Optimization Acceleration and Contact Force of Space Slider-Crank Mechanism with Spherical Clearance Joints

The clearance in traditional joints not only helps the link joints move easily but also affects the workability of the mechanical system, such as wear due to friction and the noise caused by vibration due to impact. To reduce the influence of dry friction inside the joint with clearance, lubrication for multiple joints with clearance was studied and the dynamic model of the multi-joint planar mechanism with clearance is constructed based on the dynamic accuracy reliability model and the Lagrange multiplier method for the 2DOF mechanism with 9 links [1]. The clearance model of the through-joint and the forward-joint is used to establish the dynamic equation of the 6-bar planar mechanism based on the Lagrange multiplier method [2]. This model is further confirmed by ADAM and experimentally. The Baumgarte method was used to improve the oscillation of the slider-crank mechanism caused by the translational and revolute clearances. The accuracy of the model was evaluated experimentally [3]. The experimental results also showed that the revolute clearance caused more oscillation than the translational clearance. The contact force model and the internal wear model in a rotary joint with clearance are proposed to determine the wear amount by the non-contact method. A contact force model and an internal wear model in a rotary joint with clearance are proposed to determine the wear amount by a non-contact method [4]. The measured results are compared with the numerical simulation results. The extended alpha method is based on Newton's collision law to describe the dynamic model of a spherical joint with clearance [5]. This method can avoid singularities and invariances. The time method is also used to determine the results of the alpha method. The effects of curvature radius of asperities contact speed, and roughness on dynamic response of rudder loop with revolute joint clearance were determined the Lagrange method [6]. The outcomes identified that when clearance size increased and impacted frequency nonlinearly increased, rudder angle error and contact force increased. The finite particle method is proposed to determine the influence of internal friction in revolute joints with clearance [7]. The proposed model determines that as the clearance size increases, the mechanism oscillates more strongly. As the friction coefficient increases, the oscillation of the mechanism decreases, and as the damping coefficient increases, the energy dissipation increases significantly. The fuzzy sets and fuzzy algorithms were utilized to determine the influence of clearance tolerance on the dynamic behavior of the mechanism [8]. The contact force model was applied to determine the internal contact force in the effective clearance rotary joint for small clearance sizes. The center of curvature method is applied to determine the effect of non-circular joints with clearance on the dynamic behavior of the mechanism [9]. Simulation results confirm that the center of curvature model achieves the dynamic behavior of the multi-body mechanism similar to the traditional geometric center method. The flexibility of the mechanism significantly affected the dynamic behavior of the slider-crank mechanism [10]. The Lagrange multiplier technology [11] was applied to develop a dynamic model of a mechanism with three spherical joints. The results of this model indicate that as the gap size increases the dynamic behavior of the mechanism oscillates chaotically. ADAMS and MATLAB were used to validate the results. The edited friction force model and innovated nonlinear impact force model were applied to investigate the dynamic behavior of a planar mechanical system [12]; the models were then experimentally validated. The influences of clearance size and input velocity on the dynamic response of the slider-crank mechanism were estimated by the nonlinear continuous contact force model and flexible link [13]. The revised Coulomb’s friction law and Archard’s wear law were applied to investigate a rigid-flexible coupling system's dynamic behavior and wear with clearance joints based on the NCF-ANCF formulation [14]. The influences of the clearance size and the cylinder stiffness were analyzed by a modified CF model and LuGre model [15]. The experiment was carried out to validate the effectiveness of the proposed method, as well as to demonstrate a new method to construct the manipulator, which has high computational efficiency and accurately. A dynamic equation with a Lagrange multiplier was established for the spherical clearance joints of the parallel spatial mechanism [16]. The effects of one clearance joint were approximated for the case of an ideal joint. The dynamic behavior of a mechanical system is more strongly affected than those with one clearance joint and an ideal joint. The clearance and the branched-chain elasticity introduced errors to the robot motion, which was investigated by the rigid–flexible coupling dynamic model having clearance and the spatial beam element. The wear of the spherical clearance joint was investigated [17] by the Archard wear model. The edited Flores CF model and the modified Coulomb friction model using the Newton-Euler method, and the 4^th-order Runge-Kutta method were applied to analyze the dynamic model of the parallel mechanism with the spherical clearance joint [18]. The effects of the clearance joint on the wear of the mechanical systems [19] were confirmed by Archard’s theory and Coulomb’s model. The contact collision model, friction model and the modified Coulomb friction model [20] were applied to investigate the effect of friction in the clearance joint. The transfer matrix method was applied to determine the effects of the RCJ on the rigid-flexible coupling multi-link dynamics model. The influences of clearance size on the translation clearance joint and driven speed of the slider-crank mechanism were investigated [21] using the LuGre friction model and the modified L-N contact model. Analysis of design variable on rigid and flexible dynamics of space slider crank mechanism [22]. A grey relational analysis based Taguchi method [23] was utilized to optimize dynamic of space slider-crank mechanism and the stress in the flexible connecting rod. The optimal results were verified by TOPSIS method. The spherical clearance joint model, contact force model and spherical clearance joint lubrication model [24] has been established to describe the dynamics of multi-body systems with spherical joints. The simulation results indicated that spherical clearance joint lubrication model took simulation much more time than the other. In order to reduce the effects of contact force are due to friction causes. Therefore, the improvement lubrication model in spherical clearance joint [25] proposed. The simulation results identified that the effects of clearance size and friction on the acceleration significantly reduced. The effects of the clearance size and friction in spherical clearance joint and revolute clearance joint were analyzed by the Lagrange multiplier technique and Jacobian of constraint matrix [26] and were confirmed by ADAMS and Newton–Euler method.

