Modeling and dynamics characteristics analysis of six-bar rocking feeding mechanism with lubricated clearance joint

Abstract

To research the dynamic response characteristics of the multi-link complex mechanism with lubricated clearance joint, an improved transitional lubrication model considering the roughness of the contact surface is established based on the fuzzy set model of the lubrication state. The dynamic model of a six-bar rocking feeding mechanism with lubricated clearance joint is established, and the influence of dry friction clearance and lubricated clearance on the dynamic response of the mechanism are analyzed. By comparing with the simulation results of ADAMS and Flores lubrication model, the correctness and effectiveness of the established dynamic model and the improved transition lubrication model are verified. The influence of driving speed, clearance size and number of clearances on the dynamic response of the mechanism with lubricated clearance joint is mainly studied. The simulation results show that the driving speed, clearance size and number of clearances have an impact on the dynamic characteristics of the mechanism. With the increase of driving speed and clearance value, the greater the impact on the mechanism, the increase in the number of clearances, which reduces the stability of the mechanism, and there is interaction between different clearance joints.

Appendices

Appendix 1: Kinematic analysis

According to Fig. 5b, dynamic modeling of the mechanism

$$\left[ {\begin{array}{*{20}c} {x_{1} } \\ {y_{1} } \\ \end{array} } \right] = \frac{{l_{1} }}{2}\left[ {\begin{array}{*{20}c} {\cos \theta_{1} } \\ {\sin \theta_{1} } \\ \end{array} } \right]$$

(33)

$$\left[ {\begin{array}{*{20}c} {x_{2} } \\ {y_{2} } \\ \end{array} } \right] = l_{1} \left[ {\begin{array}{*{20}c} {\cos \theta_{1} } \\ {\sin \theta_{1} } \\ \end{array} } \right] + \frac{{l_{2} }}{2}\left[ {\begin{array}{*{20}c} {\cos \theta_{2} } \\ {\sin \theta_{2} } \\ \end{array} } \right]$$

(34)

$$\left[ {\begin{array}{*{20}c} {x_{3} } \\ {y_{3} } \\ \end{array} } \right] = l_{1} \left[ {\begin{array}{*{20}c} {\cos \theta_{1} } \\ {\sin \theta_{1} } \\ \end{array} } \right] + l_{2} \left[ {\begin{array}{*{20}c} {\cos \theta_{2} } \\ {\sin \theta_{2} } \\ \end{array} } \right]$$

(35)

$$\left[ {\begin{array}{*{20}c} {x_{4} } \\ {y_{4} } \\ \end{array} } \right] = l_{1} \left[ {\begin{array}{*{20}c} {\cos \theta_{1} } \\ {\sin \theta_{1} } \\ \end{array} } \right] + l_{2} \left[ {\begin{array}{*{20}c} {\cos \theta_{2} } \\ {\sin \theta_{2} } \\ \end{array} } \right] + l_{CE} \left[ {\begin{array}{*{20}c} {\cos \theta_{3} } \\ {\sin \theta_{3} } \\ \end{array} } \right] + \frac{{l_{4} }}{2}\left[ {\begin{array}{*{20}c} {\cos \theta_{4} } \\ {\sin \theta_{4} } \\ \end{array} } \right]$$

(36)

$$x_{5} = l_{1} \cos \theta_{1} + l_{2} \cos \theta_{2} + l_{CE} \cos \theta_{3} + l_{4} \cos \theta_{4}$$

(37)

where $x_{i}$ and $y_{i} \left( {i = 1,2,3,4} \right)$ are the displacements in the x-direction and y-direction at the center of mass of the i-th rod respectively, $\theta_{1}$ is the angle of crank 1, $\theta_{2}$ is the angle of connecting rod 2, $\theta_{3}$ is the angle of rocker 3, $\theta_{4}$ is the angle of connecting rod 4, $l_{CE}$ is the length from joint C to joint E, $l_{CD}$ is the length from joint C to joint D, $x_{5}$ is the displacement of the slider 5, $l_{a}$ is the vertical distance from joint A to joint D, the length is 90 mm, $l_{b}$ is the horizontal distance from joint A to joint D, the length is 170 mm.

