Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 46, 3, pp. 679-692, Warsaw 2008 DYNAMIC STABILITY OF WEAK EQUATIONS OF RECTANGULAR PLATES Andrzej Tylikowski Warsaw University of Technology, Warsaw, Poland e-mail: aty@simr.pw.edu.pl The stability analysis method is developed for distributed dynamic pro- blemswith relaxed ssumptions imposed on solutions.Theproblem ismo- tivated by structural vibrations with external time-dependent parame- tric excitations which are controlled using surfacemounted or embedded actuators and sensors. The strong form of equations involves irregulari- ties which lead to computational difficulties for estimation and control problems. In order to avoid irregular terms resulting fromdifferentiation of force and moment terms, dynamical equations are written in a weak form. The weak form of dynamical equations of linearmechanical struc- tures is obtained using Hamilton’s principle. The study of stability of a stochasticweak system is based on examiningproperties of theLiapunov functional along a weak solution. Solving the problem is not dependent on assumed boundary conditions. Key words:weak formulation, dynamic stability, different boundary con- ditions, Liapunovmethod 1. Introduction The strong form of plate equations involves irregularities which lead to com- putational difficulties for estimation and control problems. In order to avoid irregular terms resulting from differentiation of force and moment terms, dy- namical equations are written in a weak form. The weak form of systems is useful for development of identificationmethods and general computational methods (Banks et al., 1993).We consider dynamical systemswith parametric excitations, e.g. plates with in-plane time dependent forces. The plate motion is described by partial differential equations that include time dependent co- efficients implying parametric vibrations. The response of such systems can 680 A. Tylikowski lead to a new increasing mode of oscillations and the structure dynamically buckles. The classical Liapunov technique for stability analysis of continuous elements is based on choosing or generating of a functional which is positive definite in the class of functions satisfying structure boundary conditions. The time-derivative of the Liapunov functional has to be negative in some defined sense. Almost sure stability of the beam equilibrium state in the strong for- mulation was examined by Kozin (1972). The technique of stability analysis was extended to plates and shells with in-plane or membrane time-dependent forces (Tylikowski, 1978). Uniform stochastic stability analysis of laminated beams and plates described by partial differential equations with time and space dependent variables was presented in Tylikowski and Hetnarski (1996). The weak form of a distributed controller in an active system consisting of electroded piezoelectric sensors/actuators with suitable polarization profiles (Tylikowski, 2005) is useful for the feedback theoretical developments. For the purpose of active vibration and noise control, piezoelectric devi- ces have shown great potential as elements of passive absorbers and active control systems as they are light-weight, inexpensive, small, and can be bon- ded to main structures. They can not be modeled as point force excitations, and partial differential equations should be used to describe the response of the structures driven by them. Active damping in composite structures with collocated sensors/actuators were studied by Tylikowski (2005), Tylikowski and Hetnarski (1996). Electrodes on sensors/actuators are spatially shaped to reduce the spillover