# Study of Fault Identification of Clearance in Cam Mechanism
## 1 Introduction
### Main Roller Kinematics Analysis  
The velocity decomposition of the main roller in a cam curve slot is illustrated in Figure 6.  
For the K roller group, the pure rolling of the mass transitional velocity matrix V consists of K rows and K columns:  
\[\mathrm{V}=\frac{\mathrm{V}_{\mathrm{z}}}{\mathrm{sin}\,\theta} \tag{1}\]  
where, \(\theta_{\mathrm{i}}\) is the motion pressure angle of the roller (\(\mathrm{diag}(\theta)=[\theta_{1},\theta_{2},\theta_{3},\cdots,\theta_{\mathrm{n }}]\)) and \(\mathrm{V}_{\mathrm{z}}\) is the longitudinal velocity vector of the bolt. If every main roller is pure rolling, the roller group angular velocity matrix \(\omega\) can be represented as:  
\[\omega=\frac{\mathrm{V}}{\mathrm{R}} \tag{2}\]  
where R is the radius of the main roller, \(\epsilon\) is the angular acceleration of the main roller, and it can be represented as:  
\[\epsilon=\frac{\dot{\mathrm{V}}_{\mathrm{z}}}{\mathrm{R}}\psi(\theta)+\frac{ \mathrm{V}_{\mathrm{z}}}{\mathrm{R}}\dot{\psi}(\theta) \tag{3}\]  
where  
\[\psi(\theta)=\csc\theta,\psi^{\prime}(\theta)=-\psi(\theta)\varphi(\theta) \dot{\theta} \tag{4}\]  
The variation \(\mathrm{V}_{\mathrm{z}}\) with the angular velocity of the revolving body \(\omega_{1}\) is expressed as follows:  
\[\mathrm{V}_{\mathrm{z}}=\mathrm{i}\omega_{1} \tag{5}\]  
where the letter i represents the transmission ratio of the revolving body to the bolt body and can be determined by the cam curve design. The acceleration of the bolt body is expressed as follows:  
\[\frac{\mathrm{d}\mathrm{V}_{\mathrm{z}}}{\mathrm{d}t}=\omega_{1}^{2}\frac{ \mathrm{d}\mathrm{i}}{\mathrm{d}\phi}+\mathrm{i}\alpha \tag{6}\]  
Figure 6: Velocity decomposition of the main roller with no clearance. Note: 1—cam curve slot, 2—main roller, F—the friction force, 3—the rear-end surface, and 4—the front-end surface.  
where \(\phi\) is the angle of revolution of the revolving body, and \(\alpha\) is the angular acceleration of the revolving body. The rate of change in the pressure angle matrix \(\theta\) is expressed as follows:  
\[\frac{\mathrm{d}\theta}{\mathrm{d}t}=\omega\dot{\theta} \tag{7}\]  
By substituting Equations (5) and (6) into Equation (3), the roller angular acceleration can be represented as  
\[\epsilon=\frac{\mathrm{i}\omega_{1}{}^{2}\psi(\theta)}{\mathrm{R}}\left[ \frac{\mathrm{i}}{\mathrm{i}}+\frac{\alpha}{\omega_{1}{}^{2}}-\varphi(\theta) \dot{\theta}\right] \tag{8}\]  
where  
\[\varphi(\theta)=\cot(\theta) \tag{9}\]  
Based on Equation (7), with the given cam groove curve and the angular velocity \(\omega_{1}\) of the revolving body, the angular acceleration matrix \(\epsilon\) of the roller group can be obtained under the assumption of pure rolling.  
The inertia force of the main roller is expressed as follows:  
\[\mathrm{F_{g}}=\mathrm{m}\cdot(\omega_{1}{}^{2}\frac{\mathrm{d}i}{\mathrm{d} \phi}+\mathrm{i}\alpha). \tag{10}\]  
The inertia moment \(\mathrm{M_{g}}\) is expressed as follows:  
\[\mathrm{M_{g}}=\frac{\mathrm{i}\omega_{1}{}^{2}\psi(\theta)}{\mathrm{R}}\left[ \mathrm{i}/\mathrm{i}+\alpha/\omega_{1}{}^{2}-\varphi(\theta)\dot{\theta}\right] \tag{11}\]