# Dynamic Response Analysis of a Forced Fractional Viscoelastic Beam∗

Authors: Kenan Yildirim; Sertan Alkan
Journal: Journal of Mathematics
 (2021)

---

## 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.


---

## Body

## 1. Introduction

Theory and applications of beams are very important research area due to its wide usage areas in applied sciences. Especially after starting the space adventure of the mankind, the demand to more resistant structures has great importance. Beams are generally modeled based on Euler–Bernoulli beam theory, which is called classical beam theory. The background of beam theory goes on Newton’s second law and some different aspects of beams, such as modeling, analysis of bending-buckling, and reinforcement and control, are hot topics of research papers since the beginning of the nineteenth century. The books can provide a general overview about the Euler–Bernoulli beam theory, please see [1–3]. Some important studies related to beams modeled in the sense of classical beam theory are also summarized as follows, but not limited to [4–15]. The beam systems in [1–15] have the integer order derivatives of the state function. In the beginning of 1930s, fractional derivative was introduced for describing the constitutive relation of some beam materials [16], and after 1980s, since fractional order equations have good memory and can be used to describe material properties more accurately with fewer parameters, they are considered to be good mathematical models for describing the dynamic mechanical behavior of materials [17]. In [18], the dynamic behavior of the thin plates resting on a fractionally damped viscoelastic foundation subjected to a moving point load is investigated and results show that the damping of the foundation system increases with increasing the order of the fractional derivative, which leads to a decrease in the dynamic response. In [19], the dynamic response spectra of fractionally damped viscoelastic beams subjected to concentrated moving load are presented and results reveal that with an increase in the order of the fractional derivative, the system damping of the system increases and the dynamic amplification factor (DAF) decreases, especially in the dynamic zone of the sweep parameter. In [20], the precise integration method (PIM) is extended to numerically integrate the equation of motion with fractional terms, which offers high accuracy and obtained numerical results indicate the viscoelastic dampers can enhance the seismic performance of structures significantly. In [21], the nonstationary free vibration and nonlinear dynamic behavior of the viscoelastic nanoplates are analyzed. Obtained results show that the viscoelastic model-based vibration is nonstationary unlike the elastic model. Moreover, the damping mechanism of the viscoelasticity is amplitude dependent and the contribution of the viscoelastic damping terms at higher forcing conditions becomes noticeable. On the other hand, several numerical methods are developed and employed for better analyzing the fractional mechanical systems. Widely used methods for fractional systems are finite element method [22], Galerkin method [23], variational iteration method [24], and multiscale method [25, 26]. Especially, papers existing in the literature, which include a solution method for analyzing the dynamic response of a fractional order beam system, can be shortly listed as [19, 27–30]. In [19], the authors combined Galerkin method and Newton–Raphson method for analyzing the vibration of a fractional beam equation and they compared the results for only seeing the effects of fractional or integer derivatives. In [27], the author considered the dynamic response analyzing of a fractional order viscoelastic beam by means of green function method. In [27], the author only compared the results based on changes on the fractional derivative between 0,1. In [28], the authors employed the Adomian decomposition method for solving a fractional beam equation and they only observed the effect of the order of fractional derivative. In [29], the authors used the dynamic green function method for analyzing the dynamic response in a fractional beam equation and the beam equation does not include the damping term. Results are simulated for only indicating the effects of order of fractional derivative. In [30], the author employed the green function method for a fractional viscoelastic beam system subjected to a base excitation. After obtaining the solution, the author compared the results corresponding to different fractional order derivative. By comparing the present study with the studies existing in the literature, objectives of the present study are expressed as follows:(i)
In this paper, Bernoulli collocation method is firstly employed for analyzing the fractional viscoelastic beam equation. In the literature, especially for the fractional beam systems, green function method, Galerkin method, Newton–Raphson method, Adomian decomposition method, and Bernoulli collocation method in this paper were used, but by comparing these five methods, it is clear that Bernoulli collocation method is new and has less computational process and less work.(ii)
In the literature, the authors only considered and discussed the effects of order of fractional derivative on the dynamic response. But, we discussed both the effects of the order of fractional order derivative and the effects of damping coefficient term and the effect of density of moving force load. So, it is said that the present study has wider perspective than other studies.(iii)
Also, in the literature, results are obtained for one beam system. In this paper, effects of order of fractional derivative, effects of damping coefficient term, and the effect of density of moving force load are observed and compared for two different beam systems which are polybutadiene beam and butyl B252 beam.For theoretical and experimental review about the fractional Euler–Bernoulli beams, please see [31]. 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 vt 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 [32–34]. 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 [35], stress-strain constitutive relation of a fractional viscoelastic beam is given as follows:(1)σ=Eεt+Eγ′Dtγεt=Eε+μγdγεtdtγ,in which E is the Young’s modulus of the viscoelastic beam, μγ is the damping coefficient, and Dtγ is the fractional derivative operator with the order γ with respect to t. 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 vt from the left edge to right edge of the beam, respect to x axis. In the light of [27], let us introduce the formulation of a fractional viscoelastic beam structure illustrated in Figure 1.(2)Aρ∂2wt,x∂t2+EIμγdγdtγ∂4wt,x∂x4+EI∂4wt,x∂x4=Pδx−vt,in which w is the deflection of the viscoelastic beam in C=t,x:t∈0,tf,x∈0,ℓ, t is the time variable, tf is the final time observed duration, x is the space variable, ℓ is the length of the viscoelastic beam, A is the cross-section area of the structure, ρ is the material mass density of the viscoelastic beam, I is the axial moment of inertia of the beam, P is a constant showing intensity of the external moving force load, δ is the Dirac-delta function, and vt is the velocity of the moving force load with the condition 0≤vt≤ℓ. Equation (2) is subjected to the following boundary conditions:(3)wt,x=0,wxxt,x=0atx=0,ℓ,and the following initial conditions:(4)wt,x=w0x,wtt,x=w1xatt=0,in which w0x∈H10,ℓ=w0x∈L20,ℓ:∂w0x/∂x∈L20,ℓ,w1x∈L20,ℓ. L2C means to square-integrable functions space in the manner of Hilbert in the domain C in the Lebesgue sense with the following norm and inner product:(5)η2=<η,η>,<η,ρ>C=∫CρηdC.Figure 1
Schematic of the viscoelastic beam under moving force loadP with the velocity vt.Let us assume that(6)wt,x=∑n=1Nznt2sinnπxℓ.After substituting the equations (6) into (2) and multiplying both sides of equation (2) with 2sinnπx/ℓ, integrating on 0,ℓ, we obtain the following ordinary differential equation as follows:(7)Aρzn″t+EInπ4μγdγdtγznt+EInπ4znt=P2sinnπvtℓ,n=1,…,N.Equation (7) is subjected to the following initial conditions:(8)zn0=2∫0ℓw0xsinnπxℓdx,zn′0=2∫0ℓw1xsinnπxℓdx.


