DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi 10.1155/2017/7064590 7064590 Research Article The Nonsmooth Vibration of a Relative Rotation System with Backlash and Dry Friction He Minjia 1 2 http://orcid.org/0000-0002-7864-0441 Li Shuo 2 Wang Jinjin 1 2 Lin Zhenjun 2 Liu Shuang 1 2 Avrutin Viktor 1 Liren College of Yanshan University Qinhuangdao 066004 China ysu.edu.cn 2 College of Electrical Engineering Yanshan University Qinhuangdao 066004 China ysu.edu.cn 2017 7112017 2017 02 05 2017 14 08 2017 7112017 2017 Copyright © 2017 Minjia He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate a relative rotation system with backlash and dry friction. Firstly, the corresponding nonsmooth characters are discussed by the differential inclusion theory, and the analytic conditions for stick and nonstick motions are developed to understand the motion switching mechanism. Based on such analytic conditions of motion switching, the influence of the maximal static friction torque and the driving torque on the stick motion is studied. Moreover, the sliding time bifurcation diagrams, duty cycle figures, time history diagrams, and the K-function time history diagram are also presented, which confirm the analytic results. The methodology presented in this paper can be applied to predictions of motions in nonsmooth dynamical systems.

National Natural Science Foundation of China 61673334
1. Introduction

Relative rotation system is a widespread power transmission system, which contains many nonlinear factors such as the nonlinear damping, stiffness, and backlash. These nonlinear factors can generally lead to vibration, which will reduce the transmission efficiency and performance. Therefore, the vibration of the relative rotation system is always a hot research topic.

Since the theory of rotational relativistic mechanics was first established by Carmeli in 1985 [1, 2], a lot of achievements on the relative rotation system have been obtained. Luo set up the theory of relativistic analytical mechanics of the rotational systems in [3, 4]. The Hopf bifurcation for a kind of nonlinear relative rotation system was investigated . In [8, 9], the bifurcation response equation of relative rotation system was deduced with the method of multiple scales, and the bifurcation and chaotic motions under combination resonance were investigated. The chaotic behaviors of a relative rotation nonlinear dynamical system under parametric excitation and its control were studied in [10, 11]. The papers above made a better research on the vibration of the relative rotation system, but the nonsmooth factors were rarely taken into consideration.

At present, the nonsmooth dynamics are a hot research topic. One reason for this is that the nonsmooth system widely exists in different disciplines. Mechanical engineers study the stick–slip oscillations in systems with dry friction and the dynamics of impact phenomena with unilateral constraints. Electrical circuits contain diodes and transistors, which ideally behave in a nonsmooth way. Meanwhile, there have been a lot of researches on the nonsmooth dynamics. In , a research on a piecewise linear system without damping was made, and the closed-form solution for periodic response was obtained. In , a mapping technique was developed to investigate a linear system with a single discontinuity, and the chaotic behavior was presented numerically. In , a mapping approach was adopted to investigate the periodic response and bifurcation of a piecewise linear system. In , Luo initially proposed a mapping structure for discontinuous systems. In , the idea of mapping structure was used to investigate a periodic forced piecewise linear system. In addition, the investigations by Nordmark , Błazejczyk et al. , Blazejczyk-Okolewska et al. , Luo et al. , di Bernardo et al. [21, 22], and Luo et al.  can also provide us with a plenty of meaningful conclusions.

Among the various nonsmooth factors, dry friction extensively exists in engineering, such as disk brake systems, turbine blades, and string music instruments. The discontinuity caused by dry friction forces makes the system more difficult to solve theoretically and numerically. Therefore, the friction-induced oscillations have been of great interest for a long time. In , Hartog initially made an investigation on the periodic motion of the forced linear oscillator with Coulomb and viscous damping. In , from mathematical point of view, Levitan discussed a friction oscillation model, and the stability of the periodic motion was studied. In , Shaw studied the stability for a nonstick, periodic motion through the Poincare mapping. In , Feeny researched the nonsmoothness of a Coulomb friction oscillator and presented the stick region analytically and graphically. Feeny and Moon made the experimental and numerical investigations of chaos in a dry friction oscillator. In , Feeny and Moon investigated the dynamics of a oscillator with impact and friction. In [32, 33], the stick and nonstick motions were observed, and the chaos for a nonlinear friction model was presented. In , the theoretical approach was discussed to treat models of dynamical systems involving nonsmooth nonlinearities, which was associated with differential inclusions of mainly finite dimensional dynamical systems and the introduction of maximal monotone operators (graphs) in order to describe models of impact or friction.