By calculating the second derivative of Eqs. (33) to (37), the acceleration and angular acceleration of each bar parameter can be obtained

$$\left[ {\begin{array}{*{20}c} {\ddot{x}_{1} } \\ {\ddot{y}_{1} } \\ \end{array} } \right] = - \frac{{l_{1} }}{2}\left[ {\begin{array}{*{20}c} {\cos \theta_{1} } \\ {\sin \theta_{1} } \\ \end{array} } \right]\dot{\theta }_{1}^{2}$$

(38)

$$\left[ {\begin{array}{*{20}c} {\ddot{x}_{2} } \\ {\ddot{y}_{2} } \\ \end{array} } \right] = - l_{1} \left[ {\begin{array}{*{20}c} {\cos \theta_{1} } \\ {\sin \theta_{1} } \\ \end{array} } \right]\dot{\theta }_{1}^{2} - \frac{{l_{2} }}{2}\left[ {\begin{array}{*{20}c} {\cos \theta_{2} } \\ {\sin \theta_{2} } \\ \end{array} } \right]\dot{\theta }_{2}^{2} + \frac{{l_{2} }}{2}\left[ {\begin{array}{*{20}c} { - \sin \theta_{2} } \\ {\cos \theta_{2} } \\ \end{array} } \right]\ddot{\theta }_{2}$$

(39)

$$\left[ {\begin{array}{*{20}c} {\ddot{x}_{3} } \\ {\ddot{y}_{3} } \\ \end{array} } \right] = - l_{1} \left[ {\begin{array}{*{20}c} {\cos \theta_{1} } \\ {\sin \theta_{1} } \\ \end{array} } \right]\dot{\theta }_{1}^{2} - l_{2} \left[ {\begin{array}{*{20}c} {\cos \theta_{2} } \\ {\sin \theta_{2} } \\ \end{array} } \right]\dot{\theta }_{2}^{2} + l_{2} \left[ {\begin{array}{*{20}c} { - \sin \theta_{2} } \\ {\cos \theta_{2} } \\ \end{array} } \right]\ddot{\theta }_{2}$$

(40)

$$\begin{gathered} \left[ {\begin{array}{*{20}c} {\ddot{x}_{4} } \\ {\ddot{y}_{4} } \\ \end{array} } \right] = - l_{1} \left[ {\begin{array}{*{20}c} {\cos \theta_{1} } \\ {\sin \theta_{1} } \\ \end{array} } \right]\dot{\theta }_{1}^{2} - l_{2} \left[ {\begin{array}{*{20}c} {\cos \theta_{2} } \\ {\sin \theta_{2} } \\ \end{array} } \right]\dot{\theta }_{2}^{2} + l_{2} \left[ {\begin{array}{*{20}c} { - \sin \theta_{2} } \\ {\cos \theta_{2} } \\ \end{array} } \right]\ddot{\theta }_{2} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - l_{CE} \left[ {\begin{array}{*{20}c} {\cos \theta_{3} } \\ {\sin \theta_{3} } \\ \end{array} } \right]\dot{\theta }_{3}^{2} + l_{CE} \left[ {\begin{array}{*{20}c} { - \sin \theta_{3} } \\ {\cos \theta_{3} } \\ \end{array} } \right]\ddot{\theta }_{3} - \frac{{l_{4} }}{2}\left[ {\begin{array}{*{20}c} {\cos \theta_{4} } \\ {\sin \theta_{4} } \\ \end{array} } \right]\dot{\theta }_{4}^{2} + \frac{{l_{4} }}{2}\left[ {\begin{array}{*{20}c} { - \sin \theta_{4} } \\ {\cos \theta_{4} } \\ \end{array} } \right]\ddot{\theta }_{4} \hfill \\ \end{gathered}$$

(41)

$$\begin{gathered} \ddot{x}_{5} = - l_{1} \cos \theta_{1} \dot{\theta }_{1}^{2} - l_{2} \cos \theta_{2} \dot{\theta }_{2}^{2} - l_{2} \sin \theta_{2} \ddot{\theta }_{2} - l_{CE} \cos \theta_{3} \dot{\theta }_{3}^{2} - l_{CE} \sin \theta_{3} \ddot{\theta }_{3} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - l_{4} \cos \theta_{4} \dot{\theta }_{4}^{2} - l_{4} \sin \theta_{4} \ddot{\theta }_{4} \hfill \\ \end{gathered}$$

(42)

where $\ddot{x}_{i}$ and $\ddot{y}_{i} \left( {i = 1,2,3,4} \right)$ are the x-direction and y-direction accelerations at the center of mass of the i-th rod respectively, $\dot{\theta }_{i}$ and $\ddot{\theta }_{i} \left( {i = 1,2,3,4} \right)$ are the angular velocity and angular acceleration of the i-th rod respectively, $\ddot{x}_{5}$ is the acceleration of the slider.