between circumferential modes. In order to avoid irre- gular terms resulting from the modelling, the action of piezoelements as the Dirac delta function concentrated on their edges and dynamical equations are written in a weak form. 2. Weak formulation of plate dynamic equations Consider an elastic rectangular plate of the length a, width b, thickness h, mass density ρ and bending stiffness D subjected to the in-plane time- dependent forces FX(t) and FY (t) acting in the X and Y direction, respecti- vely.Therefore, the onset of parametric vibrations is possible. In order to avoid developing modes of plate motion, viscous damping with the proportionality coefficient α is introduced. The strong form of plate dynamical equation in the transverse displacement w is given in the following form ρhw,TT +αw,T +D∆ 2w+FX(T)w,XX +FY (T)w,YY =0 (2.1) Dynamic stability of weak equations... 681 where (X,Y )∈{0,a}×{0,b}. Introducing dimensionless variables x= X b y= Y b t= T kt equation (2.1) becomes w,tt+2βw,t+∆ 2w+[fox+fx(t)]w,xx+[foy+fy(t)]w,yy =0 (2.2) where (x,y)∈{0,r}×{0,1}, r= a/b is the plate aspect ratio k2t = 1 D ρhb4 β= αkt 2ρh fox+fx(t)= 1 D b2FX(ktt) foy+fy(t)= 1 D b2FY (ktt) and the solution should be fourth time partially differentiated with respect to x and y. If the plate is simply supported at the end, the transverse displa- cement and bending moment equal zero. For clamped edges, the transverse displacement and the slope equal zero. Similarly,we can anlyse other combinations of simple boundaryconditions, i.e. a simply supported – clamped plate. We write the action integral of the plate without damping in the form A[w] = 1 2 t2 ∫ t1 r ∫ 0 1 ∫ 0 [w2,t− (w,xx+w,yy)2−2(1−ν)(w2,xy −w,xxw,yy)+ (2.3) +foxw 2 ,x+foyw 2 ,y] dxdydt where ν is Poisson’s ratio, and applying Hamilton’s principle d dε A[w+εΦ] ∣ ∣ ∣ ε=0 =0 (2.4) where ε is a real number.Adding theviscous dampingand the time-dependent components of the in-plane forces as external works, the dynamical equation can be written in the weak form as follows for all Φ r ∫ 0 1 ∫ 0 { (w,tt+2βw,t)Φ+(w,xx+νw,yy)Φ,xx+(w,yy +νw,xx)Φ,yy + (2.5) +2(1−ν)w,xyΦ,xy− [fox+fx(t)]w,xΦ,x− [foy+fy(t)]w,yΦ,y } dxdy=0 682 A. Tylikowski where Φ is a sufficiently smooth test function satisfying essential boundary conditions. There is no demand on the existence of higher derivatives than the second order. As detailed in Banks et al. (1993), usual integration by parts the terms containing derivatives of the test function Φ with respect to the in-plane variables x and y and the assumption of sufficient smoothness of the plate displacement leads to strong formulation (2.2). 3. Almost sure stability definition Dynamical equations (2.2) and (2.5) contain terms explicitely dependent on time. The time dependency of the axial forces fx(t) and fy(t) parametrically excites the plate and an increasing form of vibrations can occur. In determi- nistic parametric vibrations, the stability properties are determined from the Mathieu equation together with the corresponding Ince-Strutt diagram. If the excitation is narrow-banded or has one latent periodicity, a series of wedges on the amplitude-frequency plane can be expected, analogously to the determini- stic parametric resonance.The task ismuchmore complexwhen the stochastic excitation iswide-band and continuous systemswith an infinite number of na- tural fequencies are analysed. Due to the fact that the norms and metrics in infinite-dimensional spaces are not equivalent, the stability holds for just the norm or themetric used in the analysis. If the parametric excitation becomes random, the stability criteria