#### Abstract

In this paper, dynamic response analysis of a forced fractional viscoelastic beam under moving external load is studied. The beauty of this study is that the effect of values of fractional order, the effect of internal damping, and the effect of intensity value of the moving force load on the dynamic response of the beam are analyzed. Constitutive equations for fractional order viscoelastic beam are constructed in the manner of Euler–Bernoulli beam theory. Solution of the fractional beam system is obtained by using Bernoulli collocation method. Obtained results are presented in the tables and graphical forms for two different beam systems, which are polybutadiene beam and butyl B252 beam.

#### 1. Introduction

For theoretical and experimental review about the fractional Euler–Bernoulli beams, please see . Specifically, in the present paper, displacement analysis of a forced fractional viscoelastic beam is studied. External moving force load perfectly moves on the beam with the velocity from the left edge to the right edge of the beam. The solution of the fractional beam system is obtained by means of Bernoulli collocation method. The main advantage of the Bernoulli collocation method is that employing the Bernoulli polynomials is easier than Chebyshev, Bessel polynomials, and Haar wavelets . These advantages of Bernoulli polynomials provide us for obtaining the solution by making less computational process in shorter time. In the step of employing the Bernoulli collocation method, some external moving force loads having different load intensities are considered and also the effects of internal damping and fractional order of the derivative are searched for a fractional beam system. In the simulations, two different beam systems, which are polybutadiene beam and butyl B252 beam, are taken into account for being compared each other in the aspects of internal damping effects and resistance to effect of external moving force. Comparison results of the beam systems are presented in tables and graphics. The rest of the paper is organized as follows: in the next section, definition of the displacement analysis problem for a fractional viscoelastic beam is presented and scheme of the beam is overviewed. In the third section, short definition of the fractional derivative in the Caputo sense is introduced. In the fourth section, Bernoulli collocation method is explained and adopted to the present problem. In the fifth section, obtained results are given and discussions are made in the light of employing the Bernoulli collocation method to fractional viscoelastic beam system.

#### 2. Definition of the Problem

The motion equation of the fractional viscoelastic homogeneous beam is obtained by considering the Euler–Bernoulli beam theory by ignoring shear deformation factor and rotary inertia of the beam. The beam is considered as a uniform viscoelastic beam and mechanical energy dissipation inside the beam is modeled by fractional order differential equations. By taking into account the , stress-strain constitutive relation of a fractional viscoelastic beam is given as follows:in which is the Young’s modulus of the viscoelastic beam, is the damping coefficient, and is the fractional derivative operator with the order with respect to . The simply supported viscoelastic beam initially is at rest and nondeformed. The beam is subjected to a horizontally moving constant force load with the velocity from the left edge to right edge of the beam, respect to axis. In the light of , let us introduce the formulation of a fractional viscoelastic beam structure illustrated in Figure 1.in which is the deflection of the viscoelastic beam in , is the time variable, is the final time observed duration, is the space variable, is the length of the viscoelastic beam, is the cross-section area of the structure, is the material mass density of the viscoelastic beam, is the axial moment of inertia of the beam, is a constant showing intensity of the external moving force load, is the Dirac-delta function, and is the velocity of the moving force load with the condition . Equation (2) is subjected to the following boundary conditions:and the following initial conditions:in which . means to square-integrable functions space in the manner of Hilbert in the domain in the Lebesgue sense with the following norm and inner product:

Let us assume that

After substituting the equations (6) into (2) and multiplying both sides of equation (2) with , integrating on , we obtain the following ordinary differential equation as follows:

Equation (7) is subjected to the following initial conditions:

#### 3. The Fractional Derivative in the Caputo Sense

Definition. The Caputo definition of the fractional-order derivative iswhere is the order of the derivative and is the smallest integer greater than . For the Caputo derivative, we have

#### 4. Bernoulli Collocation Method

The recurrence relation of the Bernoulli polynomials is defined by

For , . The first few Bernoulli polynomials are

Our goal is to get the approximate solution as the truncated Bernoulli series defined bywhere denotes the Bernoulli polynomials; are the unknown coefficients for Bernoulli polynomial, and is any positive integer which possess . Let us assume that linear combination of Bernoulli polynomials equation (14) is an approximate solution of equation (7). Our purpose is to determine the matrix forms of equation (7) by using (14). Firstly, we can write Bernoulli polynomials (12) in the matrix formwhere , , , and

The matrix form of equation (14) by a truncated Bernoulli series is given by

By using equations (15) and (17), the matrix relation is expressed aswhere

By using equation (18), we obtain the following relation:

By substituting the Bernoulli collocation points given byinto equation (21), we obtainand the compact form of the relation (23) becomes

In this way, the unknown Bernoulli coefficients are obtained by solving the system. Then, these coefficients are substituted into (14), and the approximate solution is obtained. For more details, see .

#### 6. Conclusion

In this study, the Bernoulli collocation method as a new solution method for obtaining the approximate solution of a fractional viscoelastic beam model subjected to moving force load is employed. Dynamic response analysis of the fractional viscoelastic beam model is investigated for two different specific beams: polybutadiene beam and butyl B252 beam. Displacement analysis of a point on the fractional viscoelastic beams is studied for different moving force loads and also effect of the internal damping to displacement is observed for different internal damping coefficients. Moreover, dynamic response of the fractional viscoelastic beam is examined for different values of the fractional order. Obtained results are presented in tables and graphics and results reveal that Bernoulli collocation method is very effective and powerful solution method for obtaining the solution of fractional order viscoelastic beam models. After observing Figures 2 and 3, it is easy to conclude that as the moving force load increases, the displacement of a point on the beams also increases. Also, numerical results, presented in Tables 14, show that under the same moving force load with the same internal damping effect, the displacement of a point on the polybutadiene beam is greater than that corresponding to butyl B252 beam. Moreover, under the same moving force load, changes in the displacements of a point on the beams are examined in the aspect of different internal damping effects and observations made clear that butyl B252 beam better reflects the effect of internal damping to displacement of a point on the beam. By comparing polybutadiene beam and butyl B252 beam, it is concluded that polybutadiene beam is more open to destructive effects of vibrations under the same conditions with the butyl B252 beam.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

The authors completed this study and wrote and approved the final version of the manuscript.