# Developing an Assisting Device to Reduce the Vibration on the Hands of Elders
## 3 Proposed Approach
### Dynamic Analysis of Gyroscopic Modeling System
\big{]}^{T} \tag{10}\]  
\[\omega_{m}=\Big{[} \begin{array}{ccc}0&0&\dot{\phi}\end{array} \Big{]}^{T}+R_{ 3}R_{2}\omega_{x_{1}y_{1}z_{1}}=\Big{[} \begin{array}{ccc}-\dot{\phi}\cos \phi&\dot{\phi}\sin\phi\end{array} \dot{\phi}\Big{]}^{T} \tag{11}\]  
Similarly, the linear velocity of the motor is:  
\[ \begin{array}{l}\dot{p}_{m}=\big{(}\dot{p}_{m}\big{)}_{relative}+\omega_{x_{ 1}y_{1}z_{1}}\times p_{m}=\Big{[}-\big{(}r_{m}\cos\phi+y_{p}\big{)}\dot{\phi}x _{p}\dot{\phi}-r_{m}\dot{\phi}\sin\phi r_{m}\dot{\phi}\cos\phi\Big{]}^{T}\\ \big{(}\dot{p}_{m}\big{)}_{relative}=\Big{[}0-r_{m}\dot{\phi}\sin\phi r_{m}\dot {\phi}\cos\phi\Big{]}^{T}\end{array}  \tag{12}\]  
The kinetic energy \(T_{m}\) and potential energy \(V_{m}\) are:  
\[\begin{array}{l}T_{m}=\frac{1}{2}m_{m}\dot{p}_{m}^{2}+\frac{1}{2}\omega_{wm }^{T}\left[ \begin{array}{ccc}I_{xm}&0&0\\ 0&I_{ym}&0\\ 0&0&I_{zm}\end{array} \right]\omega_{m}\\ =\frac{1}{2}m_{m}.\Big{[}\big{(}r_{m}\cos\phi+y_{p}\big{)}^{2}\dot{\theta}^{2}+ \Big{(}x_{p}\dot{\phi}-r_{m}\dot{\phi}\sin\phi\Big{)}^{2}+r_{m}^{2}\dot{\phi}^{ 2}\cos^{2}\phi\Big{]}\\ +\frac{1}{2}I_{xm}\dot{\theta}^{2}\cos^{2}\phi+\frac{1}{2}I_{ym}\dot{\phi}^{ 2}\cos^{2}\phi+\frac{1}{2}I_{zm}\dot{\phi}^{2}\\ p_{m}x_{Q_{0}Y_{0}Z_{0}}=R_{1}^{T}p_{m}=\left[ \begin{array}{ccc}x_{p}\cos \theta-y_{p}\sin\theta-r_{m}\sin\theta\cos\phi\\ x_{p}\sin\theta+y_{p}\cos\theta+r_{m}\cos\theta\cos\phi\\ z_{p}+r_{m}\sin\phi\end{array} \right]\\ V_{m}=m_{m}gY_{0_{m}}=m_{m}g(x_{p}\sin\theta+y_{p}\cos\theta+r_{m}\cos\theta \cos\phi) \tag{15}\]  
Figure 5: Analysis of the system parameters for (**a**) driving motor and (**b**) flywheel. The Lagrange equation of the driving motor is then:  
\[ \begin{array}{rl}L_{m}=T_{m}-V_{m}\\ =&\frac{1}{2}\dot{\theta}^{2}\left(m_{m}r_{m}^{2}\cos^{2}\phi+2m_{m}r_{m}y_{p} \cos\phi+m_{m}x_{p}^{2}+I_{xm}\cos^{2}\phi\right.\\ &\left.+I_{ym}\sin^{2}\phi\right.\right)+\frac{1}{2}\dot{\phi}\left(m_{m}r_{m}^{ 2}+I_{zm}\right)-m_{m}r_{m}x_{p}\dot{\theta}\dot{\phi}\sin\phi\\ &\left.-m_{m}g\big{(}x_{p}\sin\theta+y_{p}\cos\theta+r_{m}\cos\theta\cos\phi \big{)}\right.\end{array}  \tag{16}\]  
As shown in Figure 5b, the flywheel that is directly connected to the motor rotates with an angular velocity \(\Omega\) and mass \(m_{f}\), and its CoG location on \(Z_{G}\) is \(r_{f}\). With \(K_{P}\) and \(D_{P}\) as the stiffness coefficient and damping ratio of the precession axis, the coordinate of CoG in \(X_{W}\), \(Y_{W}\), \(Z_{W}\) and the angular velocity in \(X_{M}\), \(Y_{M}\), \(Z_{M}\) are:  
\[ \begin{array}{rl}p_{f}=\left[x_{p}y_{p}z_{p}\right]^{T}+R_{2}^{T}R_{3}^{T} \big{[}\begin{array}{cc}0&r_{f}&0\end{array}\big{]}^{T}=\big{[}\begin{array} []{cc}x_{p}&r_{f}\cos\phi+y_{p}&r_{f}\sin\phi+z_{p}\end{array}\big{]}^{T}\\ \end{array}  \tag{17}\]  
\[ \begin{array}{rl}\omega_{f}=\Big{[}\begin{array}{cc}0&\Omega&\dot{\phi} \end{array}\Big{]}^{T}+R_{3}R_{2}\omega_{x_{1}y_{1}z_{1}}=\Big{[}\begin{array} []{cc}-\dot{\phi}\cos\phi&\Omega+\dot{\phi}\sin\phi&\dot{\phi}\end{array} \Big{]}^{T}\\ \end{array}  \tag{18}\]  
The linear velocity of the flywheel is:  
\[ \begin{array}{rl}\dot{p}_{f}=\Big{(}\dot{p}_{f}\Big{)}_{relative}+\omega_{x_ {1}y_{1}z_{1}}\times p_{f}\\ =\Big{[}-\Big{(}r_{f}\cos\phi+y_{p}\Big{)}\dot{\theta}x_{p}\dot{\theta}-r_{f} \dot{\phi}\sin\phi r_{f}\dot{\phi}\cos\phi\Big{]}^{T}\\ \Big{(}\dot{p}_{f}\Big{)}_{relative}=\Big{[}0-r_{f}\dot{\phi}\sin\phi r_{m} \dot{\phi}\cos\phi\Big{]}^{T}\end{array}  \tag{19}\]  
The kinetic energy \(T_{f}\), potential energy \(V_{f}\), and Lagrange equation of the flywheel are:  
\[