# Residual Vibration Reduction in Flexible Systems Based on Trapezoidal Velocity Profiles
## 3 Optimization of Parameters for Vibration Suppression  
### Analysis of Vibration Suppression Condition  
To analyze the amplitude of the motion-induced vibration, a standard second-order system, which is defined by the transfer function shown in Equation (7) [41], is considered in this study. The system's vibration response \(y(t)\) to a stepwise change in acceleration can be expressed as the sum of individual step responses [42], as shown in Equation (8). \[G(s)=\frac{{\omega_{n}}^{2}}{{s}^{2}+2\zeta{\omega_{n}}{s}+{\omega_{n}}^{2}} \tag{7}\]  
\[ \begin{array}{l}y(t)=\sum\limits_{i=1}^{4}{y_{i}(t)=\frac{-a}{\sqrt{1-\zeta ^{2}}}}e^{-\zeta{\omega_{n}}(t-t_{0})}\sin({\omega_{d}}(t-t_{0})+\phi)u(t-t_{ 0})+\frac{a}{\sqrt{1-\zeta^{2}}}e^{-({\omega_{n}}(t-t_{1})}\sin({\omega_{d}}( t-t_{1})+\phi)u(t-t_{1})\\ +\frac{a}{\sqrt{1-\zeta^{2}}}e^{-\zeta{\omega_{n}}(t-t_{2})}\sin({\omega_{d}}( t-t_{2})+\phi)u(t-t_{2})-\frac{a}{\sqrt{1-\zeta^{2}}}e^{-({\omega_{n}}(t-t_{3})} \sin({\omega_{d}}(t-t_{3})+\phi)u(t-t_{3})\end{array}  \tag{8}\]  
where \(\phi=\tan^{-1}\frac{\sqrt{1-\zeta^{2}}}{\zeta}\), \(\omega_{n}\) and \(\zeta\) are the undamped natural frequency and the damping ratio of the system, respectively. \(u(t)\) is the unit step function and \(\omega_{d}=\omega_{n}\sqrt{1-\zeta^{2}}\) represents the damped natural frequency. Figure 2: Triangular velocity profile. Using trigonometric angle difference formulas and auxiliary angle formulas, the residual vibration response can be written as follows from Equation (8):  
\[y(t)=\frac{-e^{-\zeta\omega_{n}t}}{\sqrt{1-\xi^{2}}}\sqrt{\left(\sum\limits_{i=0} ^{3}A_{i}e^{\zeta\omega_{n}t_{i}}\cos(\omega_{d}t_{i})\right)^{2}+\left(\sum \limits_{i=0}^{3}A_{i}e^{\zeta\omega_{n}t_{i}}\sin(\omega_{d}t_{i})\right)^{2} }\sin(\omega_{d}t-\varphi+\phi) \tag{9}\]  
\[\text{where }A_{0}=a\text{, }A_{1}=-a\text{, }A_{2}=-a\text{, }A_{3}=a\text{, and }\varphi=\tan^{-1}\sum\limits_{i=0}^{3}A_{i}e^{\zeta\omega_{n}t_{i}}\sin( \omega_{d}t_{i})\]  
Thus, the amplitude of residual vibration at motion completion time can be obtained as follows:  
\[Amp=\frac{e^{-\zeta\omega_{n}t_{3}}}{\sqrt{1-\xi^{2}}}\sqrt{\left(\sum\limits_ {i=0}^{3}A_{i}e^{\zeta\omega_{n}t_{i}}\cos(\omega_{d}t_{i})\right)^{2}+\left( \sum\limits_{i=0}^{3}A_{i}e^{\zeta\omega_{n}t_{i}}\sin(\omega_{d}t_{i}) \right)^{2}} \tag{10}\]  
For lightly damped systems, \(\zeta\) is nearly equal to zero, and the above equation can be simplified as:  
\[Amp=\left|\sum\limits_{i=0}^{3}A_{i}e^{i\omega_{n}t_{i}}\right|=a\left|\prod \limits_{i=1}^{2}\left(1-e^{i\omega_{n}t_{i}}z^{i-1}\right)\right|=4a\sin \left(\frac{\omega_{n}t_{1}}{2}\right)\sin\left(\frac{\omega_{n}t_{2}}{2}\right) \tag{11}\]  
It is shown that the amplitude of the motion-induced residual vibration can be represented as a function of the peak acceleration \(a\), time period \(t_{1}\), time period \(t_{2}\), and natural frequency \(\omega_{n}\). This implies that the time parameters can be suitably designed based on the natural frequency to optimize the vibration amplitude. Note that the vibration suppression conditions can be derived by setting the vibration amplitude to zero:  
\[Amp=0\Rightarrow\left\{ \begin{array}{c}\sin\left(\frac{\omega_{n}t_{1}}{2} \right)=0\\ \sin\left(\frac{\omega_{n}t_{2}}{2}\right)=0\end{array} \right. \tag{12}\]  
Therefore, the residual vibration becomes zero whenever one of the following conditions is satisfied:  
\[\text{C1}:\ T_{a}=t_{1}=k_{1}T_{n}\text{, \ \ }k_{1}\in\mathbb{N}^{+}\text{,} \tag{13}\]  
\[\text{C2}:\ T_{a}+T_{\varphi}=t_{2}=k_{2}T_{n}\text{, \ \ }k_{2}\in\mathbb{N}^{+} \text{,} \tag{14}\]  
where \(T_{n}=2\pi/\omega_{n}\) is the natural period of vibration. This conclusion extends the vibration suppression condition derived in [39]. The two conditions are distinguished in the vibration cancellation mechanism.