Appendix 2: Force analysis

The force analysis of crank 1

$$\left\{ {\begin{array}{*{20}l} { - F_{21x} + F_{61x} = m_{1} \ddot{x}_{1} } \\ { - F_{21y} + F_{61y} = G_{1} + m_{1} \ddot{y}_{1} } \\ {M_{b} - F_{21x} l_{1} \sin \theta_{1} + F_{21y} l_{1} \cos \theta_{1} - m_{1} g\frac{{l_{1} }}{2}\cos \theta_{1} = \left( {J_{1} + \frac{1}{2}m_{1} l_{1}^{2} } \right)\ddot{\theta }_{1} } \\ \end{array} } \right.$$

(43)

The force analysis of connecting rod 2

$$\left\{ {\begin{array}{*{20}l} {F_{21x} - F_{32x} = m_{2} \ddot{x}_{2} } \\ {F_{21y} - F_{32y} = G_{2} + m_{2} \ddot{y}_{2} } \\ {F_{21x} \frac{{l_{2} }}{2}\sin \theta_{2} - F_{21y} \frac{{l_{2} }}{2}\cos \theta_{2} - F_{32x} \frac{{l_{2} }}{2}\sin \theta_{2} + F_{32y} \frac{{l_{2} }}{2}\cos \theta_{2} = J_{2} \ddot{\theta }_{2} } \\ \end{array} } \right.$$

(44)

The force analysis of rocker 3

$$\left\{ {\begin{array}{*{20}l} {F_{32x} - F_{43x} + F_{63x} = m_{3} \ddot{x}_{3} } \\ {F_{32y} - F_{43y} + F_{63y} = G_{3} + m_{3} \ddot{y}_{3} } \\ {F_{43y} l_{CE} \cos \theta_{3} - F_{43x} l_{CE} \sin \theta_{3} + F_{63x} l_{CD} \sin \theta_{3} - F_{63y} l_{CD} \cos \theta_{3} = J_{3} \ddot{\theta }_{3} } \\ \end{array} } \right.$$

(45)

The force analysis of connecting rod 4

$$\left\{ {\begin{array}{*{20}l} {F_{43x} - F_{x} = m_{4} \ddot{x}_{4} } \\ {F_{43y} - F_{y} = G_{4} + m_{4} \ddot{y}_{4} } \\ {F_{x} \left( {\frac{{l_{4} }}{2}\sin \theta_{4} + R_{i} \sin \gamma } \right) + F_{y} \left( {\frac{{l_{4} }}{2}\cos \theta_{4} + R_{i} \cos \gamma } \right) - F_{43x} \frac{{l_{4} }}{2}\sin \theta_{4} - F_{43y} \frac{{l_{4} }}{2}\cos \theta_{4} = J_{4} \ddot{\theta }_{4} } \\ \end{array} } \right.$$

(46)

The force analysis of slider 5

$$F_{x} = m_{5} \ddot{x}_{5}$$

(47)

in Eqs. (43) to (47), $F_{{i\left( {i - 1} \right)x}}$ and $F_{{i\left( {i - 1} \right)y}} \left( {i = 2,3,4} \right)$ are the force components of the B, C and E joints, respectively, $F_{61}$ and $F_{63}$ are the acting forces of the support on the crank 1 and rocker 3, respectively, $J_{1}$, $J_{2}$, $J_{3}$ and $J_{4}$ are the moments of inertia of each member to its center of mass respectively. $m_{i} \left( {i = 1,2,3,4} \right)$ is the mass of each component respectively, $G_{i} \left( {i = 1,2,3,4} \right)$ is the gravity of each component respectively, $\omega_{b}$ is the crank speed, $M_{b}$ is the crank moment.