2.2.1 Virtual prototype model

The three-dimensional dynamic simulation model of the retraction mechanism is developed using Virtual. Lab Motion (LMS) software, considering joint clearance and node position deviation. The main parameters are outlined in Table 1.

Table 1.

Main parameters of the landing gear retraction mechanism

The process of establishing the virtual prototype model of the retraction mechanism is as follows: Firstly, while ensuring the consistency of the mechanism’s inertial mass, simplification is applied to its various components. To facilitate subsequent experiments, the main strut is simplified to a triangular framework containing a locking ring, as shown in Fig. 2. Then, the motion constraints between the components of the mechanism are clarified, with the addition of motion joints and position drivers, as depicted in Fig. 3.

Figure 2.

Retraction mechanism simulation model.

Figure 3.

Constraints among retraction mechanism components.

2.2.2 Joint clearance model

(1) Revolute joint model with clearance

The motion joints within the retraction mechanism are mostly revolute joints composed of a shaft pin and a bushing. Therefore, multiple ball-plane contact models are employed to simulate the hole-shaft clearance. As illustrated in the contact diagram in Table 2, the planes represent the hole wall structure with a diameter of D, while the multiple balls represent the shaft structure with a diameter of d, and the clearance is denoted as $Jc = D - d$ .

Table 2.

Contact modes of revolute joint

In this paper, the revolute joint model only considers the radial clearance. The description and features of the four contact modes [Reference Yan, Xiang and Zhang35] are shown in Table 2. In the table, ${\delta _{r\_f}}$ represents the radial penetration depth of the contact front surface, ${\delta _{r\_b}}$ denotes the radial penetration depth of the contact back surface, vector $e_3^k$ is defined along the axis of the bushing, and vector $e_3^l$ indicates the axis of the shaft pin.

The sign of the penetration depth can reveal the change of the contact state [Reference Flores and Ambrósio36], and contact can be detected as

(1) \begin{align}{\delta _r}\left( {q,t} \right) \cdot {\delta _r}\left( {q,t + \Delta t} \right) \le 0\end{align}

where $q$ is the generalised coordinates.

When Equation (1) is satisfied, the initiation of contact occurs at $t + \Delta t$ . Once contact between the joint members is detected, the contact mode can be ascertained based on the features listed in Table 2.

(2) Normal contact force model

The popular Lankarani-Nikravesh continuous contact force model (L-N model) [Reference Lankarani and Nikravesh37] is employed to calculate the contact force of the contact unit. The model takes into account the effects of material properties, geometry, elastic deformation, and impact velocity on the collision characteristics. Additionally, it offers a more precise account of energy dissipation and transfer throughout the collision process, thereby offering a closer approximation to the actual contact collision situation. The expression for the normal contact force F is given by

(2) \begin{align}F = {K_{st}}{\delta ^{3/2}}\left[ {1 + \frac{{3(1 - c_e^2)\dot \delta }}{{4{{\dot \delta }^{( - )}}}}} \right]\end{align}

where $\delta $ is the local relative penetration, ${c_e}$ is the restitution coefficient, $\dot {\delta} $ stands for the indentation velocity, ${\dot \delta ^{( - )}}$ is the initial indentation velocity. ${K_{st}}$ is the contact stiffness coefficient of the impact body, expressed as follows

(3) \begin{align}{K_{st}} = \frac{4}{{3\pi ({h_i} + {h_j})}}{\left[ {\frac{{{R_i}{R_j}}}{{{R_i} + {R_j}}}} \right]^{1/2}}\end{align}

in which, ${R_i}$ and ${R_j}$ represent the equivalent spherical radii for the impact between two bodies. The material parameters ${h_i}$ and ${h_j}$ can be expressed as

(4) \begin{align}{h_k} = \frac{{1 - \mu _k^2}}{{\pi {E_k}}}(k = i,j)\end{align}

Variables ${\mu _k}$ and ${E_k}$ are, respectively, the Poisson’s ratio and the Young’s modulus associated with each sphere.

