ANewNumericalProcedureforVibrationAnalysisofBeamunder ...downloads.hindawi.com/journals/jam/2020/7391848.pdf · Yassin Belkourchia 1 and Lahcen Azrar1,2 1Research Center STIS, M2CS,

Research ArticleANewNumerical Procedure for Vibration Analysis of Beam underImpulse and Multiharmonics Piezoelectric Actuators

Yassin Belkourchia 1 and Lahcen Azrar1,2

1Research Center STIS, M2CS, Department of Applied Mathematics and Informatics, ENSET, Mohammed V University,Rabat, Morocco2Department of Mechanical Engineering, Faculty of Engineering, KAU, Jeddah, Saudi Arabia

Correspondence should be addressed to Yassin Belkourchia; [email protected]

Received 23 March 2020; Revised 5 June 2020; Accepted 20 July 2020; Published 28 August 2020

Academic Editor: Yansheng Liu

Copyright © 2020 Yassin Belkourchia and Lahcen Azrar. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original workis properly cited.

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 quadraturemethod (DQM) combined with a regularization procedure for space and implicit scheme for time discretization is used. TheDQM is a simple method that can be implemented with few grid points and can give results with a good accuracy. However, theDQM presents some difficulties when applied to partial differential equations involving strong singularities. This is due to thefact that the subsidiaries of the singular functions cannot be straightforwardly discretized by the DQM. A methodologicalapproach based on the regularization procedure is used here to overcome this difficulty and the derivatives of the Dirac-deltafunction are replaced by regularized smooth functions. Thanks to this regularization, the resulting differential equations can bedirectly discretized using the DQM. The efficiency and applicability of the proposed approach are demonstrated in thecomputation of the dynamic behavior of beams for various boundary conditions and excited by impulse and Multiharmonicspiezoelectric actuators. The obtained numerical results are well compared to the developed analytical solution.

1. Introduction

Many industrial and engineering problems can be gener-ally modeled by partial differential equations. Currently,various analytical and numerical methods have been devel-oped in the last decades to deal with these equations.Often, analytical methods are preferred because they givean exact solution allowing them to get useful informationon the domain of the problem. However, analyticalmethods are almost available for simple engineering prob-lems with simple geometries. To address these weaknesses,many researchers have resorted to numerical methods.The finite difference, finite volume, and finite elementmethods are the widely used numerical methods. As thesemethods are classically part of the low order methods, it isnecessary to refine the mesh and this may require a rela-tively high computational effort.

To overcome the abovementioned difficulties of loworder methods, some researchers have thought of usinghigh order methods. The Differential Quadrature Method(DQM) is one of the most widely used methods to solvethis weakness due to many features such as high accuracyand performance. The DQM, initially presented by Bell-man et al. [1, 2] in the mid-1970s, is an amazing methodfor the direct numerical solution of partial differentialequations showing up in many fields in engineering, math-ematics, and physics [3–5]. Most applications of thismethod involve static and dynamic analysis of structuralcomponents such as beams and plates [6–8]. In addition,Bert and Malik [9] presented an analysis of thin rectangu-lar plates simply supported on two opposite ends, basedon the classical thin plate theory. Du et al. [10] introduceda generalized DQM to examine the buckling problems ofrectangular plates with internal support and variable

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bending stiffness. Wang et al. [11] investigated the vibra-tion and buckling problems of thin rectangular plates withnonlinear distributed loads along two adjacent plate edgeswith nine boundary conditions by using the differentialquadrature method. Bashan [12] implemented for thenumerical solution of nonlinear Kawahara equation viathe Crank-Nicolson-Differential Quadrature Method basedon modified cubic B-splines and Bashan et al. [13] appliedDQM based on modified cubic B-splines for numericalsolution of the complex modified Korteweg-de Vries equa-tion. Also, Bashan [14, 15] investigated numerical solu-tions of the system of coupled Korteweg-de Vriesequation based on Finite difference method and DQMand numerical solutions of the coupled Korteweg-de Vriesequation based on a combination of Crank-Nicolsonmethod and quintic B-spline based differential quadraturemethod. Ucar et al. [16] proposed a numerical approxi-mate solution of the nonlinear modified Burgers equationvia the modified cubic B-spline and DQM. Also, Tangand Wang [17] used the DQM to solve the buckling anal-ysis of symmetrically laminated rectangular plates underplane compression along two opposite edges. It has beenobserved from the evidence provided by variousresearchers that DQM is computationally productive andcan give excellent results for the problems underinvestigation.

Despite the abovementioned advantages of the DQM, itpresents certain challenges when applied to partial differen-tial equations containing singular functions such as thederivative of the Dirac-delta function. To overcome this dif-ficulty, some authors have suggested coupling the DQMand the integral quadrature method (IQM) in which this typeof problem can be handled [18]. This approach is applied tothe manipulation of the Dirac-delta function applied to theproblem of vibration of beams and rectangular plates sub-jected to a moving point load. On the other hand, Eftekhari[19] presented a methodological procedure for the numericalsolution of the moving load problem, and Eftekhari andJafari [20] combined the finite element method (FEM) andthe DQM to study the free and forced vibration and bucklingof rectangular plates. As previously mentioned in [20], thisapproach may have certain difficulties when applied to prob-lems contains a singular time-dependent function. In gen-eral, it is more favorable to find the solution of theseproblems by the DQM itself and not by combining theDQM with other techniques.

In addition, some researchers have also mentioned thedifficulty described above when similar methods such as col-location and finite difference methods are employed. In [21],it is argued that the difficulty of the collocation method indealing with such singular functions is due to the Gibbs phe-nomenon in which numerical solutions oscillate around. Tosolve this problem, a regularization of the singular functionhas been proposed in order to obtain a softer representationof the singular function and to stabilize the undesirable oscil-latory behavior of solutions close to the singularities.

Many regularization approaches have been developed bydifferent researchers in this subject. Wei et al. [22] presenteda computational approach of the regularized Dirac-delta

function based on the discrete singular convolution algo-rithm for vibration analysis of rectangular plates with partialinternal line or point supports. Burko and Khanna [23] sug-gested a regularization of the Dirac-delta function using theGaussian function. Walden [24] solved a few differentialequations with singular source terms by finite differenceand finite element methods and showed that the full orderof convergence can be achieved if the singularities are treatedin the right way. Engquist et al. [25] presented two methodsto construct consistent approximations for the regularizationof the Dirac-delta function. Rivera et al. [26] proposednumerical resolution of the hyperbolic heat equation usingsmoothed mathematical functions for approximation of theHeaviside and Dirac-delta functions.