The novel in this investigation identified as following:

•The model of space slider crank mechanism in this investigation was designed to reduce the weight and was optimized the effects of the design variables on the acceleration of the slider and the contact force in the revolute clearance joint. While the reference [22] only analyzed the effect of the design variable.

•The design model of the space slider crank mechanism in reference [23] designed the working length of the slider to be large. Because the crank was designed in the plane made with the horizontal plan of 135 degrees. While the design model of the investigation, the working length of the slider is lower. Because the crank was designed in the plane made with the horizontal plan of 90 degrees.

•The Runge-Kutta 4 rigid dynamics model in ANSYS, the grey relational analysis based on the Taguchi method [27], and the TOPSIS, WASPAS methods [28] and SAW method [29], and were used to estimate and reduce the influences of clearance size in the revolute and spherical joints with clearance and friction on the dynamic behavior of the space slider-crank mechanism. Other models, such as the analysis of variance, analysis of means, and analysis of the interaction of the design variables, were also applied to verify the design variable influence on the acceleration of the slider and the CF in the RCJ and spherical clearance joints.

•Previous studies mostly used the Lagrange equation or Neton's 2nd law to model and establish the differential equation of motion of the system and used the programming language MATLAB or Fortran to solve this system of equations. This is an effective method. However, this method requires the researcher to be proficient in using programming language to solve the problem. For researchers who do not know how to use programming languages, solving is very difficult. Different from the previous study, in this study, it is proposed to analyze and simulate the dynamics of the mechanical system using ANSYS and the optimization is performed using grey relational analysis methods - Taguchi method, TOPSIS method, SAW method, WASPAS method.

The remaining of the investigation was written as following: The clearance model in the revolute and spherical joints, the mechanism model, the Runge-Kutta 4 model, and the MEREC method are presented in Section 2. The grey relational analysis based on the Taguchi method and the TOPSIS, SAW, and WASPAS methods are presented in Section 3. The results are discussed in Section 4, and the conclusions are presented in Section 5.

3.1 Grey relational analysis

In order to achieve the optimal solution, the smallest acceleration and contact force is best. This is a multi-objective problem that cannot be achieved with the Taguchi method. Therefore, grey relational analysis [27, 40-44], is applied and performed as follows. Firstly, the design variables were selected. After, 27 models were designed by SolidWorks based on the design of experiment result of Minitab software. Finally, acceleration and contact force are recorded from the results of dynamic analysis:

Step 1: Determine the target:

The target: “The largest value is the best.”

$D_i^*=\frac{D_i^{(0)}(k)-\min D_i^0(k)}{\max D_i^{(0)}(k)-\min D_i^{(0)}(k)}$       (8)

The target: “The smallest value is the best.”

