Nonlinear Analysis of Dynamic Behavior in a High-Precision Mechanism with a Revolute Clearance Joint

Abstract

Collision and wear are common phenomena in revolute clearance joints, caused by the positional deviation between the journal and bearing centers. The freedom of motion and contact–impact characteristics are reflected in the mechanism’s movement. The penetration behavior of the clearance joint is described using modified elastic contact model combined with Coulomb’s friction. In addition, the dynamic model of a high-precision mechanism with a clearance joint is established using Largrange’s equation. A dynamic performance experiment is also conducted. The results prove the validity of the proposed method. The kinematic accuracy of this mechanism is then used to evaluate the stability and motion error in a case study. Furthermore, the influence of the clearance joint on the dynamic behavior of the high-precision mechanism is thoroughly analyzed. The results show that the fluctuation range of the slider’s dynamic repeated precision for slider is only 0.022 mm under high-speed conditions, meeting the design requirement.

1. Introduction

High-frequency contact changes the clearance value of the revolute joint. In addition, the expansion of the motion space increases the nonlinear characteristics of mechanism, especially in actual machine production [1,2,3,4,5]. A topical issue is the influence of the clearance joint on the motion trajectory of the mechanism under different working conditions. The dynamic accuracy prediction of high-precision mechanisms is an essential evaluation factor in machine design [6,7,8,9,10]. In addition, the expression of the contact feature and the modeling methods of mechanical systems should be further developed to represent the performance and trajectory of the mechanism.

The description of contact–impact feature is the first important issue in the dynamic analysis of mechanisms with revolute clearance joints [11,12,13]. Machado [14] compared the expression of normal force model, which illustrates the applied force between solids during the collision process. It was revealed that deformation and energy loss were related to the parameters of the contact elements. For low coefficient-of-restitution impact, Gharib [15] built a restitution coefficient model based on Newton’s model, Poisson’s model and the energetic model. Momentum transmission was used to describe the impulse–velocity relationship. Wang and Li [16] described the clearance joint using a spring-damping model. The deformation of the mass-spring was defined as the impact penetration, and the rebound characteristics were determined by the damping coefficient. Considering the energy balance law, Hu [17] represented the impact process with the restitution coefficient. The relationship between penetration value and restitution velocity was given, which was used as the variable for the damping coefficient. Alves [18] studied the viscoelastic contact characteristics of solids. The effects of the restitution coefficient on the hysteresis damping factor were shown, providing a reference for the contact force model. Based on the finite element method, Tian [19] described the contact between solids by elastic deformation elements, and a hemispherical lubricant was assumed in the clearance joint to represent the elastohydrodynamic behavior of the lubricated joint. For dissipative contact features, Li [20] introduced an equivalent damping factor into the contact force model, which improved the suitability of the damping coefficient. According to nonlinear contact theory, Li [21] presented the relative positional relationship of clearance joint elements using a finite element methodology. The superiority and practicality of the proposed method were demonstrated in a case study. Based on the Gauss distribution function, the surface micro-structure feature was introduced into the contact solids by Cui [22], and the contact force was illustrated using the Greenwood–Williamson model. The simulation results displayed the transient dynamic response in high-frequency collisions.

In addition, the modeling methods of multibody systems with clearance joints have received more attention [23,24,25,26]. Chen [27,28] built the constraint equation of a spherical clearance joint using the L-N contact model, and the 4UPS-RPS system was obtained by the Newton–Euler equation. The results illustrated that vibration and chaos of the mechanism would be caused by the clearance joint. Considering the influence of time-varying load, Xiao [29,30] expanded the modeling method of a double-slider-crank mechanism with a translational joint. The dynamic equations were given, and the vibration response of the mechanism was revealed. A 3D clearance joint was introduced into the kinematic model of a multibody system by Bai [31,32]. The axial deviation of the clearance joint was described in two local coordinated systems, and the motion state of the multibody system was identified by the relative displacement. Based on the LuGre friction law, Tan [33] revised the friction force model and built the dynamic equations of a slider-crank mechanism using Lagrange’s equations. Simulation results and experiment test represented the dynamic stabilization of the mechanism, which is a key element in mechanism failure. Farahan [34] investigated the dynamic response of a four-bar mechanism with a clearance joint, and the dynamic model was solved using the Runge–Kutta approach. The results were analyzed using the fast Fourier transformation method, and the frequency spectrum of the mechanism revealed the bifurcation characteristics. Zhao [35] developed error modeling for a multi-closed-loop deployable mechanism, and the angle error was added to the kinematic equation. The case study illustrated that link deviation would appear due to the clearance, and impact force existed during the motion of the mechanism. Qian [36] proposed an interesting analysis approach for predicting the stability and accuracy of a planar multibody system with a clearance joint. According to information entropy theory, singular spectrum entropy was used to predict the probability distribution of motion error. Then, the complex contact–impact feature was explained, and the nonlinear evaluation of the mechanism could be obtained. Wang [37] conducted a comparative study of planar multibody systems. The experiment test showed the fluctuation characteristics of mechanisms with clearance joints, and a smaller clearance size alleviated the stronger amplitude and frequency in the inflection point.

