# Numerical Vibration Displacement Solutions of Fractional Drawing Self-Excited Vibration Model Based on Fractional Legendre Functions

Authors: Jiaquan Xie; Yongjiang Zheng; Zhongkai Ren; Tao Wang; Guangxian Shen
Journal: Complexity
 (2019)

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## Abstract

In practice, due to the fact that the phenomenon of drawing self-excited vibration can be deemed as one of the hunting phenomena of the mechanical system, this study focuses on investigating the drawing self-excited vibration process through proposing the fractional differential equation model of hunting phenomenon of the mechanical system. The fractional Legendre functions together with their fractional differential operational matrices are used to numerically solve the model. In this way, the numerical solutions of vibration displacement of the model are obtained. At the end, the proposed model and algorithm are proved to be effective via analyzing the numerical results and phase position.


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## Body

## 1. Introduction

The self-excitation vibration exists in the natural and engineering field, which requires no external force or external action to change the structural parameters of the system but relies on the interaction of various components within the system to maintain the steady-state periodic motion [1–8]. Some self-excited oscillations are very complicated. For example, the human circulatory system is an extremely complex self-excited vibration system. The heart vibrates at a certain frequency and intensity to maintain blood flow in the blood vessels, which is a typical self-excited vibration phenomenon.The hunting phenomenon of the mechanical system exists in a low-speed drive system with strong friction force and frequently occurs in unstable and unevenly stopped motion, which sometimes is called as stick-slip, or slip. The hunting phenomenon is the main reason for the self-excited vibration of the mechanical system. Once the hunting phenomenon occurs, the mechanical transmission becomes unstable so as to be impossible to achieve accurate measurement and precision machining. To avoid the hunting phenomenon, the theory analysis and experiment research should be carried out. In order to obtain quantitative results, a simplified mechanical model with a single degree of freedom and a quantitative description of the friction force should be established.The steel strand drawing technology has been investigated for hundreds of years [9]. In the 12th century, there were blacksmiths and drawers. In the middle of the 13th century, Germany first produced the hydraulic drawing machine to promote in the world, which ushered in a new era of mechanization. For drawing machine with wire processing, when the surface reduction rate, drawing speed, and lubrication parameter are improper and exceed a reasonable range, there will be a hunting or jumping phenomenon. For example, the surface quality of the wire is not smooth, and the defect leads to the flutter accident of the puller.Hao et al. [10] carried out an in-depth theoretical analysis on the mechanism and conditions of chattering grains generated in the process of cold drawing of tapered short mandrel steel tubes. Yang [11] used the finite element method to analyze the vibration and self-excited vibration of the four-high rolling mill. Lu et al. [12] studied the phenomenon of self-excited vibration of 1420 mm cold mill, established the mathematical model of the roll and solved it with the method of multiscale perturbation, and proposed a method to solve the self-excited vibration. Erdbrink and Krzhizhanovskaya [13] used the differential evolution method to analyze the second-order time signal of the self-excited vibration equation. Aarsnes and Aamo [14] predicted the occurrence of self-excited vibrations during drilling using an infinite dimensional model. Zhang et al. [15] applied a nonlinear friction work formulation for the analysis of self-excited vibrations.Recently, fractional partial differential equations have been widely applied in various fields of science and engineering [16–18], and wave equation is one of the most important models of them. Over the past decade, the modelling and control of several classes of wave process have been researched a hot point [19, 20]. In [21], Jiang et al. analyzed the boundary control strategy for a fractional wave equation with external disturbances. In [22], the numerical approximation of the time fractional diffusion-wave equation in a semi-infinite channel by using the finite difference method and Laguerre–Legendre spectral method is carried out. In [23], time fractional diffusion-wave equation with damping has been solved by using the method of separation of variables.In this study, in order to more accurately describe the whole process of drawing self-excited vibration, a fractional differential equation model is proposed, and the fractional-order Legendre functions are utilized to obtain the numerical vibration displacement solutions.


