A rough surface damping contact model and its application in spatial variable stator vane mechanism with dry friction joint

For the mechanical system without oil lubrication, the impact or collision often occurs in the joint clearance, such as the variable stator vane (VSV) mechanism. In the dry friction joint, the damping of the contact bodies has a significant effect on the simulation stability of the tribo-dynamics calculation process. In order to investigate the effect of contact damping and joint clearance on the VSV mechanism performance, this paper proposes a damping contact model on rough surfaces to calculate the clearance contact force between the trunnion and bushing, and the spatial tribo-dynamics of VSV is established by combining this model with spatial dynamics. In addition, the effect of clearance size on the tribo-dynamics is analyzed. The results show that the contact damping must be included in the contact force model of dry friction joints, otherwise the calculation process will oscillate or even not converge, but the contact damping effect can be ignored in the case of lubricating oil. The movement of the trunnion in the bushing is affected by the adjustment drive and the aerodynamic drag, which leads to the wear concentrated on the edge of the bushing. The clearance size affects the distribution of the damping forces and the rigid forces in the contact process, and the damping forces ensure the stability of the VSV tribo-dynamics simulation process. Moreover, with the increase of clearance, the adjustment accuracy of the VSV mechanism is reduced, and the wear of the bushing is intensified. Abstract For the mechanical system without oil lubrication, the impact or collision often occurs in the joint clearance, such as the variable stator vane (VSV) mechanism. In the dry friction joint, the damping of the contact bodies has a significant effect on the simulation stability of the tribo-dynamics calculation process. In order to investigate the effect of contact damping and joint clearance on the VSV mechanism performance, this paper proposes a damping contact model on rough surfaces to calculate the clearance contact force between the trunnion and bushing, and the spatial tribo-dynamics of VSV is established by combining this model with spatial dynamics. In addition, the effect of clearance size on the tribo-dynamics is analyzed. The results show that the contact damping must be included in the contact force model of dry friction joints, otherwise the calculation process will oscillate or even not converge, but the contact damping effect can be ignored in the case of lubricating oil. The movement of the trunnion in the bushing is affected by the adjustment drive and the aerodynamic drag, which leads to the wear concentrated on the edge of the bushing. The clearance size affects the distribution of the damping forces and the rigid forces in the contact process, and the damping forces ensure the stability of the VSV tribo-dynamics simulation process. Moreover, with the increase of clearance, the adjustment accuracy of the VSV mechanism is reduced, and the wear of the bushing is intensified.