In this paper, the nonsmooth vibration of a relative rotation system with backlash and dry friction is investigated, especially the sliding bifurcation caused by the dry friction. Moreover, the analytic conditions for stick and nonstick motions are developed, and the influence of the maximum static friction torque and the driving torque on the stick motion is predicted numerically.

2. Mechanical Model

The present paper investigates a relative rotation system with backlash and dry friction, and the nonlinear model is shown in Figure 1.

Mechanical model.

By Newton’s theorem, the balance equations for the motor and load parts can be obtained as(1)Jmθ¨m+cmθ˙m+Ts=TmJlθ¨l+clθ˙l+Td=Tswith (2)θd=θm-θl,Ts=ksfθd+csθ˙d,where Jm is the motor moment of inertia, Jl is the load moment of inertia, Ts represents the transmitted shaft torque, Tm is the motor torque, Td is the load torque disturbance, cs is the inner damping coefficient of the shaft, cm is the motor viscous friction coefficient, cl is the load viscous friction coefficient, ks is the shaft elasticity, 2b is the total backlash angle, and f(θd) is a dead zone function as(3)fθd=θd-b,θdb0,θd<bθd+b,θd-b.Let x1=(θm-θl)/b, x2=θ˙m, and x3=θ˙l; we can deduce(4)x˙1=1bθ˙m-θ˙l=1bx2-x3x˙2=θ¨m=1JmTm-Ts-cmθ˙m=1JmTm-ksbfx1-csx2-x3-cmx2x˙3=θ¨l=1JlTs-Td-clθ˙l=1Jlksbfx1+csx2-x3-Td-clx3with(5)fx1=x1-1x110x1<1x1+1x1-1.Suppose α=1/b,k1=ksb/Jm,c1=csb/Jm,c2=cm/Jm,c3=csb/Jl,c4=cl/Jl,k2=ksb/Jl, and Ff=Td/Jl; then (6)x˙1=αx2-x3x˙2=-k1fx1-c1+c2x2+c1x3+Fx˙3=k2fx1+c3x2-c3+c4x3-Ff.

As for the dead zone function f(x1), it can be approximately expressed as f(x1)=0.1538x13-0.0566x1, which is shown in Figure 2.

Approximate function.

For the system model (6) above, suppose the driving torque as F=f0+f1sin(ωt) and the dry friction load torque as Ff=fssign(vl-v)-μ1(vl-v)+μ2(vl-v)3, where fs is the maximal static friction torque, μ1 is the mixed friction coefficient, μ2 is the dynamic pressure friction coefficient, vl represents the line velocity of the load parts, vl=θ˙lRl, and v is the load velocity; therefore Ff=fssign(Rlx3-v)-μ1(Rlx3-v)+μ2(Rlx3-v)3, which can be approximately described as shown in Figure 3. Then, the system model is obtained as(7)x˙1=αx2-x3x˙2=-k10.1538x13-0.0566x1-c1+c2x2+c1x3+f0+f1sinωtx˙3=k20.1538x13-0.0566x1+c3x2-c3+c4x3-fssignRlx3-v+μ1Rlx3-v-μ2Rlx3-v3.

Dry friction.

3. Nonsmooth Analysis

The dry friction load makes the relative rotation system a classical nonsmooth system, and the dynamics are obviously different from the smooth system.

For the system model in this paper, when the velocities of the load parts and the friction load are different, the friction size and direction are both changeable. The load friction direction will be opposite for the relative velocity larger and smaller than zero. Moreover, when the relative velocity is zero, the friction torque is uncertain. Therefore, this mechanical system model is a classical Filippov system, and we cannot deal with it using the conventional method.

In order to better analyze the nonsmooth dynamics, define the separation boundary as Σ=xR3x3-v/Rl=0; then the system domain will be divided into three parts as (8)Ω1=xR3x3-vRl<0Ω2=xR3x3-vRl>0and Σ=xR3x3-v/Rl=0; they are shown in Figure 4.

