#### Abstract

As the most widely used power transmission device in mechanical equipment, the vibration characteristics of gears have a very important influence on the working performance. It is of great theoretical and practical significance to study the vibration characteristics of gear system. In this paper, a gear transmission system model is set up in a forcefully nonlinear form; the continuity mapping and discontinuity mapping are utilized to analyze the nonsmooth vibration. Then, the sliding dynamics of separation boundaries is studied by using the perturbation method and the differential inclusion theory. In addition, the periodic response of gear pair system is illustrated and Floquet’s theory is presented to confirm the stability and bifurcation of periodic response. Concurrently, the maximal Lyapunov exponent is obtained to accurately determine the chaotic state in gear pair system, which is consistent with the bifurcation diagram and Poincare section. Finally, a reasonable explanation is given for the jump phenomenon in bifurcation diagram.

#### 1. Introduction

The gear system is widely used in various mechanical systems and equipment because of its compact structure and high transmission efficiency. Its own vibration characteristics can directly affect the performance and reliability of the whole system. Therefore, the study of its vibration characteristics has great significance.

Dating back to 1980s, many researchers had carried out experiments on the nonlinear vibration of gear system [1–5]; besides a certain number of complicated phenomena such as bifurcation and chaos had been observed. In [6], the nonlinear frequency response characteristics of a gear transmission system with backlash had been studied by the harmonic balance method. In [7], some experiments had been carried out on a gear pair transmission system. In addition several phenomena such as chaotic behaviors of subharmonic resonance and superharmonic resonance had been studied. In [8], a nonlinear rotor-bearing system was observed; bifurcations and the periodic responses were also investigated. Moreover, the chaotic response was checked as well through using the Lyapunov exponent and numerical methods. In [9], a research on a nonlinear model of gear transmission system including backlash, friction, and time-varying stiffness was achieved. On the contrary, the existence of bifurcation, periodic responses, and chaotic motions were studied numerically. In [10], the frequency responses of a nonlinear geared rotor-bearing system with time-varying mesh stiffness were inspected by the methods of multiple-scales and mathematical simulation. In [11], the vibration dynamic responses of a gear transmission system supported by journal bearing were studied; besides the subharmonic, periodic, and chaotic states were examined.

What is more, plenty of valuable research had been commenced on gear transmission system. But they rarely involved the nonsmooth dynamics of gear system. More precisely, the gear transmission system is a typical system with segmented properties, caused by the presence of backlash. Moreover, with the presence of backlash, the gear transmission system can be considered as one of vibration shock systems. At the beginning of this century, considerable researchers were interested in this system, especially in segmented linear systems that were stimulated by external periodic forces. In [12], the earliest research for a segmented linear system without damping was made and a closed solution of the periodic response was obtained. In [13], a mapping technique was developed to investigate a discrete linear system; besides the chaotic behavior was presented in numerical representation. In [14], a mapping approach was adopted to observe the periodic response and bifurcation of a segmented linear oscillator. In [15], a mapping structure for discontinuous system was initially proposed; the idea of mapping structure was used to investigate a periodic segmented linear system. In addition, the investigations can serve as examples in [16–18].

In all those works, a continuous mapping method was applied to transform nonsmooth gear system into segmented linear system, which can not explain the nonsmooth dynamic behavior of gear system completely. A large number of common problems can be described by discrete dynamic systems even if the problems are described by continuous dynamic systems. In this paper, the Poincare section and bifurcation diagram are obtained by numerical simulation, and the dynamic performance of gear transmission system is studied. In this paper, we construct discontinuous mapping, and combining with Floquet theory, we study the local dynamic characteristics of clearance and its front and rear, which reveal the vibration mechanism of gear transmission system in the gap nonsmooth state. The organization of this paper is as follows. Firstly, this paper briefly introduces a gear drive model with the basic dynamic response of a typical gear drive system. Then, the continuity mapping and discontinuity mapping are set up systematically. What is more, the differential inclusion theory and the method of perturbation are adopted to investigate the singularity of the sliding dynamics on separation boundaries, and the periodic response is analyzed by the mapping method. In addition, the discretization is a very important tool for analyzing the stability of the periodic motion in the gear system. The discrete-time shooting method is adopted to calculate the change of the Floquet multiplier. Then, the Floquet theory and the idea of mapping are introduced to give the methods and conditions for judging the periodic response of the system. At last, a summary of this work is presented.