Introduction
It is inevitable that there are clearances in the joints of the mechanism, which will cause motion error, vibration, wear, and even reduce the overall performance of the system. In order to study the mechanical structure with clearance, it is essential to establish a reasonable and accurate contact model, especially in the dry friction joint, there is no lubricating medium to provide buffer, and the intense impact or collision often occurs. In this case, if the contact damping of the friction pair material is not considered, the calculation of the contact force will oscillate, and even lead to the nonconvergence of the dynamic simulation process.
Most contact models are based on Hertz contact model, which only study the effect of material properties and geometric features on the contact force, without considering the hysteresis damping of material. Goldsmith [1] used linear spring and linear damping model to describe the collision process, which laid the foundation for the contact model with damping, namely Kelvin-Voigt model. After that, many scholars optimized the linear spring damping model [2][3][4], in which Lankarani and Nikravesh [4] deduced the damping coefficient through the momentum principle in the contact process and proposed a contact force model with hysteresis damping, namely the LN model. The model not only considered the effect of material properties and geometric features on the contact force, but also reflected the influence of motion state on the collision process, so it has been widely used. For example, Zhao et al. [5], Chen et al. [6], Li et al. [7] and Sun et al. [8] used the LN model to calculate the normal contact force of the clearance joints to study the dynamics of the planar mechanical system. Yan [9] and Xiang [10] calculated the contact force at the clearance joints based on the LN model to study the dynamics of the spatial linkage mechanism. The LN model is suitable for the contact of two spheres or cylinders, or the contact of they with a flat plate [11], these research objects are nonconformal contact. However, for the joints with conformal contact, the contact model should be based on the surface-to-surface contact, and in this case, the influence of surface topography cannot be ignored.
On the microscopic scale, the contact between two surfaces is actually the contact of asperities [12,13], and the contact area depends on the surface morphology, so the accurate contact model should be built on the rough surface. Greenwood and Williamson [12] used statistical method and Hertz contact theory to calculate the contact force between rough surface and rigid flat. Later, Greenwood and Tripp [13] obtained the contact calculation methods of rough surfaces, known as the GT model. Moreover, they proposed that two-rough-surface contact can be equal to equivalent single-rough-surface contact. Chang [14] proposed that the contact sphere transited from the elastic to the elastic-plastic regime after the contact interference reached the critical value. Kogut and Etsion [15] established an elastoplastic contact force model for rough surface contact namely the KE model, and clearly distinguished the elastoplastic transition regime according to critical interference. Unfortunately, these excellent contact models did not consider the damping of asperities. Chen et al. [16] and Pan et al. [17] obtained the damping dissipation factor of rough surface in the contact process by fractal method, and derived the dimensionless contact damping from the damping dissipation factor, but the dimensionless damping could not be applied to the calculation of actual contact force. Through the above studies, it can be found that using statistical model to extend the hysteresis damping of a single asperity to the whole surface is a feasible method to establish the contact damping model of rough surface.
Based on this idea, this paper established a rough surface damping contact (RSDC) model and applied it to the tribo-dynamics investigation of the variable stator vane (VSV) mechanism.
The VSV mechanism of compressor is the most important motion adjusting mechanism in the engine, which can prevent the compressor from surging and effectively improve the overall efficiency of the engine by changing the vane angle to adjust the incoming angle of attack, and is widely used in modern aeroengines [18][19][20]. 6 After long-term operation of the VSV mechanism, the rotation of the vane trunnion will inevitably wear the bushing, resulting in the gradual increase of the assembly clearance, which directly affects the motion accuracy and reliability of the mechanism. Therefore, it is of great significance to study the contact and wear of key joints of the VSV mechanism for the development of aeroengine. However, the VSV mechanism runs in a high temperature environment of more than 400 ℃, which makes the joints unable to use the lubricating medium but in dry friction conditions. As mentioned earlier, in this case, the RSDC model must be used to calculate the contact force at the joints. In addition, the bushing of VSV mechanism is generally made of polymer materials to protect the components [21], which means that the contact force of the bushing contains a more significant damping effect. It should also be noted that the trunnion rotates relative to the bushing, the tangential force is much greater than the static friction, and the tangential damping can be ignored [22,23].
The difficulty in the study of tribo-dynamics of the VSV mechanism is that it is a spatial mechanism, which leads to the interaction of clearance friction pairs is threedimensional. Therefore, the accurate analysis model of the mechanism should also be based on the spatial dimension. Compared with planar mechanism, the dynamics formulas of spatial mechanism are more various and complex. Take a torsion rod type VSV mechanism as an example, which adopts spatial multi-rod linkage to adjust vane angle, and the overall structure diagram is shown in Figure 1. The actuator drives the crank to rotate, the crank drives the drag link, and then the drag link drives the unison ring to make circular motion. Driven by the unison ring, the lever arm rotates around the axis of the vane trunnion to adjust the vane. During the process of adjustment, the rotation of the lever arm will cause the axial movement of the unison ring and make the drag link rotate in the spatial dimension. It can be seen that the whole mechanism has obvious spatial motion. Yu et al. [24] simplified the VSV mechanism to a planar mechanism for kinematic analysis, which ignored the transmission of motion in the three-dimensional space, so there were errors in the simulation results. Some scholars analyzed the overall structure of the VSV in the spatial dimension. Reitenbach et al. [25] took the engine fuel consumption as the objective function and designed the optimal setting scheme of stator vane through the aero engine simulation software tool Gas Turbine Laboratory. Wang et al. [26] studied the influence of the penny platform locations on the gas flow through experiments and numerical simulation, and proposed that optimizing the structure and size of the VSV can improve the aerodynamic performance. Tang et al. [27] combined forward and inverse kinematics design method to optimize the global dimensional scale of the VSV mechanism to improve the accuracy of the vane rotation angle. And Riesland [28] used the ADAMS software to simulate the dynamics of the VSV mechanism. However, these researches did not consider the effect of the clearance in the joints on the system, and could not describe the nonlinear dynamics of the VSV mechanism in detail. Therefore, the reasonable tribo-dynamics of the mechanism should be based on the RSDC model on the one hand, and on the other hand, it should be built in the three-dimensional space. In this paper, a rough surface damping contact model is established, and the dynamics simulation of journal bearing is carried out to verify the effect of the model under the condition of dry friction and lubrication. The rough surface damping contact model is applied to the calculate the clearance contact forces between the trunnion and bushing of the VSV mechanism, the kinematics and the dynamics are solved by the spatial coordinate system and the Newton-Euler equation respectively, and then the spatial tribo-dynamics of the VSV is established. Based on the spatial tribo-dynamics, the effect of clearance size on the system is studied and analyzed. In view of the symmetry of the VSV mechanism, the research on single-stage adjusting structure is still representative.

