Ghostscript wrapper for D:\Digitalizacja\MTS88_t26_z1_4_PDF_artykuly\04mts88_t26_zeszyt_4.pdf MECHANIKA V I I P I 'OT TEORETYCZNA TU —PL 87 I STOSOWANA 4, 26 (1988) DYNAMIC STABILITY OF ANTISYMMETRICALLY LAMINATED CROSS-PLY CYLINDRICAL SHELLS ANDRZEJ TYLIKOWSKI Warsaw University of Technology 1. Introduction Dynamic behaviour of thin laminated cylindrical shells is of great importance to engineers. The coupling between bending and tension in laminates results in the necessity to modify the classic equilibrium equations and the boundary conditions for thin uniform cylindrical shells in order to apply them to laminated shells. The reformulation of boundary conditions and the solution of the static buckling problems for the cylindrical shells was done by Almroth [1]. Numerous papers are available on free vibrations of laminated shells (see for example papers by Bert, Baker and Egle [2], Dong [3], Alam and Asnani [4]). While parametric vibrations and dynamic stability problems for uniform isotropic cylindrical shells under time-dependent membrane forces have drawn much attention, the dynamic stability of cylindrical shells has not been investigated yet. The purpose of the paper is to analyse the dynamic asymptotic stability of thin elastic cylindrical shells for cross-ply antisymmetric configuration. Membrane forces acting in the shell midsurface are assumed to be deterministic functions of time or stochastic processes with differentiable realizations. The shell consists of an even number of equal thickness orthotropic laminae laid on each other with principal material directions alter- nating at 0 and n\2 to the shell axial and circumferential directions. Using the direct Liapunov method we have derived the sufficient conditions for the asymptotic stability and the almost sure asymptotic stability. The influence of geometric and material properties of the shell as well as characteristics of loading on stability regions have been examined numerically. 2. Problem formulation Let us consider a closed elastic simply supported cylindrical shell of radius a, length / and total thickness h, a > h, I > h. The shell consists of an even number of equal thickness orthotropic layers antisymmetrically laminated with respect to.its midsurface from both the geometric and the material property standpoint. The Kirchhoff-Love hypothesis on nondeformable normal element is taken into account. Tangential, rotary and coupling 656 A. TVLIKOWSKI inertias are neglected. For the shell subjected to a concentrated load P and a uniformly distributed radial loading q, the initial membrane loads can be determined by assuming that the shell remains circular and undergoes a uniform compression circumferentially. Consequently: Nx = PlZna, N& = aq. Taking into account a linear damping in the radial direction we obtain the equations of the technical theory of thin laminated shells in terms of displacements u, v, w in tangen- tial, circumferential and radial direction, respectively [3]: ^/a2 = 0, -BlluiXXX+A12u,x/a+Bllv>&ee/a 3+A11v,(!l/a 2+Dllw,xxxx + (1) &ela 2 — 0 , = ( 0 , 1 ) X ( 0 , 2 T I ) . Internal forces and moments are expressed by the displacements as follows: Nxe = A66(v,x + u,e/a), (2) Mx = Bt! u,, - A t w,xx - £>! 2 w, e0/a 2, Me = -BnV^/a-Bnw/a-D^w^^-D^w^e/a 2, Mx9= ~2D66w,x&/a. The closed shell is assumed to be simply supported without displacement in circum- ferential direction at x — 0,1. The conditions imposed on displacements, internal forces and moments, called according to Almroth's classification S2, can be written down as: w = 0, v = 0, Nx = 0, Mx = 0 at , x = 0,1. (3) Our purpose is to investigate the stability of undisturbed shell surface u = v = w = 0 (the trivial solution). A disturbed state is estimated by means of a distance of the solution of system (1) with nontrivial initial conditions from the trivial solution. Under assumption that the membrane forces are the deterministic functions of time we will study the asympto- tic stability of trivial solution, i.e. we will derive conditions that imply: If the forces are stochastic "nonwhite" processes with sufficiently smooth realizations we shall consider the almost sure asymptotic stability which holds if a probability of event defined by (4) is equal to one: ' P { l i m | H = 0 } = l . (5> ( - • CO DYNAMIC STABILITY OF... 657 We shall examine the foregoing kinds of stability using the direct Liapunov method, which provides a significant advantage in that the conditions for stability can be obtained without the explicite solving the equations of motion. 3. Derivation of the sufficient stability conditions We construct a functional as a sum of a modified kinetic energy and the potential energy of the shell in order to apply it as a Liapunov functional: I/ a (6) +NX u, x+Ne(v, e + w)/a+Nxe(viX + u^ja)] adQ, where x «• Wlt. The functional is positive-definite since the first three terms of integrand can be rear- ranged as a sum of squares. Therefore, we can choose the square root of functional (6) as the distance used in the stability definitions. Under the previous assumptions imposed on the membrane forces the classic differentiation rule can be applied to calculate the time- derivative of functional (6). Dividing equations of motion (1) by gh and retaining for convenience the same symbols for coefficients we obtain the time-derivative of functional (6) in the following form: - J = ̂ f [2(z+pwX a -A12u,x[a-B11vieeela 3-Allv,(>la 2-D11w,xxxx+ -2(2>12+2Z>66) w, xxee/a 2 -B2i w, e e s e / a + - 2 5 1 1 w, ee/a 3 -Axx wja 2) + (7) -Mx,tw.xx-Mxz.xx-M0,tw,ee/a 2-Mez,e0la 2-2Mxe,tw.xela+ where It denotes an additional functional: a Integrating by parts, using boundary conditions (3) and periodicity conditions with respect to variable 0 we prove the following formulae: 2n ' In I j MXiXxzadQ = J MXlXz adQ- J MXtXztXadQ = -j Mxz., a o o a o + fMxz,xxadQ= JMxz.xxadQ, ad@+ J MxB,x»zadQ = 658 A. TYLIKOWSKI J M x ,,w iXX adQ = J M x z, xx adQ, J M &it w f8B ^ dQ = J MQZ i&e adQ, a a M xQit w, x& adQ — J M x @z tX@ adQ. n a I n a similar way integrating by parts we convert the functional I t to the following form: Jx ^ i J  {- (Nx, x+N xe.ela)u~(N x@, x+N e, ela)v+N ewld\ adQ. a Recognizing th e expressions in the parentheses as left h an d side expressions of the first two equation s of m otion (1) we omit them so we can write: I x = i  [N e wdQ. si J U sing th e above relations we rewrite the  time- derivative  of  functional  (7)  as: dV dt =   - 2PV+2U,  (8) wh er e:  » U  =  -   f  [(z+Pw)(N x w, xx +N e w, 0 ela 2 )+2p 2 wz+2p 3 w 2 ]adQ.  (9) a N ow  we  attem pt  to  construct  a  bou n d: U  Ś  XV,  (10) where  th e  function  X  is  t o  be  determined. P roceeding  similarly  as  Kozin  [5]  we  solve  an  additional  variational  problem  d(U~ — XV)  =   0  an d  we  o bt ain : A =   max  \ m,n= 1,2,... (11) / crur„ - 2ta)rl/ a. wh ere: k m   =  mn/ 1,  k n   — n/ a, T u   -   A lt kl+A 12 k 2 n ,  T X2   =   ~(A 12 +A 66 )k m k tt , T 22   -   A^ Substitutin g  inequality  (10)  in to  equation  (8)  we  obtain  the  differential  inequality, from  which  we  have  th e  following  estimation  of  functional  (6): F TCO« DYNAMIC  STABILITY  OF... 659 Thus,  it  immediately  follows  that  the  sufficient  stability  condition  for  the  asymptotic stability  with  respect  to  the  distance  || •  [|  = V112  is: t ft > l i m i  f X(s)ds,  (12) , - « t J or  for  the  almost  sure  asymptotic  stability,  provided  processes Nx  and N0  are  ergodic and  stationary  is: P > EX.  (13) where E  denotes  the  operator  of  the  mathematical  expectation. 4.  Results Expression  (11)  and  inequality  (13)  give us  possibility  to  obtain  the  critical  damping coefficient  guaranteeing  the  almost  sure  asymptotic  stability  as  a  function  of  laminate parameters  and  statistic  characteristics  of  membrane  forces.  In  order  to  obtain  stability regions, we choose  discrete values of force  (JVX or N@)  and  compute  Am„. Then  we choose the  largest  value  corresponding  to  the  given value  of  the  force  and  take  the  expectation numerically integrating the product of A by the probability density function.  This is accom- plished for various values of parameters by choosing the variance and varying the damping coefficient until inequality (13) will be satisfied. . Glass-epoxy Graphile-epoxy 1,0 Damping coefficient Fig. I. 660 A.  TYLIKOWSKI Numerical  calculations  are  performed  for  the  gaussian  process  with  zero  mean  and variance a2  and  the  harmonic  process  with  variance a2  = A212,  where A  denotes  its amplitude,  for  different  number  of  layers  and  the  shell  aspect  ratio a/1. The almost  sure asymptotic stability regions  as functions  of /?, a and number  of layers N  in  the  case,  when  the  shell  with all  =  1 is loaded  by the gaussian  process,  are  shown in  Fig.  1.  The  stability  regions  are  not  changed  in  going  from  the  axial  loading  to  the circumferential  one. As  the number  of layers increases  the  orthotropic  solution  is rapidly approached.  The  coupling  between  bending  and  extension  depends  on  the  orthotropic moduli  ratio EXIE2.  It  is  seen from  the figure  that  for  greater  ratios Et  to E2  the  effect of  coupling  increases. Stability  regions Glass  epoxy 4//M Harmonic  loading Gaussian  loading Shell  aspect  ratio Fig.  2 . The  dependence  of  stability  regions  as  functions  of  /?  and  the  shell  aspect  ratio all for  twolayered  shell  made  of glass-epoxy is shown in Fig. 2. It is found that the stability regions are not changed substantially in going from the gaussian process to the harmonic one. The dependence of stability regions on the direction of loading is quite essential. 5. Acknowledgment This work was supported by Grant No CPBP 02.02 coordinated by The Institute of Fundamental Technological Research of the Polish Academy of Sciences. The support is gratefully acknowledged. D YN AMIC  STABILITY  O F . . .  661 References 1.  B. O.  ALM ROTH , Influence of edge conditions on the stability of axially compressed cylindrical shells, AIAA  j : , 4  (1966)  134- 140. 2.  C. W.  BERT,  J. L.  BAKER,  D . M .  E G LE , Free vibrations of multilayer anisotropic cylindrical shells, J. Composite  M aterials,  3  (1969)  480- 499. 3.  S. B.  D O N G , Free vibration of laminated orthotropic cylindrical shells, J. Acoust.  Soc.  Am er.,  44  (1968) 1628  - 1635. 4.  N .  AL AM ,  N . T .  AS N AN I , Vibration and damping analysis of fibre reinforced composite material cylindrical shell, 3. Composite  M aterials  21  (1987)  348 -  361. 5.  F . K O Z I N , Stability of the linear stochastic systems, Lect.  N o t . in M ath . 294 (1972)  186- 229. 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