depend on the statistical characteristics of the excitation and the systems parameters. The present paper examines dynamic stability due to an action of in-plane forces in the formof stochastic physically realizable time-dependent processeswith given statistical properties.Equation (2.5) with zero initial conditions posseses the trivial solution w=w,t =0 (3.1) The trivial solution to Eq. (2.2) or (2.5) is almost sure stable if with probabi- lity 1 if the measure of distance between the perturbed solution with nonzero initial conditions and the trivial one tends to zero as time tends to infinity (Ko- zin, 1972). Usually, the measure of distance is defined by a positive-definite functional. The trivial solution is called almost sure asymptotically stable if P { lim t→∞ ‖w(·, t)‖=0 } =1 (3.2) where ‖w(·, t)‖ is ameasure of the disturbed solution wwith nontrivial initial conditions from the equilibrium state, and P is the probability measure. The Dynamic stability of weak equations... 683 almost sure stability is equivalent to the clasical Liapunov stability in linear systems. 4. Stability analysis in weak formulation In order to examine the almost sure stability of the plate equilibrium (the trivial solution), the Liapunov functional is chosen in the form V = 1 2 r ∫ 0 1 ∫ 0 [ w2,t+2βww,t+2β 2w2+(w,xx+w,yy) 2+ (4.1) +2(1−ν)(w2,xy−w,xxw,yy)−foxw2,x−foyw2,y ] dxdy The functional is positive-definite if the constant components of the in- plane forces fox and foy fullfil the static buckling condition, i.e. are sufficiently small. Therefore, themeasure of disturbed solutions is chosen as a square root of the functional V ‖w(·, t)‖= √ V (4.2) As trajectories of the solution to equations (2.5) are physically realizable, classical calculus is applied to find the time-derivative of functional (4.1). Its time-derivative is given by dV dt = r ∫ 0 1 ∫ 0 [ (w,t+βw)w,tt+βw 2 ,t+2β 2ww,t+ +w,xxw,xxt+w,yyw,yyt+2(1−ν)w,xyw,xyt+ (4.3) +ν(w,xxw,yyt+w,yyw,xxt)−foxw,xw,xt−foyw,yw,yt ] dxdy Substituting βw and w,t as the test functions in Eq.(2.5), we have two identities, respectively r ∫ 0 1 ∫ 0 [ (w,tt+2βw,t)βw+(w,xx+νw,yy)βw,xx+(w,yy+νw,xx)βw,yy + +2(1−ν)βw2,xy− (fox+fx(t))βw2,x− (foy +fy(t))βw2,y ] dxdy=0 (4.4) r ∫ 0 1 ∫ 0 [ (w,tt+2βw,t)w,t+(w,xx+νw,yy)w,xxt+(w,yy +νw,xx)w,yyt+ +2(1−ν)w,xyw,xyt− (fox+fx(t))w,xw,xt− (foy +fy(t))w,yw,yt ] dxdy=0 684 A. Tylikowski Subtracting identities (4.4), from the time-derivative of functional (4.3), we obtain the following form dV dt = 1 ∫ 0 { −βw2,t−β[w2,xx+w2,yy+2w,xxw,yy +2(1−ν)(w2,xy−w,xxw,yy)]+ (4.5) +βfoxw 2 ,x+βfoyw 2 ,y+fx(t)(w,xt+βw,x)w,x+fy(t)(w,yt+βw,y)w,y } dxdy After some algebra, we rewrite the time-derivative of functional as dV dt =−2βV +2U (4.6) where U is an auxiliary functional given by U = 1 2 1 ∫ 0 [ 2β2ww,t+2β 3w2+ (4.7) +fx(t)(w,xt+βw,x)w,x+fy(t)(w,yt+βw,y)w,y ] dxdy Now we attempt to construct a bound λV ­U (4.8) where the stochastic function λ is to be determined. In an explicite notation, the function λ has to satisfy the following equation for arbitrary functions w and w,t satisfying suitable boundary conditions r ∫ 0 1 ∫ 0 { λ[w2,t+2βww,t+2β 2w2+(w,xx+w,yy) 2+ +2(1−ν)(w2,xy−w,xxw,yy)−foxw2,x−foyw2,y]+ (4.9) −[2β3w2+2β2ww,t+fx(t)(w,xt+βw,x)w,x+fy(t)(w,yt+βw,y)w,y] } dx­ 0 It should be noticed that theway to obtain estimation (4.8) is purely alge- braic contrary to systems described by strong equations, where the derivation of stability conditions is based on integrations by parts and manipulations with higher order partial derivatives. Usually, the Liapunov stability analysis of plates is performed for all four simply supported edges (Tylikowski, 1978). Dynamic stability of weak equations... 