Compared with the demonstration model, the high-precision mechanism in a press system has distinct characteristics. Actual industrial products should maintain a stable motion trajectory under complex working conditions. However, traditional evaluation models cannot be provide an accurate prediction for high-precision mechanisms in press systems. This work focuses on demonstrating and comparing the influence of the clearance joint on the dynamic behavior of high-precision mechanisms. Based on viscoelastic constitutive contact theory, modified contact force and friction models are presented. Moreover, the contact characteristics of clearance joints are embedded into the dynamic expressions of the high-precision mechanism, which are established by the Largrangian method. Then, the engineering application is taken into account to represent the motion trajectory of this mechanism, and the influence prediction of parameters can also be conducted and discussed.

2. Mathematical Model of Mechanism

2.1. Model of Contact and Friction

The contact and friction phenomena can be found in the clearance joint, and it is necessary to calculate the motion accuracy of the mechanism. The motion state of the journal and bearing is shown in Figure 1. It is clear that the relationship between the journal and bearing can be described as separation or contact [38,39,40]. During the contact of the solids, deformation appears at the contact point. Therefore, the distance between the journal center and the bearing center can be used to distinguish the motion state and penetration ($\delta$), which is given as

$$e_{ij}=r_j^O-r_i^O$$

(1)

$$n=e_{ij}/\left|e_{ij}\right|$$

(2)

$$\delta =e-c$$

(3)

where $e_{ij}$ denotes the eccentric vector of journal and bearing, which is related to the eccentricity (e) and clearance (c).

When the contact and impact phenomena appear, the penetration is caused by the contact force [41,42]. The dissipative contact force model can be used to describe the dynamic response of contact solids, which comprises stiffness ($K_n$) and the damping coefficient (D). It is noteworthy that energy loss can be described by the hysteresis damping factor [14,43]. Then, the expression is defined as

$$F_n=K_n{\delta }^{1.5}+D\dot{\delta }$$

(4)

$$K_n={\displaystyle \frac{4}{3\left({\sigma }_i+{\sigma }_j\right)}}\sqrt{{\displaystyle \frac{R_i{R}_j}{R_i-R_j}}}$$

(5)

$${\sigma }_k={\displaystyle \frac{1-{\upsilon }_m^2}{E_m}} (m=i,j)$$

(6)

$$D={\displaystyle \frac{3K_n\left(1-c_e^2\right)}{4{\dot{\delta }}^{\left(-\right)}}}{\delta }^{1.5}$$

(7)

where $E_m$ is the elastic modulus of the contact solids, which is determined by the values for the journal ($E_i$) and the bearing ($E_j$). ${\upsilon }_m$ is Poisson’s ratio, and $c_e$ denotes the restitution coefficient. ${\dot{\delta }}^{\left(-\right)}$ is the initial impact velocity.

Then, contact force model is rewritten as

$$F_n=K_n{\delta }^{1.5}\left[1+{\displaystyle \frac{3\left(1-c_e^2\right)\dot{\delta }}{4{\dot{\delta }}^{\left(-\right)}}}\right]$$

(8)

