# Determination of Kinematic and Dynamic Characteristics of Oscillating Conveyor Mechanism
## Appendix A  
To compute the derivative of \(\vartheta_{S_{2}}\) and \(\vartheta_{S_{3}}\) with respect to \(\varphi_{1}\), let's start by carefully analyzing the expression and the steps for differentiation. This equation gives the rate of change of the magnitude of the velocity \(\vartheta_{S_{2}}\) and \(\vartheta_{S_{3}}\) with respect to \(\varphi_{1}\), considering both the geometric terms and the influence of \(u_{21}\) and \(u_{51}\) and its derivative:  
\[\frac{d\vartheta_{S_{2}}}{d\varphi_{1}}=\frac{d}{d\varphi_{1}}\bigg{[}\sqrt{ \left(L_{1}sin\varphi_{1}+L_{2}^{s}sin\varphi_{2}u_{21}\right)^{2}+\left(L_{1} cos\varphi_{1}+L_{2}^{s}cos\varphi_{2}u_{21}\right)^{2}}\omega_{1}\bigg{]}=\]\[=\frac{\omega_{1}\Big{[}\big{(}L_{1}cos\varphi_{1}+L_{2}^{*}cos\varphi_{2}u_{21}^{2 }-L_{1}sin\varphi_{1}-L_{2}^{*}sin\varphi_{2}u_{21}^{2}\big{)}+L_{2}^{*}sin \varphi_{2}\frac{du_{21}}{d\varphi_{1}}+L_{2}^{*}cos\varphi_{2}\frac{du_{21}}{d \varphi_{1}}\Big{]}}{\sqrt{\big{(}L_{1}sin\varphi_{1}+L_{2}^{*}sin\varphi_{2}u_ {21}\big{)}^{2}+\big{(}L_{1}cos\varphi_{1}+L_{2}^{*}cos\varphi_{2}u_{21}\big{)} ^{2}}}=\]  
\[=\frac{\omega_{1}\Big{[}L_{1}\big{(}cos\varphi_{1}-sin\varphi_{1}\big{)}+L_{2} ^{*}(cos\varphi_{2}-sin\varphi_{2})u_{21}^{2}+L_{2}^{*}\frac{du_{21}}{d\varphi_ {1}}\big{(}sin\varphi_{2}+cos\varphi_{2}\big{)}\Big{]}}{\sqrt{\big{(}L_{1}sin \varphi_{1}+L_{2}^{*}sin\varphi_{2}u_{21}\big{)}^{2}+\big{(}L_{1}cos\varphi_{ 1}+L_{2}^{*}cos\varphi_{2}u_{21}\big{)}^{2}}}.\]  
\[\frac{d\theta_{S_{3}}}{d\varphi_{1}}=\frac{d}{d\varphi_{1}}\Big{[}\sqrt{ (-L_{1}sin\varphi_{1}-L_{2}sin\varphi_{2}u_{21}+as_{1}cos\varphi_{1}\big{)}^ {2}+\big{(}L_{1}cos\varphi_{1}+L_{2}cos\varphi_{2}u_{21}-bu_{15}sin\varphi_{1} \big{)}^{2}}\omega_{1}\Big{]}=\]  
\[=\frac{\omega_{1}\Big{[}-L_{1}cos\varphi_{1}-L_{2}cos\varphi_{2}u_{21}^{2}-L_ {2}sin\varphi_{2}\frac{du_{21}}{d\varphi_{1}}+as_{1}cos\omega_{5}\frac{du_{2 1}}{d\varphi_{1}}-L_{1}sin\varphi_{1}-L_{2}sin\varphi_{2}u_{21}^{2}+L_{2}cos \varphi_{2}\frac{du_{21}}{d\varphi_{1}}-bsin\omega_{5}\frac{du_{21}}{d\varphi _{1}}\Big{]}}{\sqrt{\big{(}-L_{1}sin\varphi_{1}+L_{2}sin\varphi_{2}u_{21}- au_{15}cos\varphi_{1}\big{)}^{2}+\big{(}L_{1}cos\varphi_{1}+L_{2}cos \varphi_{2}u_{21}-bu_{15}sin\omega_{5}t\big{)}^{2}}}\]  
\[=\frac{\omega_{1}\Big{[}-L_{1}(sin\varphi_{1}+cos\varphi_{1})-L_{2}u_{21}^{2}( cos\varphi_{2}+sin\varphi_{2})+L_{2}\frac{du_{21}}{d\varphi_{1}}(cos\varphi_{2}- sin\varphi_{2})+\frac{du_{21}}{d\varphi_{1}}(acos\omega_{5}t-bsin\omega_{5}t )\Big{]}}{\sqrt{\big{(}-L_{1}sin\varphi_{1}+L_{2}sin\varphi_{2}u_{21}-au_{15} cos\varphi_{1}\big{)}^{2}+\big{(}L_{1}cos\varphi_{1}+L_{2}cos\varphi_{2}u_{21}- bu_{15}sin\omega_{5}t\big{)}^{2}}}.\]  
\[u_{21}=\frac{L_{1}(cos\varphi_{1}sin\varphi_{5}-cos\varphi_{5}sin\varphi_{1})}{ L_{2}(sin\varphi_{2}cos\varphi_{5}-cos\varphi_{2}sin\varphi_{5})}.\]  
Therefore, the simplified final expression for \(\frac{du_{21}}{d\varphi_{1}}\) is:  
\[\frac{du_{21}}{d\varphi_{1}} =\] \[=\frac{L_{1}}{L_{2}}\frac{\big{\{}(us_{1}cos(\varphi_{5}-\varphi_{ 1})-cos(\varphi_{5}-\varphi_{1})|sin(\varphi_{5}-\varphi_{2})-sin(\varphi_{5}- \varphi_{1})|u_{21}cos(\varphi_{5}-\varphi_{1})-u_{21}cos(\varphi_{5}-\varphi_{ 2})|)\big{\}}}{sin^{2}(\varphi_{5}-\varphi_{2})}=\] \[=\frac{L_{1}}{L_{2}}\bigg{\{}\frac{cos(\varphi_{5}-\varphi_{1})u_{ 51}-cos(\varphi_{5}-\varphi_{1})}{sin(\varphi_{5}-\varphi_{2})}-\frac{[u_{21} cos(\varphi_{5}-\varphi_{2})-u_{15}cos(\varphi_{5}-\varphi_{2})]sin(\varphi_{5}- \varphi_{1})}{sin^{2}(\varphi_{5}-\varphi_{2})}\bigg{\}}.\]  
\[u_{51}=\frac{L_{1}(cos\varphi_{1}sin\varphi_{2}-sin\varphi_{1}cos\varphi_{2})}{ L_{5}(cos\varphi_{5}sin\varphi_{2}-sin\varphi_{5}cos\varphi_{2})}.\]  
Therefore, the simplified final expression for \(\frac{du_{21}}{d\varphi_{1}}\) is:  
\[\frac{du_{51}}{d\varphi_{1}} =\] \[=\frac{\cos(\varphi_{2}-\varphi_{1})u_{21}-cos(\varphi_{2}-\varphi_ {1})}{sin(\varphi_{5}-\varphi_{2})}-\frac{sin(\varphi_{2}-\varphi_{1})[cos( \varphi_{5}-\varphi_{2})u_{21}-cos(\varphi_{5}-\varphi_{2})u_{51}]}{sin^{2}( \varphi_{5}-\varphi_{2})}.\]