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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.

Many industrial and engineering problems can be generally modeled by partial differential equations. Currently, various analytical and numerical methods have been developed in the last decades to deal with these equations. Often, analytical methods are preferred because they give an exact solution allowing them to get useful information on the domain of the problem. However, analytical methods are almost available for simple engineering problems with simple geometries. To address these weaknesses, many researchers have resorted to numerical methods. The finite difference, finite volume, and finite element methods are the widely used numerical methods. As these methods are classically part of the low order methods, it is necessary to refine the mesh and this may require a relatively high computational effort.

To overcome the abovementioned difficulties of low order methods, some researchers have thought of using high order methods. The Differential Quadrature Method (DQM) is one of the most widely used methods to solve this weakness due to many features such as high accuracy and performance. The DQM, initially presented by Bellman et al. [

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 [

In addition, some researchers have also mentioned the difficulty described above when similar methods such as collocation and finite difference methods are employed. In [

Many regularization approaches have been developed by different researchers in this subject. Wei et al. [

In the present work, 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. To establish its applicability and reliability, the proposed approach is applied here to solve the static and dynamic analyses of beams excited by piezoelectric pulses and multiharmonic actuators, where the installation of piezoelectric actuators is characterized by derivatives of a Dirac-delta function. Various numerical results are presented and compared with the analytical solution developed herein. The presented numerical results demonstrated that the proposed methodology is simple, efficient, and accurate.

In this work, a beam with the length

Beam under a PZT actuators.

As mentioned above, with regard to the derivative of the Dirac-delta function, the direct implementation of this function by the DQM is not a simple matter due to the particular characterizations associated with it. One way to solve this challenge is to replace the derivative of the Dirac-delta function with a regular and soft function. In this context, different forms of regularized derivation of the Dirac-delta function have been proposed in the literature [

For a very small

In addition, the derivatives of the regularized formulation give an approximation to the derivatives of the Dirac-delta function and are given by

Equations (

In view of Eq. (

It should be observed that decreasing of the regularization parameter

A numerical methodological approach based on the DQM combined with a regularization procedure is adapted to space and an implicit scheme for the time derivative.

The derivative with respect to the spatial variable, it is discretized by applying the DQM. The principle of this method consists of approximating the derivative of a function at any location by a weighted linear sum of the values of the function at all points of the discretization of the domain. Suppose that the function

The function values on these points are assumed to be

According to the DQM, the first and second-order derivatives on each of these nodes are given by [

The coefficients

Similarly, we can obtain higher-order derivative formulas by using the higher weighted coefficients, which are expressed in

One of the key factors in the accuracy of DQ solutions is the choice of grid points. The zeros of some orthogonal polynomials are commonly adopted as grid points. In this work, the DQM grid points are taken nonuniformly spaced and are given by the following equations [

Consider a function

As a result, form equation (

For numerical solution,

This system can be rewritten in the following matrix from

These terms are given by

The discrete classical boundary conditions of the beam at

simply supported:

clamped-clamped:

clamped-simply supported:

clamped-free:

free-free:

Similarly, the boundary conditions (

Implementing the boundary conditions into Eq. (

Equation (

Substituting this approximate expression acceleration into (

In order to demonstrate the applicability of the proposed methodological approach and its numerical implementation, multiple computational examples are investigated. Firstly, the static analysis of submitted to a piezoelectric actuator with constant voltage

In this subsection, we consider a beam with length

This problem can be modeled by the differential equation reduced from Eq. (

The vector

To ensure the validity of the proposed methodology and its implementation, the problem of a beam simply supported excited by a concentrated actuator is addressed by applying the proposed methodology for various actuator locations and two different values of the regularization parameter

In accordance with the number of grid points (

Convergence and accuracy of the dimensionless deflection of a simply supported beam under a constant voltage

Convergence and accuracy of the dimensionless deflection of a simply supported beam under a constant voltage

Figure

The results for

In Figure

Convergence and accuracy of the dimensionless deflection of a simply supported beam under a constant voltage

Figure

Convergence and accuracy of the numerical results with decreasing

Consider a beam with length ^{3}. In the first case, the beam is considered simply supported and excited by one piezoelectric actuator with harmonic excitation

The results obtained numerically in this problem demonstrated that the value of

One approach to overcome the disadvantages noted above is to express the differential equation governing the beam in a dimensionless form. Afterward, to discretize the resulting nondimensional differential equation using DQM, a procedure similar to that outlined in Section

The influence of the regularization value

Convergence and accuracy of the numerical results for different values of

Convergence and accuracy of the numerical results for different values of

On the other hand, Figure

Convergence of numerical results for normalized central deflection of a simply supported beam excited by a harmonic piezoelectric actuator.