In the tangential direction, the contact solids exhibit friction characteristics during the collision. Based on Coulomb’s friction theory, the friction force model is given as

$$F_t=-c_f{c}_d{F}_n{\displaystyle \frac{v_t}{\left|{\upsilon }_t\right|}}$$

(9)

where $c_f$ is the friction coefficient, and ${\upsilon }_t$ is the relative tangential velocity.$c_d$ denotes the dynamic correction coefficient, which is limited by the tolerances of the tangential velocity (${\nu }_0$ and ${\nu }_1$). It is given as

$$c_d=\left\{\begin{array}{c}0 if \left|{\upsilon }_t\right|\le {\upsilon }_0\\ {\displaystyle \frac{\left|{\upsilon }_t\right|-{\upsilon }_0}{{\upsilon }_1-{\upsilon }_0}} if {\upsilon }_0\le \left|{\upsilon }_t\right|\le {\upsilon }_1\\ 1 if \left|{\upsilon }_t\right|\ge {\upsilon }_1\end{array}\right.$$

(10)

2.2. Mechanical Equilibrium Equation

Figure 2 shows the schematic of the mechanism with a revolute clearance joint. A clearance joint is defined between the crank and the connecting rod, and the position of the slider changes during the motion [44]. The kinematic relationship can be expressed as

$$\left\{\begin{array}{c}{x}_c=L_{1}sin\theta \\ y_c=L_{1}cos\theta \end{array}\right.$$

(11)

$$\left\{\begin{array}{l}{\dot{x}}_c=L_1\dot{\theta }cos\theta \\ {\dot{y}}_c={-L}_1\dot{\theta }sin\theta \end{array}\right.$$

(12)

$$\left\{\begin{array}{l}{\ddot{x}}_c=L_1\ddot{\theta }cos\theta -L_1{\dot{\theta }}^{2}sin\theta \\ {\ddot{y}}_c=-L_1\ddot{\theta }sin\theta -L_1{\dot{\theta }}^{2}cos\theta \end{array}\right.$$

(13)

$$I_c\ddot{\theta }=F_{xc}{L}_{1}cos\theta +F_{yc}{L}_{1}sin\theta -T_f$$

(14)

where $L_1$ is the length of the crank, and $\theta$ is the rotation angle of the crank. The position of the clearance joint (point c) can be expressed by x_c and y_c. I_c is the inertia of the crank. $T_f$ is the friction torque.

The geometrical parameters of the slider and the clearance joint are shown as

$$\left\{\begin{array}{l}{y}_p=L_{1}cos\theta -L_{2}cos\varnothing \\ {\dot{y}}_p={-L}_1\dot{\theta }sin\theta +L_2\dot{\varnothing }sin\varnothing \\ {\ddot{y}}_p=-L_1\ddot{\theta }sin\theta {-L}_1{\dot{\theta }}^{2}cos\theta +L_2\ddot{\varnothing }sin\varnothing +L_2{\dot{\varnothing }}^{2}cos\varnothing \end{array}\right.$$

(15)

$$\left\{\begin{array}{l}{x}_b={-L}_{2}sin\varnothing \\ y_b=y_p+L_{2}cos\varnothing \end{array}\right.$$

(16)

$$\left\{\begin{array}{l}{\dot{x}}_b={-L}_2\dot{\varnothing }cos\varnothing \\ {\dot{y}}_b={{\dot{y}}_p-L}_2\dot{\varnothing }cos\varnothing \end{array}\right.$$

(17)

$$\left\{\begin{array}{l}{\ddot{x}}_b={-L}_2\ddot{\varnothing }cos\varnothing +L_2{\dot{\varnothing }}^{2}sin\varnothing \\ {\ddot{y}}_b={\ddot{y}}_p-L_2\ddot{\varnothing }sins\varnothing -L_2{\dot{\varnothing }}^{2}cos\varnothing \end{array}\right.$$

(18)

where L₂ and $\varnothing$ are the length and the rotation angle of the connecting rod.

Then, velocity characteristics of the slider and the connecting rod can be given as

$${\left[\begin{array}{c}{\dot{x}}_b\\ {\dot{y}}_b\\ {\dot{y}}_p\\ \dot{\varnothing }\end{array}\right]}_{4\times 1}={\left[\begin{array}{cc}0& -L_{2}cos\varnothing \\ 1& -L_{2}sin\varnothing \\ 1& 0\\ 0& 1\end{array}\right]}_{4\times 2}{\left[\begin{array}{c}{\dot{y}}_p\\ \dot{\varnothing }\end{array}\right]}_{2\times 1}$$

(19)

And the kinetic energy equation of the mechanism is established by

$$T={\displaystyle \frac{1}{2}}\left[{\dot{y}}_p \dot{\varnothing }\right]{\left[K\right]}^T\left[M\right]\left[K\right]\left[\begin{array}{c}{\dot{y}}_p\\ \dot{\varnothing }\end{array}\right]$$

