Nonlinear vibrations of rectangular plates with linearly varying thickness

Simplified nonlinear governing differential equations proposed by Berger for static cases and extended by Nash and Modeer for dynamic cases are used to analyse the title problem. Steady-state harmonic oscillations are assumed and the time variable is eliminated by a Kantorovich averaging method. The enclosure or comparison theorem of Collatz is then applied to the reduced equations to obtain the upper and lower bounds for the fundamental nonlinear frequency of simply-supported rectangular plates with linearly varying thickness. The fundamental eigenvalues are given for several taper and aspect ratios.Nomenclature a, b dimensions of plates - A_i series coefficients - D Eh³/12(1–²) flexural rigidity - D₀ Eh₀³/12(1–²) - E Young's modulus - h thickness, h₀(1+x) - h₀ thickness parameter - N_x, N_y stress resultants in the X and Y directions - N (N_x+N_y)/(1+) - P₁, P₂, ... parameters - Q₁, Q₂, ... parameters - R[X, (A/h₀)²] bounding function - t time - u, v in-plane displacements - lateral deflections of plate - X=x/a dimensionless co-ordinate - x, y rectangular co-ordinates - y_n(X) series related to - thickness taper ratio - parameter in the neighbourhood of - error-function associated with differential equation - eigenvalue relating to frequency - Poisson's ra-tio - plate material specific weight - (X) function related to plate deflection - (X) admissible functions - circular frequency