Meriam, Kraige & Bolton — Introduction to Three-Dimensional Dynamics of Rigid Bodies: Problem 7_15

7/15 The robot shown has five degrees of rotational freedom. The x-y-z axes are attached to the base ring, which rotates about the z-axis at the rate \(\omega_1\). The arm \(O_1O_2\) rotates about the x-axis at the rate \(\omega_2 = \dot{\theta}\). The control arm \(O_2A\) rotates about axis \(O_1-O_2\) at the rate \(\omega_3\) and about a perpendicular axis through \(O_2\) which is momentarily parallel to the x-axis at the rate \(\omega_4 = \dot{\beta}\). Finally, the jaws rotate about axis \(O_2-A\) at the rate \(\omega_5\). The magnitudes of all angular rates are constant. For the configuration shown, determine the magnitude \(\omega\) of the total angular velocity of the jaws for \(\theta = 60^\circ\) and \(\beta = 45^\circ\) if \(\omega_1 = 2 \text{ rad/s}\), \(\dot{\theta} = 1.5 \text{ rad/s}\), and \(\omega_3 = \omega_4 = \omega_5 = 0\). Also express the angular acceleration \(\alpha\) of arm \(O_1O_2\) as a vector.