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Dynamic Analysis of Gear Pairs with the Effects of Stick-slip

A stick-slip dynamic model of a two-gear set is proposed. In this model, time-varying mesh stiﬀness, tooth separation, friction between the gear teeth surfaces and potential stick-slip are considered. Considering the contributions of the vibration velocity to the gear teeth sliding velocity, transition conditions between the stick state and slip state are revealed. Dynamic analysis shows that, when stick appears, the vibration amplitudes are dramatically increased, and tooth separation occurs. Additionally, stick also induces half-frequency components of the mesh frequency and causes chaos. Parametric studies show that heavy loads and rough tooth surfaces increase the probability of stick and increase the time of the stick state over a single mesh period. However, due to the complex nonlinearity, the existence of a certain relationship between the gear rotation speed and the stick time is not conﬁrmed.


Introduction
Friction between mating teeth dramatically affects gear dynamics. On the one hand, friction is one of the vibration excitations in the off-line-of-action direction [1]. Friction increases the magnitude of dynamic transmission error(DTE) [2], weakens the reduction effect  [3], and significantly affects high-frequency parts at low speed [4]. With increasing friction coefficients, the spectral amplitudes of the meshing components are increased [5]. On the other hand, friction suppresses gear vibrations like a damping [6] and acts as one of the main sources of power loss [7] [8]. The meshing force fluctuations, vibrations in the line-of-action [1] and gear angular displacements [4] [9] are reduced by friction. Friction also reduces the large oscillations at certain resonant conditions [10] and causes impulses at the pitch points where the friction changes its direction [11]. Additionally, friction has some other effects on gear dynamics. Liu and Parker [12] found that different types of instabilities have different sensitivities to the friction moment and bending effect. The torsional response is probably less sensitive to tooth friction than the bending response [13]. The effect of friction during approach and recessing is converse on the time-varying mesh stiffness [14].Another important potential hazard of gear tooth friction is scuffing of gear teeth [8] [15] [16] [17]. Scuffing may damage the designed tooth profile, induce additional transmission errors and induce additional excitation of vibrations.
The interactions between friction and other factors make the nonlinear vibration of the gear more complicated [4]. For helical gear sets, the effect of friction on teeth contact stress varies at different contact positions, which induces nonlinearity [18]. For herringbone gear sets, the effects of friction may be covered by the effects of profile errors [1]. Due to the interaction of the mesh force and friction force, tooth profile modification may even amplify the motions in the off-line-of-action [2]. The dissipation mechanism of tooth friction is also substantially influenced by tooth profile modifications [7]. Stick-slip phenomena, which is caused by friction, may lead to negative damping that can cause self-excited oscillations of mechanical structures in certain instances [19].Under the effects of wheel-set stick-slip, the gear transmission system of high-speed trains has a larger superharmonic response and loses its stability because of Hopf bifurcation [20]. A prominent characteristic of gear tooth friction is the reversal of its direction near the pitch point. Repeatedly changing of the friction direction induces significant oscillatory bearing forces [21]. Usually, the contributions of the vibration velocity to the actual relative sliding velocity are ignored. Under this approximate treatment, as shown in Fig.1, the relative sliding velocity between the mating gear teeth is monotonic as the gear rotates, and the direction of the relative sliding velocity reverses when the contact line exactly passes through the pitch point. However, the vibration velocity induces fluctuations in the relative sliding velocity, and the direction-reversing point is not exactly at the pitch point. Furthermore, there even exists more than one direction-reversing point near the pitch point, as shown in the magnified figure of Fig.1. The repeated changes of the sliding velocity direction means the repeated reversals of the friction direction. In the areas of the magnified figure of Fig.1, the value of the sliding velocity is close to zero, and the two gear tooth surfaces seem to bond together. It is not clear whether stick phenomenon exist in this area. Furthermore, the effects of the stick phenomenon on the dynamic character of the gear transmission are also still not clear.
In this study, the gear dynamics with the effects of the stick-slip phenomenon are studied. A stick-slip dynamic model of a two-gear set is proposed. In this model, time-varying mesh stiffness, tooth separation, gear teeth friction and the potential stick-slip phenomenon are considered. Considering the contributions of the vibration velocity to the gear teeth sliding velocity, transition conditions between the stick and slip states are revealed. Discussion of the dynamic response of gear transmissions under the stick-slip effect offers a compre-hensive understanding of gear scuffing from a dynamic perspective.The remainder of the paper is organized as follows. In the next section, the modeling details and the governing equations of motion for a two-gear set with stick slip are presented. The effects of stick-slip on the dynamic responses and parametric study are discussed in Section 3. Concluding remarks are given in the last section.