(20)

where ${\left[K\right]}_{4\times 2}$ and ${\left[M\right]}_{4\times 4}$ are the velocity-coefficient matric and the mass matrix, which are defined as

$$\left[K\right]=\left[\begin{array}{cc}0& -L_{2}cos\varnothing \\ 1& -L_{2}sin\varnothing \\ 1& 0\\ 0& 1\end{array}\right]$$

(21)

$$\left[M\right]=\left[\begin{array}{cccc}{m}_2& 0& 0& 0\\ 0& m_2& 0& 0\\ 0& 0& m_p& 0\\ 0& 0& 0& { I}_2\end{array}\right]$$

(22)

where $m_2$ and $m_p$ are the masses of the connecting rod and the slider. ${ I}_2$ denotes the moment of inertia of the connecting rod.

Considering the effects of the element masses, the potential energy of this mechanism is given by

$$V=-m_p\mathrm{g}{y}_p-{\displaystyle \frac{1}{2}}{m}_{1}g{L}_{1}cos\theta -{\displaystyle \frac{1}{2}}{m}_{2}g\left(y_p-L_2\mathrm{c}\mathrm{o}\mathrm{s}k\right)$$

(23)

$$\tan \varnothing ={\displaystyle \frac{L_2\sin k-L_1\cos \theta }{y_p-L_1\cos \theta -L_1\cos k}}$$

(24)

Then, the motion equation of the high-precision mechanism is expressed as

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial \left(T-V\right)}{\partial \dot{q_i}}}\right)-{\displaystyle \frac{\partial \left(T-V\right)}{\partial q_i}}=Q_i^{nc}$$

(25)

where $q_i$ and $Q_i^{nc}$ are the ith independent coordinate component and the generalized force component.

Considering the influence of slider acceleration, the dynamic equation of the mechanism is written using the Lagrange method [45]:

$$\left[\begin{array}{ll}{m}_p+m_2& {-L}_2{m}_{2}sin\varnothing \\ {-L}_2{m}_{2}sin\varnothing & { I}_2+L_2^2{m}_2\end{array}\right]\left[\begin{array}{c}{\ddot{y}}_p\\ \ddot{\varnothing }\end{array}\right]+\left[\begin{array}{cc}0& -\dot{\varnothing }{L}_2{m}_{2}cos\varnothing \\ 0& 0\end{array}\right]\left[\begin{array}{c}{\dot{y}}_p\\ \dot{\varnothing }\end{array}\right]=\left[\begin{array}{c}{F}_{yb}-F_e\\ -F_{xb}{L}_{2}cos\varnothing -F_{yb}{L}_{2}sin\varnothing \end{array}\right]$$

(26)

where $F_e$ is the external load.

3. Dynamic Response Analysis

High-precision press systems are used to manufacture precision components, and the transmission mechanism can illustrate the effects of the clearance joint on the motion trajectory of multibody systems. The structure parameters of this mechanism should be listed by length (L₁ = 0.015 m, L₂ = 0.35 m) and mass (m_crank = 453.87 kg, m_rod = 412.53 kg and m_slider = 2700.71 kg). The density (7800 kg/m³), Poisson’ ratio (0.3) and restitution coefficient (0.9) are defined as the material parameters of the elements. The contact parameters include the maximum damping coefficient (0.1), the maximum penetration value (0.1 mm) and the friction coefficient (0.1). The restitution, damping and friction coefficients are taken from published data [20,43]. In Figure 3, the dynamic response experiment is conducted on the mechanism, and the slider acceleration is collected at different rotational speeds. The acceleration sensor (KISTLER) is installed below the slider, and the data cable is connected with the data acquisition card (NI). The data acquisition system is built in Labview, which performs filtering and noise reduction. Significantly, the acceleration sensor should satisfy the requirement of the maximum measuring range.