In the present work, a numerical procedure based on thecombination of the DQM with the regularization of thederivatives of the Dirac-delta function is elaborated for thenumerical solution of the vibration response of the beamunder the impulse and multiharmonic piezoelectric actua-tors. Based on this regularization, the DQM can be applieddirectly to discretize the resulting partial differential equa-tions. To establish its applicability and reliability, the pro-posed approach is applied here to solve the static anddynamic analyses of beams excited by piezoelectric pulsesand multiharmonic actuators, where the installation ofpiezoelectric actuators is characterized by derivatives of aDirac-delta function. Various numerical results are presentedand compared with the analytical solution developed herein.The presented numerical results demonstrated that the pro-posed methodology is simple, efficient, and accurate.

2. Mathematical Formulation

2.1. Equilibrium Equation. In this work, a beam with thelength L, cross-section area A, Young’s modulus E, momentof area I, and density ρ excited by Na piezoelectric actuatorsis considered as shown in Figure 1. Based on Euler-Bernoullibeammodel, the transverse motion is modeled by the follow-ing partial differential equation [27]

EI∂4w x, tð Þ

∂x4+ ρA

∂2w x, tð Þ∂t2

= f x, tð Þ, ð1Þ

where

f x, tð Þ = b〠Na

i=0

∂2Mai

∂x2= b〠

Na

i=1Eipd

i31hip + h

2 Vi tð Þ δ′ x − xia1

� �h− δ′ x − xia2

� �i,

ð2Þ

in whichwðx, tÞ is the transverse displacement of the beam, xis the axial coordinate, t is the time, b and h are the width andthickness of the beam, respectively. Na is the number of thepiezoelectric actuators. Ma

i is the force moment induced bythe piezoelectric actuator i, and di31, E

ip, and hip are the piezo-

electric strain constant, Young’s modulus, thickness of thei-th actuator, respectively. δ′ð:Þ is the derivative of the

2 Journal of Applied Mathematics

Dirac-delta function, xia1, xia2 are the coordinates of the

two ends of i-th actuator and ViðtÞ is the voltage appliedon the i-th actuator.

2.2. Regularization Procedure. As mentioned above, withregard to the derivative of the Dirac-delta function, the directimplementation of this function by the DQM is not a simplematter due to the particular characterizations associated withit. One way to solve this challenge is to replace the derivativeof the Dirac-delta function with a regular and soft function.In this context, different forms of regularized derivation ofthe Dirac-delta function have been proposed in the literature[22]. In this work, the Hermite polynomials are used [28]. Anapproximation of the distribution delta is constructed usingthe usual Hermite polynomials H2n as follows

δM x − x0ð Þ = 1αexp − x − x0ð Þ2

2α2

!〠M2

n=0

−14

� �n 1ffiffiffiffiffiffi2π

pn!H2n

x − x0ffiffiffi2

pα

� �:

ð3Þ

For a very small α, the function δMðx − x0Þ becomes iden-tical to the functional delta when the degree of polynomialMis fixed.

limα→0

δM x − x0ð Þ = δ x − x0ð Þ ð4Þ

In addition, the derivatives of the regularized formulationgive an approximation to the derivatives of the Dirac-deltafunction and are given by

δlð ÞM x − x0ð Þ = 2−l/2

αl+1exp − x − x0ð Þ2

2α2

!〠M2

n=0

−14

� �n

× −1ð Þl 1ffiffiffiffiffiffi2π

pn!H2n+l

x − x0ffiffiffi2

pα

� �:

ð5Þ

Equations (3) and (5) mean that the differentiations havebeen transformed into an algebraic process in the approxi-mate representation. This important feature of this approxi-mate representation allows it to be a powerfulcomputational tool for solving various partial and ordinarydifferential equations with singularities. The parameter αmust be as small as possible with fixed M.

In view of Eq. (5), the excitation force can be rewritten as

f x, tð Þ = b〠Na

i=0Eipd

i31hip + h

2 Vi tð Þ1

α2ffiffiffi2

p exp − x − xia2� �22α2

!"

� 〠M/2

m=0

−14

� �m

× 1ffiffiffiffiffiffi2π

pm!

H2mx − xia2ffiffiffi

2p

α

� �−

1α2

ffiffiffi2

p exp

� − x − xia1� �22α2

!〠M2

m=0

−14

� �m


pm!

H2mx − xia1ffiffiffi

2p

α

� �:

ð6Þ

It should be observed that decreasing of the regulariza-tion parameter α yields a more accurate representation. Asthe DQM is a higher-order method, this can considerablyincrease the cost of calculation, especially when the value αis too small. Consequently, it is necessary to find a suitablevalue of α for the numerical accuracy and efficiency of thecalculation.

3. Solution Procedure

A numerical methodological approach based on the DQMcombined with a regularization procedure is adapted to spaceand an implicit scheme for the time derivative.

3.1. Differential Quadrature Method. The derivative withrespect to the spatial variable, it is discretized by applyingthe DQM. The principle of this method consists of approxi-mating the derivative of a function at any location by aweighted linear sum of the values of the function at all pointsof the discretization of the domain. Suppose that the functionwðxÞ is sufficiently smooth over the interval ½x1, xN �. In thisinterval, N distinct nodes are defined.

x1 < x2 <⋯ < xN : ð7Þ

The function values on these points are assumed to be

w1,w2 ⋯ ,wN: ð8Þ

According to the DQM, the first and second-order deriv-atives on each of these nodes are given by [29].

dw xið Þdx

≈ 〠N

j=1aijwj i = 1, 2,⋯,N , ð9aÞ

d2w xið Þdx2

≈ 〠N

j=1bijwj i = 1, 2,⋯,N: ð9bÞ

The coefficients aij and bij are the weighting coeffi-cients of the first and second-order derivatives with

h

z

xi

a1

xi

a2

x

hi

p

Figure 1: Beam under a PZT actuators.