Stick model and slip model
In this section, a stick-slip dynamic model of a twogear set is proposed. In this model, time-varying mesh stiffness, tooth separations, friction between the mating gear teeth and the potential stick-slip phenomenon is considered. This stick-slip dynamic model is the combination of a stick model and a slip model. The slip model in this current study is the traditional 6 degrees of freedom (DOF) model considering two translational and rotational motions. The stick model is an evolution of the slip model based on instantaneous dynamic conditions. Fig.2 shows the traditional slip model with 6 DOF. In this model, two translational motions and the rotational motion of each gear are considered. The components are modeled as rigid bodies and the masses are concentrated to the corresponding centroid. A linear spring is used to model the compliance of the gear mesh.In Fig.2, k b1 and k b2 represent the translational support stiffness, k r2 is the rotational stiffness, and k m is the mesh stiffness; x and y denote the translational motions, and u denotes the rotational motions.

Slip model
As shown in Fig.3, S 1 S 2 is the mesh line and point P is the pitch point. M j represents the moving contact point between the jth mating gear teeth. For spur gears, j = 1 and 2, v rj1 and v rj2 are the velocities of the mesh teeth at mesh point M j , respectively. The direction of v rj1 is perpendicular to line o 1 M j , and the direction of v rj2 is perpendicular to line o 2 M j . The components of v rj1 and v rj2 on the y axis are equal during the whole mesh process;α j1 and α j2 are the pressure angles of the driving gear and driven gear, respectively, and f f j represents the friction forces, whose direction is perpendicular to line S 1 S 2 . The sliding friction f f j21 is positive when it is directed to the left, and f f j12 is positive when it is directed to the right.
Under the running state, the gear teeth start to contact near S 1 , and the contact point M j moves towards S 2 . Suppose there is no reversal of the friction direction before M j arrives at point P s and also no reversal of friction direction after M j passes by point P e . When M j moves in S 1 P s , the components of v rj2 on the x axis are larger than those of v rj1 . The direction of the friction force applied on the 1 st gear tooth f f j21 is to the left. In addition, the reaction force f f j12 points to the right. After M j passes by point P e , f f j21 points to the right and f f j12 to the left. When M j is in P s P e , the friction oscillates, and the mating teeth may transit between the slip and stick states. The motion equations of the gear pair under the slip state can be expressed as Eqs. (1) and (2).
where m represents mass, c represents damping and J represents the moment of inertia; r b1 and r b2 are the corresponding base radii; T r is the transmitted torque applied on the driving gear; ǫ represents the contact ratio; and δ is the compressive deflections of the gear mesh.
The matrix form of Eq.(1) and Eq.(2) could be expressed as, where M is the mass matrix, K b is the support stiffness matrix.
K m is the time-varying mesh stiffness matrix, as expressed in Eq. (6). k m (t) in Eq. (7) is the time-varying mesh stiffness. θ(δ) is the tooth separation function, as expressed in Eq. (8).
F f and F t are respectively the friction force vector and applied torque vector, as expressed in Eq.(9) and Eq. (10).
The sliding friction force could be expressed as, where v r j is the relative sliding velocity between the jth mating gear teeth, as expressed in Eq. (12). (13) where ω 1 and ω 2 are the rotational speed.

Stick-model
As shown in Fig.1, there may exist more than one direction reversal of friction when the moving point M j is near the pitch point. At the direction-changing point, if the force that attempts to slide one tooth surface over the other is less than the maximum possible friction force, the surfaces of the meshing gear teeth stick together.The preconditions for the stick condition is where f f 21 and f f 12 are the forces attempting to slide one tooth surface over the other, which can be obtained by the instantaneous conditions.
Comparing the motion equations of the slip-model and stick-model, one can find that the stick-model has one DOF less than the slip-model. This is because x 2 has a certain relationship with x 1 under the stick-state, and x 2 is eliminated. These stick-model equations could be expressed in matrix forms as where the subscript t denotes the matrices and vectors in stick-model.
The gear set is in the stick state or slip state depending on Eq. (14). If both of the conditions in Eq. (14) are satisfied, the gear set is in the stick state. Otherwise, the slip state domains the gear set. In the process of numerical integration, a status check based on Eq. (14) is required at each substep calculation.