Now, three actuators with the same parameters exciting the beam by various harmonic excitation are considered and the piezoelectric actuators are located at

Time response for normalized central deflection of S-S and C-C beam excited by three actuators with different harmonics (

Time-space solution of normalized deflection of S-S beam excited by three actuators with different harmonics (

Numerical results 3D for normalized deflection of C-C beam excited by three actuators with excitation harmonic different.

Numerical results for normalized central deflection of C-F beam excited by three actuators with excitation harmonic different.

Numerical results 3D for normalized deflection of C-F beam excited by three actuators with excitation harmonic different.

This subsection focuses on the dynamic response of beams under various types of piezoelectric impulse excitations. The beam is under a short duration excitation and the maximum response is reached in a very short time. The transient response is considered as well as the permanent one. The used beam parameters are the same to those described in subsection

The piezoelectric actuator excites the beam by an impulse that is described by a half-cycle sinusoidal load. The voltage function

Central displacement of a simply supported beam subjected to a sinusoidal piezoelectric impulse for

Figure

Time-space response of S-S beam subjected to a sinusoidal impulse for

In this case, the piezoelectric actuator excites the beam by a rectangular impulse and the excitation function

Figure

Central displacement of a simply supported beam subjected to rectangular piezoelectric impulse voltage for

Time-space normalized deflection of S-S beam subjected to rectangular piezoelectric impulse voltage for

In this case, the forcing voltage

These three steps functions can also be written as a single function

where

Central displacement of a simply supported beam subjected to symmetric triangular piezoelectric impulse for

Time-space normalized deflection of S-S beam subjected to Symmetric triangular piezoelectric impulse for

There are many various fields of engineering and physics, whose governing partial differential equations with strong singularities like the derivative of the Dirac-delta function. For instance, the behavior of structures under piezoelectric patches can be modelled mathematically by the derivative of the function Dirac-delta. The direct discretization of the derivative of the Dirac-delta function using point discretization techniques like the DQM is not a facile task and special processing is required. In this work, the regularization procedure of the derivative of the Dirac-delta function by the distributed functionals using the Hermite polynomials combined with DQM and a time Implicit scheme was elaborated for the numerical solution of the dynamic behavior of beams with various boundary conditions excited by impulse and harmonic piezoelectric actuators. The DQM combined with regularization procedure was elaborated developed to the numerical solution for space discretization and implicit scheme for time discretization was used. Analytical solutions are also derived for these problems to validate the proposed approach. The location of the actuators is described by the derivatives of the Dirac-delta function that are regularized. The obtained numerical results proved that this proposed methodology is efficient and appropriate. Its main advantage is its simplicity ability to consider an arbitrary number of piezoelectric patches as well its high accuracy. Most importantly, the numerical results reveal that the methodological approach can be used as an efficient tool for many physical and mechanical phenomena modeled by partial differential equation with strong singular coefficients and excitation.

Under the piezoelectric actuator, the motion governing equation can be expressed as

For a simply supported beam, the solution can be expressed by the following Fourier series:

Substituting the solution into Equation (

with

The following algebraic equation is resulted.

Letting

Thus, the displacement response can be obtained by substituting Equation (

The motion governing equation of this problem can be expressed as

The analysis is organized in two phases.

In this phase

During this phase, the system starting at

Assuming the solution in the form of mode superposition, the transverse deflection of a free simply supported beam can be written as

Substituting the solution into equation (

The general solution of equation (

The constants of integration

Equation (

Thus, the free vibration displacement of beam for

The code source data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

The authors would like to acknowledge the financial support of the CNRST and the Moroccan Ministry of Higher Education and Scientific Research with the project PPR2/06/2016, as well as to the DSR at the King Abdulaziz University, Jeddah, Saudi Arabia.