(3) Tangential friction model

The friction force is typically modeled using the classical Coulomb friction model, which relates the friction force to the normal contact force through the coefficient of friction. The tangential friction force $f$ is given by

(5) \begin{align}f = - {\mu _f}F\frac{{{v_\tau }}}{{\left| {{v_\tau }} \right|}}\end{align}

where ${\mu _f}$ is the dynamic friction coefficient and ${v_\tau }$ is the relative tangential velocity.

Due to the discontinuity of the Coulomb friction model, it fails to handle cases where the tangential velocity is zero during the contact collision process, leading to low stability in simulation calculations. To address this, many improved Coulomb friction models [Reference Threlfall38–Reference Bhalerao and Kurien Issac41] have been proposed. In this paper, the Rooney-Deravi friction model [Reference Rooney and Deravi39] is adopted to simulate the tangential friction, with the expression provided below

(6) \begin{align}f = - {\mu _f}F\frac{{{v_\tau }}}{{\left| {{v_\tau }} \right|}}\tanh \left( {\frac{{5{v_\tau }}}{{2{v_r}}}} \right)\end{align}

where ${v_r}$ represents a transition velocity that is independent of both friction force and tangential velocity.

The contact force model is illustrated in Fig. 4.

Figure 4.

Friction model.

2.2.3 Node position deviation model

A massless dummy object is added to the supports of the suspension rocker arm and resistance support rod, with translational joints introduced between the support and the dummy object in both horizontal and vertical directions, actuated by position drivers to simulate node position deviation, as depicted in Fig. 5.

Figure 5.

Node position deviation simulation diagram.

From the functional analysis of the retraction mechanism, it is evident that there are a total of seven joint clearances (Jc1∼Jc7) and four node position deviations (Adx1, Adz1, Adx2, Adz2) affecting the locking position, as illustrated in Fig. 2. These joint clearances and node position deviations are parameterised for further analysis.

This study uses the locking position deviation $\Delta $ , which represents the relative distance between the locking ring and the predetermined locking position when the retraction mechanism is retracted, as an indicator of kinematic accuracy. Therefore, sensor axis systems are established on both the locking ring and the locking hook, allowing the dynamic changes in distance between them to be monitored in real-time using motion sensors in the software. Furthermore, motion sensors can track the rotation angles and normal loads of seven joints during the retraction process.

2.3.1 Effects of node position deviation on kinematic accuracy

The variation of locking position offset with node position deviation within the range of −4 to 4 mm is illustrated in Fig. 6 and Fig. 7 while keeping other influencing parameters of the retraction mechanism constant. It’s noteworthy that the locking position offset in the Y direction is negligible and thus not presented.

Figure 6.

Variation of locking position offset in X direction with node position deviation.

Figure 7.

Variation of locking position offset in Z direction with node position deviation.

The following conclusions can be drawn: node position deviation significantly affects the locking position offset in the Z direction but has minimal impact on the locking position offset in the X direction. Specifically, when each node position deviation is 1.6 mm, the X-directional positive locking position offset is less than 0.63 mm, and the absolute value of the Z-directional locking position offset is less than 3.2 mm.

Aiming to mitigate the locking position offset, which is vital for the security of the retraction mechanism, the assembly accuracy of Adx1, Adx2 and Adz2 should be handled with the utmost care during the assembly of the landing gear system. During the maintenance phase, it is crucial to closely monitor the node position deviation of these places to prevent the mechanism locking failure caused by the excessive deformation of the fuselage structure.

2.3.2 Effects of joint clearance on kinematic accuracy

(1) Effect analysis of single joint clearance

The variation of locking position offset with joint clearance within the range of 0.01 to 3 mm is illustrated in Figs 8 and 9, while keeping other influencing parameters of the retraction mechanism constant. The locking position offset in the Y direction does not exceed 0.01 mm and therefore is not displayed.

Figure 8.

Variation of locking position offset in X direction with joint clearance.

Figure 9.

Variation of locking position offset in Z direction with joint clearance.

