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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

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 [

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 [

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 [

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.

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

Mechanical model.

By Newton’s theorem, the balance equations for the motor and load parts can be obtained as

As for the dead zone function

Approximate function.

For the system model (

Dry friction.

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

System domain.

According to the discussion above, in the subregions

However, when the system flow arrives at the separation boundary,

From the analysis above, the vector field can be described by a set-valued vector field as

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

Sliding bifurcation type.

In the case presented in Figure

A different bifurcation event, which we shall call sliding bifurcation of type II or switching-sliding, is depicted in Figure

The fourth and last case is the so-called multisliding bifurcation, shown in Figure

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

When the following condition is satisfied

As for the sliding region, it is defined as

The system flow leaves the separation boundary and enters the subregion

The system flow leaves the separation boundary and enters the subregion

For the first 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.

In this section, we investigate the sliding dynamics by using the analytic conditions listed above. Firstly, select the system parameters as

Then, in order to better study the sliding dynamics, choose the separation boundary

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

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

Time history and

From the above investigation, a conclusion is obtained as follows: when the maximal static friction torque is smaller

Next, we investigate the influence of the driving torque

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

Sliding bifurcation.

Sliding time duty cycle.

To better understand the influence of the driving torque

Time history and

By observing the numerical conclusions above, we can predict that the sliding region decreases with the increase of

Time history diagram.

By observing Figure

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

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.

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

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

^{3}topology. I. The Klein-Gordon and Schrödinger equations