$D_i^*=\frac{\max D_i^{(0)}(k)-D_i^0(k)}{\max D_i^{(0)}(k)-\min D_i^{(0)}(k)}$         (9)

where, $D_i^{(0)}(k)$ are the output values. $D_i^{(0)}(k)$ are the acceleration and CF which were obtained by the Runge-Kutta 4 rigid dynamics model in ANSYS.

Step 2: Calculation deviation:

$\Delta_{0 i}=\left\|D_0^*(k)-D_i^*(k)\right\|$       (10)

$\Delta_{\min }=\max _{\forall j \in i} \min _{\forall k}\left\|D_0^*(k)-D_j^*(k)\right\|$      (11)

$\Delta_{\max }=\max _{\forall j \in i} \max _{\forall k}\left\|D_0^*(k)-D_j^*(k)\right\|$       (12)

Step 3: Estimate the grey relational coefficient (GRC) ($\gamma$) as follows:

$\gamma_i(k)=\frac{\Delta_{\min }+\xi \Delta_{\max }}{\Delta_{0 i}+\xi \Delta_{\max }}$       (13)

where, $\xi \in[0,1]$ is the distinguishing coefficient, typically with a value of 0.5.

Step 4: Compute GRG $\psi_i$ as follows:

$\psi_i=\sum_{k=1}^n w_k \gamma_i(k)$       (14)

where, n=27, where w_k is determined in Section 2.4.

Step 5: Determine the maximum value of GRG as the optimal value. The case where this maximum value is achieved is the optimal case.

The signal to noise ratio obtained by “the maximum value is better” [45-48]:

$S / N=-10 \log \left(\frac{1}{n} \sum_{i=1}^n \frac{1}{y_i^2}\right)$       (15)

where, y_i is GRG values:

Determine the forecasted value of GRG:

$\mu_G=G_m+\sum_{i=1}^q\left(\mathrm{G}_0-G_m\right)$       (16)

Compute CI value at $\alpha=0.05$ by employing Eq. (17):

$C I_{C E}= \pm \sqrt{F_\alpha(1, f e) V e\left(\frac{1}{n_{e f f}}+\frac{1}{R_e}\right)}$          (17)

The $F_\alpha(1, f e)$ was obtained from Table B-2 in reference [49].

3.2 TOPSIS method

The decision-making method based on multiple criteria is the technique for order preference by similarity to ideal solution (TOPSIS) invented in 1980. This method helps select an optimal model based on many criteria such as the required slider speed being small, the contact force inside the rotary joint and ball joint of the system must be small, and the system must be durable enough. With different model sizes, different criteria are achieved. It is necessary to determine an optimal model with many criteria or multiple goals. To achieve the desired advantages, the TOPSIS method is very useful in this. In order the optimal case, the TOPSIS method [28, 50-54] is utilized and is implemented:

Step 1: Determine the normalized values of the objective

$n_{i j}=\frac{a_{\mathrm{ij}}}{\sqrt{\sum_{i=1}^n a_{i j}^2}}$      (18)

where, a_ij the acceleration and CF archived by the Runge-Kutta 4 rigid dynamics model in ANSYS.

Step 2: Determine the weighted normalized values of the acceleration and CF

$v_{i j}=\mathrm{w}_k n_{i j}$            (19)

where, w_k is determined in Section 2.4.

Step 3: Determine the biggest and smallest of b_ij

$b^{+}=\left(b_1^{+}, b_2^{+}, \ldots b_n^{+}\right)$        (20)

$b^{-}=\left(b_1^{-}, b_2^{-}, \ldots b_n^{-}\right)$        (21)

Step 4: Determine the values $S_i^{+}$ and $S_i^{-}$ of the optimal criteria

$S_i^{+}=\sqrt{\sum_{j=1}^n\left(b_{i j}-b_j^{+}\right)^2}$       (22)

$S_i^{-}=\sqrt{\sum_{j=1}^n\left(b_{i j}-b_j^{-}\right)^2}$          (23)

Step 5: Determine the values for CC_i

$C C_i=\frac{S_i^{-}}{S_i^{+}+S_i^{-}}$       (24)

Step 6: Determine the biggest value of CC_i as the optimal values. The case where this maximum value is achieved is the optimal case.

