Question 6.7: 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. 6.8 and identify the mass, damping, and stiffness matrices.
Assuming that the motion in the horizontal direction can be neglected, and applying D’Alembert’s principle, the differential equations of motion can be written as
m\ddot{y} = −c_{1}(\dot{y} − a \dot{θ}) − k_{1}(y − aθ) − c_{2}(\dot{y} + b \dot{θ}) − k_{2}(y + bθ)
I \ddot{θ} = c_{1}(\dot{y} − a \dot{θ})a + k_{1}(y − aθ)a − c_{2}(\dot{y} + b \dot{θ})b − k_{2}(y + bθ)b
In developing these equations, we assumed that the weight and the static deflections at the equilibrium position can be eliminated from the dynamic equations by using the static equations of equilibrium. The above differential equations can be rewritten as
m\ddot{y} + (c_{1} + c_{2})\dot{y} − (c_{1}a − c_{2}b) \dot{θ} + (k_{1} + k_{2})y − (k_{1}a − k_{2}b)θ = 0
I \ddot{θ} + (c_{1}a^{2} + c_{2}b^{2}) \dot{θ} − (c_{1}a − c_{2}b)\dot{y} + (k_{1}a^{2} + k_{2}b^{2})θ − (k_{1}a − k_{2}b)y = 0
which can be written in the following matrix form
\begin{bmatrix} m & 0 \\ 0 & I \end{bmatrix} \begin{bmatrix} \ddot{y} \\ \ddot{θ} \end{bmatrix} + \begin{bmatrix} c_{1}+c_{2} & −(c_{1}a − c_{2}b) \\−(c_{1}a − c_{2}b) & c_{1}a^{2} + c_{2}b^{2} \end{bmatrix} \begin{bmatrix} \dot{y} \\ \dot{θ} \end{bmatrix} + \begin{bmatrix} k_{1}+k_{2} & k_{2}b − k_{1}a \\k_{2}b − k_{1}a & k_{1}a^{2} + k_{2}b^{2} \end{bmatrix} \begin{bmatrix} y \\ θ \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}where the mass matrix M, the damping matrix C, and the stiffness matrix K can be identified as
M =\begin{bmatrix} m & 0 \\ 0 & I \end{bmatrix}, C =\begin{bmatrix} c_{1}+c_{2} & c_{2}b-c_{1}a \\ c_{2}b-c_{1}a & c_{1}a^{2} + c_{2}b^{2} \end{bmatrix},
K = \begin{bmatrix} k_{1}+k_{2} & k_{2}b − k_{1}a \\k_{2}b − k_{1}a & k_{1}a^{2} + k_{2}b^{2} \end{bmatrix}