System domain.

According to the discussion above, in the subregions Ω1 and Ω2, the equations of motion are as follows, respectively:(9)F1x=x˙1=αx2-x3x˙2=-k10.1538x13-0.0566x1-c1+c2x2+c1x3+f0+f1sinωtx˙3=k20.1538x13-0.0566x1+c3x2-c3+c4x3+fs+μ1Rlx3-v-μ2Rlx3-v3,F2x=x˙1=αx2-x3x˙2=-k10.1538x13-0.0566x1-c1+c2x2+c1x3+f0+f1sinωtx˙3=k20.1538x13-0.0566x1+c3x2-c3+c4x3-fs+μ1Rlx3-v-μ2Rlx3-v3.

However, when the system flow arrives at the separation boundary, Ff is not a determined value but a range as Ff-fs,fs; we cannot obtain the corresponding equation of motion. Therefore, the differential inclusion theory is introduced in this section to deal with it.

From the analysis above, the vector field can be described by a set-valued vector field as(10)x˙Fx=F1xxΩ1co¯F1x,F2xxΣF2xxΩ2,where co¯F1(x),F2(x) is a vector field along the separation boundary, denoted by Fs(x). From the convexity of the set-valued vector field, we have(11)Fsx=λF2x+1-λF1x,where λ0,1; then the system vector field can be described by a differential inclusion as follows:(12)x˙Fx=F1xxΩ1,λ=0FsxxΣ,λ0,1F2xxΩ2,λ=1.Since the vector field Fs(x) determines the system flow along the separation boundary, we have nΣTFs(x)=0 with nΣT=(0,0,1); then we get (13)λ=nΣTF1xnΣTF1x-F2x. By using (11) and (13), the vector field Fs(x) is obtained as(14)Fsx=x˙1=αx2-x3x˙2=-k10.1538x13-0.0566x1-c1+c2x2+c1x3+f0+f1sinωtx˙3=0.Then we can investigate the sliding dynamics along the separation boundary.

4. Sliding Bifurcation

The model in the present work is a classical Filippov system, and its typical nonsmooth phenomenon is the sliding bifurcation. According to the existing literatures, the sling bifurcation can be mainly divided into four types as follows.

Figure 5(a) depicts the scenario we term as sliding bifurcation of type I.

Sliding bifurcation type.

In the case presented in Figure 5(b), instead, a section of trajectory lying in region Ω1 or Ω2 grazes the boundary of the sliding region from above (or below). Again, this causes the formation of a section of sliding motion which locally tends to leave Σ. We term this transition as a grazing-sliding bifurcation.

A different bifurcation event, which we shall call sliding bifurcation of type II or switching-sliding, is depicted in Figure 5(c). This scenario is similar to the sliding bifurcation of type I shown in Figure 5(a).

The fourth and last case is the so-called multisliding bifurcation, shown in Figure 5(d). It differs from the scenarios presented above, since the segment of the trajectory which undergoes the bifurcation lies entirely within the sliding region Σ^.

Under the influence of the dry friction, the system flow may enter different subregions, and when the flow arrived at the separation boundary, it may stay on the boundary; then some type of sliding bifurcation may occur. Otherwise the system flow leaves the boundary. As for the different cases above, we give the corresponding analytic conditions in the following.

When the following equation is satisfied (15)nΣTF1xnΣTF2x>0xΣ.the system flow will cross the separation boundary Σ. Meanwhile, when nΣTF1(x)<0,nΣTF2(x)<0, the system flow will enter the subregion Ω1, and when nΣTF1(x)>0,nΣTF2(x)>0, the system flow will enter the subregion Ω2.

When the following condition is satisfied(16)nΣTF1x>0nΣTF2x<0xΣ.the system flow will be sliding on the separation boundary. The stick motion occurs.

As for the sliding region, it is defined as(17)Σ^=xΣ:0λx1with(18)Σ^=Σ^+Σ^-,where Σ^-=xΣ:λ(x)=0,Σ^+=xΣ:λ(x)=1. When λ(x)=0, we have Fs(x)=F1(x) and nΣTF1(x)=0; when λ(x)=1, we have Fs(x)=F2(x),nΣTF2(x)=0. As for the stick motion, it may end in two cases as follows.