Rough surface damping contact model
In this section, the damping characteristics of asperities are considered in the study of surface contact, and a rough surface damping contact (RSDC) model is proposed, the expression of contact pressure is as follows: Where the contact pressure is composed of rigid pressure and damping pressure , and ̇ represent the relative interference and the interference velocity of the asperities, respectively. The elastic-plastic contact model proposed by Kogut and Etsion is used to obtain the rigid pressure . According to the KE model, the contact process is divided into elastic stage, elastic-plastic stage and plastic contact stage by the critical interference , then can be expressed as [29,30]: where is the elastic contact pressure, 1 and 2 are the elastic-plastic contact pressure, and is the plastic contact pressure, specifically expressed as: where is the hardness of the softer material, is the hardness coefficient can be obtained by = 0.454 + 0.41 , and ν is the Poisson's ratio of the softer material.
is the equivalent areal asperity density, is the equivalent curvature radius of asperity, Poisson ratios of the contact surfaces, respectively. (z * ) is the asperity height probability density function. δ * =z * −h * , z * and h * are the dimensionless asperity height and separation of surfaces respectively. and its dimensionless form * can be expressed as: Lankarani and Nikravesh established a contact force model with hysteresis damping. For two spheres with an equivalent radius of R, the contact force follows the relation according to the LN model: . e is the coefficient of restitution and defined as = −̇+ − , ̇− and ̇+ represent the velocity of initial contact and separation, respectively.
Extend the contact force of the two spheres to the rough surfaces covered with asperities, and then the damping pressure of the surfaces with Gaussian distribution of asperities is obtained: In the LN model, usually given a value of restitution coefficient e to calculate the contact force. However, e depends on the hysteresis characteristic of material, so it is necessary to clarify the value of e. The damping loss factor μ can describe the hysteresis characteristic, and it can be expressed by the ratio of the dissipated energy and the elastic strain energy during the entire contact process: where and can be obtained from the work done by the elastic force and the plastic force in the KE model, respectively. Then and on the unit area can be expressed as [17] : In addition, the dissipated energy and the elastic strain energy are respectively equal to the consumption of kinetic energy and the residual kinetic energy during the contact process, so the damping loss factor μ can also be expressed by the following formula: Then the coefficient of restitution e can be obtained from the damping loss factor . Therefore, the damping pressure on the rough surfaces can be modified as: By introducing Eqs. (2) and (10)  Therefore, the RSDC model should be more accurate in the study of conformal contact friction pairs.