## 3. The Fractional Derivative in the Caputo Sense

Definition. The Caputo definition of the fractional-order derivative is(9)Dγfx=1Γn−γ∫0xfntx−tγ+1−ndt,n−1<γ≤n,n∈ℕ,where γ>0 is the order of the derivative and n is the smallest integer greater than γ. For the Caputo derivative, we have(10)DγC=0,Cis constant,Dγxq=0,forq∈ℕ0andq<γ,Γq+1Γq+1−γxq−γ,forq∈ℕ0andq≥γorq∉ℕandq>γ.


## 4. Bernoulli Collocation Method

The recurrence relation of the Bernoulli polynomials is defined by(11)Bnx=2xBn−1x+Bn−2x.Forn≥3., B1x=1,B2x=2x. The first few Bernoulli polynomials are(12)B1x=1,(13)B2x=x−12,B3x=x2−x−16,B4x=x3−32x2+x2.⋮Our goal is to get the approximate solution as the truncated Bernoulli series defined by(14)yx=∑n=1N+1cnBnx,where Bnx denotes the Bernoulli polynomials; cn1≤n≤N+1 are the unknown coefficients for Bernoulli polynomial, and N is any positive integer which possess N≥m. 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 form(15)Bx=TxM,where Bx=B1xB2x⋯BN+1x, Tx=1xx2x3…xN, C=c1c2⋯cN+1T, and(16)M=1−12160−13001420−13001−1120−160160001−3210−120230001−2530−76000001−52520−73000001−37200000001−7214300000001−4000000001.The matrix form of equation (14) by a truncated Bernoulli series is given by(17)yx=BxC.By using equations (15) and (17), the matrix relation is expressed as(18)yx≅yNx=TxMC,yγx≅yNγx=TxXγxDγMC,y′′x≅yN′′x=TxD2MC,where(19)Xγx=0,x1−γ,x2−γ,…,xN−γ,(20)D=010000⋯0002000⋯0000300⋯0000040⋯0000005⋯0000000⋯0⋮⋮⋮⋮⋮⋱⋱N000000⋯0,D0=100000⋯0010000⋯0001000⋯0000100⋯0000010⋯0000001⋯0⋮⋮⋮⋮⋮⋱⋱0000000⋯1,T=Tx0Tx1⋮TxN=1x0…x0N1x1…x1N1⋮…⋮1xN…xNN,Dγ=000⋯00Γ2Γ2−γ0⋯000Γ3Γ3−γ⋯0⋮⋮⋮⋱⋮000⋯ΓNΓN−γ.By using equation (18), we obtain the following relation:(21)Ykx=TxDkMC.By substituting the Bernoulli collocation points given by(22)xi=a+b−aiN,i=0,1,…,N,into equation (21), we obtain(23)Ykxi=TxiDkMC,k=0,γ,2.and the compact form of the relation (23) becomes(24)Yk=TDkMC,k=0,γ,2.In this way, the unknown Bernoulli coefficientscn,n=1,2,…,N+1 are obtained by solving the system. Then, these coefficients are substituted into (14), and the approximate solution is obtained. For more details, see [36].