The comparison of simulation and experiment is shown in Figure 4. Obvious fluctuation is found during the motion period of the high-precision mechanism, especially near the end of the displacement. This indicates that changes in motion direction could strengthen the contact and impact in the clearance joint. Then, the reverse contact force enhances the transient energy variation, and the original movement trajectory cannot be maintained, which means that the dynamic behavior is unpredictable. At higher rotational speeds, frequency and contact characteristics of the high-precision mechanism are obvious. Based on the enlarged view (100 rpm), the deviation between simulation results and test data is only 0.226 m/s² (7.459%) at the peak point. When the rotational speed increases to 200 rpm, this deviation value is 0.611 m/s² (9.518%). And the mean value of this deviation is between 0.042 m/s² and 0.168 m/s². The simulation results reveal a similar variation inclination of the test data, and the effectiveness of the proposed model is verified.

Relatively speaking, the slider velocity has the obvious contact response of the high-precision mechanism with a clearance joint. In the ideal joint, the slider velocity follows a smooth route during the motion period, even if it is located at the extreme position. With a smaller clearance size, the slider velocity can reveal the variation in motion state. The gradient of slider velocity also changes, and the folding feature gradually becomes clear. With the growth of clearance size, the curve of slider velocity shows a deformation point. When the rotational speed is 300 rpm, the obvious fluctuation and high-frequency variation in the slider velocity appear.

The coupled effects of clearance size and rotational speed on the movement trajectory of the high-precision mechanism are shown in Figure 5, Figure 6, Figure 7 and Figure 8. Clearance size and rotational speed are important parameters in the nonlinear dynamic analysis. Although the clearance size is smaller in the high-precision press system, wear can enlarge the clearance during operation. In order to show the nonlinear characteristics of the high-precision mechanism clearly, the maximum clearance size is defined as 0.5 mm and the rotational speed is selected based on actual working conditions [10,21,27]. The increase in clearance size strengthens the collision feature of the slider position at the extreme position. At higher rotational speeds, this phenomenon is more pronounced. When the clearance size is less than 0.5 mm, there is a slight deviation of the slider position between the revolute clearance joint and the ideal joint at 50 rpm. When the clearance size is 0.5 mm, the deviation value shows a different trend, and significant fluctuation appears under the same condition. Moreover, the curves of the slider position between the revolute clearance joint and the ideal joint become complete separated under high-speed conditions. The results show that a larger clearance size extends the motion range of the revolute joint, and a higher movement speed increases the inertia force of the element. With the higher speed (300 rpm) and larger clearance size (0.5 mm), the fluctuation and deformation can be found in the curve of the slider position, highlighting the negative influence of this coupled factor.

The slider acceleration can reflect the dynamic response of a high-precision mechanism with a revolute clearance joint. Compared with the ideal joint, obvious fluctuation is found throughout the entire movement period of the revolute clearance joint. Moreover, a stronger peak value can be found at the extreme position. When the machine operates at low speed, a higher transient impact force is generated, and larger deformation occurs in the revolute joint elements. As the rotation speed exceeds 100 rpm, a larger clearance size (0.5 mm) causes a longer vibration period of slider acceleration during system movement. The decay rate of vibration slows down under the same condition, which is related to the energy loss of contact and impact. In addition, it is noteworthy that the larger clearance size (0.5 mm) decreases the vibration frequency of the slider acceleration. This phenomenon can be explained by the fact that an increase in clearance size enlarges the relative motion range between the journal and bearing, extending the motion period and reducing the number of contacts.

As an important evaluation indicator, the dynamic repeated precision at the extreme position represents the manufacturing performance of high-precision press systems. Clearance size and driving speed have a significant impact on the dynamic repeated precision at the extreme position (Figure 9). A larger clearance size not only reduces the dynamic repeated precision, but also causes a drift in the extreme position. The expansion of the motion range of the revolute joint element is a key factor contributing to the decline in dynamic repeated precision. under different working speeds, the average deviation value of the dynamic repeated precision shows a slight decrease. However, the distribution range of the extreme position becomes larger at higher speeds. The dispersion of the extreme position indicates that the nonlinear characteristics of the high-precision mechanism are more pronounced. When the clearance size is 0.05 mm, the fluctuation range of dynamic repeated precision is only 0.0073 mm, but this value increases to 0.0261 mm (0.5 mm). A similar phenomenon appears when changing rotation speed, and the fluctuation range reaches 0.022 mm under high-speed conditions (300 rpm). And motion trajectory of the slider deviates obviously from the planned route.