Spatial kinematics model of the VSV mechanism
The single-stage VSV with one vane can be simplified as a spatial multi-rod mechanism, as shown in Figure 2. The rod 1 (AS1) is the driving crank, which is connected with the casing through the rotary joint A, and the rod 2 (S1S2) is the drag link, which is connected with the rod 1 and the unison ring (rod 3) through the spherical joints S1 and S2 respectively. The rod 3 is composed of S1B and BC which are fixedly connected. One end of the rod 3 is connected with the outer casing through the cylindrical joint B, and the other end is connected with the lever arm rod 4 (S3E) through the pin (CD) and the spherical joint S3, wherein S3 can move along the axis of the pin.
Finally, the rod 4 is fixedly connected with the vane, the trunnion of the vane is connected with the casing through the rotary joint E, and the bushing is between the trunnion and the casing.
Regarding all joints as ideal pairs, the degree of freedom of the mechanism can be obtained by the G-K formula: Where M is the degree of freedom of the mechanism, d is the common constraint, n is the number of components (including the frame), g is the number of kinematic pairs, is the degree of freedom of joint , ν is the number of redundant constraints, and is the number of local degrees of freedom. In this mechanism, d = 6, n = 5, g = 6, f1=1, f2=3, f3=3, f4=2, f5=4, f6=1, ν=0, ξ=1, so M = 1.

Fig 2 Schematic diagram of spatial multi-rod mechanism
In Figure 2, the global coordinate system O-XYZ is established, in which the origin is at point O, the Y-axis is along the casing axis, the Z-axis is vertically upward along the trunnion axis, and the X-axis is perpendicular to the OYZ plane. The local coordinate system 1 -1 1 1 takes point 1 as the origin, the 1 -axis is fixedly connected to rod 1, the 1 -axis is along the axis of the casing, and the 1 -axis is perpendicular to the 1 1 1 plane. The local coordinate system 2 -2 2 2 takes point 2 as the origin, and the 2 -axis is fixed on the BC rod, the 2 -axis is along the axis of the casing, and the 2 -axis is perpendicular to the 2 2 2 plane.
During the operation of the VSV mechanism, the rod 1 rotates only around the 1axis, the angle is , the rod 2 rotates and moves in space, and the angle with the 1 -axis is , and the angle between its projection on the o1'x1'y1' plane and x1'-axis is , where o1'x1'y1' is the plane that 1  According to the motion characteristics of the rod 1, the transformation matrix from the local coordinate system 1 -1 1 1 to the global coordinate system O- the relationship between 2 and 2 1 is as follows According to the motion characteristics of the rod 3, the transformation matrix . Simplify Eq. (16) and let: Eq. (17) is solved by dichotomy method to get , and then the vane rotation angle is obtained from the following equation: The detailed derivation process can be seen in Appendix 1.
The rotation velocity , rotation acceleration of body are obtained by numerical method, that is, = , = .

Spatial dynamics model of the VSV mechanism
According to Newton-Euler equation, the forces acting on the components of VSV are described, and the spatial moments can be derived from the following formula: where x, y and z are the distances of the force action point relative to the center of rotation.
As shown in Figure 3, the rod 1 rotates under the action of the driving moment , and is subjected to the constraint forces , , and constraint moments , at the rotary joint A, and the constraint forces 21 , 21 , 21 from the rod 2 at the spherical joint S1. At the center of mass 1 , it is subjected to inertial forces 1 , 1 , gravity 1 and inertial moment 1 . The relationships between the forces are as follows: In the global coordinate system, the coordinates of A and S1 relative to the center of mass 1 are ( 1 , 1 , 1 ) and ( 1 1 , 1 1 , 1 1 ) respectively, then the moment acting on 1 can be expressed as: The rod 2 rotates and moves in spatial dimension, as shown in Figure 4, and is restrained by − 21 , − 21 , − 21 from the rod 1 and 32 , 32 , 32 from the rod 3, respectively. At the center of mass 2 , the rod 2 is subjected to inertial forces In the global coordinate system, the coordinates of S1 and S2 relative to the center of mass 2 are ( 1 2 , 1 2 , 1 2 ) and ( 2 2 , 2 2 , 2 2 ) respectively, then the moment acting on 2 can be expressed as: 4. In addition, the rod 2 applies constraint forces − 32 , − 32 and − 32 to the rod 3, and the rod 3 is subjected to inertial forces 3 , 3 , 3 , gravity 3 and inertial moment 3 at the center of mass 3 . The relationships between the forces are as follows: In the global coordinate system, the coordinates of S2, S3 and the cylindrical pair B relative to the center of mass 3 are ( 2 3 , 2 3 , 2 3 ) , ( 3 3 , 3 3 , 3 3 ) and ( 3 , 3 , 3 ) respectively, then the moment acting on 3 can be expressed as: The calculation formula of friction moment caused by the trunnion rotation is as follows: where and are the radius and the height of bushing, is the circumference coordinate, is the radius of the lever arm support surface, is the distance between the contact point and , which is positive along the Z-axis and negative on the contrary.
is the coefficient of friction between the trunnion and the bushing. The detailed description of these symbols can refer to Figure   7.