## 5. Simulation Results and Discussion

Bernoulli collocation method for obtaining the solution of fractional viscoelastic beam equation is employed. Hence, displacement analysis of a forced fractional viscoelastic beam is investigated by taking into account the different moving force loads, different values of internal damping coefficient, and different values of fractional order of derivative. Obtained results are simulated and presented in the tables and graphical forms. The velocity, from left to right, of the external moving force on beamvt is considered as sinπt. In order to observe the dynamic response of the viscoelastic beams under the different intensity of external moving force, the intensity constant of the external moving force load on the beam is involved to computation as P=1,25,50. Also, the values in Tables 1–6 are computed on x=0.5, which is the middle point of the fractional viscoelastic beams. Observed duration of time is tf=1. In the first case, forced displacement analysis of a polybutadiene beam is observed for different values of moving force load and results are presented in Table 1. The length and material density of the fractional viscoelastic beam are taken into account as ℓ=1m and ρ=160kg/m3, respectively. The cross-sectional area A is 0.72m2, moment of inertia J is 0.14/12, and Young’s modulus E is 8.15×105 for a fractional viscoelastic polybutadiene beam. Also, the order of fractional derivative γ is evaluated as 0.528 for the results in Figure 2 and Tables 1 and 3. By observing Figure 2, it is concluded that while the intensity of the external moving load force increases, namely, P is 1 to 25 and 50, the displacement of the fractional viscoelastic polybutadiene beam also increases. Also, parallel observation results to Figure 2 are obtained by taking into account Table 1. For example, on the moment t=0.5, the amount of the displacement of the polybutadiene beam is measured as 0.001 1 for P=1, 0.028 for P=25, and 0.056 for P=50. This observation is valid the entire time interval t=0,…,1 for polybutadiene beam. Also, the effect of internal damping on the displacement is presented in Table 3 for polybutadiene beam. The internal damping coefficient is evaluated from 0.2 to 1 and by examining Table 3; it reveals that when internal damping coefficient increases, the displacement of the polybutadiene beam decreases under the same conditions. The effect of the fractional order to system is observed from Table 5 and it can be concluded that while increasing the values of the fractional derivative, the value of the displacement is decreasing. In the second case, a butyl B252 beam is taken into account by the coefficients; the cross-sectional area A is 0.72m2, moment of inertia J is 0.14/12, and Young’s modulus E is 1.05×106. The order of fractional derivative γ is considered as 0.519 for Figure 3 and Tables 2 and 4. By checking Figure 3, it is easy to see that displacements corresponding to much bigger intensity of moving force load are much bigger. For example, on the moment t=0.5, while P=1 to P=25,50, corresponding displacements are calculated as 0.00089, 0.022, and 0.044, respectively. This is effective along the observation duration. In Table 4, some results related to the effect of internal damping are presented and internal damping coefficient is included in the computation as 0.2 to 1. After looking at Table 4, it is concluded that while internal damping coefficient decreases, the displacement of the butyl B252 beam increases and relation between the effects of internal damping and displacements is inversely proportional. The relation between the displacement and fractional order in the system is vice versa. As understood from Table 6, while decreasing the values of the fractional derivative, the value of the displacement is increasing. These observation results of the present study are also compatible with the results existing in the literature. By taking into account Tables 1–6 and Figures 2 and 3 and comparing these two kinds of fractional viscoelastic beams, it is seen that the polybutadiene beam has more greater displacements than butyl B252 beam under same conditions. Also, the effect of internal damping coefficient is more visible on the butyl B252 beam according to polybutadiene beam. These observations make clear that butyl B252 beam is stronger and preferable than the polybutadiene beam.Table 1