685 In order to extend the field of possible applications, let us assume the fol- lowing combinations of boundary conditions: a) c-c-c-c, b) c-c-c-s, c) c-c-s-c, d) c-c-s-s, e) s-c-s-s, f) s-s-s-s, shown in Fig.1, where ”s” denotes a simply supported edge, and ”c” denotes a clamped edge. Contrary to the Levy me- thod of determination of displacements of a rectangular plate with two simply supported opposite edges, the proposed technique of determining the stability domains can be applied to plates with arbitrary combinations of simply sup- ported and clamped edges. In the first combinations of boundary conditions for plates with all four edges simply supported, we have w(x,y,t) = ∞ ∑ m,n=1 wmn(t)sin mπx r sin(nπy) (4.10) If the plate withmixed simply supported-clamped edges is considered, the plate displacement is written in the following form w(x,y,t) = ∞ ∑ m,n=1 wmn(t)Xm (x r ) Yn(y) (4.11) where Xm and Yn are beam functions depending on boundary conditions for x=0, r and y=0, 1, respectively. For example, if all four edges of plate are clamped, the beam functions in Eq. (4.11) have the following form Xm = ( sin βnx r − sinhβnx r ) (cosβn− coshβn)+ −(sinβn− sinhβn) ( cos βnx r − coshβnx r ) (4.12) Yn =(sinγny− sinhγny)(cosγn− coshγn)+ −(sinγn− sinhγn)(cosγny−coshγny) Integrating, we have the following equality r ∫ 0 X2m,xx dx=κmβ 2 m r ∫ 0 X2m,x dx 1 ∫ 0 Y 2n,yy dy=χnγ 2 n 1 ∫ 0 Y 2n,y dy (4.13) Using the orthogonality condition, we can also write r ∫ 0 X2m,x dx= β2m κm r ∫ 0 X2m dx 1 ∫ 0 Y 2n,y dy= γ2n χm 1 ∫ 0 Y 2n dy (4.14) 686 A. Tylikowski where the first few constants κm and χn used in equalities (4.13) and (4.14) are given in Table 1 for the most commonly used boundary conditions: a clamped-clamped beam (c-c), and a clamped-simply supported beam (c-s). The constants tend to 1 as n → ∞. In order to unify the notations in the case of the simply supported beam, we assume βn = nπ and κn = 1. Using properties of the functions Xm and Yn, we have r ∫ 0 1 ∫ 0 w,xt(x,y,t)w,x(x,y,t) dx= ∞ ∑ m,n=1 wmn,t(t)wmn(t) r ∫ 0 X2m,x dx 1 ∫ 0 Y 2n dy (4.15) Substituting Eq. (4.14)2, yields ∞ ∑ m,n=1 wmn,t(t)wmn(t) r ∫ 0 X2m,x dx 1 ∫ 0 Y 2n dy= (4.16) = ∞ ∑ m,n=1 β2m κm wmn,t(t)wmn(t) r ∫ 0 X2m dx 1 ∫ 0 Y 2n dy Similarly, we have r ∫ 0 1 ∫ 0 w,xx(x,y,t)w,yy(x,y,t) dxdy= (4.17) = ∞ ∑ m,n=1 ∞ ∑ i,j=1 wmn(t)wij(t) r ∫ 0 Xm,xxXi dx 1 ∫ 0 YnYj,yy dy Integrating by parts the terms in the right-hand-side of Eq. (4.17) for simply supported or clamped edges, we have r ∫ 0 Xm,xxXi dx=Xm,xXi ∣ ∣ ∣ r 0 − r ∫ 0 Xm,xXi,x dx=−δmi r ∫ 0 X2m,x dx (4.18) where δmi denotes the Kronecker delta function 1 ∫ 0 YnYj,yy dy=Yn,yYj ∣ ∣ ∣ 1 0 − 1 ∫ 0 Yn,yYj,y dx=−δnj 1 ∫ 0 Y 2n,y dy (4.19) Dynamic stability of weak equations... 687 Substituting Eqs. (4.18) and (4.19) into Eq. (4.17), yields r ∫ 0 1 ∫ 0 w,xx(x,y,t)w,yy(x,y,t) dxdy= r ∫ 0 1 ∫ 0 w2,xy dxdy (4.20) Table 1.Numbers κn m, n 1 2 3 4 5 6 7 c-s 1.3396 1.1649 1.1086 1.0809 1.0645 0.90205 1.0537 c-c 1.8185 1.3392 1.2224 1.1648 1.1309 1.10857 1.09276 Combining Eqs. (4.9) and (4.20) and substituting expansion (4.11), we have ∞ ∑ m,n=1 { [λw2mn,t+2β(λ−β)wmn,twmn+2β2(λ−β)w2mn] r ∫ 0 X2m dx 1 ∫ 0 Y 2n dy+ +λw2mn ( r ∫ 0 X2m,xxdx 1 ∫ 0 Y 2ndy+2 r ∫ 0 X2m,xdx 1 ∫ 0 Y 2n,ydy+ r ∫ 0 X2mdx 1 ∫ 0 Y 2n,yydy ) + (4.21) − [(λfox+βfx(t))w2mn+fx(t)wmn,twmn] r ∫ 0 X2m,x dx 1 ∫ 0 Y 2n dy+ − [(λfoy +βfy(t)]w2mn+fy(t)wmn,twmn] r ∫ 0 X2m dx 1 ∫ 0 Y 2n,y dy } ­ 0 Using Eqs. (4.14)-(4.17), yields ∞ ∑ n=1 { λw2mn,t+ ( 2β(λ−β)−fx(t) β2m κm −fy(t) γ2n χn ) wmn,twmn+ + [ 2β2(λ−β)+λ ( β4m+2 β2m κm γ2n χn +γ4n ) − (λfox+βfx(t)) β2m κm + (4.22) − (λfoy+βfy(t)) γ2n χn ] w2mn } r ∫ 0 X2m dx 1 ∫ 0 Y 2n dy­ 0 Therefore, the variational inequality is reduced to an infinite system of quadratic inequalities λw2mn,t+2Amn(λ,t)wmn,twmn+Bmn(λ,t)w 2 mn ­ 0 (4.23) 688 A. Tylikowski where Amn(λ,t) =β(λ−β)−fx(t) β2m 2κm −fy(t) γ2n 2χn Bmn(λ,t)= 2β 2(λ−β)+λg ( β4m+2 β2m κm γ2n χn +γ4ng ) + −[λfox+βfx(t)] β2m κm − [λfoy +βfy(t)] γ2n χn The unknown function λ is determined from the zero determinant condi- tion for all m and n of the form ∣ ∣ ∣ ∣ ∣ λ Amn(λ,t) Amn(λ,t) Bmn(λ,t) ∣ ∣ ∣ ∣ ∣ =0 (4.24) After some algebraic manipulations, (4.24) yields λ= max m,n=1,2,... λmn (4.25) where λmn = ∣ ∣ ∣β2+fx(t) β2 m 2κm +fy(t) γ2 n 2χn ∣ ∣ ∣ √ β4m+2 β2 m κm γ2 n χn +γ4n−fox β2 m κm −foy γ 2 n χn +β2 (4.26) Combining inequality (4.6) and (4.8), yields an upper estimation of the time-derivative of the functional dV dt =−2βV +2U ¬−2(β−λ)V (4.27) Integratingwith respect to time, yields the followingupperestimation of V V (t)¬V (0)exp [ −2 ( β− 1 t t ∫ 0 λ(τ) dτ ) t ] (4.28) Therefore lim t→∞ ‖w(·, t)‖=0 (4.29) if the exponent in (4.28) is negative β­ lim t→∞ 1 t t ∫ 0 λ(τ) dτ (4.30) Dynamic stability of weak equations... 689 Additionally, assuming the ergodicity of in-plane forces, we substitute the time-averaging in (4.30) by the average over a probabilistic space lim t→∞ 1 t t ∫ 0 λ(τ) dτ =E{λ} (4.31) Finally, the trivial solution toEq. (2.5) is almost surely stablewith respect to themeasure of distance defined as the square root of functional (4.1), if the transcendental inequality is fulfilled β­E{λ} (4.32) Therefore, stability domains for the clamped-clamped plate and the clamped-simply supported plate are defined as follows β­E { max m,n=1,2,... [ ∣ ∣ ∣ β2+fx(t) β2 m 2κm +fy(t) γ2 n 2χn ∣ ∣ ∣ √ β4m+2 β2 m γ2 n κmχn +γ4n−fox β2 m κm −foy γ 2 n χn +β2 ]} (4.33) We mention that Eq. (4.33) is a generalization of the similar form as the analytical formula defining the stability region obtained by Kozin (1972) for the simply supportedbeam in the strong formulation. It shouldbe emphasized that formulae (4.32) and(4.33) areobtainedwithoutdealingwith the thirdand fourth order spatial derivatives. If thedensity functions for the time-dependent componenets of in-plane forces are known, numerical integration can be used to evaluate Eqs. (4.32) and (4.33) for different values of parameters such as for example the variance σ2x and σ 2 y. As the damping coefficient β is involved in the right hand side of the equations they have to be solved in an iterative way. The structure of stability conditions for all boundary conditions is simi- lar. They contain different values of constants admitting Eqs. (4.13)-(4.14)1. The stability domains are examined for quadratic plates uniaxially loaded foy = fy(t) = 0 with each combination of simple and clamped supports, cf. Fig.1. The dependence of stability regions on different boundary conditions is shown in Fig.2 for fox = 0 on the plane β, σ 2 for the Gaussian in-plane force. The influence of boundary conditions on the shape and size of stability domains is rather weak. The effect is more severe (Fig.3) for fox = 39 cor- responding to the critical loading of the quadratic plate with all four edges simply supported, Fig.1f. The influence of the constant component fox =64 corresponding to the critical loading (Fig.1d) and fox =83 corresponding to the critical loading (Fig.1b) on stability domains is shown in Fig.4 andFig.5, respectively. 