3Journal of Applied Mathematics

respect to x, respectively. The coefficients aij and bij aregiven as follows [29].

aij =1

xj − xi� � YN

k=1,k≠i,j

xi − xkxj − xk

i, j = 1, 2,⋯,N i ≠ j,

ð10aÞ

aii = − 〠N

j=1,j≠iaij i = 1, 2,⋯,N , ð10bÞ

bij = 2 aijaii −aij

xj − xi

" # i, j = 1, 2,⋯,N i ≠ j, ð10cÞ

bij = − 〠N

j=1,j≠ibij i = 1, 2,⋯,N: ð10dÞ

Similarly, we can obtain higher-order derivative formu-las by using the higher weighted coefficients, which are

expressed in eðmÞij to avoid confusion. They are character-

ized by induction [4].

e mð Þij =m aije

m−1ð Þii −

e m−1ð Þii

xi − xj

! i, j = 1, 2,⋯,N i ≠ j,m = 2, 3,⋯,N − 1,

ð11aÞ

e mð Þii = − 〠

N

j=1,j≠ie mð Þij i = 1, 2,⋯,N: ð11bÞ

One of the key factors in the accuracy of DQ solutionsis the choice of grid points. The zeros of some orthogonalpolynomials are commonly adopted as grid points. In thiswork, the DQM grid points are taken nonuniformlyspaced and are given by the following equations [6]

xi = x1 +12 1 − cos i − 1

N − 1π� ��

xN − x1ð Þ: ð12Þ

3.2. Numerical Accuracy of the DQM. Consider a functionwðxÞ, which is approximated by the Lagrange interpola-tion polynomial of degree N − 1. The error for the r − thorder derivative approximation of this function at pointxi can be obtained as [29].

E rð Þ w xið Þ½ � = w Nð Þ ξð ÞM rð Þ xið ÞN!

, i = 1, 2,⋯,N , ð13Þ

where

M xð Þ =YNj=1

x − xj� �

: ð14Þ

As a result, form equation (13), the accuracy of theDQM can be directly proportional to N . This means that

a result can be achieved with very high accuracy even ifthe number of grid points N is small.

3.3. DQM Analogues. For numerical solution, n grid pointswith coordinates x1, x2,⋯, xn in the x-direction are consid-ered. Applying the quadrature rule, to Eq. (1), the followingordinary differential system is obtained

EI 〠n

k=1e 4ð Þik wk tð Þ + ρA

∂2wi tð Þ∂t2

= f i tð Þ, i = 1,⋯, n: ð15Þ

This system can be rewritten in the following matrix from

K½ � W tð Þf g + M½ � €W tð Þ �= F tð Þf g, ð16Þ

where ½M� and ½K� are the resulting mass and stiffness matri-ces of the beam, fWðtÞg and f €WðtÞg are the displacementand acceleration vectors, and fFðtÞg is the charge vectorapplied by the piezoelectric actuators.

These terms are given by

K½ � = EI e½ � 4ð Þ, M½ � = ρA I½ �,W tð Þf g = w x1, tð Þ w x2, tð Þ⋯w xn, tð Þ½ �T ,€W tð Þ �

= €w x1, tð Þ €w x2, tð Þ⋯ €w xn, tð Þ½ �T ,ð17Þ

F tð Þf g = bffiffiffi2

pα2

〠Na

i=0Eipd

i31hip + h

2 Vi tð Þ exp − x1 − xia2� �2

2α2

!"

� 〠M2

m=0

−14

� �m


pm!

H2mx1 − xia2ffiffiffi

2p

α

� �

− exp − x1 − xia1� �2

2α2

!〠M2

m=0

−14

� �m


pm!

H2mx1 − xia1ffiffiffi

2p

α

� �⋯ exp − xn − xia2

� �22α2

!

� 〠M2

m=0

−14

� �m


pm!

H2mxn − xia2ffiffiffi

2p

α

� �

− exp − xn − xia1� �2

2α2

!〠M2

m=0

−14

� �m


pm!

H2mxn − xia1ffiffiffi

2p

α

� �T,

ð18Þ

where ½I� is the identity matrix of order n × n and ½e�ð4Þ is theDQM weighting coefficient matrix of the fourth-orderderivative.


The discrete classical boundary conditions of the beam atx = 0 and x = 1, using the DQ method, can be written as fol-lowing:

w1 tð Þ = 0, 〠n

k=1e n0ð Þ1k wk tð Þ = 0, ð19Þ

wn tð Þ = 0, 〠n

k=1e n1ð Þnk wk tð Þ = 0, ð20Þ

where n0 and nl can be chosen from 1, 2, or 3. The selection ofthe values n0 and nl can have the following traditionalboundary conditions [6]:

(i) simply supported: n0 = 2; n1 = 2(ii) clamped-clamped: n0 = 1; n1 = 1(iii) clamped-simply supported: n0 = 1; n1 = 2(iv) clamped-free: n0 = 1; n1 = 3(v) free-free: n0 = 2; n1 = 3

12–30

–20

Perc

ent e

rror

in d

imen

sionl

ess d

eflec

tion

at x

/L =

1/2

–10

0

10

20

30

14 16 18 20 22 24Number of grid points (n)

26 28 30

xa/L = (0.2,0.5)

xa/L = (0.4,0.7)

xa/L = (0.6,0.9)

(a)

12–30

–20

Perc

ent e

rror

in d

imen

sionl

ess d

eflec

tion

at x

/L =

1/5

–10

0

10

20

40

30


26 28 30

xa/L = (0.2,0.5)

xa/L = (0.4,0.7)

xa/L = (0.6,0.9)

(b)

12–30

–20

Perc

ent e

rror

in d

imen

sionl

ess d

eflec

tion

at x

/L =

2/3

–10

0

10

20

40

30


26 28 30

xa/L = (0.2,0.5)

xa/L = (0.4,0.7)

xa/L = (0.6,0.9)

(c)

12–30

–20

Perc

ent e

rror

in d

imen

sionl

ess d

eflec

tion

at x

/L =

3/4

–10

0

10

20

50

40

30


26 28 30

xa/L = (0.2,0.5)

xa/L = (0.4,0.7)

xa/L = (0.6,0.9)

(d)

Figure 2: Convergence and accuracy of the dimensionless deflection of a simply supported beam under a constant voltage V0 applied byactuator at various locations for (α = 0:25) at x/L = 1/2 (a), x/L = 1/5 (b), x/L = 2/3 (c), and x/L = 3/4 (d).