## 2. Fractional Dynamic Model of Mechanical System Hunting Phenomenon

Although some studies considered mechanical system hunting phenomenon [24], to our best knowledge, the fractional-order dynamic model to characterize hunting phenomenon of mechanical system is first presented, as shown in Figure 1. The diagram of drawing plastic working [25] is depicted in Figure 2. Figure 2 shows the deformation of drawing long parts from coarse to fine in the drawing process.Figure 1
The diagram of fractional dynamic model of mechanical system hunting phenomenon.Figure 2
The diagram of drawing plastic working.
(a)
(b)The fractional differential equation of mechanical system hunting phenomenon is given as follows:(1)mu¨+cDtαu−Dtαu0−φu˙+ku−u0=0,0<α≤1,where Dtα denotes the fractional differential operator defined in the Caputo sense [26, 27], m is the quality of the slider, c is the fractional damping coefficient, k is the spring stiffness coefficient, u0 is the displacement at drive end, u is the slider displacement, and φu˙ is the frictional force.If the slider is driven at constant speed, setting the speed of the driving end asv, then the displacement u0 of the driving end T can be expressed as(2)u0=vt.Substituting equation (2) into Equation (1), we get(3)mu¨+cDtαu−vDtαt−φu˙+ku−u0=0.Converting the slider displacementu into the relation of relative displacement x1 as(4)u=x1+u0=x1+vt,then we get(5)mx¨1+cDtαx1+kx1−φx˙1+v=0.Introducing the dimensionless timeτ and defining the dimensionless displacement x and damping ratio ξ:(6)τ=ω0tω0=km,x=ω0x1v,ξ=c2km,the sliding block motion equation can be simplified as(7)x″+2ξDαx+x−φx′+1vmω0v=0,where x′,x″,andDαx are the first, second, and α derivatives of x with respect to τ.The Karnopp friction model is employed to describe frictionφx′+1v. The difference between static and dynamic friction is(8)ΔF=Fs−Fk=ΔμNΔμ=μs−μk.The dimensionless dynamic friction coefficientf and dimensionless static-dynamic friction drop d are defined as follows:(9)f=μkNmω0v,d=ΔμNmω0v.The sliding block motion equation is simplified again as(10)x″+2ξDαx+x=−fsgnx′+1,where sgnx is the sign function, namely,(11)sgnx=1,x>0,0,x=0,−1,x<0.Segmentation analysis:(a)
Sliding section before rolling:(12)x″+2ξDαx+x=f,0<α≤1.(b)
Sliding section after rolling:(13)x″+2ξDαx+x=−f,0<α≤1.From equations (12) and (13), the general form of fractional differential equation model of mechanical system hunting phenomenon is established:(14)y″x+2ξDxαyx+yx=fx,0<α≤1,where ξ are constants. With the initial conditions,(15)y0=δ1,y′0=δ2.


## 3. Computing Method

### 3.1. Definitions

The generalized fractional-order Legendre functions (GFLFs), devoted byFlihαt, is defined by introducing the change of variable t=xh. Then, the GFLFs have recurrence formula as follows [28]:(16)Fli+1hαt=2i+12t/hα−1i+1Flihαt−ii+1Fli−1hαt,i=1,2,…,where Fl0hαt=1andFl1hαt=2t/hα−1.The analytical form of the GFLFsFlihαx of degree iα is given by(17)Flihαx=∑s=0ibs,itsαhsα,i=1,2,….


### 3.2. Functions Approximation

Supposeyx∈L20,h, it can be expanded in terms of the GFLFs as follows [29]:(18)yx=∑i=0∞ciFlihαx,where ci are obtained by(19)ci=α2i+1hα∫01Flihαxfxωlαdx,i=1,2,…,where the weight function ωlαx=xα−1.If we consider truncated series in equation (13), we obtain(20)yx≈ymx=∑i=0m−1ciFlihαx=CTΦx,where C=c0,c1,…,cm−1TandΦx=Fl0hαx,Fl1hαx,…,Flm−1hαx.


### 3.3. The GFLF Fractional-Order Operational Matrix of Derivative

The derivative of the function vectorΦx can be approximated as follows:(21)DαΦx=PαΦx,where Pα is called the GFLF operational matrix of the derivative.Theorem 1.
SupposePα is the m×m GFLF operational matrix of Caputo fractional derivatives of order α, β>α/2, β∉ℕ, then the elements of Pα are given as follows [30]:(22)piji,j=1m−1,m−1=2j+1βh−α∑s=0i∑r=0jbr,jbs,i′Γsβ+1Γsβ−α+11s+r+1β−α,where bs,i′=0,sβ∈N0andsβ<α,bs,i′=bs,i,sβ∉N0andsβ≥αorsβ∈N0andsβ≥α.


