The Nonsmooth Vibration of a Relative Rotation System with Backlash and Dry Friction

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 themotion switchingmechanism. Based on such analytic conditions of motion switching, the influence of themaximal 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 theK-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.


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 [5][6][7].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 [12], a research on a piecewise linear system without damping was made, and the closed-form solution for periodic response was obtained.In [13], a mapping technique was developed to investigate a linear system with a single discontinuity, and the chaotic behavior was presented numerically.In [14], a mapping approach was adopted to investigate the periodic response and bifurcation of a piecewise linear system.In [15], Luo initially proposed a mapping structure for discontinuous systems.In [16], the idea of mapping structure was used to investigate a periodic forced piecewise linear system.In addition, the investigations by Nordmark [17], Błazejczyk et al. [18], Blazejczyk-Okolewska et al. [19], Luo et al. [20], di Bernardo et al. [21,22], and Luo et al. [23][24][25][26] 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 [27], Hartog initially made an investigation on the periodic motion of the forced linear oscillator with Coulomb and viscous damping.In [28], from mathematical point of view, Levitan discussed a friction oscillation model, and the stability of the periodic motion was studied.In [29], Shaw studied the stability for a nonstick, periodic motion through the Poincare mapping.In [30], 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 [31], 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 [34], 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.

Mechanical Model
The present paper investigates a relative rotation system with backlash and dry friction, and the nonlinear model is shown in Figure 1.
By Newton's theorem, the balance equations for the motor and load parts can be obtained as with where   is the motor moment of inertia,   is the load moment of inertia,   represents the transmitted shaft torque,   is the motor torque,   is the load torque disturbance,   is the inner damping coefficient of the shaft,   is the motor viscous friction coefficient,   is the load viscous friction Figure 2: Approximate function.
For the system model ( 6) above, suppose the driving torque as  =  0 +  1 sin() and the dry friction load torque as 3 , where   is the maximal static friction torque,  1 is the mixed friction coefficient,  2 is the dynamic pressure friction coefficient, V  represents the line velocity of the load parts, V  = θ    , and V is the load velocity; therefore 3 , which can be approximately described as shown in Figure 3.Then, the system model is obtained as

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 = { ∈  3 |  3 − V/  = 0}; then the system domain will be divided into three parts as and Σ = { ∈  3 |  3 − V/  = 0}; they are shown in Figure 4.
According to the discussion above, in the subregions Ω 1 and Ω 2 , the equations of motion are as follows, respectively: However, when the system flow arrives at the separation boundary,   is not a determined value but a range as   ∈ [−  ,   ]; 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 where { 1 (),  2 ()} is a vector field along the separation boundary, denoted by   ().From the convexity of the setvalued vector field, we have where  ∈ [0, 1]; then the system vector field can be described by a differential inclusion as follows: Since the vector field   () determines the system flow along the separation boundary, we have   Σ   () = 0 with   Σ = (0, 0, 1); then we get By using ( 11) and ( 13), the vector field   () is obtained as Then we can investigate the sliding dynamics along the separation boundary.

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.
In the case presented in Figure 5(b), instead, a section of trajectory lying in region Ω 1 or Ω 2 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 condition is satisfied the system flow will be sliding on the separation boundary.
The stick motion occurs.
Case 1.The system flow leaves the separation boundary and enters the subregion Ω 1 .
Case 2. The system flow leaves the separation boundary and enters the subregion Ω 2 .
For the first case, it should satisfy the following: and for the second case, it should satisfy the following: Then, by using the analytic conditions listed above, we can predict the sliding bifurcation for our system, which is presented in the next section.

Numerical Predictions
In this section, we investigate the sliding dynamics by using the analytic conditions listed above.Firstly, select the system parameters 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.
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   =   Σ   (),  ∈ {1, 2}, and choose   = 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.
From the above investigation, a conclusion is obtained as follows: when the maximal static friction torque is smaller (  = 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  0 on the sliding dynamics.Choose the system parameters as Then, we obtain the sliding bifurcation and the sliding time duty cycle as shown in Figures 9 and 10.
To better understand the influence of the driving torque  0 on the sliding dynamics, select  0 = 3.5, 4, 4.5 to make a detailed investigation; then the corresponding time history and K-function are shown in Figure 11.
By observing the numerical conclusions above, we can predict that the sliding region decreases with the increase of  0 at first, then the sliding region increases, and at last the sliding region may disappear through the grazingsliding bifurcation.To verify the prediction, choose  0 = 4.52, 4.531, 4.54; then the corresponding dynamics are shown in Figure 12.
By observing Figure 12, when  0 = 4.532, the grazingsliding bifurcation occurs; then the sliding region disappears, which is consistent with our prediction.

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.