Results and discussion
In this chapter, an example gear set is applied to eliminate the effects of stick-slip on the dynamic responses. The main parameters are shown in Table.1. Unlike frictionless gear models, constant input and output torques cannot be assumed since energy dissipation due to the tooth friction [13].  As shown in Fig.4, the first natural frequency of the slip model is 32 Hz, which is associated with the 'rigid' mode. The rigid mode has pure rotational motions of both gears. The 3 rd and 5 th modes have pure translational motions of Gear1 and Gear2, respectively. The 2 nd , 4 th and 6 th modes have combined motions of the translation and rotation.
The first natural frequency of the stick model, as shown in Fig.5, approximately equals the first natural frequency of the slip model. However, these two modes are very different in their corresponding modal shapes. There are similar differences between the second modes, and between the 4 th mode of the slip model and the 3 rd mode of the stick model. In the stick state, these two gears are bonded in the x direction. Therefore, there is no relative displacement between x 1 and x 2 in any of the modal shapes of the stick model. Therefore, in the stick model, there is no corresponding mode to the 3 rd and 5 th modes of the slip model. The results of modal analysis reveal that stick effect has dramatic influence on both natural frequencies and modal shapes.

Comparisons of the dynamic responses
Once stick occurs, the vibration conditions of the gear set vary between the stick state and the slip state. The constant transitions between these states certainly induce impacts.  Fig.6, one can see that the relative velocity is close to zero and that there exists a small fluctuation near the pitch point. Actually, the relative velocity changes its direction several times near the pitch point, as shown in the magnified inset in Fig.6. These repeated changes in the direction of the relative velocity are also reflected in the friction direction, as shown in Fig.10. Fig.7(a) shows the corresponding velocities of the stick-slip model. The solid points denote the stick state. It is clear that stick occurs in every single mesh period. However, the stick situation in any specific mesh period is different from that in its adjacent mesh periods. The specific situation is the same as that in every other mesh period. Fig.7(b) and (c) show the details in rectangles A and B of Fig.7(a) Fig. 7 Velocities of the stick-slip model four times in the single mesh period marked A. Under the stick state, the vibration velocity is opposite to the sliding velocity with equal value. Therefore, the relative velocity exactly equals zero. Stick occurs twice in the single mesh period marked B, and the relative velocity changes its direction rapidly out of the scope of stick. These repeated changes in the directions of relative velocity are also reflected in the friction direction shown in Fig.11.
The corresponding mesh forces of the slip model and stick-slip model are shown in Figs.8 and 9. In Fig.8, mesh forces fluctuate from 25 kN to 60 kN. These fluctuations are mainly caused by the varying mating mesh tooth number, and the mesh force fluctuations are the same among all mesh periods. However, as shown in Fig.9(a), the mesh force fluctuations of the stick-slip model are the same every other mesh period, and the mesh forces between adjacent mesh periods are much different. In the mesh period marked A, mesh forces fluctuate from 0 to 280 kN. The mesh forces fluctuate from 0 to 140 kN in the mesh period marked B. Con- Mesh Force [N] 10 4 Contact Loss Contact Loss Fig. 9 Mesh force of stick-slip model tact loss occurs when the mesh force equals 0. Details on mesh forces in rectangles A and B of Fig.9(a) are shown in Fig.9(b) and (c), respectively. Similar to the stick effect, contact loss occurs four times in mesh period A and twice in mesh period B. Comparisons of the mesh forces of slip and stick-slip models show that, the excitation effect of stick causes contact loss, and the interactions of stick and contact loss dramatically increase the fluctuations of mesh forces. As shown in Fig.6, in the slip model, the relative velocity changes its direction frequently near the pitch point, causing frequent direction changes of the corresponding friction. Over the whole period, friction always consists of sliding friction. Actually, the mating teeth are not always able to break through the frictional resistance at every direction-changing point. The state between the mating teeth depends on whether the instantaneous state satisfies Eq. (14). The mating gear teeth are bonded together once Eq. (14) is satisfied, and friction should be recognized as static friction. Fig.11(a) shows the corresponding friction of the stick-slip model. Similar to the mesh force, fluctuations in friction are dramatically increased by stick. The friction in the adjacent mesh periods are much different. Details of periods A and B in Fig.11(a) are shown in Fig.11(b) and (c), respectively. In period A, stick and contact loss occur alternatively, and there exists short slip between stick and contact loss. The interaction of stick, slip and contact loss leads to a much more complex dynamic friction. There are four and two transitions among these three states in periods A and B,respectively. Under the stick state, the mating teeth are bonded together and the friction equals the force that tries to pull them out. Under the slip state, the mating teeth slide one from another, and friction is recognized as sliding friction. As shown in Fig.11(c), the direction of friction under the slip state changes frequently as the direction of the relative velocity changes frequently. Under the contact loss state, the mating teeth lose their contact, and no mesh force or friction exists between them.
Finally, comparisons of DTE are shown in Fig.12. Under the alternative effects of stick, slip and contact loss, the fluctuation in DTE obtained by the stick-slip model is much larger than that obtained by the slip model. The DTE obtained by stick-slip has much more half frequency components of mesh frequency. While the amplitudes of ω m ,2ω m and 3ω m of DTE obtained by stick-slip are slightly lower than that obtained by slip model. It means that parts of the dynamic energy  Fig. 11 Friction of the stick-slip model is transmitted from the multiple components to half components of mesh frequency.

