Technische Universit¨atChemnitz Sonderforschungsbereich 393 Numerische Simulation auf massiv parallelen Rechnern S. I. Solov0¨ev Eigenvibrations of a plate with elastically attached load Preprint SFB393/03-06 Abstract This paper presents the investigation of the nonlinear eigenvalue problem de- scribing natural oscillations of a plate with elastically attached load. We study properties of eigenvalues and eigenfunctions and prove the existence theorem for this eigenvalue prob- lem. Theoretical results are illustrated by numerical experiments. Key Words nonlinear eigenvalue problem, eigenvibrations of a plate, natural oscilla- tions, eigenvalue, eigenfunction AMS(MOS) subject classification 74H20, 74H45, 49R50, 65N25, 47J10, 47A75, 35P05, 35P30 Preprint-Reihe des Chemnitzer SFB 393 ISSN 1619-7178 (Print) ISSN 1619-7186 (Internet) SFB393/03-06 February 2003 Contents 1 Introduction 1 2 Variational statement of the problem 2 3 Parameter eigenvalue problems 4 4 Existence of eigensolutions 12 5 Nonlinear biharmonic eigenvalue problem 14 6 Conclusion 16 References 16 Current address of the author: Sergey I. Solov0¨ev Fakult¨atf¨urMathematik TU Chemnitz 09107 Chemnitz, Germany [email protected] Address of the author: Sergey I. Solov0¨ev Faculty of computer science and cybernetics Kazan State University Kremlevskaya 18 420008 Kazan, Russia [email protected] 1 Introduction 1 1 Introduction Problems on eigenvibrations of mechanical structures with elastically attached loads have important applications. A survey of results in this direction is presented in [1]. An an- alytical method for solving some problems of this class is also described and analyzed in [1]. This method can be applied only in particular cases when we are known analytical formulae for eigenvalues and eigenfunctions of mechanical structures without loads. In the present paper, to treat the general case we propose a new approach for describing and finding eigensolutions of problems on eigenvibrations of mechanical structures with elastically attached loads. To introduce our approach we shall study the problem on eigenvibrations of a plate with elastically attached load. First let us describe eigenvibrations of the plate-spring-load system. We shall investi- gate the flexural vibrations of an isotropic elastic thin clamped plate with middle surface occupying the plane domain Ω with the boundary Γ [4]. Assume that ρ = ρ(x) is the volume mass density, D = D(x) = Ed3=12(1 ν2) is the flexural rigidity of the plate, E = E(x), ν = ν(x), d = d(x), are Young modulus,− Poisson ratio, and the thickness of the plate at the point x Ω, respectively, 0 < ν < 1=2. Assume that a load of mass M 2 is joined by an elastic spring with the stiffness coefficient K at the point x0 Ω, i.e., a harmonic oscillator with the vibration frequency ! = pσ, σ = K=M, is attached2 at the point x0 Ω of the plate. Denote2 by w(x; t) the vertical deflection of the plate at the point x Ω at time t, by ξ(t) the vertical displacement of the load of mass M at time t. These functions2 satisfy the following equations (see, for example, [1]): Lw(x; t) + ρ(x)d(x)wtt(x; t) + Mξtt(t)δ(x x0) = 0; x Ω; t > 0; − 2 w(x; t) = @nw(x; t) = 0; x Γ; t > 0; (1) 2 Mξtt(t) + K(ξ(t) + w(x0; t)) = 0; t > 0; where δ(x) is the delta function of Dirac, @n is the outward normal derivative on Γ, L is the differential operator defined by the relation: Lw = @11D(@11w + ν@22w) + @22D(@22w + ν@11w) + 2@12D(1 ν)@12w; − where @ij = @i@j, @i = @=@xi, i; j = 1; 2. The eigenvibrations of the plate-spring-load system are characterized by the functions w(x; t) and ξ(t) of the form: w(x; t) = u(x)v(t); x Ω; ξ(t) = c0u(x0)v(t); t > 0; 2 where v(t) = a0 cospλt + b0 sinpλt, t > 0; a0, b0, c0, and λ are constants. From the third equation of (1), we conclude that c0 = σ=(λ σ), σ = K=M. The first two equations of (1) lead to the following nonlinear eigenvalue problem:− find values λ and nontrivial functions u(x), x Ω, such that 2 2 2 Variational statement of the problem λσ Lu + Mδ(x x0)u = λ ρd u; x Ω; λ σ − 2 (2) − u = @nu = 0; x Γ: 2 The present paper is devoted to the investigation of nonlinear eigenvalue problem (2). In Section 2 we state the variational formulation for differential eigenvalue problem (2). In Section 3 we introduce parameter linear eigenvalue problems and study their properties. These parameter eigenvalue problems are used for proving the existence theorem in Section 4. In Section 5 we consider the nonlinear biharmonic eigenvalue problem and demonstrate numerical experiments. Similar results have been established in [24] for a beam with elastically attached load and in [23], [12], for a cylindrical shell with elastically attached load. In this paper we use an analysis analogous to [16]. 2 Variational statement of the problem By R denote the real axis. Let Ω be a plane domain with a Lipschitz-continuous boundary 2 Γ. As usual, let L2(Ω) and W2 (Ω) denote the real Lebesgue and Sobolev spaces, equipped with the norms : 0 and : 2: j j k k 1=2 2 1=2 0 2 1 2! u 0 = Z u dx ; u 2 = u i ; j j k k X j j @Ω A i=0 where 2 1=2 2 1=2 u = @ u 2! ; u = @ u 2! ; 1 X i 0 2 X ij 0 j j i=1 j j j j ij=1 j j ◦ 2 2 @i = @=@xi, @ij = @i@j, i; j = 1; 2. Denote by W2 (Ω) the space of functions u from W2 (Ω) such that u = @nu = 0 on Γ, @nu is the outer normal derivative of u along the boundary Γ. ◦ 2 Put Λ = (0; ), H = L2(Ω), V = W (Ω). Note that the space V is compactly 1 2 embedded into the space H, any function from V is continuous on Ω. The semi-norm : 2 j j is a norm over the space V , which is equivalent to the norm : 2. Assume that L (Ω) is the space of measurable real functionsk k u bounded almost every- where on Ω with the1 norm u 0; = ess: sup u(x) : j j 1 x Ω j j 2 Note that there exists c0 such that 2 v 0; c0 v 2 v W2 (Ω): j j 1 ≤ j j 8 2 Introduce the numbers K > 0, M > 0, σ = K=M. Define functions E, ν, ρ, and d from L (Ω), for which there exist positive numbers E1, E2, ρ1, ρ2, d1, d2, such that 1 E1 E(x) E2; 0 < ν(x) < 1=2; ≤ ≤ ρ1 ρ(x) ρ2; d1 d(x) d2; ≤ ≤ ≤ ≤ 2 Variational statement of the problem 3 for almost all x Ω. Set 2 Ed3 D = : 12(1 ν2) − Define the bilinear forms a : V V R, b : H H R, c : V V R, and the functions ξ(µ), µ Λ, ζ(µ), µ Λ,× by the! formulae: × ! × ! 2 2 a(u; v) = Z D[(@11u + @22u)(@11v + @22v) + Ω +(1 ν)(2@12u@12v @11u@22v @22u@11v)] dx; − − − b(u; v) = Z ρd uv dx; Ω c(u; v) = Mu(x0)v(x0); u; v V; σ 2 ζ(µ) = ; µ Λ; σ µ 2 µσ− ξ(µ) = ; µ Λ; µ σ 2 − where x0 is the fixed point on Ω. Consider the following differential eigenvalue problem: find λ Λ, u V 0 , Lu H, such that 2 2 nf g 2 Lu + ξ(λ)Mδ(x x0)u = λ ρd u: − This differential problem is equivalent to the following variational eigenvalue problem: find λ Λ, u V 0 , such that 2 2 n f g a(u; v) + ξ(λ) c(u; v) = λ b(u; v) v V: (3) 8 2 The number λ that satisfies (3) is called an eigenvalue, and the element u is called an eigenelement of problem (3) corresponding to λ. The set U(λ) that consists of the eigenele- ments corresponding to the eigenvalue λ and the zero element is a closed subspace in V , which is called the eigensubspace corresponding to the eigenvalue λ. The dimension of this subspace is called a multiplicity of the eigenvalue λ. A pair λ and u is called the eigensolution or the eigenpair of problem (3). Remark 1 Equation (3) can be written in the following equivalent form: a(u; v) = λ(b(u; v) + ζ(λ) c(u; v)) v V: 8 2 4 3 Parameter eigenvalue problems 3 Parameter eigenvalue problems Put Λ1 = (0; σ), Λ2 = (σ; ). 1 Let us write problem (3) for λ = σ as two following problems on the intervals Λ1 and 6 Λ2. Find λ Λ1, u V 0 , such that 2 2 n f g a(u; v) = λ(b(u; v) + ζ(λ) c(u; v)) v V: (4) 8 2 Find λ Λ2, u V 0 , such that 2 2 n f g a(u; v) + ξ(λ) c(u; v) = λ b(u; v) v V: (5) 8 2 For problems (4) and (5) we introduce parameter linear eigenvalue problems for fixed parameter µ Λ. Find '(µ)2 R, u V 0 , such that 2 2 n f g a(u; v) = '(µ)(b(u; v) + ζ(µ)c(u; v)) v V: (6) 8 2 Find (µ) R, u V 0 , such that 2 2 n f g a(u; v) + ξ(µ)c(u; v) = (µ)b(u; v) v V: (7) 8 2 Define the subspace V0 = v : v V; v(x0) = 0 of the space V .