Similarly, the boundary conditions (19) can also beexpressed in a matrix form

KBB½ � WB tð Þf g + KBI½ � WI tð Þf g = 0, ð21Þ

where the subscripts B and I indicate the grid pointsused for writing the quadrature analog of the boundaryconditions and the governing differential equation, respec-tively. It is noted that the dimensions of matrices ½KBB�and ½KBI � are 4 × 4 and 4 × ðn − 4Þ, respectively.

Implementing the boundary conditions into Eq. (16)leads to the following system of ordinary differential equa-tions

~K�

WI tð Þf g + ~M�

€WI tð Þ �

= FI tð Þf g, ð22Þ

wherein½~K� = ½KII � − ½KIB�½KBB�−1½KBI � and ½ ~M� = ½MII �; where

½~K�, ½ ~M�, and ½KIB� are matrices of order ðn − 4Þ × ðn − 4Þ,ðn − 4Þ × ðn − 4Þ, and ðn − 4Þ × 4, respectively.

12–150

–100

Perc

ent e

rror

in d

imen

sionl

ess d

eflec

tion

at x

/L =

1/2

–50

0

50

100

200

150


26 28 30

xa/L = (0.2,0.5)

xa/L = (0.4,0.7)

xa/L = (0.6,0.9)

(a)

12–250

–200

–150

–100

Perc

ent e

rror

in d

imen

sionl

ess d

eflec

tion

at x

/L =

1/5

–50

0

50

100

150


26 28 30

xa/L = (0.2,0.5)

xa/L = (0.4,0.7)

xa/L = (0.6,0.9)

(b)

Number of grid points (n)12

–150

–100

Perc

ent e

rror

in d

imen

sionl

ess d

eflec

tion

at x

/L =

2/3

–50

0

50

100

200

150

14 16 18 20 22 24 26 28 30

xa/L = (0.2,0.5)

xa/L = (0.4,0.7)

xa/L = (0.6,0.9)

(c)

12–100

Perc

ent e

rror

in d

imen

sionl

ess d

eflec

tion

at x

/L =

3/4

–50

0

100

50

250

200

150


26 28 30

xa/L = (0.2,0.5)

xa/L = (0.4,0.7)

xa/L = (0.6,0.9)

(d)

Figure 3: Convergence and accuracy of the dimensionless deflection of a simply supported beam under a constant voltage V0 applied byactuator at different locations for (α = 0:08) at x/L = 1/2 (a), x/L = 1/5 (b), x/L = 2/3 (c), and x/L = 3/4 (d).


Equation (22) can be solved by using various step-by-step time integration schemes. In this study, the timederivative is discretized using the centered finite differencescheme, then

∂2WI

∂t2= Wi+1

I − 2WiI +Wi−1

I

Δt2: ð23Þ

Substituting this approximate expression accelerationinto (22), one gets:

~K�

Wi+1I + ~M

� Wi+1I − 2Wi

I +Wi−1I

Δt2= Fi+1

I + initial conditions:�

ð24Þ

4. Numerical Results and Discussion

In order to demonstrate the applicability of the proposedmethodological approach and its numerical implementation,multiple computational examples are investigated. Firstly,the static analysis of submitted to a piezoelectric actuatorwith constant voltage V0 is considered to demonstrate the

30–2

35 40 45Perc

ent e

rror

in d

imen

sionl

ess d

eflec

tion

at x

/L =

1/4

Number of grid points (n)

–1.5

–1

–0.5

0

0.5

1

xa/L = (0.3,0.6)

𝛼 = 0.08𝛼 =0.125𝛼 = 0.15

(a)

30–6

35 40 45Perc

ent e

rror

in d

imen

sionl

ess d

eflec

tion

at x

/L =

1/4

Number of grid points (n)

–4

–2

0

2

4

6

8

10

12

xa/L = (0.54,0.84)

𝛼 = 0.08𝛼 =0.125𝛼 = 0.15

(b)

25–5

35 40 45Perc

ent e

rror

in d

imen

sionl

ess d

eflec

tion

at x

/L =

1/3


–4

–3

–2

–1

0

1

2

xa/L = (0.3,0.6)

𝛼 = 0.08𝛼 =0.125𝛼 = 0.15

(c)

25–6

35 40 45Perc

ent e

rror

in d

imen

sionl

ess d

eflec

tion

at x

/L =

1/3


–4

–2

0

2

4

6

8

10

12

xa/L = (0.54,084)

𝛼 = 0.08𝛼 =0.125𝛼 = 0.15

(d)

Figure 4: Convergence and accuracy of the dimensionless deflection of a simply supported beam under a constant voltage V0 applied byactuator at x/L = 1/4 for xa/L = ð0:3,0:6Þ (a), xa/L = ð0:54,0:84Þ (b), and at x/L = 1/3 for xa/L = ð0:3,0:6Þ (c), xa/Lð0:54,0:84Þ (d) fordifferent values of α.


feasibility and effectiveness of the proposed procedure. Thisnumerical example is introduced to further verify the exacti-tude and convergence of the proposed approach. The secondexample concerns the study of a beam exited by multiharmo-nic piezoelectric actuators. The third example is focused onthe study of a beam excited by various types of impulse pie-zoelectric actuators. Three boundary conditions, namely,simply supported edges (S-S), clamped edges (C-C), andclamped free edges (C-F) conditions are considered in thesecond and third examples. In the present computation, theparameter M is taken as M = 4 for all examples. The analyti-cal solutions of simply supported beam exited by harmonicand impulse excitation using a piezoelectric actuator andfor impulse excitation by a piezoelectric actuator is developedand given in the Appendices A and B. The accuracy of theproposed method can be verified by comparing the calcu-lated results with those of the developed analytically ones.

