Nonlinear Vibrations of Orthotropic Plates Carrying Concentrated Mass

Abstract

Following Berger’s approximate method for large deflections, nonlinear transverse free vibrations of an orthotropic right isosceles triangular plate, simply supported along its edges and carrying a concentrated mass, have been investigated in this technical brief.

A New Approach for Solving the Undamped Helmholtz Oscillator for the Given Arbitrary Initial Conditions and Its Physical Applications

“…Early descriptions of many of them date back to, at least, 1892, when the book by Greenhill [1] appeared, presenting a variety of such problems: a simple pendulum, catenaries, and a uniform chain that rotates steadily with a constant angular velocity about an axis to which the chain is fixed at two points. Later applications include nonlinear plasma oscillations [2], Duffing oscillators [3], rigid plates satisfying the Johansen yield criterion [4], nonlinear transverse vibrations of a plate carrying a concentrated mass [5], a beam supported at an axially oscillating mount [6], doubly periodic cracks subjected to concentrated forces [7], surface waves in a plasma column [8], coupled modes of nonlinear flexural vibrations of a circular ring [9], dualspin spacecrafts [10], spacecraft motion about slowly rotating asteroids [11], nonlinear vibration of buckled beams [12], a nonlinear wave equation [13], deep-water waves with two-dimensional surface patterns [14], oscillations of a body with an orbital tethered system [15], and nonlinear mathematical models of DNA [16,17]. Numerical studies of phase spaces, stability analysis, solution by means of finite differences, application of the Bernoulli wavelet method for estimating a solution of linear stochastic integral equations, existence of periodic solutions, and numerical simulations can be found in [18][19][20][21][22].…”

Jacobi elliptic functions: A review of nonlinear oscillatory application problems

“…Early descriptions of many of them date back to at least 1892, when the book by Greenhill [5] appeared, presenting a variety of such problems: a simple pendulum, catenaries, Watt's governor, Lintearia, a uniform chain that rotates steadily with a constant angular velocity about an axis to which the chain is fixed at two points, etc. Later applications include nonlinear plasma oscillations [6], Duffing oscillators [7], rigid plates satisfying the Johansen yield criterion [8], nonlinear transverse vibrations of a plate carrying a concentrated mass [9], a beam supported at an axially oscillating mount [10], doubly periodic cracks subjected to concentrated forces [11], surface waves in a plasma column [12], coupled modes of nonlinear flexural vibrations of a circular ring [13], dual-spin spacecrafts [14], spacecraft motion about slowly rotating asteroids [15], nonlinear vibration of buckled beams [16], a nonlinear wave equation [17], deep-water waves with two-dimensional surface patterns [18], oscillations of a body with an orbital tethered system [19], etc.…”

Approximate Solutions to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Numerical Solution

“…The additional equations are '(0) = ' 0 and ' 0 (0) = _ ' 0 . The semi-analytical solution (12) to the i.v.p. (3) could be recovered some special cases.…”

“…Case (1) For ( ; ! 2 ; F ) = (0; 0; 0:1), the semi-analytical solution (12) to the forced undamped Du¢ng oscillator with constant coe¢cients is plotted against the numerical solutions using both the RK and Chebyshev collocation methods as elucidated in Fig. 1a.…”

“…Case (2) For ( ; ! 2 ; F ) = (0; 1; 0:1), the comparison between the semi-analytical solution (12) and both the RK and Chebyshev collocation numerical solutions to the forced undamped Du¢ng equation with variable coe¢cients is considered as shown in Fig. 1b.…”