4. Nonlinear Characteristics Analysis

The motion state of a high-precision mechanism is a key concern in the dynamic evaluation of multibody systems. The center trajectory of the revolute clearance joint can reveal the motion state of the high-precision mechanism. Moreover, Poincaré portraits of the slider position represent the evolution characteristics of the high-precision mechanism with a revolute clearance joint. Therefore, the motion trajectory and Poincaré portraits are employed to conduct the nonlinear dynamic analysis of the high-precision mechanism.

Figure 10 shows the influence of clearance size on the motion trajectory of the revolute clearance joint. The simulation results show that the journal almost moves along the contour line of the bearing when the clearance is small (c < 0.1 mm). An obvious impact phenomenon of the revolute joint elements is found at the extreme position, and penetration occurs between the contact elements. When the clearance increases (c = 0.2 mm), the motion trajectory of the revolute joint center becomes more complex. Meanwhile, the impact velocity and contact frequency increase significantly. And then, the movement route becomes unpredictable, which is an important factor leading to the nonlinear feature of the high-precision mechanism.

With the change in rotation speed, the motion trajectory of the revolute joint center is shown in Figure 11. A lower rotation speed increases the complexity of the motion route for the revolute joint center. Although a higher rotation speed can improve the initial contact velocity, the larger inertia force can maintain the motion state of the revolute joint elements. Then, the contact feature of the revolute joint elements occupies almost the entire motion period. Because the constraint conditions are weak, the penetration value of the contact elements is more significant at lower rotation speeds. The increase in free movement leads to separation between the journal and the bearing.

Figure 12 and Figure 13 show the Poincaré portraits of the high-precision mechanism with a revolute clearance joint. The structure feature (clearance) and the working environment (rotation speed) strongly affect the movement curve of the high-precision mechanism. The strange attractor of the slider position represents different patterns, including periodic and irregular features. When the clearance is small (c < 0.2 mm), most strange attractor points are concentrated in the central area. Although some points are separated, the discrepancy and deviation are small, and the main motion state remains periodic. However, the limited region of the strange attractor expands noticeably, and the structure characteristics become more complex. As a result, the motion of the high-precision mechanism with a revolute clearance joint cannot be predicted effectively. When the multibody system operates at high speed, the distribution of the stranger attractor becomes more concentrated. Although the slider velocity increases, the structural pattern of the Poincaré portraits becomes simple. However, lower speed increases the complexity of the structural pattern in the Poincaré portraits. Meanwhile, the nonlinear characteristics of the high-precision mechanism are enhanced, and the system enters an irregular movement state. This reduces the dynamic repeated accuracy, which is an important factor affecting the manufacturing accuracy of the high-precision press systems.

5. Conclusions

In this work, an applied approach is proposed to conduct the dynamic behavior and movement characteristics of a high-precision mechanism with a clearance joint. The relationship between the extrusion load and the contact characteristics is presented, and the stiffness coefficient is obtained. Then, the dissipative contact force model is introduced into the Largrane’s equations of the high-precision mechanism. The nonlinear dynamic analysis of the high-precision mechanism is performed using the proposed model. Under different conditions (clearance size and rotation speed), the engineering application study is conducted to illustrate the practicality of the proposed method.

The dynamic response analysis of the high-precision mechanism shows that the coupled effects of clearance size and rotation speed cannot be ignored. An increase in clearance size separates the movement trajectory of the high-precision mechanism with a revolute clearance joint from that of the ideal joint. A higher rotation speed not only increases contact velocity, but also increases contact frequency. With regard to the coupled effects of clearance size and rotation speed, the movement characteristics are sensitive to variations in the basic conditions. A dynamic performance experiment of the high-precision mechanism is also conducted, and the deviation value is only 0.611 m/s², confirming that the proposed method provides an accurate prediction. Furthermore, the complex nonlinear behavior reveals that the movement trajectory of the high-precision mechanism with a clearance joint becomes unpredictable. Meanwhile, the larger inertial force can restrain the expansion of kinematic instability for the high-precision mechanism under specific conditions.

Moreover, the dynamic repeated precision at the extreme position is the main indicator for evaluating the machining accuracy and stability of press systems, which is different from traditional mechanisms. Therefore, the applied prediction of the high-precision mechanism can be evaluated using the dynamic repeated precision under specific operating conditions. Unfortunately, the lubricating effects cannot be considered in this study, although lubrication would improve the stability of high-precision mechanisms. In the future, the dynamic model of a high-precision mechanism with a lubricated revolute joint will be developed.