#### 2. The Mechanical Model

When both the bearing support of the entire system and the stiffness of the drive shaft are large for the gears in the transmission system, the torsional vibration model can be simplified as the form shown in Figure 1.

If the vibration between teeth is ignored in the gear system, the time-varying stiffness and static transmission error of the basic oscillation frequency are equal to the gear meshing frequency:

and represent the number of teeth in the driving wheel and the driven wheel, respectively. This means that the time-varying stiffness and static transmission error of the system can be expressed in a form of Fourier to Fourier series:

Through Newton’s theorem, the balance equation of driving wheel and driven wheel can be written as

and are, respectively, the rotational inertia of the driving gear and the driven gear, and are, respectively, the angular displacement of driving gear and driven gear, and are, respectively, the radius of base circle of driving gear and the driven gear, and and are, respectively, the load torque of driving wheel and wheel. In addition, is the time-varying meshing stiffness, is the transmission error, and is the mesh damping.

By assuming that , (3) and (4) can be transformed as

; ; .

Because the time-varying stiffness and static transmission error of the basic oscillation frequency are equal to the gear meshing frequency , the mesh stiffness and the static transmission error terms can be expressed in the form of Fourier series. Taking the first harmonic,

is the average stiffness, and is the fluctuation amplitude. Assuming that , , , , , , and replacing with , then the dynamic model can be simplified as follows:

And , , ,

From (7), the mechanical model is set up. Because the gear system is a typical nonsmooth system, the previous method about smooth is not applicable. Therefore, a mapping method is utilized to analyze the nonsmooth system.

#### 3. The Construction of Mapping

For the gear model in this paper, due to the backlash, time-varying stiffness, and static transmission error, a variety of complex motion patterns may occur between the driving wheel and the driven wheel, so it is necessary to establish different mappings to study each movement. There are three main cases and corresponding mapping methods.

*(1) The Meshing State. *In this case, the system flow does not pass through the constraint surface. But in a smooth area, the operating state of the system is continuous and smooth, and it can be analyzed by continuous smooth theory.

*(2) The Collision State. *In this case, the system flow moves from one subinterval through the constraint surface to another subinterval. The Jacobi matrix of the system has a jump when the flow crosses over the constraint surface. A discontinuous mapping is necessary to compensate for the system flow jump, and then the composite mapping method can be used to analyze the whole system.

*(3) The Edge State. *In this case, the teeth of driving wheel are in contact with the teeth of driven wheel at a relative zero speed, which is a critical situation. The system flow is tangent to a constraint in a subinterval. Edge theory can be used to study this condition; besides the discontinuous mapping is used to analyze the local characteristics.

##### 3.1. The Basic Mapping

For the gear pair system (7), the phase space is divided into three subdomains by two separation boundaries and the corresponding phase space is defined as

The two constraint surfaces are defined as

In order to establish the mapping, the two constraints can be further divided, , , and the four subsets are defined as

From the four subsets, all the six basic mappings are defined as shown in Figure 2.

##### 3.2. Discontinuous Mapping

Discontinuous mapping is a conversion relationship that represents the flow of the system between two adjacent subintervals. This transformation is to compensate for the discontinuity of the system flow at the constraint surface. Considering the gear pair system in this paper, it is used to compensate for the discontinuity of the teeth during the transition between the disengaged state and the meshing state. The motion between the driving wheel teeth and the driven wheel teeth mainly includes three cases; the relationship between the system flow and the constraint surface is different from the cases (2) and (3), which are the collision state and the edge state. Therefore, it is necessary to introduce two types of discontinuous mappings, which are noncritical discontinuous mapping and critical discontinuous mapping.

