\frac{K(t)}{K_e(t)}= \begin{bmatrix} sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & r_1.sin(\alpha) & -sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & r_2.sin(\alpha) \\ sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & r_1.cos(\alpha) & -sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & r_2.cos(\alpha) \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ r_1.sin(\alpha) & r_1.cos(\alpha) & 0 & r_1^2 & -r_1.sin(\alpha) & -r_1.cos(\alpha) & 0 & r_1.r_2 \\ -sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & -r_1.sin(\alpha) & sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & -r_2.sin(\alpha) \\ -sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & -r_1.cos(\alpha) & sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & -r_2.cos(\alpha) \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ r_2.sin(\alpha) & r_2.cos(\alpha) & 0 & r_1.r_2 & -r_2.sin(\alpha) & -r_2.cos(\alpha) & 0 & r_2^2 \\ \end{bmatrix}¶ M = \begin{bmatrix} m_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & m_1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & I_{11} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I_{12} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & m_2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & m_2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & I_{22} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & I_{21} \\ \end{bmatrix} \\¶ Ks= \begin{bmatrix} k_{x1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & k_{y1} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & k_{\theta1}& -k_{\theta1} & 0 & 0 & 0 & 0 \\ 0 & 0 & -k_{\theta1} & k_{\theta1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & k_{x1} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & k_{y1} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & k_{\theta2} & -k_{\theta2} \\ 0 & 0 & 0 & 0 & 0 & 0 & -k_{\theta2} & k_{\theta2} \\ \end{bmatrix}¶ C = \eta M+\beta K_{mean}¶ $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.¶ $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ $r_{bij}$ the base radius of pinions $K_e$ is the varying meshing stiffness mass $m_i = \pi r^2_{i}\rho$ for both pinions $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia ¶ $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$  \\ $r_{bij}$ the base radius of pinions \\ $K_e$ is the varying meshing stiffness mass \\ $m_i = \pi r^2_{i}\rho$ for both pinions \\ $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia ¶ \begin{cases} &\frac{\partial Ep_1}{\partial x_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{x}_1}\right)=m_1\ddot{x}_1+k_{x1}x_1+sin(\alpha)K_e(t)\delta _1(t)=0 \\ &\frac{\partial Ep_1}{\partial y_1}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{y}_1}\right)=m_1\ddot{y}_1+k_{y1}y_1+cos(\alpha)K_e(t)\delta _1(t)=0 \\ &\frac{\partial Ep_1}{\partial \theta_{11}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{11}}\right)=I_{11}\ddot{\theta}_{11}+k_{\theta 1}(\theta_{11}-\theta_{12})=Cm \\ &\frac{\partial Ep_1}{\partial \theta_{12}}+\frac{d}{dt}\left(\frac{\partial Ec_1}{\partial \dot{\theta}_{12}}\right)=I_{12}\ddot{\theta}_{12}+k_{\theta 1}(\theta_{12}-\theta_{11})+K_e(t)r_{b12}\delta_1(t)=0 \\ &\frac{\partial Ep_1}{\partial x_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{x}_2}\right)=m_2\ddot{x}_2+k_{x2}x_2-sin(\alpha)K_e(t)\delta _1(t)=0 \\ &\frac{\partial Ep_1}{\partial y_2}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{y}_2}\right)=m_2\ddot{y}_2+k_{y2}y_2-cos(\alpha)K_e(t)\delta _1(t)=0 \\ &\frac{\partial Ep_1}{\partial \theta_{22}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{22}}\right)=I_{22}\ddot{\theta}_{22}-k_{\theta 2}(\theta_{21}-\theta_{22})=-Cr \\ &\frac{\partial Ep_1}{\partial \theta_{21}}+\frac{d}{dt}\left(\frac{\partial Ec_2}{\partial \dot{\theta}_{21}}\right)=I_{21}\ddot{\theta}_{21}-k_{\theta 2}(\theta_{21}-\theta_{22})+K_e(t)r_{b21}\delta_1(t)=0 \end{cases}¶ With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia.¶ With $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia. ¶ \noindent $\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ and $r_{bij}$ the base radius of pinions, $K_e$ is the varying meshing stiffness, mass $m_i = \pi r^2_{i}\rho$ for both pinions, and $I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia. ¶ [M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\ \{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}¶ \begin{equation*} [M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\ \{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\} \end{equation*}¶ \begin{equation*} &[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\ &\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\} \end{equation*}¶ \begin{equation*} &[M]{\ddot{q}}+C{\dot{q}}+([Ks]+[K(t)]){q}={F_0} \ &{q}={x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}} \end{equation*} ¶ [M]{\ddot{q}}+C{\dot{q}}+([Ks]+[K(t)]){q}={F_0} \\ {q}={x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}} ¶ \noindent[M]{\ddot{q}}+C{\dot{q}}+([Ks]+[K(t)]){q}={F_0} \\ {q}={x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}} ¶ M = \begin{bmatrix} m_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & m_1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & I_{11} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I_{12} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & m_2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & m_2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & I_{22} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & I_{21} \\ \end{bmatrix} \\¶ \frac{K(t)}{K_e(t)}= \begin{bmatrix} sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & r_1.sin(\alpha) & -sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & r_2.sin(\alpha) \\ sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & r_1.cos(\alpha) & -sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & r_2.cos(\alpha) \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ r_1.sin(\alpha) & r_1.cos(\alpha) & 0 & r_1^2 & -r_1.sin(\alpha) & -r_1.cos(\alpha) & 0 & r_1.r_2 \\ -sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & -r_1.sin(\alpha) & sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & -r_2.sin(\alpha) \\ -sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & -r_1.cos(\alpha) & sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & -r_2.cos(\alpha) \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ r_2.sin(\alpha) & r_2.cos(\alpha) & 0 & r_1.r_2 & -r_2.sin(\alpha) & -r_2.cos(\alpha) & 0 & r_2^2 \\ \end{bmatrix}¶ Ks= \begin{bmatrix} k_{x1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & k_{y1} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & k_{\theta1}& -k_{\theta1} & 0 & 0 & 0 & 0 \\ 0 & 0 & -k_{\theta1} & k_{\theta1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & k_{x1} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & k_{y1} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & k_{\theta2} & -k_{\theta2} \\ 0 & 0 & 0 & 0 & 0 & 0 & -k_{\theta2} & k_{\theta2} \\ \end{bmatrix}¶ \frac{K(t)}=K_e(t) \begin{bmatrix} sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & r_1.sin(\alpha) & -sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & r_2.sin(\alpha) \\ sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & r_1.cos(\alpha) & -sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & r_2.cos(\alpha) \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ r_1.sin(\alpha) & r_1.cos(\alpha) & 0 & r_1^2 & -r_1.sin(\alpha) & -r_1.cos(\alpha) & 0 & r_1.r_2 \\ -sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & -r_1.sin(\alpha) & sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & -r_2.sin(\alpha) \\ -sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & -r_1.cos(\alpha) & sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & -r_2.cos(\alpha) \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ r_2.sin(\alpha) & r_2.cos(\alpha) & 0 & r_1.r_2 & -r_2.sin(\alpha) & -r_2.cos(\alpha) & 0 & r_2^2 \\ \end{bmatrix}¶ K(t)=K_e(t) \begin{bmatrix} sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & r_1.sin(\alpha) & -sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & r_2.sin(\alpha) \\ sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & r_1.cos(\alpha) & -sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & r_2.cos(\alpha) \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ r_1.sin(\alpha) & r_1.cos(\alpha) & 0 & r_1^2 & -r_1.sin(\alpha) & -r_1.cos(\alpha) & 0 & r_1.r_2 \\ -sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & -r_1.sin(\alpha) & sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & -r_2.sin(\alpha) \\ -sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & -r_1.cos(\alpha) & sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & -r_2.cos(\alpha) \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ r_2.sin(\alpha) & r_2.cos(\alpha) & 0 & r_1.r_2 & -r_2.sin(\alpha) & -r_2.cos(\alpha) & 0 & r_2^2 \\ \end{bmatrix}¶ C = \eta M+\beta K_{mean}¶ \noindent[M]\ddot{q}+C\dot{q}+([Ks]+[K(t)]){q}={F_0} \\ {q}={x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}} ¶ \noindent \[M\]\ddot{q}+\[C\]\dot{q}+(\[Ks\]+\[K(t)\]){q}={F_0} \\ {q}={x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}} ¶ {q}={x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}} ¶ \left[M \right]\ddot{q} + \left[C \right]\dot{q} + (\left[Ks \right]+\left[K(t) \right])q = \left\{F_0 \right\}¶ \varepsilon = \frac{RMS-RMS_{mean}}{RMS_{mean}}\times 100¶