The following conclusions can be drawn: the impact of joint clearances on the locking position offset in the X direction is less significant than their impact on the locking position offset in the Z direction. Among the joint clearances, Jc3 has the most significant influence on the locking position offset, causing a positive locking position offset of 0.41 mm in the X direction and a negative locking position offset of 2.62 mm in the Z direction when its value is set to 1.6 mm. Joint clearances Jc1, Jc2 and Jc7 also have a notable impact on the locking position offset in the Z direction, causing a negative locking position offset of over 2.25 mm when their value is set to 1.6 mm. However, the influence of joint clearances Jc4, Jc5 and Jc6 on the locking position offset is relatively minor.

(2) Coupled effect analysis of joint clearance

In practical scenarios, joint clearances at various locations are subjected to varying degrees of wear, which may involve error compensation or accumulation. Hence, it is essential to conduct a coupled effect analysis of joint clearances based on the simulation model. Following engineering expertise, two joint clearances are selected from the seven positions, each set at a value of 1.60 mm, while the remaining joint clearances are set to 0.01 mm, resulting in a total of 21 conditions. The notation ‘1&2’ denotes the joint clearances Jc1 and Jc2 acting simultaneously. The locking position offsets obtained from simulations under coupled clearance conditions are illustrated in Figs 10 and 11. The locking position offset in the Y direction, not exceeding 0.05 mm, is omitted from the presentation.

Figure 10.

Locking position offset in X direction under coupled clearance conditions.

Figure 11.

Locking position offset in Z direction under coupled clearance conditions.

The following conclusions can be drawn: when the joint clearance Jc3 acts in conjunction with Jc1, Jc2 and Jc7, the positive X-directional locking position offsets are 0.096 mm, 0.106 mm and 0.273 mm, respectively, all less than when Jc3 acts alone. Joint clearances Jc1, Jc2 and Jc7 contribute to reducing the positive X-directional locking position offset. When joint clearances Jc1, Jc2, Jc3 and Jc7 are coupled pairwise, error accumulation occurs, resulting in negative Z-directional locking position offsets exceeding 3.6 mm, greater than when acting individually. When joint clearances Jc4, Jc5 and Jc6 are coupled, the negative Z-directional locking position offsets are all less than 1.6 mm.

In the range of 0 to 3 mm, the values of two joint clearances are sequentially altered, and the total locking position offset is simulated, as depicted in Fig. 12. The total locking position offset under each condition exhibits a linearly increasing relationship with the joint clearance size. The total locking position offset is particularly pronounced when joint clearances Jc1, Jc2, Jc3 and Jc7 are coupled pairwise, with the ‘3&7’ condition being especially prominent. In order to guarantee the locking safety of the retraction mechanism, the fit accuracy of joints 1, 2, 3 and 7 should be focused on in the landing gear design stage. During the maintenance process, it is essential to give priority to inspecting the clearance growth at these joints and to replace the bearings and bushings promptly based on the inspection results.

Figure 12.

Total locking position offset under coupled clearance conditions.

2.4.1 Calculation of joint wear depth

In the prediction process of wear in components such as cams, revolute joints and gears, the Archard model is widely utilised [Reference Alakhramsing, de Rooij, van Drogen and Schipper42–Reference Wang, Zhou, Lei and Liu45]. This model offers the ability to predict wear rates based on factors such as relative motion, load and material hardness, making it highly versatile across various materials and operating conditions. Hence, the Archard wear model is employed to calculate the wear volume of the seven revolute joints, expressed as

(7) \begin{align}V = KF(t){L_s}/H\end{align}

here, V represents the wear volume, ${L_s}$ denotes the relative sliding distance of the motion joints, K is the wear coefficient, $F(t)$ represents the normal load between the motion joints, and H indicates the material’s Brinell hardness.

Assuming uniform wear in the motion joints, the wear volume is given by:

(8) \begin{align}V = \left[ {\pi {{\left( {r + h} \right)}^2} - \pi {r^2}} \right]b \approx 2\pi rhb\end{align}

where r denotes the radius of the shaft or hole, b represents the contact length between the shaft and hole, and h denotes the radius change of the shaft or hole (i.e. the joint wear depth).

The relative sliding distance of the motion joints per unit time is calculated as

(9) \begin{align}{L_s} = r\omega (t)dt\end{align}

here, $\omega (t)$ represents the relative angular velocity of the motion joints during the retraction or extension process, measured in rad/s.