Wear model
In this study, it is assumed that wear occurs only on the surface of the bushing due to the material is soft. According to Archard wear model [31], the wear depth ℎ of the bushing can be expressed as: where is the wear coefficient, is the hardness of the bushing, and is the relevant step time. Here, the material of bushing is polyimide, the reference value of hardness is 65 HV, and the wear coefficient is assumed as a constant = 1.1 × 10 −15 2 −1 , the trunnion is made of stainless steel. The wear load is the average contact pressure of the vane in a rotation period , which can be expressed as: , and is the sliding speed of the trunnion, = • .
Therefore, the separation distance ℎ between the trunnion and the bushing can be expressed as: where is the radial clearance, that is, = − .

Numerical method
The tribo-dynamics of the VSV spatial mechanism contains many nonlinear ordinary differential equations (ODEs). To solve these ODEs, the first step is to express

Application and analysis of the RSDC model 4.1 RSDC model under dry friction condition
In this section, the proposed RSDC model is used to simulate a simple physical object: journal bearing under the condition of dry friction and steady loads, in order to show the beneficial effect of the model in dynamics calculation. The motion diagram of the journal in the bearing can be referred to Figure 7 (b). Here, it is assumed that the journal does not rotate, and the center of the journal is subjected to the stable loads = 200 in the X direction and = 100 in the Y direction respectively, so the journal only deviates ( , ) in the X and Y directions without deflection ( , ).
The radial clearance between bearing and journal is set as 20 μm, and the basic parameters of journal and bearing correspond to the trunnion and bushing in VSV mechanism respectively. It is specially noted that the trunnion and the vane are integrated, and their total mass is used as the mass of the journal for simulation. The values of each parameter are shown in Table 1.  However, the calculation result of KE model is oscillatory, and with the calculation process, the oscillation intensifies. Gradually, the deviation of the journal is larger than the radial clearance, which means that the journal penetrates the bearing. The variation of journal deviations is also reflected in the contact forces, as shown in Figure 8 (b).
During the initial contact, the contact forces and increase sharply due to the high-speed impact of journal, but the RSDC model still maintains the stability of the contact forces. In the KE model, only the interference of the surface asperities provides the rigid contact forces, which causes the distance of the journal rebound to be greater than the distance of the initial state, so the interference is more serious when they contact again. Eventually, the journal will penetrate the bearing and produce extremely large contact force. This phenomenon can be attributed to the damping dissipates the kinetic energy of the journal in the contact process, and the steady-state equilibrium can be formed only when the energy decays. In other words, the contact model without damping will satisfy the conservation of kinetic energy or even increase the kinetic energy, so the dynamics of the system is difficult to be stable.

