147 lines
No EOL
9.5 KiB
Text
147 lines
No EOL
9.5 KiB
Text
\frac{K(t)}{K_e(t)}=
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\begin{bmatrix}
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sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & r_1.sin(\alpha) & -sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & r_2.sin(\alpha) \\
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sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & r_1.cos(\alpha) & -sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & r_2.cos(\alpha) \\
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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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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 \\
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-sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & -r_1.sin(\alpha) & sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & -r_2.sin(\alpha) \\
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-sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & -r_1.cos(\alpha) & sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & -r_2.cos(\alpha) \\
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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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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 \\
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\end{bmatrix}¶
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M =
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\begin{bmatrix}
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m_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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0 & m_1 & 0 & 0 & 0 & 0 & 0 & 0 \\
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0 & 0 & I_{11} & 0 & 0 & 0 & 0 & 0 \\
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0 & 0 & 0 & I_{12} & 0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 & m_2 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 & 0 & m_2 & 0 & 0 \\
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0 & 0 & 0 & 0 & 0 & 0 & I_{22} & 0 \\
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0 & 0 & 0 & 0 & 0 & 0 & 0 & I_{21} \\
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\end{bmatrix} \\¶
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Ks=
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\begin{bmatrix}
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k_{x1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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0 & k_{y1} & 0 & 0 & 0 & 0 & 0 & 0 \\
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0 & 0 & k_{\theta1}& -k_{\theta1} & 0 & 0 & 0 & 0 \\
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0 & 0 & -k_{\theta1} & k_{\theta1} & 0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 & k_{x1} & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 & 0 & k_{y1} & 0 & 0 \\
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0 & 0 & 0 & 0 & 0 & 0 & k_{\theta2} & -k_{\theta2} \\
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0 & 0 & 0 & 0 & 0 & 0 & -k_{\theta2} & k_{\theta2} \\
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\end{bmatrix}¶
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C = \eta M+\beta K_{mean}¶
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$\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ 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.¶
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$\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$
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$r_{bij}$ the base radius of pinions
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$K_e$ is the varying meshing stiffness
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mass $m_i = \pi r^2_{i}\rho$ for both pinions
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$I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia
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¶
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$\delta_i(t)=(x_1-x_2)sin(\alpha)+(y_1-y_2)cos(\alpha)+\theta_{12}r_{b1}+\theta_{21}r_{b2}$ \\
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$r_{bij}$ the base radius of pinions \\
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$K_e$ is the varying meshing stiffness mass \\
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$m_i = \pi r^2_{i}\rho$ for both pinions \\
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$I_{ij} = \frac{1}{2}m_{ij}r_{ij}^2$ for pinions and input/output inertia
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¶
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\begin{cases}
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&\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 \\
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&\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 \\
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&\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 \\
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&\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 \\
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&\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 \\
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&\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 \\
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&\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 \\
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&\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
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\end{cases}¶
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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.¶
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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.
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¶
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\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.
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¶
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[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
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\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}¶
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\begin{equation*}
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[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
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\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
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\end{equation*}¶
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\begin{equation*}
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&[M]\{\ddot{q}\}+C\{\dot{q}\}+([Ks]+[K(t)])\{q\}=\{F_0\} \\
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&\{q\}=\{x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}\}
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\end{equation*}¶
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\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*}
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¶
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[M]{\ddot{q}}+C{\dot{q}}+([Ks]+[K(t)]){q}={F_0} \\
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{q}={x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}}
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¶
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\noindent[M]{\ddot{q}}+C{\dot{q}}+([Ks]+[K(t)]){q}={F_0} \\
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{q}={x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}}
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¶
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M =
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\begin{bmatrix}
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m_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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0 & m_1 & 0 & 0 & 0 & 0 & 0 & 0 \\
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0 & 0 & I_{11} & 0 & 0 & 0 & 0 & 0 \\
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0 & 0 & 0 & I_{12} & 0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 & m_2 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 & 0 & m_2 & 0 & 0 \\
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0 & 0 & 0 & 0 & 0 & 0 & I_{22} & 0 \\
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0 & 0 & 0 & 0 & 0 & 0 & 0 & I_{21} \\
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\end{bmatrix} \\¶
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\frac{K(t)}{K_e(t)}=
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\begin{bmatrix}
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sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & r_1.sin(\alpha) & -sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & r_2.sin(\alpha) \\
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sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & r_1.cos(\alpha) & -sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & r_2.cos(\alpha) \\
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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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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 \\
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-sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & -r_1.sin(\alpha) & sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & -r_2.sin(\alpha) \\
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-sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & -r_1.cos(\alpha) & sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & -r_2.cos(\alpha) \\
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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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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 \\
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\end{bmatrix}¶
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Ks=
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\begin{bmatrix}
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k_{x1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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0 & k_{y1} & 0 & 0 & 0 & 0 & 0 & 0 \\
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0 & 0 & k_{\theta1}& -k_{\theta1} & 0 & 0 & 0 & 0 \\
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0 & 0 & -k_{\theta1} & k_{\theta1} & 0 & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 & k_{x1} & 0 & 0 & 0 \\
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0 & 0 & 0 & 0 & 0 & k_{y1} & 0 & 0 \\
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0 & 0 & 0 & 0 & 0 & 0 & k_{\theta2} & -k_{\theta2} \\
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0 & 0 & 0 & 0 & 0 & 0 & -k_{\theta2} & k_{\theta2} \\
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\end{bmatrix}¶
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\frac{K(t)}=K_e(t)
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\begin{bmatrix}
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sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & r_1.sin(\alpha) & -sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & r_2.sin(\alpha) \\
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sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & r_1.cos(\alpha) & -sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & r_2.cos(\alpha) \\
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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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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 \\
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-sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & -r_1.sin(\alpha) & sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & -r_2.sin(\alpha) \\
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-sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & -r_1.cos(\alpha) & sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & -r_2.cos(\alpha) \\
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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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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 \\
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\end{bmatrix}¶
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K(t)=K_e(t)
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\begin{bmatrix}
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sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & r_1.sin(\alpha) & -sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & r_2.sin(\alpha) \\
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sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & r_1.cos(\alpha) & -sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & r_2.cos(\alpha) \\
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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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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 \\
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-sin(\alpha)^2 & -sin(\alpha).cos(\alpha) & 0 & -r_1.sin(\alpha) & sin(\alpha)^2 & sin(\alpha).cos(\alpha) & 0 & -r_2.sin(\alpha) \\
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-sin(\alpha).cos(\alpha) & -cos(\alpha)^2 & 0 & -r_1.cos(\alpha) & sin(\alpha).cos(\alpha) & cos(\alpha)^2 & 0 & -r_2.cos(\alpha) \\
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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
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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 \\
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\end{bmatrix}¶
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C = \eta M+\beta K_{mean}¶
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\noindent[M]\ddot{q}+C\dot{q}+([Ks]+[K(t)]){q}={F_0} \\
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{q}={x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}}
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¶
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\noindent \[M\]\ddot{q}+\[C\]\dot{q}+(\[Ks\]+\[K(t)\]){q}={F_0} \\
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{q}={x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}}
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¶
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{q}={x_1,y_1,\theta_{11},\theta_{12},x_2,y_2,\theta_{22},\theta_{21}}
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¶
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\left[M \right]\ddot{q} + \left[C \right]\dot{q} + (\left[Ks \right]+\left[K(t) \right])q = \left\{F_0 \right\}¶
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\varepsilon = \frac{RMS-RMS_{mean}}{RMS_{mean}}\times 100¶ |