By combining Equations (7) to (9), the joint wear depth can be obtained as

(10) \begin{align}h = \frac{{KF(t)}}{{2\pi bH}}\omega (t)dt\end{align}

Thus, the joint wear depth within one retraction cycle is determined as

(11) \begin{align}\Delta h = 2\int_0^{{t_r}} {\frac{{KF(t)}}{{2\pi bH}}\omega (t)dt} = \int_0^{{t_r}} {\frac{{KF(t)}}{{\pi bH}}\omega (t)dt} \end{align}

in which, ${t_r}$ represents the time taken by the mechanism for retraction or extension, measured in seconds.

Based on the dynamic model established in Section 2.2, the normal loads and relative angular velocities of those joints in the retraction mechanism are obtained, as shown in Figs 13 and 14.

Figure 13.

Schematic of normal load for each joint.

Figure 14.

Schematic of relative angular velocity for each joint.

According to Equation (11), the joint wear depth during one retraction cycle is calculated, as shown in Fig. 15. It can be observed that joints 1, 2, 6 and 7 exhibit relatively significant wear depths.

Figure 15.

Schematic of joint wear depth.

2.4.2 Effects of node position deviation on joint wear depth

The node position deviation of the retraction mechanism affects the motion and loading of the joints, thereby influencing the wear depth. Therefore, analysing the variation of wear depth in each joint during one retraction cycle with respect to the node position deviation is essential, as depicted in Fig. 16.

Figure 16.

Variation of wear depth in joints 1 to 7 with node position deviation.

The following conclusions can be drawn: joint 7 exhibits consistently greater wear depth compared to other joints, with the highest sensitivity to node position deviation Adx2, followed by Adz1. Joint 6 demonstrates higher sensitivity to node position deviations Adx1 and Adx2. With an increase in node position deviation Adx2, the wear depth of joints 1, 2 and 7 gradually increases, while that of joint 6 decreases gradually.

2.4.3 Effects of joint clearance on joint wear depth

Joint clearances have a significant impact on the load-bearing characteristics and joint wear of the mechanism. Therefore, analysing the variation of joint wear depth with joint clearance during one retraction cycle is crucial, as illustrated in Fig. 17.

Figure 17.

Variation of wear depth in joints 1 to 7 with joint clearance.

Figure 18.

Assembly schematic diagram of test prototype. (a) Test bench (b) Test prototype assembly.

The following conclusions can be drawn: joint 7 exhibits consistently greater wear depth compared to other joints, with the highest sensitivity to joint clearance Jc7 and the second-highest sensitivity to Jc3. Joint 6 demonstrates the highest sensitivity to Jc7, followed by Jc3 and Jc1. With an increase in Jc7, joint 6 exhibits a gradual increase in wear depth, while joint 7 shows a fluctuating decrease. As for Jc3, joints 1, 2, 6 and 7 experience a gradual decrease in wear depth, while joints 3, 4 and 5 show minimal variation.

Furthermore, compared to the initial state, the variation amplitude of wear depth in each joint during one retraction cycle under the influence of node position deviation and joint clearance is small. Thus, the order of wear rates among the joints remains constant, with joints 1, 2, 6 and 7 still accumulating wear at a relatively faster pace. Over time, the continuous accumulation of wear will gradually diminish the load-bearing capacity of the components. To prevent component failure due to wear, special attention must be given to the wear condition of joints 1, 2, 6 and 7, and regular maintenance and inspections should be carried out.

3.2.1 Test for initial condition and node position deviation

In the initial condition, the variation curves of the rotation angle ${\theta _1}$ of the suspension rocker arm and the rotation angle ${\theta _2}$ of the main strut with respect to the actuator stroke L are depicted in Fig. 21. When the retraction mechanism has vertical and horizontal node position deviations, the rotation angles of the suspension rocker arm and the main strut with respect to the actuator stroke vary as depicted in Fig. 22. It can be observed that there is a good agreement between the experimental and simulated motion curves. The mean relative error for the rotation angles of the suspension rocker arm and the main strut is less than 10%.

Figure 21.

Motion curves under the initial condition.