690 A. Tylikowski Fig. 1. Rectangular plate under uniaxial in-plane loading and boundary conditions Fig. 2. Stability domains of a plate subjected to the uniaxial in-plane Gaussian parametric excitation for different boundary conditions, f ox =0 Fig. 3. Stability domains of a plate subjected to the uniaxial in-plane harmonic parametric excitation for different boundary conditions, f ox =39 Although, the stability regions are similar qualitatively for different boun- dary conditions, the diferences of critical values of σ2 (the force variance) are significant for compressive forces close to critical loadings. Therefore, a more Dynamic stability of weak equations... 691 Fig. 4. Stability domains of a plate subjected to the uniaxial in-plane harmonic parametric excitation for different boundary conditions, f ox =64 Fig. 5. Stability domains of a plate subjected to the uniaxial in-plane harmonic parametric excitation for different boundary conditions, f ox =83 carefull analysis of boundary conditions is needed in the analysis of dynamic stability of continuous systems. 5. Conclusions The stability analysis method is developed for distributed dynamic problems with relaxed assumptions imposed on solutions. The Lyapunov method can be used to stability analysis of equations in weak formulation. The results are obtained in the frame of the distributed parameter approach without earlier discretization or truncation.Without any viscous damping, theplatemotion is unstable due to the parametric excitation. The stability domains are presented for plates with each combination of simple and clamped supports. Stabilitydomainsofplates compressedby forces close to thecritical loading substantially depend on the assumed boundary conditions. Stability results obtained for a plate with simply supported boundary conditions in strong 692 A. Tylikowski formulations and stability conditions obtained for simply supported plates described by strong equations are also valid under weak formulation. References 1. Banks H.T., Fang W., Silcox R.J., Smith R.C., 1993, Approximation methods for control of structural acousticsmodels with piezoelectric actuators, Journal of Intelligent Material Systems and Structures, 4, 98-116 2. Graff K.F., 1975, Wave Motion in Elastic Solids, Dover Publications, New York, 155-158 3. Kozin F., 1972, Stability of the linear stochastic system, Lecture Notes in Mathematics, 294, 186-229 4. Tylikowski A., 1978, Stability of nonlinear rectangular plate,ASME Journal of Applied Mechanics, 45, 583-585 5. Tylikowski A., 2005, Stabilization of plate parametric vibrations via distri- buted control, Journal of Theoretical and Applied Mechanics, 43, 695-706 6. Tylikowski A., Hetnarski R.B., 1996, Thermally induced instability of laminated beams and plates, ASME Journal of Applied Mechanics, 63, 884-890 Dynamiczna stateczność słabych równań płyt prostokątnych Streszczenie W pracy rozszerzonomożliwości analizy stabilności układów ciągłych na układy z osłabionymiwarunkami nakładanymi na rozwiązania.Układy aktywnego tłumienia drgań cienkościennych elementów płytowychmogą zawierać elementy piezoelektrycz- ne oddziaływującenakonstrukcję.Wuproszczonymmodeluoddziaływanie to sprowa- dza się do działaniamomentów gnących lub sił rozłożonychna krawędziach elementu piezoelektrycznego.Wprowadzenie dystrybucji δ-Diraca i jej pochodnej prowadzi do analitycznego zapisu obciążenia i wprowadza nieregularności do rozwiązania zadania drgańwymuszonychukładu ciągłego.Słabąpostać równańpłyty otrzymano zapomo- cą zasady Hamiltona. Badanie staeczności stochastycznych układów w formie słabej jest oparte na analizie funkcjonału Lapunowa wzdłuż słabego rozwiązania. Rozwią- zanie zadania jest niezależne od przyjętych warunków brzegowych. Manuscript received February 22, 2008; accepted for print March 28, 2008