The noncritical discontinuous mapping is to compensate for the discontinuity of the system’s flow through the constraint surface. The figure is shown in Figure 3.

is the time that the undamaged line reaches the interface ; is the time that the disturbed trajectory reaches the constraint surface ; is the initial disturbance value; represents the disturbance value in the subspace before crossing the boundary ; is the disturbance value in the subspace after crossing the boundary . Assuming that , represents the transition from the subspace across the interface to the subspace . Make the trajectory in front of the first-order Taylor expansion.

is the time which reaches the constraint surface after the trajectory reaches the interface. According to the analysis of the transition point conditions ,

Then

If it is reversible, then

Equations (16) and (17) are the noncritical discontinuous mapping. When the state of motion of the gear system transitions between teeth and meshing happens, it is necessary to introduce the above discontinuous mapping.

The critical discontinuous mapping is to compensate for discontinuities when the system flow near the edge of the wipe traverses the interface.

The discontinuous mapping of the critical situation is the discontinuous mapping when the edge is bifurcated. For the vector field segmented smooth system, the vector field is given as follows:

. Define the switching plane or interface as follows: . The switching surface corresponds to the constraint surface of the model. The interface divides the phase space of the system into two parts, , . Assuming that, in , the motion is determined by the flow ; in , the motion is determined by the flow . The phase space is shown in Figure 4.

If bifurcation occurs in the system at the edge of , it should meet the analytical conditions:

In general, the bifurcation point can be converted to by coordinate transformation. In this paper, the analytical form of the discontinuity map is ignored. We just give the conclusion that the local mapping in the neighborhood of grazing point has a 3/2-type singularity.

##### 3.3. Local Singularity

The sliding dynamics along the separation boundaries will be investigated in this section by using the perturbation method and the differential inclusion theory.

For (7), suppose , , then

whereand the system can also be expressed as a uniformwhere ,

,

.

, represent system parameters. In order to obtain the sliding dynamics along the separation boundaries, the differential inclusion theory will be introduced.

For (22), it can also be expressed as

The set-valued vector field is convex and continuous with respect to the parameter *λ* contained in the closed interval . The following property holds for the convex set of the vector field.

, and and are the input and output vector fields, respectively. is a vector field along the separation boundary.

From the convexity of the set-valued vector field, we have

The sliding motion is along the separation boundary, which indicates the vector field is along the boundary. So from which we have

For our system, the separation boundaries are , so the normal vector of separation boundaries is ; then we get

From (26) and (27), we can obtain . However, from the convexity, the parameter is required. Therefore, a perturbation parameter is introduced for a new separation boundary near the original boundaries ; then

Finally, the vector field on the new separation boundaries can be determined as , so the sliding dynamics along the separation boundaries can be investigated by

With initial condition at , the foregoing equation gives

Since is very small, (31) can approximately describe the sliding dynamics along the separation boundaries. From (30), if , all the sliding motions on the two boundaries, respectively, approach the two static balance points (i.e., (±1, 0)) as . However, for given , the sliding motions on the two boundaries, respectively, go away from the two static balance points. So the two balance points are like saddles as shown in Figure 5. But for , the sliding dynamics along the separation boundaries are undetermined.

From the analytical conditions for grazing motions, the grazing bifurcation conditions on the separation boundaries for the flows of this nonsmooth system are

Therefore, in the neighborhoods of the two equilibrium points, the local topological structures can be sketched in Figure 6.

In order to verify the rationality of the topology, several grazing trajectories will be obtained. Herein, we make use of these grazing trajectories to approximately describe the sliding dynamics along the separation boundaries.

Choose the system parameters as , , , , where represents the constant forcing parameter and denotes the initial phase.

Consider the parameters , , initial state (1,0.3); , , initial state (1,0.4), respectively. Two grazing trajectories can be obtained as shown in Figure 7. The grazing points are both (1, 0). In the neighborhoods of (1, 0), the system trajectories conform to the topology above.

Suppose , , initial state (1,0.3); , , initial state (1,-0.3), respectively. We can also obtain two grazing trajectories in the points (-1, 0), (1, 0), respectively. As shown in Figure 8, they both conform to the topology above.

As shown in Figure 9, several grazing trajectories are obtained to draw on the same coordinates to approximately describe the sliding motions along the separation boundaries and to verify the topology in the neighborhood of grazing points.