Figure 22.

Motion curves under node position deviation.

3.2.2 Test for joint clearance

Based on the dynamic simulation analysis results, joint clearances Jc2, Jc3, Jc4 and Jc6 on the retraction mechanism are selected as test variables. Different-diameter shafts are used to simulate joint clearances, with each joint clearance being tested at four levels: 0.2, 0.8, 1.2 and 2.0 mm, respectively. The diameters of the holes at joints J2 and J3 are set to 50 mm, with shaft sizes of 49.8, 49.2, 48.8 and 48.0 mm. The diameters of the holes at joints Jc4 and Jc6 are set to 25 mm, with shaft sizes of 24.8, 24.2, 23.8 and 23.0 mm, respectively.

To streamline testing procedures and minimise time investment, an orthogonal test design method is employed, supplemented with three additional sets of typical test conditions. According to the test design table for the retraction mechanism shown in Table 3, periodic cyclic tests are conducted under each set of conditions, and the average locking position offset is measured.

Table 3.

Test design table and test results

During the testing process, abnormal locking is observed in six test conditions, as depicted in Fig. 23. Under the remaining conditions, the retraction mechanism’s locking proceeded normally, as shown in Fig. 24. In two of these test conditions, the mechanism reaches a critical locking state with measured Z-directional locking position offsets of −3.91 mm and −3.95 mm. In the other four test conditions, the locking ring is distant from the designated locking position, resulting in unsuccessful locking. The measured Z-directional locking position offsets in these cases are −4.21 mm, −4.33 mm, −4.87 mm and −5.18 mm, respectively.

Figure 23.

Abnormal locking.

Figure 24.

Normal locking.

In the aforementioned tests, the X-directional locking position offsets are all no higher than 0.74 mm, causing minimal impact on the abnormal locking of the mechanism, and hence, they are not considered significant. Therefore, it can be concluded that the negative Z-directional offset is the primary cause of the failure in the mechanism’s kinematic accuracy, with a lower limit of −3.9 mm for the Z-directional locking position offset.

As shown in Fig. 25, the locking position offsets obtained from tests and simulations exhibit consistent trends across different conditions, with relatively close numerical agreement. The average relative error of Z-directional locking position offset is 2.22%, while that of X-directional locking position offset is 11.7%. This suggests a reasonable degree of accuracy in the simulation model.

Figure 25.

Comparison chart of locking position offset. (a) Projection in X direction. (b) Projection in Z direction.

4.2.1 Neural network model training

Five hundred sets of input data conforming to the distribution of node position deviations and joint clearances are randomly generated. These inputs are processed through the dynamic simulation model to compute corresponding output data, including the Z-directional locking position offset and the wear depths of the seven joints. In order to improve the training performance of the neural network, a BP neural network is constructed by using normalised sample data. The network adopts a three-layer structure, with the hidden layer employing the tangent sigmoid (tansig) function as the transfer function and the output layer using the pure linear (purelin) function. To prevent overfitting, the sample data fed into the network is divided into three sets: 80% for the training dataset and 10% each for the validation and test datasets. The Levenberg-Marquardt training algorithm is applied, with a maximum of 1,000 training epochs, a learning rate of 0.001, and a target minimum error threshold of 1×10^-6. Training is automatically halted when the model’s training error falls below this threshold.

The training performance of the neural network’s output parameters, which include the Z-directional locking position offset and the wear depths of joints 2, 6 and 7, is illustrated in Fig. 30. Notably, the neural network model achieves a regression coefficient as high as 0.99042 and a maximum mean square error of 9.7×10^-4. Moreover, the Z-directional locking position offset error histogram, shown in Fig. 31, demonstrates that the majority of errors fall within a narrow range, indicating a high level of accuracy.

Figure 30.

Comparison between actual and predicted values.

Figure 31.

Error histogram.

The trained neural network effectively captures the complex relationship between node position deviations, joint clearances, and the retraction mechanism’s performance. Therefore, the completion of neural network training is considered successful.