RSDC model under lubrication condition
In the actual working condition, there is lubricating oil in the bearing, and the oil film force provided by the lubricating oil and the contact force provided by the bearing play a supporting role together. Not only that, the lubricating medium can also provide effective damping effect. In order to prove the damping effect of lubricating oil, the journal bearing under the action of stable loads is still selected as the simulation object in this section, and the load values and the parameters of journal bearing are derived from Section 4.1. The average Reynolds equation is used to calculate the oil film pressure, the expression is as follows: where is the average lubricant pressure, the meaning of h is the same as that in Eq. (48), and it can also indicate the oil film thickness here. is the viscosity of lubricating oil, = 0.01 Pa · s. and are the pressure flow factors, is the shear flow factor [35,36], and is the contact factor [37]. shows that the contact models have little influence on the journal motion in the presence of lubricating medium. Under the action of external loads, the journal compresses the oil film, so the lubricating oil rapidly produces the support forces. When the journal starts to contact with the bearing, the contact forces are generated, and the oil film forces are reduced accordingly, as shown in Figure 9 (b) and (c). It can be seen from Figure 9 (b) that when the RSDC model is used to calculate the contact forces, the oil film forces decrease rapidly after contact, while when the KE model is used, the oil film forces decrease slowly. It is also shown in Figure 9 (c) that in the initial stage of contact, the contact forces produced by the RSDC model are significant, while those produced by the KE model are very small. This is because the contact forces in the RSDC model is not only from the deformation of the asperities, but also from the impact velocity.
However, the contact forces in the KE model only depend on the former, so the contact forces are not obvious when the interferences are just generated. Even so, the calculation process of the two models is stable under the action of lubricating oil, and the final contact forces are balanced with the external loads. From these results, it can be seen that the damping effect of lubricating oil is more significant than that of friction pairs materials, which is due to its viscosity. Therefore, the dynamics simulation process of clearance pair with lubricating medium can be stable even if rigid contact model is used. However, the joints of the VSV mechanism are dry friction condition, so the calculation of contact forces should adopt the model with damping.

Rotation angle and velocity of the vane
The material parameters of the trunnion and the bushing can be seen in Table 1, and the structural parameters of the VSV mechanism are shown in Table 2.  In this study, given the input rotation speed = 20 sin 1 (°⋅ −1 ) and the starting angle 0 = −16 ° of the crank to simulate the tribo-dynamics of the VSV mechanism, as shown in Figure 10. The rotation angle of the vane in a complete cycle is calculated according to Eq. (18), and the rotation speed is obtained by differentiating the time with the rotation angle. The results are shown in Figure 11, it can be seen that the rotation range of the vane is 60° to 120°. In addition, the direction 26 of aerodynamic drag will change when the vane reverses. The specific value of aerodynamic drag is shown in Table 3.

Dynamics analysis of the trunnion
Based on the tribo-dynamics and the RSDC model established in this paper, the contact forces between the trunnion and the bushing with a clearance of 20 μm are calculated. As shown in Figure 12, the contact forces and are periodic during the two cycles of adjustment, where presents a positive and negative transformation with the direction of vane rotation, but is almost always negative due to the effect of the main aerodynamic resistance . In addition, is always larger than , which indicates that the interference distance of the asperities is more serious in X direction. The contact forces include rigid forces ( , ) , damping forces ( , ), and the rigid forces are dominant. When the rotation direction of the vane is changed, the action direction of the driving forces and the resistance moment will be reversed, and the interference velocity and distance will also vary accordingly, which cause the fluctuation of rigid forces and damping forces. This phenomenon can be seen in the enlarged pictures in Figure 12. However, the rigid forces and damping forces compensate each other to make the contact forces stable, so that the system still maintains the balance of the forces at the moment of intense contact. In the cyclic operation, the movement of the trunnion in the bushing is shown in Figure 13. Obviously, the deviations of the top center , the middle center , and the bottom center of the trunnion in the X direction are more obvious than that in the Y direction, that is, the contact areas are concentrated in the X direction, which also proves the performance of the contact forces in Figure 12. It can be seen from the continuous trajectories of the three centers that a tilt angle always exists in the whole movement process, which is due to the deflection of the trunnion around the Xaxis caused by the main aerodynamic drag . In brief, the driving forces of the unison ring acting on the lever arm make the trunnion deviate obviously in the X direction, and makes the trunnion deflect around the X-axis.
The deviation velocities of the three centers , , and in the X and Y directions are shown in Figure 14, in which the velocities of the trunnion are small in the continuous adjustment process, but when the vane changes the direction of rotation, the velocities will increase suddenly, corresponding to the variations of displacement in Figure 13. In Figure 14 (a), the velocities of the three centers are approximate in value and consistent in direction, which is due to the adjustment function of the mechanism. However, there are obvious differences in the velocity of the three centers in the Y direction, and the velocity of and is opposite, as shown in Figure 14 (b), which is due to the moment generated by the main aerodynamic drag on the center , resulting in the deflection of the trunnion around the X-axis.