#### 4. Analysis of Periodic Response

For the gear transmission system, due to the existence of time-varying period stiffness and periodic excitation, the system must have periodic motion under certain system parameters’ combination. Therefore, the external load is used as the conversion parameter, and the periodic response of the system is analyzed by the mapping method.

##### 4.1. The First Case Periodic Motions

The system parameter adopts common parameter, which can be accessed from [2, 18]. Select the system parameters as follows: , , , . The system flow is always within the range of intervals (Figure 2). The phase diagram is shown in Figure 10. The dynamics is constrained by the equation:

where ; ; .

##### 4.2. The Second Case Periodic Motion

For the gear transmission system, the second case periodic motion that corresponds to the collision state of the gear can be divided into two cases: single-sided impact periodic motions and double-sided impact periodic motions.

###### 4.2.1. Single-Sided Impact Periodic Motions

When , other parameters remain the same. In this case, the maximum value of the corresponding periodic flow is bigger than 1, the minimum value is between the two constraint surfaces . The gear system operates in a single-sided impact periodic motions, and the teeth are transitioning between the two states of disengagement and engagement. The phase diagram is shown in Figure 11.

For the convenience of research, assuming the initial state of the periodic motion is a fixed point on the interface , as shown in Figure 12. At this point the system flow consists of two parts; one is in the interval and the other is in the interval . For the trajectory in , the dynamic is constrained by (33), and the corresponding basic mapping is ; for the trajectory in , the dynamic is determined by (34), and the corresponding basic mapping is .

In this case, the system flow moves through the constraint surface. In order to establish the periodic mapping corresponding to periodic motion, it is necessary to introduce discontinuous mappings and to compensate for the discontinuity of the system flow and then use the mapping compound rule to obtain the periodic mapping .

Suppose , ; then the system (7) can be expressed as follows:

In interval ,In interval ,According to the noncritical discontinuous mapping,Therefore, the periodic mapping is

###### 4.2.2. Double-Sided Impact Periodic Motions

When , other parameters remain the same. A steady periodic state can be obtained as shown in Figure 13. In this case, the maximum value is bigger than 1 and the minimum value is smaller than -1. This state corresponds to the double-sided impact.

As shown in Figure 14, the initial state still selects the fixed point on interface . Adopting the similar method above, the periodic mapping could be obtained as

As , then

#### 5. Stability Analysis

In this section, these mapping structures will be used to analyze the periodic motion stability of the system through Floquet theory. At the same time, in order to judge whether the system enters the chaotic state, the maximum Lyapunov exponent spectrum of the system is obtained. That paper takes the double-sided impact case as an example to demonstrate the mapping method of stability analysis. Its corresponding periodic mapping is . The stability and bifurcation for periodic motion can also be confirmed through the periodic mapping which is corresponding to Jacobi matrix. By the chain rule, the Jacobi matrix can be expressed as follows:

For , , due to the time-varying stiffness of the gear system and backlash, has difficulty in getting the analytical form of its system flow, so the analysis of each mapping form can not be got. But they can be obtained through numerical method, such as the shooting method. At the same time, the corresponding Jacobi matrix can be yielded.

Suppose that the Jacobi matrix of periodic motion has been obtained, then the eigenvalues of a fixed point for this periodic mapping can be demonstrated as

represents the trace of ; denotes the determinant of .

. If , (43) can be demonstrated through

; ; .

For , the period-1 motion will be stable if they are located in the unit circle. But if one of them is located outside the unit circle, then the period-1 motion will be unstable. This means that only if , the periodic motion of the system is stable.

If one eigenvalue is -1 and the other is within the unit circle, the periodic doubling bifurcation is going to occur.

If one eigenvalue is +1 and the other is within the unit circle, the saddle-node bifurcation is going to occur.

Therefore, an improved shooting method is adopted to calculate the variation of the Floquet multiplier when the system changes with the external load parameters. The principle of the shooting method is to convert the two-point boundary value problem into the initial value problem. When the shooting method is used to calculate the Floquet multiplier, the Floquet multiplier is discrete and parameterized in the time domain. According to the change of Floquet multiplier, the stability of the system’s periodic motion and the way of instability are predicted. The Floquet multiplier map and the corresponding bifurcation diagram in the interval [0.075,0.086] are also presented as shown in Figure 15.

