5/ If a block D of negligible size and of mass m is attached at C, and the bell crank of mass M is given a small angular displacement of $\theta$, the natural period of oscillation is $\tau_1$. When D is removed, the natural period of oscillation is $\tau_2$. Determine the bell crank's radius of gyration about its center of mass, pin B, and the spring's stiffness k. The spring is unstretched at $\theta=0$, and the motion occurs in the horizontal plane. no mass in FBD
Ans:
$m_D = m$
$m_{ABC} = M$
$\tau_1 = \frac{2\pi}{\omega_{n1}}$
$\tau_2 = \frac{2\pi}{\omega_{n2}}$
$k_B = \sqrt{\frac{I_B}{M}} = ? k = ?$
$k_B = a\sqrt{\frac{m}{M}\frac{(\tau_1^2 - \tau_2^2)}{(\tau_1^2 - \tau_2^2)}}$
$k = \frac{4\pi^2}{(\tau_1^2 - \tau_2^2)}m$
initial
final
Kinetic Energy: $\frac{1}{2}(M+m)V^2 + \frac{1}{2}I_B\omega^2$
$V = \omega \cdot a$
$\omega = \dot{\theta}$
$\frac{1}{2}(M+m)a^2\dot{\theta}^2 + \frac{1}{2}I_B\dot{\theta}^2$
Potential Energy: $mg~a\theta + \frac{1}{2}k(a\theta)^2$
Total Energy: $\frac{1}{2}(M+m)a^2\dot{\theta}^2 + \frac{1}{2}I_B\dot{\theta}^2 + mga\theta + \frac{1}{2}ka^2\theta^2$
$\frac{d}{d\theta}$
$2 \cdot \frac{1}{2}(M+m)a^2\dot{\theta}\ddot{\theta} + 2 \cdot \frac{1}{2}I_B\dot{\theta}\ddot{\theta} + mga\dot{\theta} + 2 \cdot \frac{1}{2}ka^2\theta\dot{\theta} = 0$
$((M+m)a^2 + I_B)\dot{\theta}\ddot{\theta} + (mga + ka^2\theta)\dot{\theta} = 0$
$(M\dot{a}^2\ddot{\theta} + ma^2\ddot{\theta} + I_B\ddot{\theta} + mga + ka^2\theta)\dot{\theta} = 0$
$((M+m)a^2 + I_B)\ddot{\theta} + mga + ka^2\theta = 0$
