Exercise n1 on kinematics and dynamics

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APPLIED MECHANICS Exercise n.1 The mechanical system illustrated in Fig. 1 moves in a vertical plane. The disk of centre O is rolling, without any slippage, on an inclined plane (left side of Fig. 1). The disk has a mass m1 and a mass moment of inertia JO. The bar OA is hinged at both ends, to the disk and a slider, respectively. The bar, whose barycentre is G1, has a mass m 2 and a mass moment of inertia J G1. The slider, having a mass m 3 and a barycentre G 2, moves on an inclined plane (right side of Fig. 1) with velocity v and acceleration a. A force F is applied to the top of the slider. The kinetic friction coefficient between the slider and the supporting plane is fk. The rolling friction coefficient and the static friction coefficient between the disk and the plane are fr and fs respectively. All the geometrical parameters are assumed to be known. You are asked to describe the procedure to: 1 evaluate the angular velocity and angular acceleration of the disk; 2 evaluate the absolute velocity and acceleration of the barycentre G of the bar OA; 3 evaluate the drive torque M1 applied to the disk; 4 evaluate the internal actions transmitted by the bar to the slider, at point A; 5 write the expression of the kinetic energy of the system; 6 describe the procedure to verify that no slippage occurs between disk and plane. Figure 1 System configuration Kinematic analysis Figures 2-a and 2-b show a kinematic model of the system. This model is based on position vectors that describe the absolute and relative position o f some significant points of the considered system. The vector 3 33 i e ϑ ρρ= is able to model the absolute position of point A, fixed to the slider. The attitude angle 3ϑ is constant while the magnitude 3ρ is a time-varying quantity. We assume that the…