Case 1.

The system flow leaves the separation boundary and enters the subregion Ω1.

Case 2.

The system flow leaves the separation boundary and enters the subregion Ω2.

For the first case, it should satisfy the following:(19)nΣTF1x=0nΣTF2x<0xΣand for the second case, it should satisfy the following:(20)nΣTF1x>0nΣTF2x=0xΣ.

Then, by using the analytic conditions listed above, we can predict the sliding bifurcation for our system, which is presented in the next section.

5. Numerical Predictions

In this section, we investigate the sliding dynamics by using the analytic conditions listed above. Firstly, select the system parameters as(21)α=10,k1=10,k2=10,c1=0.2,c2=0.1,c3=0.2,c4=0.2,μ1=0.1,Rl=1,v=10,μ2=0.0005,f0=3,f1=1.5,ω=10.

Then, in order to better study the sliding dynamics, choose the separation boundary Σ as the Poincare section. =(x1,x2,x3)Tx3-v/Rl=0, and the Poincare mapping is as .

Choose the maximal static friction torque as the researched parameter; then we obtain the sliding bifurcation diagram and the sliding time duty cycle as shown in Figures 6 and 7.

Sliding time bifurcation diagram.

Sliding time duty cycle.

From the bifurcation and the duty cycle diagram, we can conclude as follows: when the maximal static friction torque is smaller, there is no sliding bifurcation and the stick motion cannot be observed. As it increases, the sliding bifurcation occurs, and the sliding region shows the tendency of increase. When the static friction increased to a certain extent, the system flow always stayed on the separation boundary, which means that the stick motion occurs.

To better study the influence of maximal static friction torque on the sliding dynamics and verify the analytic conditions, define Kα=nΣTFα(x),α1,2, and choose fs=1.2,1.5,1.8,2.0,2.5; then we can obtain the corresponding time history diagram and K-function diagram as shown in Figure 8.

Time history and K-function diagram.

From the above investigation, a conclusion is obtained as follows: when the maximal static friction torque is smaller fs=1.2, the system flow crosses the separation boundary periodically; there exist no stick motions. As it increases, the sliding bifurcation occurs and the sliding region increases with the static friction torque increasing.

Next, we investigate the influence of the driving torque f0 on the sliding dynamics. Choose the system parameters as(22)α=10,k1=10,k2=10,c1=0.2,c2=0.1,c3=0.2,c4=0.2,μ1=0.1,Rl=1,v=10,μ2=0.0005,f1=1.5,fs=1.5,ω=10.

Then, we obtain the sliding bifurcation and the sliding time duty cycle as shown in Figures 9 and 10.

Sliding bifurcation.

Sliding time duty cycle.

To better understand the influence of the driving torque f0 on the sliding dynamics, select f0=3.5,4,4.5 to make a detailed investigation; then the corresponding time history and K-function are shown in Figure 11.

Time history and K-function.

By observing the numerical conclusions above, we can predict that the sliding region decreases with the increase of f0 at first, then the sliding region increases, and at last the sliding region may disappear through the grazing-sliding bifurcation. To verify the prediction, choose f0=4.52,4.531,4.54; then the corresponding dynamics are shown in Figure 12.

Time history diagram.

By observing Figure 12, when f0=4.532, the grazing-sliding bifurcation occurs; then the sliding region disappears, which is consistent with our prediction.

6. Conclusion

In this paper, we investigate the nonsmooth vibration of a relative rotation system with backlash and dry friction, especially the sliding bifurcation dynamics. The analytic conditions for the sliding dynamics are also obtained. Then, by using these conditions, we research the influence of the maximal static friction and the driving torque on the sliding dynamics. Moreover, the corresponding bifurcation diagram, duty cycle, time history diagram, and K-function are presented to predict the sliding bifurcation numerically, which are consistent with the analytic conditions.

The present work not only makes us have a deeper understanding of the sliding bifurcation dynamics, but also provides us with a method to study the nonsmooth vibration of the nonsmooth systems.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The authors gratefully appreciate the support from National Natural Science Foundation of China (Grant no. 61673334).

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