# Determination of Kinematic and Dynamic Characteristics of Oscillating Conveyor Mechanism
## Appendix B  
The equation describes the differentiation of the moment of inertia \(I_{k}(\varphi_{1})\) with respect to the angular displacement \(\varphi_{1}\), accounting for various components of the system, such as link lengths, angular velocities, and the kinematic ratios between the joints. Let's break down the terms and explain each part of the differentiation:  
\[\frac{dI_{k}(\varphi_{1})}{d\varphi_{1}}=2m_{2}\sqrt{\big{(}L_{1}sin\varphi_{1}+ L_{2}^{*}sin\varphi_{2}u_{21}\big{)}^{2}+\big{(}L_{1}cos\varphi_{1}+L_{2}^{*}cos \varphi_{2}u_{21}\big{)}^{2}}\omega_{1}.\]  
\[\cdot\frac{\Big{\{}\omega_{1}\big{[}L_{1}(cos\varphi_{1}-sin\varphi_{1})+L_{2}^{*}u _{21}^{2}(cos\varphi_{2}-sin\varphi_{2})\big{]}+L_{2}^{*}\frac{du_{21}}{d \varphi_{1}}(sin\varphi_{2}+cos\varphi_{2})\Big{\}}}{\sqrt{\big{(}L_{1}sin\varphi_{1 }+L_{2}^{*}sin\varphi_{2}u_{21}\big{)}^{2}+\big{(}L_{1}cos\varphi_{1}+L_{2}^{*}cos \varphi_{2}u_{21}\big{)}^{2}}}+2l_{52}\frac{L_{1}}{L_{2}}\frac{sin(\varphi_{5}- \varphi_{1})}{sin(\varphi_{2}-\varphi_{1})}.\]  
\[\cdot\frac{L_{1}}{L_{2}}\bigg{[}\frac{us_{1}\cos(\varphi_{5}-\varphi_{1})-cos( \varphi_{5}-\varphi_{1})}{sin(\varphi_{5}-\varphi_{2})}-\frac{[u_{21}cos(\varphi_{5}- \varphi_{2})-u_{15}cos(\varphi_{5}-\varphi_{2})]sin(\varphi_{5}-\varphi_{1})}{sin^{2}( \varphi_{5}-\varphi_{2})}\bigg{]}+\]  
\[+2m_{3}\omega_{1}^{2}\Big{[}-L_{1}(cos\varphi_{1}+sin\varphi_{1})-L_{2}u_{21}^{2}( cos\varphi_{2}-sin\varphi_{2})+L_{2}\frac{du_{21}}{d\varphi_{1}}(cos\varphi_{2}-sin \varphi_{2})+\frac{du_{51}}{d\varphi_{1}}(acos\omega_{5}t-bsin\omega_{5}t) \Big{]}\]\[+2(I_{\rm SF}+I_{6G})\frac{L_{1}}{L_{5}}\bigg{\{}\frac{cos(\varphi_{1}-\varphi_{2})u _{21}-cos(\varphi_{1}-\varphi_{2})}{sin(\varphi_{2}-\varphi_{5})}-\frac{[\cos( \varphi_{1}-\varphi_{2})u_{21}-cos(\varphi_{1}-\varphi_{2})sin(\varphi_{2}- \varphi_{1})]}{sin^{2}(\varphi_{2}-\varphi_{5})}\bigg{\}}\frac{L_{1}\frac{sin( \varphi_{2}-\varphi_{1})}{sin(\varphi_{2}-\varphi_{5})}}{L_{5}\frac{sin( \varphi_{2}-\varphi_{5})}{sin(\varphi_{2}-\varphi_{5})}}\]  
\[=2m_{2}\omega_{1}^{2}\bigg{[}L_{1}(cos\varphi_{1}-sin\varphi_{1})+L_{2}^{*}u_{ 21}(cos\varphi_{2}-sin\varphi_{2})+L_{2}^{*}\frac{du_{21}}{d\varphi_{1}}(sin \varphi_{2}-cos\varphi_{2})\bigg{]}+\]  
\[+2I_{\rm S2}\frac{L_{1}^{2}}{L_{2}^{2}}\frac{sin(\varphi_{5}-\varphi_{1})}{sin (\varphi_{2}-\varphi_{5})}\bigg{\{}\frac{u_{51}cos(\varphi_{1}-\varphi_{5})- cos(\varphi_{5}-\varphi_{1})}{sin(\varphi_{2}-\varphi_{5})}-\frac{[u_{21} cos(\varphi_{5}-\varphi_{2})-u_{51}cos(\varphi_{5}-\varphi_{2})]sin( \varphi_{5}-\varphi_{1})}{sin^{2}(\varphi_{5}-\varphi_{2})}\bigg{\}}-\]  
\[-2m_{3}\omega_{1}^{2}\bigg{[}L_{1}(cos\varphi_{1}+sin\varphi_{1})+L_{2}^{*}u_{ 21}^{2}(cos\varphi_{2}-sin\varphi_{2})-L_{2}\frac{du_{21}}{d\varphi_{1}}(sin \varphi_{2}-cos\varphi_{2})-\frac{du_{51}}{d\varphi_{1}}(a\alpha cos\omega_{5 }t-bsin\omega_{5}t)\bigg{]}\]  
\[+2(I_{4F}+I_{5G})\frac{L_{1}^{2}}{L_{4}^{2}}\frac{sin(\varphi_{2}-\varphi_{1})} {sin(\varphi_{2}-\varphi_{5})}\bigg{\{}\frac{cos(\varphi_{1}-\varphi_{2})u_{ 21}-cos(\varphi_{1}-\varphi_{2})}{sin(\varphi_{2}-\varphi_{5})}-\frac{[\cos( \varphi_{1}-\varphi_{2})u_{21}-cos(\varphi_{1}-\varphi_{2})]sin(\varphi_{2}- \varphi_{1})}{sin^{2}(\varphi_{2}-\varphi_{5})}\bigg{\}}.\]  
The equation shows the dynamics of a oscillating conveyor mechanism, where the rate of change of the moment of inertia \(I_{k}(\varphi_{1})\) depends on several terms related to joint angular displacements, velocities, and the applied torques.