Plane Dynamics of Rigid Bodies Prepared by Dr. Hassan Fadag. Lecture XI Plane Dynamics of Rigid Bodies

i) Plane Kinematics of Rigid Bodies Prepared by Dr. Hassan Fadag. i) Plane Kinematics of Rigid Bodies

Introduction The relationships governing the displacement, velocity, and acceleration of particles (points) as they moved along straight or curved path were developed earlier. In rigid-body kinematics, same relationships are used but must also account for the rotational motion of the body. This means that rigid-body kinematics involves both linear and angular displacements, velocities, and accelerations. Rigid Body: a system of particles for which the distance between the particles remain unchanged. However, this an ideal case, since all solid materials change shape to some extent when forces are applied to them. Nevertheless, if the movements associated with the changes in shape are very small compared with the movements of the body as a whole, then the assumption of rigidity is acceptable. Rigid Body Plane Motion: occurs when all parts of the body move in parallel planes. For convenience, the plane of motion is considered to be the plane that contains the mass center, and the body will be treated as a thin slab whose motion is confined to the plane of the slab.

Introduction – Cont.

Rotation The rotation of a rigid body is described by its angular motion. All lines on a rigid body in its plane of motion have the same angular displacement, velocity and acceleration. Angular-Motion Relations: Rotation about a Fixed Axis: For constant angular acceleration

Absolute & Relative Motion Analysis In Absolute Motion Analysis: the geometric relations (linear and angular variables) will be defined in a quite straightforward manner; then, the time derivatives of these quantities will involve both linear and angular velocities and accelerations. If the geometric configuration is complex, the relative motion analysis will be considered. Relative Motion Analysis: Relative Velocity due to Rotation: Instantaneous Center of Zero Velocity: this point has a zero velocity. A body may be considered to be in pure rotation about an axis normal to the plane of motion, passing through this point. This axis called the instantaneous axis of zero velocity, and the intersection of this axis with the plane of motion is known as the instantaneous center of zero velocity.

Absolute & Relative Motion Analysis – Cont. Relative Acceleration due to Rotation:

Motion Relative to Rotating Axes In particles, when we describe the relative motion analysis, a non-rotating axes were used. The use of this type of axes facilitates the solution of many problems in kinematics where motion is generated within a system or observed from a system which itself is rotating. Relative Velocity: Relative Acceleration:

Exercises

Exercise # 1 The right-angle bar rotates clockwise with an angular velocity which is decreasing at the rate of 4 rad/s2. Write the vector expressions for the velocity and acceleration of point A when w = 2 rad/s. x

Exercise # 2 Crank CB oscillates about C through a limited arc, causing crank OA to oscillate about O. When the linkage passes the position shown with CB horizontal and OA vertical, the angular velocity of CB is 2 rad/s counterclockwise. For this instant, determine the angular velocities of OA and AB. 250 mm

Exercise # 3 Crank CB has a constant counterclockwise angular velocity of 2 rad/s in the position shown during a short interval of its motion. Determine the angular acceleration of the links AB and OA for this position. 250 mm

Exercise # 4 At the instant represented, the disk with the radial slot is rotating about O with a counterclockwise angular velocity of 4 rad/s which is decreasing at the rate of 10 rad/s2. The motion of slider A is separately controlled, and at this instant, r = 150 mm, r. = 125 mm/s, and r. . = 2025 mm/s2. Determine the absolute velocity and acceleration of A for this position.

ii) Plane Kinetics of Rigid Bodies Prepared by Dr. Hassan Fadag. ii) Plane Kinetics of Rigid Bodies

Introduction The kinetics of rigid bodies treats the relationships between the external forces acting on a body and the corresponding translational and rotational motions of the body.

Force, Mass, and Acceleration General Equations of Motion: Plane Motion Equations: Thus, the controlling equations in plane motion are:

Force, Mass, and Acceleration – Cont. Alternative Moment Equations: when point P becomes a point O fixed in an inertial reference system and attached to the body (or body extended), then aP = 0, and the equation reduces to For System of Interconnected Bodies:

Force, Mass, and Acceleration – Cont. Translation: Fixed-Axis Rotation: For fixed-axis rotation, it is generally useful to apply a moment equation directly about the rotation axis O. General Plane Motion:

Exercises

Exercise # 5 The vertical bar AB has a mass of 150 kg with center of mass G midway between the ends. The bar is elevated from rest at q = 0 by means of the parallel links of negligible mass, with a constant couple M = 5 kN·m applied to the lower link at C. Determine the angular acceleration a of the links as a function of q and find the force B in the link DB at the instant when q = 30°.

Exercise # 6 The pendulum has a mass of 7.5 kg with center of mass at G and has a radius of gyration about the pivot O of 295 mm. If the pendulum is released from rest at q = 0, determine the total force supported by the bearing at the instant when q = 60°. Friction in the bearing is negligible.

Exercise # 7 A car door is inadvertently left slightly open when the brakes are applied to give the car a constant rearward acceleration a. Derive expressions for the angular velocity of the door as it swings past the 90° position and the components of the hinge reactions for any value of q. The mass of the door is m, its mass center is a distance r from the hinge axis O, and the radius of gyration about O is kO.

Work & Energy Kenitic Enegy: a) Translation: b) Fixed-axis Rotation: c) General Plane Motion: Potential Energy & the Work-Energy Equation: Power:

Exercise # 8 In the mechanism shown, each of the two wheels has a mass of 30 kg and a centroidal radius of gyration of 100 mm. Each link OB has a mass of 10 kg and may be treated as a slender bar. The 7-kg collar at B slides on the fixed vertical shaft with negligible friction. The spring has a stiffness k = 30 kN/m and is contacted by the bottom of the collar when the links reach the horizontal position. If the collar is released from rest at the position q = 45° and if friction is sufficient to prevent the wheels from slipping, determine (a) the velocity vB of the collar as it first strikes the spring and (b) the maximum deformation x of the spring.

Impulse & Momentum Linear Impulse-Linear Momentum: Angular Impulse-Angular Momentum:

Exercise # 9 The sheave E of the hoisting rig shown has a mass of 30 kg and a centroidal radius of gyration of 250 mm. The 40-kg load D which is carried by the sheave has an initial downward velocity v1=1.2 m/s at the instant when a clockwise torque is applied to the hoisting drum A to maintain essentially a constant force F = 380 N in the cable at B. Compute the angular velocity w2 of the sheave 5 seconds after the torque is applied to the drum and find the tension T in the cable at O during the interval. Neglect all friction.