A Method of Hamiltonian System for the Vibration Problem of Cracked Plates with Piecewise Boundaries

This paper focuses on analytical solutions for studying the vibrations of partially cracked orthotropic plates with piecewise boundaries using the symplectic elasticity method. Crack compliance coefficients are derived from the line spring model and introduced into the governing equations. In this paper, by establishing the Hamiltonian system and solving the Hamiltonian canonical equations, the vibration problem of cracked plates with piecewise boundaries can be reduced to finding the symplectic eigenvalues and symplectic eigensolutions. In the Hamiltonian system, the problem is transformed into seeking the solution of an algebraic equation system by the aid of symplectic eigensolutions and the adjoint symplectic orthogonal relations between the eigensolutions. Thus, for orthotropic plates with cracks under piecewise boundary conditions, this work presents a set of analytical solutions. Specifically, expressions for the free vibration behavior are derived. Furthermore, the steady-state response is determined. Finally, the transient dynamic response is also formulated and solved. The numerical results show that this method is effective and the natural frequency obtained in this paper is consistent with experimental results. A parametric investigation is performed, with the aim of comprehensively investigating the dynamic behavior of cracked plates subjected to piecewise boundaries. Significant findings include how aspect ratio, piecewise boundary conditions, and crack length affect structural vibration characteristics. For the plates containing cracks under piecewise boundaries, the frequency-response and the time-history curves for the forced vibration are plotted. These curves clearly illustrate the corresponding vibration characteristics of the plates. Some of the results can provide benchmarks for validating numerical or approximate methods. The method presented in this paper can provide a pathway for solving similar problems in the field.