The Stationary Vibrations of a Rectangular Plate Subjected to Stress Prescribed Partially at the Circumference

The stationary periodical problem of a vibrating rectangular plate, stressed at a segment while fixed elsewhere at one of its edges, is considered. Using the finite Fourier transformation, the problem is converted to a singular integral equation that in turn can be reduced to an infinite system of algebraic equations. The truncation of the algebraic system is justified. KEYWORDSANDPHRASES. Stationary vibrations, mixed boundary value problems, Elastodynamics. 1980 AMS SUBJECT CLASSIFICATION CODE. 35A22, 35C25, 73C35. 1. CONVERSION OF THE GOVERNING EQUATIONS INTO A SINGULAR INTEGRAL EQUATION The boundary conditions considered in this problem are o-(x,O;t)f-P’e’’, (P-const.), Ixl <c (1.1) (x,o;t)--o, c lxl <t (1.2) x(x,0;t) 0, Ixl <t (1.3) "(x,-1;t)=O, Ixl <:t (1.4) x(x,-1;t) 0, Ixl < (1.5) Expressed in terms of the longitudinal and transversal potentials t and xp, respectively, the stresses and displacements satisfy (Nowacki 1]) the followng equations v V25+ (1.6) oy-2 1-2v Oy OxOy 0.__9 0xp (1.7) v "Oy x’ +----(1.8) OxOy Oy Ox where v is the Poisson’s ration, and a and are the Lam6’s constants. The functions and xp satisfy the equation 582 M.G. EL SHEIKH, A.H. KHATER AND D.K. CALLEBAUT where cl and c2 are the propagation velocities of longitudinal and transversal waves respectively. The stationary solutions of the problem can be expressed in the form O0-(x,y;t)--e’’OO(x,y), -(x,y;)-e’(x,y) Oy eiOy, e’v and e where