4.2.2 Kinematic accuracy reliability assessment

Based on the random variable distributions shown in Table 4, a large number of sample points are generated using the Monte Carlo method. These samples are then input into the trained neural network model to obtain the corresponding locking position offsets and joint wear depths. By substituting these output parameters into Equation (15), the variation curve of the kinematic accuracy reliability ${\hat P_r}$ against the number of retraction cycles n is obtained, as shown in Fig. 32. For retraction cycles ranging from 0 to 6400, the kinematic accuracy reliability of the mechanism remains above 0.999. However, as the number of retraction cycles exceeds 10,000, the reliability of the mechanism sharply declines due to the cumulative effects of joint clearances. At 11,000 retraction cycles, the kinematic accuracy reliability of the mechanism drops to 0.795.

Figure 32.

Time-dependent reliability of retraction mechanism kinematic accuracy.

4.3.1 Sensitivity analysis of reliability

Given the numerous joint clearances in the retraction mechanism, conducting sensitivity analysis on kinematic accuracy reliability for each joint clearance is necessary to enhance optimisation design efficiency. This analysis helps in selecting appropriate joint clearance parameters as design variables.

Reliability sensitivity is defined as the partial derivative of the failure probability ${P_f}$ with respect to the j-th distribution parameter $\theta _{{x_i}}^{(j)}$ of the random variable ${x_i}$ [Reference Feng, Zhenzhou, Lijie and Shufang51], which can be obtained as follows

(16) \begin{align}\frac{{\partial {P_f}}}{{\partial \theta _{{x_i}}^{(j)}}} = \int_F {\frac{{\partial {f_X}(x)}}{{\partial \theta _{{x_i}}^{(j)}}}dx} = E\left[ {\frac{{{I_F}(x)}}{{{f_X}(x)}}\frac{{\partial {f_X}(x)}}{{\partial \theta _{{x_i}}^{(j)}}}} \right]\end{align}

where ${f_X}(x)$ represents joint probability density function, $F$ is the failure domain, $E[\! \cdot \!]$ denotes the expectation value of operator.

Using the Monte Carlo method, the estimated value of reliability sensitivity is given by

(17) \begin{align}\frac{{\partial {{\hat P}_f}}}{{\partial \theta _{{x_i}}^{(j)}}} = \frac{1}{N}\sum\limits_{k = 1}^N {\left[ {\frac{{{I_F}({x_k})}}{{{f_X}({x_k})}}\frac{{\partial {f_X}({x_k})}}{{\partial \theta _{{x_i}}^{(j)}}}} \right]} \end{align}

When input variables are independent and normally distributed, the reliability sensitivity estimates are

(18) \begin{align}\frac{{\partial {{\hat P}_f}}}{{\partial {\mu _{{x_i}}}}} = \frac{1}{N}\sum\limits_{k = 1}^N {\left[ {{I_F}({x_k}) \cdot \frac{{{x_{ki}} - {\mu _{{x_i}}}}}{{\sigma _{{x_i}}^2}}} \right]}\end{align}

(19) \begin{align}\frac{{\partial {{\hat P}_f}}}{{\partial {\sigma _{{x_i}}}}} = \frac{1}{N}\sum\limits_{k = 1}^N {\left\{ {{I_F}({x_k}) \cdot \frac{1}{{{\sigma _{{x_i}}}}}\left[ {{{\left( {\frac{{{x_{ki}} - {\mu _{{x_i}}}}}{{{\sigma _{{x_i}}}}}} \right)}^2} - 1} \right]} \right\}} \\[3pt]\nonumber\end{align}

Figure 33.

Failure probability at 11,000 cycles variation with mean of clearance.

By sequentially changing the mean and standard deviation of each initial joint clearance, the failure probability of the mechanism is calculated at 11,000 retraction cycles. This procedure is illustrated in Figs 33 and 34. The conclusion drawn is that the mechanism’s failure probability gradually increases as the mean of the initial joint clearances increases, with Jc7 having the greatest impact on failure probability. The failure probability curves with respect to the mean and standard deviation of joint clearances Jc1, Jc2, Jc3 and Jc7 are concave curves. As the mean and standard deviation of the clearances increase, the rate of increase in the mechanism’s failure probability accelerates. The failure probability changes relatively smoothly with variations in the mean and standard deviation of Jc4, Jc5 and Jc6. The mean value of initial joint clearances has a greater influence on failure probability than the standard deviation.

