with intensities at. As realiza- tions of the Wiener processes are not differentiable the Ito calculus has to be applied in the stability analysis (see e.g. CURTAIN and FALB [9]). Taking functional (2) we calculate its differential dV= lloclv, -Ku-Dv- ^ ? , £ , « ) + 2 ( 1 - a ) ( v + Du, -Ku- £ /=i 1=1 ) . (12) On integrating with respect to t from s to rs(t), where rd(0 = min{T,,, 0 , ra = fafif: \\u\\ > d > 0} and rearranging the integrand it follows that *• >(<) i = V(s)-2J {u<.v, Dv> + (\-aXKu, Du} + (l-a)(Dw, ^? qtLtu)- J (v + CL-a)Du,^atLludw^. (13) (i We now take the conditional average of equation (13) remembering that the second integral is a stochastic one, so the conditional average of it is equal to zero rtdD EV{xa(t)) = V(s)-2E f ja»> and proceeding similarly to the proof of the Chebyshev inequality we have the following chain of inequalities V(s) > EV(rd(t)) = f V(r6(t))P(dy) > / V((rs(t))P(dy) > 3 2i»{sup||«|| > Ó], where y e (F, P, P), i.e. y belongs to the probability space F with a — algebra /? and pro- bability measure P. Setting s ^ O w e conclude that the trivial solution of equation (11) is uniformly stochastically stable with respect to measure || • || = F1'2, if the following inequality is satisfied for every u e Y t i (\-OL)(DU,KU+ ^ qthu)- ±- ̂ oKLtU^tU} > 0. (15) l-l If a is arbitrarily small but positive, we shall obtain the largest stability region as a function of damping parameter and intensities V / **> / / -** 0,02 0,04 0,06 0,08 0,10 A DAMPING COEFFICIENT Fig. 1. where the variance of the sinusoidal process is equal to the half of the amplitude squared. As the second numerical example we take the beam compressed by a Gaussian process. The dependence of the stability regions on the retardation time X, variance a and constant load q is shown in Fig. 2. STABILITY OF VISCOELASTIC SYSTEMS 133 1600 1400 1200 7 0 0 0 - 800 ^.600 ^.400 200 — GAUSSIAN PR OCEi j y _—-— / / / y / / / y 0,02 QOt 0,06 0,08 0,10 A DAMPING COEFFICIENT Fig. 2. 6. Uniform Stochastic Stability of a Yiscoelastic Beam Compressed by the Gaussian white noise If the load acting in the beam axis is a broad-band Gaussian process we model the excitation by means of a white noise of intensity a and rewrite equation (17) in the Itó differential form du ~ vdt, dv = -{u""+Uv""+qu")dt-au"dw, (19) where w is the standard Wiener process. Using the functional V defined by (2) we obtain the simpler form of the general stability condition 1 2 , J [X(u""Y - Xq(u"'Y - ^ (u")2\dx > 0. Finally the condition for the uniform stochastic stability of the undeflected beam is given by 7, > o2l4n2(n2-q). 134 A. TYLIKOWSKI 7. Uniform Stochastic Stability of a Plane Bending Form of a Viscoelastic Thin­Walled Double­Tee Beam Let us consider the f lexural-torsional stability of a thin-walled double-tee simply sup- ported beam subjected to broad-band Gaussian couples m acting on both ends in the plane of greater bending stiffness. Assuming the technical theory of thin-walled beams we neglect rotatory inertia terms and an influence of transverse forces on displacements of the beam and describe the displacement state by the axis displacements and the angle of torsion. As we are going to examine the stability of the plane form we can omit the equation of motion in the plane of the couples and describe the deviations from the plane state by the transverse displacement u of the beam axis and the angle of torsion "-e3 = -(e2"+qu")dt-(ni"dw. We can now identify the operators r«.(.)"" O e2{ r o o"]L D = 2XK. The functional is specified as follows V= J {a(^2 + V 2) + ( l - « ) ((v + 2Xe1u"") 2 + (f + 2Xe2y""-2ke3cp") 2) + e1(u"") 2 + e2('") 2 +0«i u"" 0. (24) Integrating by parts and using boundary conditions (23) one can show that i i / u"" 0, where a2 is to be determined. Taking into account the extremal property of minimal eigenvalue of boundary problem (23) we obtain the following inequality i / {[(el - (ex + e2y/2oc 2)^-e23/2a 2 -o2/2X\(u")2 + o + {(e2n 2 + e3y-q 2U2-G2l2X]( 0 . (25) Setting the first coefficient of integrand equal to zero and solving for the coefficient a2 we find Substituting a2 into inequality (25) we obtain the sufficient condition for the uniform stochastic stability with respect to measure || • || = V112 136 A. TYLIKOWSKI where - i /2n"e\{e2n q" V (ei+e2) 2n The condition (26) generates a stability region shown in Fig. 3. Fig. 3. 