# Dynamics Analysis of Rotating Cantilever Beams with Free End Mass
## Appendix A  
The velocities of the point \(P\) and the free end mass \(m\) in details are:  
\[\dot{\mathbf{r}}_{P}=\dot{\mathbf{R}}(\mathbf{r}_{a}+\mathbf{r}_{x})+\mathbf{ R}\dot{\mathbf{r}}_{x}=\Omega\begin{bmatrix}-\sin\theta&-\cos\theta\\ \cos\theta&-\sin\theta\end{bmatrix}\begin{bmatrix}a+x+u\\ w\end{bmatrix}+\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}\begin{bmatrix}\frac{\partial u}{\partial t }\\ \frac{\partial u}{\partial t}\end{bmatrix} \tag{11}\]  
\[\dot{\mathbf{r}}_{m}=\dot{\mathbf{R}}(\mathbf{r}_{a}+\mathbf{r}_{L})+\mathbf{ R}\dot{\mathbf{r}}_{L}=\Omega\begin{bmatrix}-\sin\theta&-\cos\theta\\ \cos\theta&-\sin\theta\end{bmatrix}\begin{bmatrix}a+L+u(L)\\ w(L)\end{bmatrix}+\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}\begin{bmatrix}\frac{\partial u(L)}{ \partial t}\\ \frac{\partial u(L)}{\partial t}\end{bmatrix} \tag{12}\]  
where \(\Omega=\dot{\theta}\).  
With regard to the rotational transformation matrix  
\[\dot{\mathbf{R}}^{T}\dot{\mathbf{R}}=\Omega^{2}\begin{bmatrix}1&0\\ 0&1\end{bmatrix},\quad\mathbf{R}^{T}\mathbf{R}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix} \tag{13}\]  
Thus, the kinetic energy in detail is given as:  
\[\begin{split} T_{P}&=\frac{1}{2}\int_{0}^{L}\rho A\dot{ \mathbf{r}}_{P}^{T}\dot{\mathbf{r}}_{P}\mathrm{d}x\\ &=\frac{1}{2}\int_{0}^{L}\rho A\Bigg{\{}\left(\frac{\partial u}{ \partial t}\right)^{2}+\left(\frac{\partial w}{\partial t}\right)^{2}+2 \Omega\bigg{[}(a+x)\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial t }-w\frac{\partial u}{\partial t}\bigg{]}\\ &\quad+\Omega^{2}\Big{[}(a+x)^{2}+u^{2}+w^{2}+2(a+x)u\Big{]} \Bigg{\}}\mathrm{d}x\\ & T_{m}&=\frac{1}{2}mi_{m}^{T}\dot{\mathbf{r}}_{m}\\ &=\frac{1}{2}m\Bigg{\{}\left(\frac{\partial u(L)}{\partial t} \right)^{2}+\left(\frac{\partial w(L)}{\partial t}\right)^{2}+2\Omega\bigg{[} (a+L)\frac{\partial w(L)}{\partial t}+u(L)\frac{\partial w(L)}{\partial t}- w(L)\frac{\partial u(L)}{\partial t}\bigg{]}\\ &\quad+\Omega^{2}\Big{[}(a+L)^{2}+u(L)^{2}+w(L)^{2}+2(a+L)u(L) \Big{]}\Bigg{\}}\\ \end{split} \tag{14}\]Applying Hanilton's principle yields:  
\[\delta\int_{t_{1}}^{t_{2}}T_{p}\mathrm{d}t =\frac{1}{2}\int_{t_{1}}^{t_{2}}\int_{0}^{L}\rho A\bar{r}_{p}^{2} \bar{r}_{p}\mathrm{d}x\mathrm{d}t \tag{10}\] \[=\frac{1}{2}\int_{t_{1}}^{t_{2}}\int_{0}^{L}\rho A\left\{2\bigg{[} \frac{\partial^{2}u}{\partial t^{2}}-\Omega^{2}(a+x+u)-2\Omega\frac{\partial w} {\partial t}-\dot{\Omega}w\bigg{]}\delta u\right.\] \[\left.\quad+2\bigg{[}\frac{\partial^{2}w}{\partial t^{2}}-\Omega^ {2}w+2\Omega\frac{\partial u}{\partial t}-\dot{\Omega}(a+x+u)\bigg{]}\delta w \bigg{\}}\mathrm{d}x\mathrm{d}t\right.\] \[\delta\int_{t_{1}}^{t_{2}}T_{m}\mathrm{d}t =\frac{1}{2}\int_{t_{1}}^{t_{2}}m\bar{r}_{m}^{2}\bar{r}_{m} \mathrm{d}t\] (11) \[=\frac{1}{2}\int_{t_{1}}^{t_{2}}m\Bigg{\{}2\bigg{[}\frac{\partial ^{2}u(L)}{\partial t^{2}}-\Omega^{2}(a+L+u(L))-2\Omega\frac{\partial w(L)}{ \partial t}-\dot{\Omega}w(L)\bigg{]}\delta u\] \[\left.\quad+2\bigg{[}\frac{\partial^{2}w(L)}{\partial t^{2}}- \Omega^{2}w(L)+2\Omega\frac{\partial u(L)}{\partial t}-\dot{\Omega}(a+L+u(L)) \bigg{]}\delta w\bigg{\}}\mathrm{d}t\right.\] \[\delta\int_{t_{1}}^{t_{2}}H\mathrm{d}t =\frac{1}{2}\int_{t_{1}}^{t_{2}}\int_{0}^{L}\bigg{[}EA\sigma \varepsilon+EI\bigg{(}\frac{\partial^{2}w}{\partial x^{2}}\bigg{)}^{2}\bigg{]} \mathrm{d}x\mathrm{d}t\] (12) \[=\frac{1}{2}\int_{t_{1}}^{t_{2}}\int_{0}^{L}\bigg{\{}2 EA\bigg{[} \bigg{(}\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial u}{\partial x} \frac{\partial^{2}u}{\partial x^{2}}\bigg{)}\delta\frac{\partial u}{\partial x }+\bigg{(}\frac{\partial u}{\partial x}\frac{\partial^{2}w}{\partial x^{2}} \bigg{)}\delta\frac{\partial w}{\partial x}\bigg{]}\] \[\left.\quad+2 EI\frac{\partial^{4}w}{\partial x^{4}}\delta\frac{ \partial^{2}w}{\partial x^{2}}\right\}\mathrm{d}x\mathrm{d}t\]  
Then, the analytical equations are derived by the following equation:  
\[\delta\int_{t_{1}}^{t_{2}}\big{(}T_{p}+T_{m}-H\big{)}\mathrm{d}t=0 \tag{13}\]