Effect of clearance size
The effect of clearance size on the VSV mechanism is studied in the range of 20 μm to 40 μm. In one cycle, the rotation angle and rotation velocity of the vane with different clearance sizes are shown in Figure 16. Figure 16 (a) shows the variation of rotation angle with time, from which it can be seen that the error of rotation angle caused by the clearance of 20 μm is about 0.02° compared with the ideal pair. Figure   16 (b) shows that the difference of rotation velocity between the clearance of 20 μm and the ideal pair is about 2.5×10 -4 °/s. According to the error of single joint clearance, the error of angle and velocity are about 0.14 ° and 1.75×10 -3 °/s respectively when all joints of the mechanism are considered clearance. Obviously, as the clearance size increases gradually, the error also increases. It can be seen from Figure 16  As mentioned previously, there is a reversal of the vane direction in a complete cycle, and the contact area of the trunnion in the bushing changes accordingly, that is to say, the two surfaces in contact will separate and collide again. As shown in Figure   17, the contact forces, the damping forces, and the rigid forces oscillate in different degrees at the moment of collision. The contact forces and of different clearance sizes are almost equal, but when the clearance is greater than 30 μm, slight oscillations also occur in the continuous adjustment process, as shown in figures (a) and (b), which indicates that the large clearances may weaken the stability of the system.
The damping forces are strongly dependent on the initial contact velocity, and the contact velocity increases with clearance size, so the effect on the damping forces is more obvious. In figures (c) and (d), when the vane is turned to 90 °, the aerodynamic component force changes direction, causing slight relative motion between trunnion and bushing, the relative motion velocity of large clearance is greater, so the damping force is higher than that of small clearance. When the two surfaces collide again after separation, the contact velocity is violent and the interference is serious, so the damping forces and rigid forces increase suddenly, as shown in figures (c) ~ (f). In this process, the damping forces play a role in preventing excessive interference.
Moreover, after reaching the maximum interference distance, the two surfaces begin to separate, and the damping forces also acts in the opposite direction to weaken the The trajectory of the trunnion center and the deflection angle of the trunnion are shown in Figure 18. In a movement cycle, the deviation distances and increase with the increase of the clearance size, as shown in Figure 18 (a). Then, after the separation of the two contact surfaces, the velocity of the next collision in larger clearance will be greater, resulting in more significant damping force. Figure 18 (b) shows the deflection angle of the trunnion, which still follows the rule that the larger the clearance is, the larger the deflection angle is. In addition, the deflection angle is always positive, which is the effect of aerodynamic drag, while the deflection angle is alternating positive and negative, which is the effect of mechanism adjustment. The wear depth ℎ of the bushing is shown in Figure 19. For each size of clearance, the wear of the mechanism running for 100 to 400 cycles is simulated. The (1) For dry friction pairs, the contact damping must be considered in the contact model, otherwise the calculation process will oscillate and not converge. This is quite different from the case with lubricating oil, that is to say, when there is lubricating oil, the contact damping is usually allowed to be omitted, mainly because the lubricating oil provides a more significant damping effect.
(2) During the operation of the VSV mechanism, the driving and aerodynamic drag lead to the deviation and deflection of the trunnion, so the contact area is concentrated on the edge of the bushing, and then wear occurs. In the dry friction pair of the trunnion and the bushing, the damping forces slow down the extrusion and separation of the asperities on the two surfaces, avoid the sudden variation of contact forces, and ensure the stability of the calculation process.
(3) With the increase of clearance size, the contact forces in the calculation process are more likely to fluctuate, and the distribution of the damping forces and the rigid forces is also affected. In addition, the increase of clearance reduces the adjustment accuracy and aggravates the wear.
and from Eqs. (14) and (15), it can be concluded that: