Abstract
The dynamic behavior of structures with piezoelectric patches is governed by partial differential equations with strong singularities. To directly deal with these equations, well adapted numerical procedures are required. In this work, the differential quadrature method (DQM) combined with a regularization procedure for space and implicit scheme for time discretization is used. The DQM is a simple method that can be implemented with few grid points and can give results with a good accuracy. However, the DQM presents some difficulties when applied to partial differential equations involving strong singularities. This is due to the fact that the subsidiaries of the singular functions cannot be straightforwardly discretized by the DQM. A methodological approach based on the regularization procedure is used here to overcome this difficulty and the derivatives of the Dirac-delta function are replaced by regularized smooth functions. Thanks to this regularization, the resulting differential equations can be directly discretized using the DQM. The efficiency and applicability of the proposed approach are demonstrated in the computation of the dynamic behavior of beams for various boundary conditions and excited by impulse and Multiharmonics piezoelectric actuators. The obtained numerical results are well compared to the developed analytical solution.
Highlights
Many industrial and engineering problems can be generally modeled by partial differential equations
Despite the abovementioned advantages of the differential quadrature method (DQM), it presents certain challenges when applied to partial differential equations containing singular functions such as the derivative of the Dirac-delta function
A numerical procedure based on the combination of the DQM with the regularization of the derivatives of the Dirac-delta function is elaborated for the numerical solution of the vibration response of the beam under the impulse and multiharmonic piezoelectric actuators
Summary
Many industrial and engineering problems can be generally modeled by partial differential equations. Despite the abovementioned advantages of the DQM, it presents certain challenges when applied to partial differential equations containing singular functions such as the derivative of the Dirac-delta function To overcome this difficulty, some authors have suggested coupling the DQM and the integral quadrature method (IQM) in which this type of problem can be handled [18]. A numerical procedure based on the combination of the DQM with the regularization of the derivatives of the Dirac-delta function is elaborated for the numerical solution of the vibration response of the beam under the impulse and multiharmonic piezoelectric actuators Based on this regularization, the DQM can be applied directly to discretize the resulting partial differential equations. The presented numerical results demonstrated that the proposed methodology is simple, efficient, and accurate
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