4.1. Beams under a Piezoelectric Actuator with ConstantVoltage. In this subsection, we consider a beam with length

L, Young’s modulus E, moment of area I under one PZTactuator of thickness hp, the piezoelectric strain constantd31, and Young’s modulus Ep with a constant voltage V0.The PZT actuator is assumed to be perfectly bonded and itsstiffness is neglected due to its limited contribution to thebehavior of the beam, as shown in Figure 1.

This problem can be modeled by the differential equationreduced from Eq. (1) as follows

EI∂4w xð Þ∂x4

= f xð Þ, ð25Þ

where f ðxÞ is given by Eq. (6) in which VðtÞ =V0. By usingthe DOM procedure, Eq. (25) is reduced to the followingalgebraic equation

K½ � Wf g = Ff g, ð26Þ

0–0.07

–0.06

–0.05

–0.04

–0.03

W/W

cs

–0.02

–0.01

0

0.2 0.4 0.6 0.8 1x/L

xa/L = (0.2,0.5)

xa/L = (0.6,0.9)

0–0.07

–0.06

–0.05

–0.04

–0.03

W/W

cs

–0.02

–0.01

0

0.2 0.4 0.6 0.8 1x/L

0–0.05

–0.045

–0.04

–0.035

–0.03

–0.025

–0.02

–0.015

–0.01

–0.005

0

W/W

cs

0.2 0.4 0.6 0.8 1x/L

xa/L = (0.4,0.7)

𝛼 = 0.08Exact 𝛼 = 0.15

𝛼 = 0.25

Figure 5: Convergence and accuracy of the numerical results with decreasing α of the dimensional deflection of a simply supported beamunder a constant voltage V0 applied by actuator for different locations ðn = 51Þ.


where ½K�, fWg, and fFg are given by

K½ � = EI e½ � 4ð Þ,Wf g = w x1ð Þ w x2ð Þ⋯w xnð Þ½ �T ,

Ff g = bEpd31hp + h

2 V0 χf g,ð27Þ

The vector χ is given by:

χf g = 1ffiffiffi2

pα2

exp − x1 − xa2ð Þ22α2

!〠M2

m=0

−14

� �m24


pm!

H2mx1 − xa2ffiffiffi

2p

α

� �− exp − x1 − xa1ð Þ2

2α2

!

� 〠M2

m=0

−14

� �m


pm!

H2mx1 − xa1ffiffiffi

2p

α

� �⋯ exp

� − xn − xa2ð Þ22α2

!〠M2

m=0

−14

� �m


pm!

H2mxn − xa2ffiffiffi

2p

α

� �

− exp − xn − xa1ð Þ22α2

!〠M2

m=0

−14

� �m


pm!

H2mxn − xa1ffiffiffi

2p

α

� �T:

ð28Þ

To ensure the validity of the proposed methodologyand its implementation, the problem of a beam simplysupported excited by a concentrated actuator is addressedby applying the proposed methodology for various actua-tor locations and two different values of the regularization

parameter α = 0:25 and α = 0:08, respectively. The errorvariations used in numerical solutions is defined by

wDQM −wExact��

wExact× 100: ð29Þ

In accordance with the number of grid points (n), theerror variations for different locations ðxa1, xa2Þ of theactuator are shown in Figures 2 and 3.

Figure 2 shows the variance of the percent error in thenumerical results for α = 0:25 and for different actuatorlocations. It is clearly noticed that the results obtainedconverge uniformly towards their final values. On theother hand, the numerical precision of the results is notvery good because the maximum error in the numericalresults is around 10%. Therefore, relying on the case α =0:25 does not allow a good estimate of the original deriv-ative of the Dirac-delta function.

The results for α = 0:08 is illustrated in Figure 3. It canbe clearly seen from Figure 3 that the error in this case ishigh and that its response is oscillating when the grid-point number is small. On the other hand, this trend insolutions is explained by the inaccurate representation ofthe regularized derivative of the Dirac-delta function inthe discretized model for a few points on the grid. Inaddition, the percent error converges quickly to zero byaugmenting the grid-point number. However, from thenumerical results shown in Figure 3, we can also see thatthe error is influenced by the location of the actuator.We can also observe that the solutions at different pointson the beam are of the same order of precision. On thecontrary, when comparing the results of Figures 3 and 2,we notice that when higher values of α are utilized inthe methodology approach, a more convergent tendencyof solutions is achieved, especially for a few grid points.

0–0.04

–0.03

–0.02

–0.01

0

0.01

0.02

0.03

0.04W

cd/W

cs

0.2 0.4 0.6 0.8 1t

xa = (0.061,0.0914)

𝛼 = 0.15Exact 𝛼 = 0.05

𝛼 = 0.065

0–0.06

–0.04

–0.02

0

0.02

0.04

0.06

Wcd/W

cs

0.2 0.4 0.6 0.8 1t

xa = (0.0203,0.0508)

Figure 6: Convergence and accuracy of the numerical results for different values of α for the case where the dimensional differential equationof motion of the beam is discretized using the DQM (n = 31).


Therefore, when smaller values of α are utilized, a greaterprecision of the solution is obtained at a relatively largernumber of grid points.

In Figure 4, the error percentage is presented accordingto the number of grid points for different values of α andthe location of the actuator xa. It is observed that when thegrid-point number increases, the error in the numerical out-comes tends to a constant value. This constant value is due tothe error of the regularization procedure. Also, we canobserve clearly that the value of this error can be decreasedconsiderably by decreasing the value by α.

Figure 5 presents the convergence of the dimensionlessdeflection of the beam normalized by the static deflection

wcs = bEbd13L3ðhp + hÞV0/ð2EIÞ of a simply supported

beam under a constant voltage V0 applied by an actuatorfor different locations and with different values of the reg-ularization parameter α. By comparing the results withthose of exact solutions, one can conclude that the solu-tions have good accuracy with small values of the regular-ization parameter α.

4.2. Multiharmonic Piezoelectric Actuation. Consider a beamwith length L equal to 10:16 cm, width b = 0:635 cm, thick-ness h = 0:635 cm, Young’s modulus E = 2:068 × 1011 Pa,mass density ρ = 10686:9 kg/m3. In the first case, the beamis considered simply supported and excited by one

𝛼 = 0.08

0 0.2 0.4 0.6 0.8 1t

xa/L = (0.2,0.5)

–0.06

–0.04

–0.02

0.02

0

0.04

0.06

Wcd/W

cs

n = 25Exact n = 27

n = 31

Figure 8: Convergence of numerical results for normalized central deflection of a simply supported beam excited by a harmonic piezoelectricactuator.