3.3 SAW method

In order to increase the reliability of the Grey-Taguchi and TOPSIS methods, the SAW (Simple Additive Weighting) Method commonly used in decision making for multi-objective problems [52] and was carried out step by step:

Step 1: Determine the normalized values of every criterion

The target: “The largest value is the best.”

$n_{i j}=\frac{a_{i j}}{\max a_{i j}}$          (25)

$n_{i j}=\frac{\min u_{i j}}{u_{i j}}$       (26)

where, a_ij the acceleration and CF which are archived by the Runge-Kutta 4 rigid dynamics model in ANSYS.

Step 2: Determine the sum of the weight normalized values

$v_i=\sum_{j=1}^n w_k \cdot n_{i j}$            (27)

where, w_k is determined in Section 2.4.

Step 3: Determine the biggest value v_i as the optimal value. The case where this maximum value is achieved is the optimal case.

3.4 WASPAS method

Besides, optimum results were also performed using the WASPAS method [28, 49, 55, 56]. WASPAS is also one of the most effective decision-making methods for selecting the optimal case that meets multiple criteria or multiple objectives, was performed as follows:

Step 1: Using Eq. (25) and Eq. (26) to determine $n_{i j}$, next determine v_ij, Q_i, and P_i.

$v_{i j}=w_k \cdot n_{i j}$            (28)

where, w_k is determined in Section 2.4.

$Q_i=\sum_{j=1}^n v_{i j}$       (29)

$Q_i=\sum_{j=1}^n v_{i j} P_i=\prod_{j=1}^n\left(v_{i j}\right)^{w_i}$       (30)

Step 2: Determine A_i.

$A_i=\lambda \cdot Q_i+(1-\lambda) \cdot P_i, \lambda=0.9$          (31)

Step 3: Determine biggest value A_i as the optimal value. The case where this maximum value is achieved is the optimal case.

The acceleration and CF which are two criteria that need to be achieved when designing a space slider crank mechanism with design variables being crank length, clearance size, coefficient friction and input velocity. The optimal case is selected as the highest value of the grey relational grade, and the optimal result of the Grey-Taguchi method is determined by the TOPSIS method, SAW method and WASPAS method. Applying the Taguchi method by performing 27 cases is relatively sufficient to determine the optimal case.

Based on the results of the Runge-Kutta 4 rigid dynamics model in ANSYS, the design variables significantly affected on the acceleration and the CF in the RCJ and imperfect spherical and translation joints. The output of the GRA based on the Taguchi method confirmed that the crank length and driving speed significantly affect the acceleration of the slider and CF in the clearance joints. In contrast, clearance size and friction slightly influence the acceleration and CF. Outcomes of the interaction analysis, ANOVA, and regression analysis also confirmed the model outcomes. In addition, the optimal values are compared with predicted values; all of the outputs are in good agreement with the GRA based on the Taguchi method, with an error of less than 4%. The optimal values obtained for the acceleration of the slider and the CF were 32.72 m/s² and 1.3452 kN, respectively. The TOPSIS, SAW, and WASPAS methods also verified the optimal values. All four methods provide outputs that identify the first case as optimal; the combination design includes a crank length of 80 mm, a clearance size of 0.1 mm, a friction coefficient of 0.01, and an input velocity of 800 rpm. To reduce vibration in the mechanical system, acceleration and force must be limited. The proposed method also achieves a reliable assessment result and is confirmed by the decision-making criteria. The deviation between the predicted value and the optimal value is very low. It indicated that the results of the proposed optimization method are reliable. The factors that affected on the vibration of the mechanical system are the selected design variables. The limitation of this study is that the optimal case is within the limit of 27 cases. Compared to other methods, the optimal level of design variables and optimal cases are outside the 27 cases. That means it cannot be ruled out that the optimal case is something else. Because of the limitations of the proposed method in the analysis and discussion section. Therefore, future research will focus on theoretical and experimental research to confirm the results of ANSYS simulation analysis and the results of the optimization methods proposed.