Question 4.7.2: A machine part consists of three levers connected by lightwe......
A machine part consists of three levers connected by lightweight linkages. A vibration model of this part is given in Figure 4.19. Use the Lagrange method to obtain the equation of vibration. Take the angles to be the generalized coordinates. Linearize the result and put it in matrix form.
The kinetic energy is
T=\frac{1}{2} m_1 l^2 \dot{\theta}_1^2+\frac{1}{2} m_2 l^2 \dot{\theta}_2^2+\frac{1}{2} m_3 l^2 \dot{\theta}_3^2
The potential energy becomes
Applying the Lagrange equation for i=1 yields
m_1 l^2 \ddot{\theta}_1+m_1 g l \sin \theta_1-\alpha k_1\left(\alpha \theta_2-\alpha \theta_1\right)=0
For i=2, the Lagrange equation yields
m_2 l^2 \ddot{\theta}_2+m_2 g l \sin \theta_2+\alpha k_1\left(\alpha \theta_2-\alpha \theta_1\right)-\alpha k_2\left(\alpha \theta_3-\alpha \theta_2\right)=0
and for i=3, the Lagrange equation becomes
m_3 l \ddot{\theta}_3+m_3 g l \sin \theta_3+\alpha k_2\left(\alpha \theta_3-\alpha \theta_2\right)=0
These three equations can be linearized by assuming \theta is small so that \sin \theta \sim \theta. Note that this linearization occurs after the equations have been derived. In matrix form this becomes
where x (t)=\left[\begin{array}{lll}q_1 & q_2 & q_3\end{array}\right]^T=\left[\begin{array}{lll}\theta_1 & \theta_2 & \theta_3\end{array}\right]^T is the generalized set of coordinates.