8. Conclusions The applicability of the Liapunov method has been extended to linear Voigt-Kelvin continuous systems subjected to timedependent deterministic as well as stochastic para- metric excitations. Two different dynamical models have been used, the first when the excitations are deterministic processes or stochastic nonwhite processes, the second one is applicable to describing the Gaussian white excitations. The class of Liapunov functio- nals useful for analysing both asymptotic stability and uniform stochastic stability has been proposed. Obtaining asymptotic stability and almost sure asymptotic stability criteria for the first model has been reduced to solving an auxiliary variational problem. The expli- cit stability criteria for stability of an Euler beam compressed by a periodic or stochastic force and a thin-walled double-tee beam bending by two broad -band Gaussian couples. have been obtained as an application of the derived theory. References 1. S. L. D E LEEUW, Buckling criterion for linear vi&coetastic column, AIAA Journal 1 (1963) 2665-2666. 2. J. GENIN, J. S. MAYBEE, On the asymptotic stability of solutions of a linear viscoelastic beam, J. of The Franklin Inst. 293 (1972) 191—197. 3. R. H. PLAUT, Asymptotic stability and instability criteria for some elastic systems by Liapunov's direct method, Quart. Appl. Math. 29 (1972) 535 - 540. 4. J. A. WALKER, M. W. DIXON, Stability of the general plane membrane adjacent to a supersonic airstream, J. Appl. Mech. 40 (1973) 395 - 398. STABILITY OF VISCOELASTIC SYSTEMS 137 5. F . K O Z I N , Stability of the linear stochastic system, Lect. N o t . in M ath . 294 (1972) 186 -  229. 6.  S. T.  ARIARATN AM,  D . S. F .  TAM M , Stochastic problems in stability of structures, in :  Applications  of Statistics,  P. R.  KR I SH N AI AH   (ed.),  N orth- H olland Publishing  Company  (1977)  43 -  53. 7.  A.  TYLIKOWSKI, Stability of a nonlinear rectangular plate, J. Appl.  M ech. 45  (1978)  583 -  585. 8.  R. H .  PLAU T,  E. F .  IN F AN TE, On the stability of some continuous systems subjected to random excita- tions, J.Appl.  M ech.  37  (1970)  623- 627. 9.  R. F .  CU RTAIN ,  P. L.  F ALB, Itd's lemma in infinite dimensions, J.  M ath .  Anal.  Appl.  31 ](197O) 434- 448. 10.  A.  TYLIKOWSKI, Dynamic stability of plane bending form of thin- walled double- tee beams,  m) P olish), Kwartalnik  M echanika  AG H   2  (1983)  N o  4,  5- 18. 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Streszczenie S t r e s z c z e n i e D YN AM ICZN A  STATECZN OŚĆ LEPKOSPRĘ Ż YSTYCH  U KŁAD ÓW P OD D AN YC H  D Z I AŁ AN I U Z ALEŻ N EGO  OD   CZASU   OBCIĄ Ż EN IA W pracy  pokazano moż liwość  zastosowania  bezpoś redniej  metody Lapunowa  do  badania  statecznoś ci liniowych  lepkosprę ż ystych  ukł adów  cią gł ych  poddanych  dział aniu zależ nego  od  czasu  deterministycznego lub  stochastycznego  wymuszenia  parametrycznego.  Wprowadzono  klasę   funkcjonał ów  Lapun owa  wygod- nych  w  analizie  statecznoś ci  róż nych  ukł adów  cią gł ych.  Efektywnie  otrzymano  dostateczne  warunki asymptotycznej  statecznoś ci,  prawie  pewnej  asymptotycznej  statecznoś ci  i  jednostajnej  statecznoś ci  st o - chastycznej  nieodkształ conych  postaci  (rozwią zań  trywialnych)  ukł adów  Voigta- Kelvina.  Jako  przykł ady zbadano dynamiczną   stateczność prę ta Eulera ś ciskanego  okresową   lub  stochastyczną   siłą  oraz dynamiczną stateczność  pł askiej  postaci  zginania  cienkoś ciennego  prę ta  pod  dział aniem  szerokopasmowych  n o rm al- nych  momentów. Praca wpł ynę ł a do Redakcji dnia 24 maja 1985 roku.