Some values ofwt,x for P=1,25,50 (for a polybutadiene beam).
twP=1wP=25wP=500.10.000 025 20.000 632 00.001 264 00.20.000 176 10.004 402 50.008 805 00.30.000 463 70.011 593 40.023 186 80.40.000 805 80.020 145 20.040 290 40.50.001 115 00.027 877 20.055 754 50.60.001 350 90.033 774 50.067 549 00.70.001 529 40.038 236 10.076 472 10.80.001 709 40.042 735 60.085 471 20.90.001 933 00.048 326 20.096 652 51.00.002 026 30.050 657 40.101 315 0Table 2
Some values ofwt,x for P=1,25,50 (for a butyl B252 beam).
twP=1wP=25wP=500.10.000 020 20.000 505 30.001 010 70.20.000 140 70.003 517 70.007 035 40.30.000 370 10.009 253 10.018 506 40.40.000 642 10.016 053 00.032 106 10.50.000 886 50.022 164 50.044 328 90.60.001 070 90.026 772 40.053 544 80.70.001 207 90.030 198 10.060 396 20.80.001 345 00.033 624 90.067 249 80.90.001 515 90.037 899 20.075 798 41.00.001 581 70.039 544 70.079 089 3Table 3
Some values ofwt,x for different values of μ for P=1 (for a polybutadiene beam).
tμ0.20.40.60.81.00.10.000 024 900.000 024 240.000 023 620.000 023 030.000 022 480.20.000 170 500.000 161 040.000 152 420.000 144 540.000 137 320.30.000 440 420.000 402 520.000 369 540.000 340 700.000 315 370.40.000 748 720.000 659 750.000 586 200.000 524 910.000 473 410.50.001 011 320.000 856 730.000 735 830.000 640 020.000 563 110.60.001 194 990.000 973 520.000 810 050.000 687 000.000 592 560.70.001 323 260.001 044 560.000 850 350.000 711 060.000 608 300.80.001 459 320.001 136 460.000 922 480.000 774 520.000 667 970.90.001 645 410.001 287 750.001 058 630.000 902 750.000 790 711.00.001 713 620.001 340 710.001 112 910.000 964 620.000 862 85Table 4
Some values ofwt,x for different values of μ for P=1 (for a butyl B252 beam).
tμ0.20.40.60.81.00.10.000 019 900.000 019 350.000 018 850.000 018 370.000 017 910.20.000 136 090.000 128 330.000 121 260.000 114 820.000 108 940.30.000 350 960.000 319 920.000 293 020.000 269 580.000 249 070.40.000 595 320.000 522 710.000 463 020.000 413 520.000 372 120.50.000 801 770.000 676 170.000 578 620.000 501 790.000 440 450.60.000 943 910.000 764 930.000 633 940.000 536 040.000 461 360.70.001 040 910.000 817 110.000 662 680.000 552 790.000 472 220.80.001 459 320.000 886 190.000 717 380.000 601 550.000 667 970.90.001 285 910.001 002 830.000 823 340.000 701 970.000 518 571.00.001 333 740.001 041 330.000 864 750.000 750 700.000 615 00Table 5
Some values ofwt,x for different values of γ for P=1 (for a polybutadiene beam).
tγ0.20.40.60.81.00.10.000 025 280.000 025 280.000 025 280.000 025 270.000 025 270.20.000 176 110.000 176 100.000 176 090.000 176 070.000 176 040.30.000 463 790.000 463 760.000 463 710.000 463 630.000 463 490.40.000 805 960.000 805 880.000 805 750.000 805 560.000 805 260.50.001 115 390.001 115 230.001 114 990.001 114 650.001 114 160.60.001 351 460.001 351 200.001 350 840.001 350 340.001 349 690.70.001 530 100.001 529 730.001 529 250.001 528 640.001 527 880.80.001 710 240.001 709 780.001 709 200.001 708 520.001 707 720.90.001 933 990.001 933 450.001 932 810.001 932 090.000 790 711.00.002 027 310.002 026 710.002 026 060.002 025 380.001 931 32Table 6
Some values ofwt,x for different values of γ for P=1 (for a butyl B252 beam).
tγ0.20.40.60.81.00.10.000 020 210.000 020 210.000 020 210.000 020 210.000 020 210.20.000 140 710.000 140 710.000 140 700.000 140 690.000 140 660.30.000 370 160.000 370 140.000 370 110.000 370 060.000 369 970.40.000 642 210.000 642 160.000 642 080.000 641 960.000 641 770.50.000 886 760.000 886 660.000 886 510.000 886 290.000 885 980.60.001 071 190.001 071 020.001 070 790.001 070 480.001 070 070.70.001 208 330.001 208 090.001 207 790.001 207 410.001 206 930.80.001 345 490.001 345 200.001 344 840.001 344 410.001 343 920.90.001 516 530.001 516 200.001 515 800.001 515 360.001 514 891.00.001 582 390.001 582 020.001 581 620.001 581 210.001 580 82Figure 2
Displacements of a polybutadiene beam forP=1,25,50.
(a)(b)(c)Figure 3
Displacements of a butyl B252 beam forP=1,25,50.
(a)(b)(c)


## 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 Figures2 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 1–4, 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.



---

*Source: 3920937-2021-12-15.xml*