An artificial small perturbation parameter and nonlinear plate vibrations

Nonlinear natural in-plane vibrations of a rectangular plate are studied using three small parameters. Firstly, the nonlinearity is assumed to be small. Then, a solution to a problem of the zeroth order (linear) is sought in the form of an asymptotic series with respect to the ratio of stiffness characteristics. ternal resonance, vibration modes are coupled via an infinite system of nonlinear algebraic equations, and the artificial small parameter approach is proposed to solve the obtained system. Analytical formulas for the amplitude–frequency characteristics are derived and the solutions are compared with numerical results.