Question 6.6: Determine the matrix differential equations of motion of the...
Determine the matrix differential equations of motion of the damped two degree of freedom system shown in Fig. 6.7.
Let θ_{1} and θ_{2} be the system degrees of freedom. Without loss of generality, we assume that θ_{2} > θ_{1} and \dot{θ_{2}} > \dot{θ_{1}}. We also assume that the angular oscillation is small such that the motion of the masses in the vertical direction can be neglected. From the free body diagram shown in the figure and by taking the moments about the fixed point, one can show that the differential equation of motion of the first mass is given by
m_{1}l^{2} \ddot{θ_{1}} = ka^{2}(θ_{2} − θ_{1}) + ca^{2}( \dot{θ_{2}} − \dot{θ_{1}}) − m_{1}glθ_{1}Similarly, for the second mass, one has
m_{2}l^{2} \ddot{θ_{2}} = −ka²(θ_{2} − θ_{1}) − ca^{2}( \dot{θ_{2}} − \dot{θ_{1}}) − m_{2}glθ_{2}
These differential equations can be rewritten as
m_{1}l^{2} \ddot{θ_{1}} + ca^{2} \dot{θ_{1}} − ca^{2} \dot{θ_{2}} + (ka^{2} + m_{1}gl) θ_{1} − ka^{2}θ_{2} = 0
m_{2}l^{2} \ddot{θ_{2}} + ca^{2} \dot{θ_{2}} − ca^{2} \dot{θ_{1}} + (ka^{2} + m_{2}gl) θ_{2} − ka^{2}θ_{1} = 0
which can be written in matrix form as
\begin{bmatrix} m_{1}l^{2} & 0 \\ 0 & m_{2}l^{2} \end{bmatrix} \begin{bmatrix} \ddot{θ_{1}} \\ \ddot{θ_{2}} \end{bmatrix} +\begin{bmatrix} ca^{2} & -ca^{2} \\ -ca^{2} & ca^{2} \end{bmatrix} \begin{bmatrix} \dot{θ_{1}} \\ \dot{θ_{2}} \end{bmatrix}+ \begin{bmatrix} (ka^{2} + m_{1}gl) & -ka^{2} \\ -ka^{2} & (ka^{2} + m_{2}gl) \end{bmatrix} \begin{bmatrix} θ_{1} \\ θ_{2} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}
This equation can be written in compact form as
M\ddot{θ} + C\dot{θ} + Kθ = 0
where \ddot{θ}, \dot{θ}, and θ are the vectors
\ddot{θ}= \begin{bmatrix} \ddot{θ_{1}} \\ \ddot{θ_{2}} \end{bmatrix}, \dot{θ}= \begin{bmatrix} \dot{θ_{1}} \\ \dot{θ_{2}} \end{bmatrix}, θ= \begin{bmatrix} θ_{1} \\ θ_{2} \end{bmatrix}
and M, C, and K are, respectively, the mass, damping, and stiffness matrices defined as
M= \begin{bmatrix} m_{1}l^{2} & 0 \\ 0 & m_{2}l^{2} \end{bmatrix}, C= \begin{bmatrix} ca^{2} & -ca^{2} \\ -ca^{2} & ca^{2} \end{bmatrix}, K= \begin{bmatrix} ka^{2} + m_{1}gl & -ka^{2} \\ -ka^{2} & ka^{2} + m_{2}gl \end{bmatrix}