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This paper presents a new approach for analyzing the free vibration of thin plates with arbitrary piecewise smooth curvilinear contour under various boundary conditions. It can also be applied to plates with cracks. This approach is based on the transformation of energy integral expressions and the domain decomposition technique. Furthermore, boundary conditions are modeled by using linear springs to restrain the plate edges. For obtaining the expressions of energy integral, the arbitrarily shaped domain of integration is divided into several trapezoid domains with curved sides by a set of parallel lines passing through the intersection points of contour curve segments. Then, Jacobi orthogonal polynomials are introduced as the admissible functions, so that the repeated integrals in the energy expressions are reduced to definite integrals analytically. At this point, the calculation method of the energy functional is determined by the equation forms of the curve segments of the plate contour. When the equations of the curve segments allow the integrands to have analytic primitive functions, the energy functional has an analytical solution. Otherwise, the Gauss–Legendre method is used to obtain the numerical solution. Accordingly, the arbitrarily shaped plate with cracks is decomposed into several arbitrarily shaped subdomains based on the cracks. Each subdomain is handled according to the above procedure. The continuity conditions at the interconnecting interfaces of the subdomains are realized by linear springs. The plate without or with crack is modeled by setting the spring stiffness to infinity or 0. The accuracy of the proposed method is verified by comparing the obtained results with the published results. Furthermore, the vibration characteristics of plates with various shapes, such as astroid-shaped plates and cracked elliptical plates, are investigated. These new results can serve as benchmarks for further studies on the vibration of plates.

The presence of part-through cracks with limited length is one of the prevalent defects in plate structures. The vibrational response of plates can be used to investigate the effect of crack presence. In this paper the modified line spring method (MLSM) is used to develop a nonlinear multi degree of freedom model of part-through cracked rectangular plate. After a convergence study in time series domain, tuning of the chaotic signal is conducted in two steps: (i) crossing of the Lyapunov exponents’ spectrums for the nonlinear model of the plate and the chaotic signal which assures the effectiveness of the model filter on the Lyapunov dimension of chaotic signal and (ii) varying the tuning parameter of chaotic signal to find a span in which tangible sensitivity in statistical parameters can be observed. Standard deviation, skewness and kurtosis are proposed as features to analyze the time series response of cracked plate. Damage characteristics such as: length, angle, location and depth of crack are considered as parameters to be varied in order to scrutinize the response of plates. Results show that by implementation of a tuned chaotic interrogator signal, tangible sensitivity and also near to monotonic behavior of statistical parameters versus damage characteristics are achievable.