𝛼 = 0.07Exact 𝛼 = 0.09

𝛼 = 0.1

0–0.04

–0.03

–0.02

–0.01

0

0.01

0.02

0.03

0.04

Wcd/W

cs

Wcd/W

cs

0.2 0.4 0.6 0.8 1t

xa = (0.6,0.9)

0 0.2 0.4 0.6 0.8 1t

–0.06

–0.04

–0.02

0.02

0

0.04

0.06xa/L = (0.2,0.5)

Figure 7: Convergence and accuracy of the numerical results for different values of α for the case where the dimensionless differentialequation of motion of the beam is discretized using the DQM (n = 31).


piezoelectric actuator with harmonic excitation Ω = 9:8695.In the following cases, the beam is considered SS, C-C, andC-F excited by three piezoelectric actuators with differentharmonic excitation and the same property. The dynamicresponse at the center of the beam,wcd , is evaluated for differ-ent locations of the piezoelectric actuator and normalized bythe static deflection wcs = ðbEpL

3d31ðhp + hÞÞ/ð2EIÞ. Thetime step used to solve the resulting dynamic equationsis Δt = 0:001, which is sufficient as the implicit timescheme is used to assure the stability of the algorithm.

The results obtained numerically in this problem demon-strated that the value of α at which the convergence isachieved depends significantly on the length value of thebeam. In particular, for small beams, the numerical conver-gence was obtained with a small specific value of α.

One approach to overcome the disadvantages notedabove is to express the differential equation governing thebeam in a dimensionless form. Afterward, to discretize theresulting nondimensional differential equation using DQM,a procedure similar to that outlined in Section 2 is also used.Then, after this manipulation, larger values of α could beused in the proposed procedure. By inserting the dimension-less variable X = x/L, the associated governing differentialequation of the dynamic response of the beam can be rewrit-ten from Eq. (1) as follows

∂4w X, tð Þ∂X4 + η1

∂2w X, tð Þ∂t2

= 〠Na

i=1ηi2 sin Ωit

� �δ′ X − Xi

a1� �h

− δ′ X − Xia2

� �i,

ð30Þ

where

η1 =ρhL4

EI, ηi2 = L3bEi

pdi31hip + h

2EI , Xia1 =

xia1L

, Xia2 =

xia2L

:

ð31Þ

Ωi is the harmonic associated to the i-th actuator. Theintroduction of the regularized Dirac-delta derivative (4) inEq. (30) leads to

∂4w X, tð Þ∂X4 + η1

∂2w X, tð Þ∂t2

= 〠Na

i=1ηi2 sin Ωit

� � 1α2

ffiffiffi2

p exp − X − Xia2

� �22α2

!"

× 〠M2

n=0

−14

� �n


pn!H2n

X − Xia2ffiffiffi

2p

α

� �

− exp − X − Xia1

� �22α2

!〠M2

n=0

−14

� �n


pn!H2n

X − Xia1ffiffiffi

2p

α

� �:

ð32Þ

The influence of the regularization value α on the pre-cision and the consistency of the simulation results for twodifferent actuator locations is illustrated in Figures 6 and7. For the comparison, analytical solutions developed inthe appendix A are used, when only one actuator is con-sidered ðNa = 1Þ. The numerical results obtained are pre-sented in two separate cases: (1) the case where thedimensional differential equation of the beam is taken intoaccount (see Eq. (1)), and (2) the case where the dimen-sionless differential equation is examined (see Eq. (32)).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1–0.06

–0.05

–0.04

–0.03

–0.02

–0.01

0

0.01

0.02

Wcd/W

cs

S–S beamC–C beam

t

Ω1 = 𝜋, Ω2 = 2𝜋, Ω3 = 9.8695

Figure 9: Time response for normalized central deflection of S-S and C-C beam excited by three actuators with different harmonics(Ω1,Ω2,Ω3).


Based on Figures 6 and 7, the results, of case (1) are notvery satisfactory in terms of precision and convergence.It can be seen that the numerical results obtained in thecase (2) are more precise than the one obtained in case(1), mainly in terms of numerical convergence. Hence, itshould be noted that when the case (2) is taken intoaccount, better accuracy is achieved.

On the other hand, Figure 8 shows the convergence ofsolutions in accordance with the number of points in the grid(n). It is observed that the results of the proposed methodol-ogy converge uniformly and coincide with the analyticalsolutions.

Now, three actuators with the same parameters excit-ing the beam by various harmonic excitation are consid-ered and the piezoelectric actuators are located atx1a/L = ð0:1,0:3Þ, x2a/L = ð0:4,0:5Þ, and x3a/L = ð0:7,0:9Þ forα = 0:062 and n = 31. As the presented DQM allows con-sidering various boundary conditions, beams with simplysupported (S-S), clamped (C-C), and clamped free (C-F)are investigated. The central displacement wcd for S-Sand C-C beam normalized by the static deflection wcs is

presented in Figure 9 and the obtained 3D plot is shownthe Figure 10 for S-S beam. The time-space responses ofC-C are plotted in Figures 11. Also the numerical resultfor the C-F beam, it is shown in Figures 12 and 13. Fromthe numerical results presented in this section, it can beconcluded that the proposed approach is very well suitablefor the problem of partial differential equations involvingsingular functions like the Dirac-delta function and itsderivatives. These numerical tests demonstrated that theproposed procedure is reliable and accurate for differentboundary conditions.

4.3. Impulse Piezoelectric Actuation. This subsection focuseson the dynamic response of beams under various types ofpiezoelectric impulse excitations. The beam is under a shortduration excitation and the maximum response is reachedin a very short time. The transient response is considered aswell as the permanent one. The used beam parameters arethe same to those described in subsection 4.2 and the usednumerical procedure is based on the DQM and implicit timescheme.

–0.061

–0.05

–0.04

0.8 1

–0.03

–0.02

0.80.6

–0.01

0.6

0

x/L0.4

0.01

0.4

0.02

0.2 0.2

0 0

W(x

,t)/W

cs

t

0.8

Figure 10: Time-space solution of normalized deflection of S-S beam excited by three actuators with different harmonics (Ω1,Ω2,Ω3).