### 3.4. Solving Process

Substituting equations (18) and (21) into equation (14), we have(23)y″x≈CTd2Φxdx2≈CTP2Φx,(24)Dxαyx≈CTdαΦxdxα≈CTPαΦx,(25)fx≈FTΦx,where the coefficient matrices F can be obtained by equation (19). Substituting equations (23)–(25) and (18) into equation (14), we get(26)CTP2Φx+2ξCTPαΦx+CTΦx=FTΦx.Simplifying equation (26), we obtain(27)CTP2+2ξCTPα+CT=FT.For equation (15), we have(28)CTΦ0=δ1,CTPΦ0=δ2.Equation (27) together with equation (28) constitutes a system of linear algebraic equations. These unknown coefficients ci can be obtained by solving the system. A flowchart of the algorithm is given in Figure 3.Figure 3
Algorithm flowchart.


## 3.1. Definitions

The generalized fractional-order Legendre functions (GFLFs), devoted byFlihαt, is defined by introducing the change of variable t=xh. Then, the GFLFs have recurrence formula as follows [28]:(16)Fli+1hαt=2i+12t/hα−1i+1Flihαt−ii+1Fli−1hαt,i=1,2,…,where Fl0hαt=1andFl1hαt=2t/hα−1.The analytical form of the GFLFsFlihαx of degree iα is given by(17)Flihαx=∑s=0ibs,itsαhsα,i=1,2,….


## 3.2. Functions Approximation

Supposeyx∈L20,h, it can be expanded in terms of the GFLFs as follows [29]:(18)yx=∑i=0∞ciFlihαx,where ci are obtained by(19)ci=α2i+1hα∫01Flihαxfxωlαdx,i=1,2,…,where the weight function ωlαx=xα−1.If we consider truncated series in equation (13), we obtain(20)yx≈ymx=∑i=0m−1ciFlihαx=CTΦx,where C=c0,c1,…,cm−1TandΦx=Fl0hαx,Fl1hαx,…,Flm−1hαx.


## 3.3. The GFLF Fractional-Order Operational Matrix of Derivative

The derivative of the function vectorΦx can be approximated as follows:(21)DαΦx=PαΦx,where Pα is called the GFLF operational matrix of the derivative.Theorem 1.
SupposePα is the m×m GFLF operational matrix of Caputo fractional derivatives of order α, β>α/2, β∉ℕ, then the elements of Pα are given as follows [30]:(22)piji,j=1m−1,m−1=2j+1βh−α∑s=0i∑r=0jbr,jbs,i′Γsβ+1Γsβ−α+11s+r+1β−α,where bs,i′=0,sβ∈N0andsβ<α,bs,i′=bs,i,sβ∉N0andsβ≥αorsβ∈N0andsβ≥α.


## 3.4. Solving Process

Substituting equations (18) and (21) into equation (14), we have(23)y″x≈CTd2Φxdx2≈CTP2Φx,(24)Dxαyx≈CTdαΦxdxα≈CTPαΦx,(25)fx≈FTΦx,where the coefficient matrices F can be obtained by equation (19). Substituting equations (23)–(25) and (18) into equation (14), we get(26)CTP2Φx+2ξCTPαΦx+CTΦx=FTΦx.Simplifying equation (26), we obtain(27)CTP2+2ξCTPα+CT=FT.For equation (15), we have(28)CTΦ0=δ1,CTPΦ0=δ2.Equation (27) together with equation (28) constitutes a system of linear algebraic equations. These unknown coefficients ci can be obtained by solving the system. A flowchart of the algorithm is given in Figure 3.Figure 3
Algorithm flowchart.


