# Design Procedure for Motion Profiles with Sinusoidal Jerk for Vibration Reduction
## 3 Parameter Design for Sinusoidal Jerk Profile
### Analysis of Vibration Characteristics
\begin{array}{l}y_{res}(t)=y(t|t\geq t_{\gamma})\\ =-\frac{1}{\omega_{d}}\left[\begin{array}{l}\int_{t_{0}}^{t_{1}}\frac{A_{peak }}{2}\Bigl{(}1-\cos\frac{\pi}{T_{1}}(\tau-t_{0})\Bigr{)}\sin(\omega_{d}(t-\tau ))d\tau+\int_{t_{1}}^{t_{2}}A_{peak}\sin(\omega_{d}(t-\tau))d\tau\\ +\int_{t_{1}}^{t_{3}}\frac{A_{peak}}{2}\Bigl{(}1+\cos\frac{\pi}{T_{1}}(\tau-t _{2})\Bigr{)}\sin(\omega_{d}(t-\tau))d\tau-\int_{t_{4}}^{t_{5}}\frac{A_{peak }}{2}\Bigl{(}1-\cos\frac{\pi}{T_{1}}(\tau-t_{4})\Bigr{)}\sin(\omega_{d}(t- \tau))d\tau\\ -\int_{t_{5}}^{t_{4}}A_{peak}\sin(\omega_{d}(t-\tau))d\tau-\int_{t_{6}}^{t_{2} }\frac{A_{peak}}{2}\Bigl{(}1+\cos\frac{\pi}{T_{1}}(\tau-t_{6})\Bigr{)}\sin( \omega_{d}(t-\tau))d\tau\end{array}\right]\end{array} . \tag{19}\]  
After a series of mathematical operations involving integration by parts and the sum-to-product trigonometric identity, the above expression becomes the following:  
\[y_{res}(t)=\frac{-4\pi^{2}A_{peak}}{\omega_{d}{}^{2}\Bigl{(}\pi^{2}-(\omega_{ d}T_{1})^{2}\Bigr{)}}\cos\Bigl{(}\frac{\omega_{d}T_{1}}{2}\Bigr{)}\sin\Bigl{(} \frac{\omega_{d}T_{1}}{2}+\frac{\omega_{d}T_{2}}{2}\Bigr{)}\sin\Bigl{(}\omega_ {d}\Bigl{(}T_{1}+\frac{T_{2}}{2}+\frac{T_{3}}{2}\Bigr{)}\Bigr{)}\cos\Bigl{(} \omega_{d}\Bigl{(}(t-t_{0})-\Bigl{(}2T_{1}+T_{2}+\frac{T_{3}}{2}\Bigr{)} \Bigr{)}\Bigr{)}. \tag{20}\]  
In order to eliminate the residual vibrations, the amplitude of the residual-vibration response must equal zero. Hence, settling \(y_{res}(t)=0\) yields the vibration suppression conditions:  
\[\left\{ \begin{array}{l}\pi^{2}-(\omega_{d}T_{1})^{2}\
eq 0\\ \cos\Bigl{(}\frac{\omega_{d}T_{1}}{2}\Bigr{)}\sin\Bigl{(}\frac{\omega_{d}T_{1} }{2}+\frac{\omega_{d}T_{2}}{2}\Bigr{)}\sin\Bigl{(}\omega_{d}\Bigl{(}T_{1}+ \frac{T_{2}}{2}+\frac{T_{3}}{2}\Bigr{)}\Bigr{)}=0\end{array} \right.. \tag{21}\]  
Without a loss of generality, the initial time (\(t_{0}\)) is assumed to be zero. Let \(\mathbb{N}^{+}\) represent the set of positive integers. Then, the solution of (21) can be found as follows:  
\[ \begin{array}{l}\text{C1}:T_{1}=t_{1}=\Bigl{(}k_{1}+\frac{1}{2}\Bigr{)}T_{d},\ \ \ k_{1}\in\mathbb{N}^{+},\\ \\ \text{or C2}:T_{1}+T_{2}=t_{2}=k_{2}T_{d},\ \ \ k_{2}\in\mathbb{N}^{+},\\ \\ \text{or C3}:2T_{1}+T_{2}+T_{3}=t_{4}=k_{3}T_{d},\ \ \ k_{3}\in\mathbb{N}^{+}, \end{array}  \tag{22}\]  
where \(T_{d}=2\pi/\omega_{d}\) is the vibration period of the system. According to the above analysis, if any one of the three conditions ((22), (23), or (24)) is satisfied, complete residual-vibration elimination can be achieved for an undamped system. Hereinafter, these conditions are denoted as C1, C2, and C3, respectively. Interestingly, although all of them result in a vibrationless stop, they display certain fundamentally distinct properties that impact the transient vibration during motion, as illustrated in Figure 5. These properties will be clarified in the following paragraphs. In a similar way, the dynamic response during the constant acceleration segments \([t_{1},\ t_{2}]\) and \([t_{5},\ t_{6}]\) can be derived as follows:  
\[ \begin{array}{rl}y(t|t_{1}\leq t<t_{2})=-\frac{1}{\omega_{d}}\Big{[}\int_{t_{ 0}}^{t_{1}}\frac{A_{peak}}{2}\Big{(}1-\cos\frac{\pi}{T_{1}}(\tau-t_{0})\Big{)} \sin(\omega_{d}(t-\tau))d\tau+\int_{t_{1}}^{t_{2}}A_{peak}\sin(\omega_{d}(t- \tau))d\tau\Big{]}\\ =-\frac{A_{peak}}{\omega_{d}^{2}}\Big{[}1-\frac{\pi^{2}}{\pi^{2}-(\omega_{d}T _{1})^{2}}\cos\Big{(}\frac{\omega_{d}T_{1}}{2}\Big{)}\cos\Big{(}\omega_{d} \Big{(}(t-t_{0})-\frac{T_{1}}{2}\Big{)}\Big{)}\Big{]}\end{array} , \tag{25}\]  
\[