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1t

–0.5

0

0.5

1

1.5

2

2.5

Ω1 = 𝜋, Ω2 = 2𝜋, Ω3 = 9.8695.

Wcd/W

cs

Figure 12: Numerical results for normalized central deflection of C-F beam excited by three actuators with excitation harmonic different.

–0.011

–0.005

0

0.8 1

0.005

0.6 0.8

0.01

0.015

0.60.4

0.02

0.40.20.2

0 0

W(x

,t)/W

cs

t

x/L

0.8

Figure 11: Numerical results 3D for normalized deflection of C-C beam excited by three actuators with excitation harmonic different.


4.3.1. Sinusoidal Piezoelectric Impulse. The piezoelectric actu-ator excites the beam by an impulse that is described by ahalf-cycle sinusoidal load. The voltage function VðtÞ isexpressed as

V tð Þ =V0 sin Ωtð Þ for 0 ≤ t ≤ td ,0 for t > td:

(ð33Þ

td is the time duration and Ω is the excitation frequency.The fundamental period T is T = 2π

ffiffiffiffiffiffiffiffiffiffiffiffiρh/EI

p, and the time

duration td/T is too small (td < <T). Figure 14 shows thenormalized displacement wcd/wcs at the middle of the spanof a simply supported beam with respect to the normal-ized time t/T for Ω = π, td = 1:5T , the parameter of regu-larization α = 0:062, and the location of actuator fixed atxa/L = ð0:6,0:9Þ. These results are obtained by the presentmethod and analytical solution giving in the appendix Bwhere wcs = ðV0bEpL

3d31ðhp + hÞÞ/ð2EIÞ.Figure 14 demonstrates that the results of the proposed

methodology converge uniformly and the results areidentical to analytical ones. It is clearly shown that themaximum amplitude is reached in the time impulseduration (tmax ≃ 0:72T) and the response amplitudewmax/wcs ≃ 9ðwFree/wcsÞ. This large amplitude may dam-age the structure and then can be used to design safe

0 1 2 3 4t/T

–0.04

–0.035

–0.03

–0.025

–0.02

–0.015

–0.01

–0.005

0

0.005

Wcd/W

cs

Numerical solutionExact solution

Figure 14: Central displacement of a simply supported beamsubjected to a sinusoidal piezoelectric impulse for td = 1:5T , Ω =π, xa/L = ð0:6,0:9Þ.

–2

–1

1

0

1

0.8

2

1

3

4

0.6 0.8

5

6

0.6

7

0.4

8

0.40.2 0.2

0 0

W(x

,t)/W

cs

t

x/L

0.8

0.6 0.8

Figure 13: Numerical results 3D for normalized deflection of C-F beam excited by three actuators with excitation harmonic different.


microbeams with piezoelectric patches. The associatedtime-space plot is given in Figure 15 and shows thetime-space response behavior.

4.3.2. Rectangular Piezoelectric Impulse. In this case, thepiezoelectric actuator excites the beam by a rectangularimpulse and the excitation function VðtÞ is expressed as

V tð Þ =V0 for 0 ≤ t ≤ td,

0 for t > td:

(ð34Þ

Figure 16 shows the normalized displacement w/wcsat the middle of the span of a simply supported beamwith respect to the normalized time t/T for Ω = π,parameter of regularization α = 0:062, and the locationof the actuator is fixed at xa/L = ð0:6,0:9Þ the timeand space deflection is shown in Figure 17.

4.3.3. Symmetric Triangular Piezoelectric Impulse. In thiscase, the forcing voltage VðtÞ is expressed as

V tð Þ =

2V0ttd

for t ≤ 0,

2V0 td − tð Þtd

for t > td2 ,

0 for t > td:

8>>>>>><>>>>>>:

ð35Þ

These three steps functions can also be written as a singlefunction

V tð Þ = 2V0ttd

−4V0 t − td/2ð Þ

tdH t −

td2

� �+ 2V0 t − tdð Þ

tdH t − tdð Þ,

ð36Þ

0 1 2 3 4t/T

–0.08

–0.06

–0.04

–0.02

0

0.02

0.04

Wcd/W

cs

Figure 16: Central displacement of a simply supported beamsubjected to rectangular piezoelectric impulse voltage for td =1:5T ,xa/L = ð0:6,0:9Þ.

–0.05

–0.04

0

–0.03

–0.02

0.2

–0.01

0

0.4

0.01

x/L 40.6 3.53

t/T

2.50.8 21.510.51 0

W(x

,t)/W

cs

0.2

0.4

Figure 15: Time-space response of S-S beam subjected to a sinusoidal impulse for td = 1:5T ,xa/L = ð0:6,0:9Þ.


where Hðx − aÞ is the Heaviside function: Hðx − aÞ = 1for x ≤ a and Hðx − aÞ = 0 for x < a. The numerical resultsfor this case are presented in Figure 18 with respect to thenormalized time t/T for parameter of regularization α =0:062 and location of the actuator is fixed at xa/L =ð0:6,0:9Þ. Again, the time-space response is shown inFigure 19.

5. Conclusion

There are many various fields of engineering and physics,whose governing partial differential equations with strongsingularities like the derivative of the Dirac-delta function.For instance, the behavior of structures under piezoelectricpatches can be modelled mathematically by the derivativeof the function Dirac-delta. The direct discretization ofthe derivative of the Dirac-delta function using point dis-cretization techniques like the DQM is not a facile taskand special processing is required. In this work, the regu-larization procedure of the derivative of the Dirac-deltafunction by the distributed functionals using the Hermitepolynomials combined with DQM and a time Implicitscheme was elaborated for the numerical solution of thedynamic behavior of beams with various boundary condi-tions excited by impulse and harmonic piezoelectric actua-tors. The DQM combined with regularization procedurewas elaborated developed to the numerical solution forspace discretization and implicit scheme for time discreti-zation was used. Analytical solutions are also derived forthese problems to validate the proposed approach. The

location of the actuators is described by the derivativesof the Dirac-delta function that are regularized. Theobtained numerical results proved that this proposedmethodology is efficient and appropriate. Its main advan-tage is its simplicity ability to consider an arbitrary num-ber of piezoelectric patches as well its high accuracy.