## 4. Numerical Simulation

In this section, several numerical experiments are presented to verify the established model and the provided algorithm. The parameters of the following examples are actual physical parameters. In the following numerical examples,m represents the number of discrete terms of the polynomial; that is, the larger the value of m is, the more discrete terms of the polynomial, the closer the numerical result approaches to the analytical result, but the corresponding calculation amount will also increase. In order to obtain effective numerical results and save calculation time, we take m=4.Case 1.
Considering the following fractional differential equations model of mechanical system hunting phenomenon withξ=0.1:(29)y″+2ξD0.5y+y=2+0.2×2Γ2.5x1.5+x2,y0=y′0=0.
The analytical solution of this problem isy=x2. When m=4, the numerical and analytical solutions are shown in Figure 4. In this case, the analytical result is given to verify the effectiveness of the proposed algorithm. Figure 4 shows that the proposed method can achieve a good agreement between numerical and analytical results.Figure 4
Numerical and analytical results withm=4.Case 2.
Considering the second-order wave equation withξ=0.1,fx=−1. The analytical result of this problem is y=e−ξxc1cos1−ξ2x+c2sin1−ξ2x−f, here c1=c2=1,ξ=0.1,andfx=−1. The numerical results with m=4,5 and analytical results are shown in Figure 5. The main reason why the numerical solutions do not coincide the analytical solutions at the beginning is that the initial conditions are homogeneous in this example. However, when m takes different values, the obtained numerical results are consistent, which proves that the proposed scheme is effective:(30)y″+2ξy′+y=fx.Figure 5
Analytical solutions and numerical vibration displacement solutions withm=4,5.Case 3.
Considering the fractional-order wave equation withfx=−1:(31)y″+2ξDαy+y=fx.
Whenm=4andα=0.5, the numerical results with ξ=0.15,0.6 are shown in Figure 6. Figure 6 shows that the numerical results are frequency invariant attenuation waveforms. When m=4andξ=0.15, the numerical results with α=0.5,0.7,0.9,and1 are shown in Figure 7. When m=4andα=1, the phase diagram with ξ=0.15 and ξ=0.6 is shown in Figure 8. Figure 8 shows that images with ξ=0.6 attenuate faster than those with ξ=0.15, which is consistent with the results in Figure 6. Figures 7, 10, and 13 show that the numerical vibration displacement solutions approach to the solution (α=1) with the fractional order α gradually approximating to 1. These show that the fractional order model has memory and can extract short-term sequence details arbitrarily.Figure 6
Numerical vibration displacement solutions withξ=0.15and0.6.Figure 7
Numerical vibration displacement solutions withα=0.5,0.7,0.9,and1.Figure 8
The phase diagram withξ=0.15 and ξ=0.6.Case 4 (see [31]).
Considering the fractional-order wave equation, in this case, we take the friction forcef=4001−e−20x+100sin6πx. When m=4andα=0.5, the numerical results with ξ=0.01,0.15,and0.6 are shown in Figure 9. Figure 9 shows that the vibration displacement solutions are still attenuated waveform, and the waveform is equiperiodic when ξ=0.01. Moreover, the waveforms no longer change regularly with ξ increasing. When m=4andξ=0.15, the numerical results with α=0.5,0.7,0.9,and1 are shown in Figure 10. When m=4 and α=1, the phase diagram with ξ=0.01, ξ=0.15, and ξ=0.6 is shown in Figure 11. Figure 11 shows that the waveform attenuation is the fastest when ξ=0.6. From Figures 6, 9, and 12, it can be concluded that the larger the value of ξ takes, the faster the waveform attenuates:(32)y″+2ξDαy+y=−4001−e−20x+100sin6πx.Figure 9
Numerical vibration displacement solutions withξ=0.01,0.15,and0.6.Figure 10
Numerical vibration displacement solutions withα=0.5,0.7,0.9,and1.Figure 11
The phase diagram withξ=0.01, ξ=0.15, and ξ=0.6.Figure 12
Numerical vibration displacement solutions withξ=0.01,0.15,and0.6.Case 5 (see [31]).
Considering the following fractional-order wave equation, in this case, we take the friction forcef=200e−xcos2πx. When m=4andα=0.8, the numerical results with ξ=0.01,0.15,and0.6 are shown in Figure 12. Figure 12 shows that the vibration displacement solutions are still attenuated waveform, and the waveform is equiperiodic when ξ=0.01. Moreover, the waveforms no longer change regularly with ξ increasing. The numerical results show that there is a large fluctuation at the beginning, and the fluctuation process tends to be smooth with x increasing. When m=4andξ=0.15, the numerical results with α=0.6,0.7,0.8,0.9,and1 are shown in Figure 13. When m=4 and α=1, the phase diagram with ξ=0.01, ξ=0.15, and ξ=0.6 is shown in Figure 14.(33)y″+2ξDαy+y=−200e−xcos2πx.Figure 13
Numerical vibration displacement solutions withα=0.6,0.7,0.8,0.9,and1.Figure 14
The phase diagram withξ=0.01, ξ=0.15, and ξ=0.6.


## 5. Conclusions

In this paper, a fractional differential equation model of drawing self-excited vibration is proposed based on the mechanical system hunting phenomenon. The fractional-order Legendre function method is utilized to obtain the vibration displacement solutions of the given model, and the established model and proposed algorithm are analyzed from several specific numerical experiments.



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