Nonlinear dynamic behaviors of clamped laminated shallow shells with one-to-one internal resonance

This paper investigates one-to-one internal resonance of laminated shallow shells with rigidly clamped edges. It is assumed that the natural frequencies ω2 and ω3 of two asymmetric (second and third) vibration modes have the relationship ω2≈ω3. The displacements are expressed by using eigenvectors for linear vibration modes calculated by the Ritz method. Applying Galerkin's procedure to the equation of motion, nonlinear differential equations are derived. By considering the first vibration mode in addition to the two asymmetric vibration modes, quadratic nonlinear terms expressing the interaction between the asymmetric and the first modes appear in the differential equations. Shooting method is used to obtain the steady-state response when the driving frequency Ω is near ω2. The dynamic characteristics of the shells with the internal resonance are discussed.