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An approach is presented to investigate the nonlinear vibration of stiffened plates. A stiffened plate is divided into one plate and some stiffeners, with the plate considered to be geometrically nonlinear, and the stiffeners taken as geometrically nonlinear Euler beams. Lagrange equation and modal superposition method are used to derive the dynamic equilibrium equations of the stiffened plate according to the energy of the system. Besides, the effect caused by boundary movement is transformed into equivalent excitations. The primary parametric resonance–primary resonance of the stiffened plate is studied by using homotopy analysis method. Numerical examples for stiffened plates with different thicknesses of the plates are presented to discuss the amplitude–frequency and amplitude–excitation relationship of the primary parametric resonance–primary resonance. In addition, the analysis on how the damping coefficients and the transverse excitations influence amplitude–frequency curves is also carried out. Some nonlinear vibration characteristics of stiffened plates are obtained, which are useful for engineering design.

An approach is presented to study the nonlinear forced vibration of a stiffened plate. The stiffened plate is divided into one plate and some stiffeners, with the plate considered to be geometrically nonlinear, and the stiffeners taken as geometrically nonlinear Euler beams. Assuming the displacement of the stiffened plate, Lagrange equation and modal superposition method are used to derive the dynamic equilibrium equations of the stiffened plate according to energy of the system. A stiffened plate with four clamped edges subjected to harmonic excitation is studied by means of the method of multiple scales; the first approximation solutions of the double-modal motion of the system are obtained. Numerical examples for different stiffened plates are presented to discuss the steady response of the primary resonance and the amplitude–frequency relationship; and some nonlinear forced vibration characteristics of the stiffened plate are obtained, which are useful for engineering design.