Figure 34.

Failure probability at 11,000 cycles variation with standard deviation of clearance.

When the mean and standard deviation of each initial joint clearance remain consistent, the reliability sensitivity of the clearance parameters during retraction cycles is calculated, as shown in Figs 35 and 36. The conclusion can be drawn that with an increase in the number of retraction cycles, the reliability sensitivity of each clearance mean gradually increases, with the sensitivity of Jc1, Jc2, Jc3 and Jc7 relatively higher. The reliability sensitivity range of each clearance standard deviation is relatively small. When the number of retraction cycles is less than 6200, the reliability sensitivity of both the mean and standard deviation of clearances is less than 0.01.

Figure 35.

Reliability sensitivity of clearance mean during retraction cycles.

Table 5.

Dimensional tolerances for holes and shafts, mm

Figure 36.

Reliability sensitivity of clearance standard deviation during retraction cycles.

According to the above analysis results, it is evident that when accounting for the dynamic evolution of wear clearances in each joint, the clearances Jc1, Jc2, Jc3 and Jc7 exert a significant influence on the kinematic accuracy failure probability of the retraction mechanism. This observation corroborates the conclusion reached in section 2.3.2, which emphasises the necessity to focus on the fit accuracy of joints 1, 2, 3 and 7 in the design phase. Ensuring the meticulous design and maintenance of these critical joints can substantially mitigate the risk of operational failures due to wear clearances, thereby enhancing the overall reliability of the mechanism.

4.3.2 Reliability optimisation model

(2) Objective function

The objective function aims to minimise the accuracy cost [Reference Sfantsikopoulos, Diplaris and Papazoglou54] of the hole-shaft fit at various joints in the retraction mechanism, as represented by

(20) \begin{align}\min {C_F} = {C_S} + {C_H} = \frac{{{C_{OS}} \cdot i{{(D)}^{{r_c} + 1}}}}{{{{({T_S} + {\lambda _S}{T_S})}^{{r_c}}}}} + \frac{{{C_{OH}} \cdot i{{(D)}^{{r_c} + 1}}}}{{{{({T_H} + {\lambda _H}{T_H})}^{{r_c}}}}}\end{align}

where, ${C_F}$ is the fit accuracy cost, ${C_S}$ is the shaft accuracy cost, ${C_H}$ is the hole accuracy cost. $i(D)$ is the ISO standard tolerance factor for the nominal diameter D, ${r_c}$ is a cost sensitivity to the tolerance exponent. ${C_{OS}}$ and ${C_{OH}}$ are process capability coefficients depending on the machining setup. ${T_S}$ and ${T_H}$ are dimensional tolerance intervals of the shaft and hole, respectively. ${\lambda _S}$ and ${\lambda _H}$ are geometric tolerance coefficients of the shaft and hole, respectively.

(3) Constraint conditions

The kinematic accuracy reliability of the retraction mechanism is set as the constraint, ensuring that the reliability still meets the requirement at the expected number of retraction cycles (n=11,000), represented as:

(21) \begin{align}{P_r}^n = 1 - P(g(X,Y(t)) \le 0) \geqslant {P_{re}}\end{align}

where ${P_{re}}$ is the specified kinematic accuracy reliability threshold, set to 0.99.

(4) Optimisation results

Using the test design method, the joint clearance design scheme is determined based on the reliability-based optimisation model of the retraction mechanism. The kinematic accuracy reliability at 11,000 cycles and the accuracy cost of the retraction mechanism are computed. Four optimisation schemes satisfying reliability constraints are obtained as shown in Table 6. Among them, the accuracy cost of scheme 1 and scheme 2 is lower, while the kinematic accuracy reliability of scheme 1 is slightly higher than that of scheme 2. Therefore, scheme 1 is chosen as the best optimisation scheme, achieving the best balance between cost-effectiveness and reliability. The time-dependent reliability curves after optimisation are shown in Fig. 37.

Table 6.

Optimisation schemes

Figure 37.

Time-dependent reliability curves after optimisation.