Effects of working conditions on the stick states
In this section, the effects of key parameters, including rotation speed, friction coefficient and applied torque, on the occurrence of stick is discussed. We define two parameters to illustrate the degree of stick and contact loss. The intercept for a sufficiently long period of time in steady-state vibration is denoted by t t . The stick ratio r k is defined as the ratio of the stick time t k in this intercepted period to t t . The contact loss ratio r l is the ratio of the corresponding contact loss time t l to t t .  Fig.13 shows the effects of the rotation speed on stick ratio and contact loss ratio. For the example gear set in this study, stick occurs only when the rotation speed in the low range of [0,300]r/min. Stick occurs in three single ranges in the range of [0,300]r/min, and the trend of the contact loss ratio versus rotation speed is similar to that of the stick ratio. However, there is no certain relation between the stick ratio and rotation speed. Fig.14 shows the bifurcation diagrams with varying rotation speeds. Without considering the stick phenomenon, as shown by the bifurcation diagrams of the slip model in Fig.14(a), the dynamic response is steady and periodic. However,the dynamic response is chaotic in the ranges with the stick effect, as shown by the bifurcation diagrams of the stick-slip model in Fig.14(b). The dynamic response obtained by the stick-slip model   The friction coefficient µ is another key parameter, and Fig.15 shows the effects of µ on the stick ratio and on contact loss ratio. In the range of µ ⊂ [0, 0.01], the stick ratio and contact loss ratio remain at almost 0. As µ increases above 0.01,the stick ratio monotonically increases. The trend of the contact loss ratio is similar to that of the stick ratio.
The applied torque has similar effects to those of the friction coefficient on stick ratio and on the contact loss ratio, as shown in Fig.16. When the applied torque is in the range of [0,3000]Nm, the stick ratio and contact loss ratio remain almost at 0. As the applied torque increasing in ranges larger than 3000, the stick ratio monotonically increases. The trend of the contact loss ratio is similar to that of the stick ratio.
Based on these parametric studies, designers should avoid choosing parameters in the designing process that may later result in stick.

Conclusions
In this paper, the stick-slip phenomenon is studied between mating teeth in a two-gear set. A stickslip dynamic model of a two-gear set is proposed, and time-varying mesh stiffness, tooth separation, friction and the potential stick-slip phenomenon are considered. Considering the contributions of the vibration velocity to the sliding velocity, transition conditions between the stick and slip states are revealed. An example gear set is applied, and the results of numerical integration show that the stick phenomenon has a dramatic effect on gear dynamics. The traditional slip model is not able to capture the stick phenomenon and is not appropriate under conditions with potential stick phenomena. The discussion on the dynamic response of gear transmissions under the stick-slip effect offers a comprehensive understanding of gear scuffing from a dynamic perspective, and several conclusions are obtained.
1. The stick phenomenon has dramatic effects on gear dynamics. The mating gear teeth are bonded together once stick occurs, and the modal shapes are dramatically changed. The stick phenomenon involves excitation and induces considerable fluctuations in the DTE, mesh forces and friction. The traditional slip model, which ignores the stick effect, is not able to capture realistic vibration characteristics in the presence of sticks. 2. Stick causes contact loss, and the interaction effects of stick and contact loss lead to much more complex nonlinear gear dynamics. Under the effects of stick, there exists more than one contact loss in a mesh period, which is different from the assumption in previous studies. 3. Stick occurs only at low rotation speeds. However, there is no certain relationship between the occurrence of stick and the rotation speed. Stick causes the gear dynamic response to be chaotic. Out of the stick speed ranges, the dynamic response is periodic, similar to that obtained by the slip model. 4. Stick is more likely to occur under heavy load and poor lubrication conditions. The stick ratio increases with increasing applied torque and with increasing friction coefficient. The contact loss ratio has similar growth trends with the stick ratio.