**(a) Real part of eigenvalues**

**(b) Imaginary part of eigenvalues**

**(c) Eigenvalue magnitude**

**(d) Local bifurcation diagram**

From Figure 15, we can deduce that when both eigenvalues are located in the unit circle, which implies that corresponding period-1 motion is stable. When is at the vicinity of 0.0785, one of the two eigenvalues jump out the unit circle from -1. Meanwhile the other is still within the unit circle, which indicated the period doubling bifurcation takes place.

By using the Floquet theory, we can merely get the stability and bifurcation of periodic motions. For judging the chaos state, other effective methods need to be introduced. As the Lyapunov exponent spectrum is one of the most precise tools to determine the chaos state, therefore, the maximum Lyapunov exponent spectrum is presented in Figure 16.

From Figure 16, we can deduce that when , the maximum Lyapunov exponents are negative, which implies the system does not enter the chaos state. When , the maximum Lyapunov exponents are positive, which implies the system is under chaos state. By comparing the result with the bifurcation diagram and Poincare section, consistent conclusions can be obtained. For the sake of demonstrating this system and ensuring the conclusion above, the system bifurcation of the gear system with external load parameters is shown in Figure 17. Enlarging the 0.06 - 0.08 part of bifurcation diagram, we can get the local bifurcation diagram as demonstrated in Figure 18.

The bifurcation diagram is used in the process of simulation. Bifurcation refers to small and continuous changes in the parameters of the system. As a result, the nature or topological structure of the system suddenly changes.

Poincare is a method for discretizing continuous systems and the Poincare map can replace the order continuous system by using the discrete mapping of order . The Poincare map is used to reduce the order of the system, and the Poincare map builds a bridge between the continuous system and the discrete system.

In order to demonstrate the transition process in detail, suppose =0.08, 0.078,0.064,0.0624,0.058, respectively; both the Poincare section and the corresponding phase diagram are gained as demonstrated in Figure 19.

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**#### 6. Conclusions

Firstly, for common gear systems, a nonlinear vibration model with backlash, time-varying stiffness, and static transmission error is established. The nonsmooth characteristics of the system are analyzed theoretically. According to the model, the state of motion is summarized into three main cases. For the period motion, the corresponding Poincare mapping is established. In order to analyze the stability of the periodic motion of the system, the discrete-time shooting method was adopted to calculate the variation of the Floquet multiplier. Then, the Floquet theory and the idea of mapping are used to give methods and conditions for judging the stability of the system’s periodic response. At the same time, to judge the chaotic state of the system, the maximum Lyapunov exponent of the system is obtained. Finally, in order to verify the rationality of the above method, the global bifurcation diagram of the system is gained. Through the comparison, the same conclusion can be got, the whole process of the system from the periodical bifurcation to the chaos is given, and a reasonable explanation for the jumping phenomenon in the bifurcation diagram is given.

Through the study of the gear system we can get the following conclusions. When the external load of the gear system is large, it is in a fully engaged state; as the load decreases, the state of motion changes from full meshing to unilateral collisions and bilateral collisions. In addition unilateral collisions and bilateral collisions occur alternately. As the load continues to decrease, the stable periodic motion of the system begins to lose its stability, and the motion state enters chaotic state through periodic bifurcation. This conclusion has important practical value, which can guide us in the actual project to select reasonable load parameters.

As a complex nonlinear system, the gear system not only includes backlash, time-varying stiffness, and static transmission error of these nonsmooth factors, but also includes many other nonsmooth factors. Therefore, how to improve the system's nonsmooth characteristics in the further is an important research direction. In addition, how to avoid chaos by taking effective measures is also a future research tendency.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by Proceedings of the 2017 International Conference on Advances in Construction Machinery and Vehicle Engineering, the China Postdoctoral Science Foundation funded project (Grant no. 2017M620096), and the Natural Science Foundation of Hebei Province, China (Grant no. E2017203144).