0 1 2 3 4–0.04

–0.035

–0.03

–0.025

–0.02

–0.015

–0.01

–0.005

0

0.005

t/T

Wcd/W

cs

Figure 18: Central displacement of a simply supported beamsubjected to symmetric triangular piezoelectric impulse for td = 1:5T , xa/L = ð0:6,0:9Þ.

–0.1

–0.08

–0.06

0

–0.04

–0.02

0

0.02

0.04

0.5

43.532.521.51 10.50

W(x

,t)/W

cs

x/L

t/T

Figure 17: Time-space normalized deflection of S-S beam subjected to rectangular piezoelectric impulse voltage for td = 1:5T , xa/L =ð0:6,0:9Þ.


Most importantly, the numerical results reveal that themethodological approach can be used as an efficient toolfor many physical and mechanical phenomena modeledby partial differential equation with strong singular coeffi-cients and excitation.

Appendix

A.Analytical Solution for S-S Beam Exited byOne Piezoelectric Harmonic Actuator

Under the piezoelectric actuator, the motion governing equa-tion can be expressed as

EI∂4w x, tð Þ

∂x4+ ρh


= f x, tð Þ, ðA:1Þ

where

f x, tð Þ = bEpd31hp + h

2 sin Ωtð Þ δ′ x − xa1� �

− δ′ x − xa2� �h i

:

ðA:2Þ

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

w x, tð Þ = 〠∞

n=1f n sin

nπxL

sin Ωtð Þ: ðA:2Þ

Substituting the solution into Equation (A.5), multiply-ing both sides

withÐ L0 sin ðn∗πx/LÞdx and using the mode orthogonal-

ity, namely,

ðL0sin n∗πx

Lsin nπx

Ldx =

12 for n = n∗,

0 for else:

8<: ðA:3Þ

The following algebraic equation is resulted.

EI2

n∗πL

� �4f ∗n −

12 ρhΩ

2 f ∗n

= bEpd31hp + h

2n∗πL

cos n∗πxa2L

− cos n∗πxa1L

� :

ðA:4Þ

Letting n = n∗ one obtains f n as

f n =2γnM0

γ4n − ρhΩ2 cos γnxa2 − cos γnxa1½ �, ðA:5Þ

where γn = nπ/L and M0 = ð1/2ÞbEpd31ðhp + hÞ.

–0.05

–0.04

0

–0.03

–0.02

0.2

–0.01

0

0.4

0.01

0.643.50.8 32.521.511

0.50

.20.4

W(x

,t)/W

cs

x/L

t/T

Figure 19: Time-space normalized deflection of S-S beam subjected to Symmetric triangular piezoelectric impulse for td = 1:5T , xa/L =ð0:6,0:9Þ.


Thus, the displacement response can be obtained bysubstituting Equation (A.5) into Equation (A.2) as follows:

w x, tð Þ = 〠∞

n=1

2γnM0γ4n − ρhΩ2 cos γnxa2 − cos γnxa1½ � sin nπx

LsinΩt:

ðA:6Þ

B.Vibration of S-S Beam Exited by a SinusoidalPiezoelectric Impulse

The motion governing equation of this problem can beexpressed as

EI∂4w x, tð Þ

∂x4+ ρh


=f x, tð Þ for t ≤ td ,0 for t ≥ td ,

(

ðB:1Þ

where

f x, tð Þ = bEpd31hp + h

2 V0 sin Ωtð Þ δ′ x − xa1� �

− δ′ x − xa2� �h i

:

ðB:2Þ

The analysis is organized in two phases.

B.1.Phase I. In this phase t ∈ ½0, td� the excitation is harmonicand the response includes the transient and permanent solu-tions, and is given by equation (A.6)

w x, tð Þ = 〠∞

n=1

2γnM0γ4n − ρhΩ2 cos γnxa2 − cos γnxa1½ �

� sin nπxL

sinΩt for t ≤ td:

ðB:3Þ

B.2.Phase II. During this phase, the system starting at t = td ,the system is in free vibration with initial conditions qðtdÞand _qðtdÞ at the end of phase I.

Assuming the solution in the form of mode superposi-tion, the transverse deflection of a free simply supportedbeam can be written as

w x, tð Þ = 〠∞

n=1sin nπx

Lqn tð Þ for t ≥ td: ðB:4Þ

Substituting the solution into equation (B.10) for phaseII, we get

€qn tð Þ + ω2nqn tð Þ = 0, ðB:5Þ

where

ωn =nπL

� �2 ffiffiffiffiffiffiEIρh

s: ðB:6Þ

The general solution of equation (B.12) can be written as:

qn tð Þ = A cos ωn t − tdð Þ + B sin ωn t − tdð Þ, ðB:7Þ

_qn tð Þ = −ωnA sin ωn t − tdð Þ + ωnB sin ωn t − tdð Þ: ðB:8ÞThe constants of integration A and B can be determined

knowing the initial displacement qðtdÞ and the initial velocity_qðtdÞ at time t = td . Substituting these initial conditions intoequation (B.7) and equation (B.8) one gets

A = q tdð Þ, B = _q tdð Þωn

: ðB:9Þ

Equation (B.7) can then be written as

qn tð Þ = q tdð Þ cos ωn t − tdð Þ + _q tdð Þωn

sin ωn t − tdð Þ: ðB:10Þ

Thus, the free vibration displacement of beam for t ≥ tdcan be obtained by substituting Equation (B.10) into Equa-tion (B.4):

w x, tð Þ = 〠∞

n=1Mn sinΩtd cos ωn t − tdð Þð

+ Ω

ωncosΩtd sin ωn t − tdð Þ

�sin nπx

L,

ðB:11Þ

where

Mn =2γnM0

γ4n − ρhΩ2 cos γnxa2 − cos γnxa1½ �: ðB:12Þ

Data Availability

The code source data used to support the findings of thisstudy are available from the corresponding author uponrequest.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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

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ANewNumericalProcedureforVibrationAnalysisofBeamunder ...downloads.hindawi.com/journals/jam/2020/7391848.pdf · Yassin Belkourchia 1 and Lahcen Azrar1,2 1Research Center STIS, M2CS,