PLANE MOTION OF RIGID BODIES: FORCES AND ACCELERATIONS

This module entails the linear and angular accelerations of a rigid body and how it is affected by the mass and mass moment of inertia, how to determine whether a body slips or tips and if a wheel rolls with or without slip using rigid-body kinetics principles, and on how to apply appropriate kinetic equations and kinematics relationships to solve kinetics problems for a rigid body.

kinetics of a rigid body

equations of motion for a rigid body

Shown in the right figure are the equations for translational equation of motion and rotational equation of motion.
Continuing the discussion, the term HG is referred simply as the angular momentum of the rigid body about its mass center G. From these, the equations express that the system of the external forces and moments is equipollent to the system consisting of the vector mâ attached at G and the couple of moment ḢG

angular momentum of a rigid body in plane motion

The rate of change of the angular momentum of the rigid body is represented by a vector in the same direction as α.
The angular momentum HG of the rigid body about its mass center and the rate of change of angular momentum about G are shown in the right figure.
The equations remain valid in the case of the plane motion of rigid bodies that are symmetrical with respect to a reference plane (or, more generally, bodies that have a principal centroidal axis of inertia perpendicular to a reference plane). However, they do not apply in the case of nonsymmetrical bodies or in the case of three-dimensional motion.

plane motion of a rigid body

Consider a rigid body with a mass m moving under the action of several external forces F1, F2, F3, . . . contained in the plane of the body.
The equations as seen in the figure show that the acceleration of the mass center G of the rigid body and its angular acceleration α can be obtained once the resultant of the external forces acting on the body and their moment resultant about G is determined.
The motion of the rigid body is completely defined by the resultant force and resultant moment about G acting on the body.
Since the motion of a rigid body depends only upon the resultant and resultant moment of the external forces acting on it, it follows that two systems of forces that are equipollent (i.e., that have the same resultant and the same moment resultant) are also equivalent.
The external forces acting on a rigid body are equivalent to the inertial terms of the various particles forming the body .

examples of rigid body motion

Translation - In the case of a body in translation, the angular acceleration of the body is equal to zero and its inertial terms reduce to the vector ma attached at G.
Centroidal Rotation - When a rigid body, or more generally, a body symmetrical with respect to a reference plane, rotates about a fixed axis perpendicular to the reference plane and passing through its mass center G, we say that the body is in centroidal rotation. Since the acceleration a is identically equal to zero, the inertial terms of the body reduce to the couple Iα
General Plane Motion - In the general case of the plane motion of a rigid body, the resultant of the external forces acting on the body does not pass through the mass center of the body.

The mass center G of a rigid body in plane motion moves as if the entire mass of the body were concentrated at that point, and as if all the external forces act on it.

solution of problems involving the motion of a rigid body

Problems involving the motion of a rigid body can be solved by following these simple steps:

1.Isolating the body

2. Defining the axes

3. Replacing constraints with support forces

4. Adding applied forces and moments, as well as body forces to the diagram

5. Labeling the free-body diagram with dimensions

Shown in the left figure are free body diagram and kinetic diagram for a pendulum with external moment applied.

systems of rigid bodies

For systems involving multiple rigid bodies, the general equation of motion is written as shown in the right.
If no more than three unknowns are involved,the free-body and kinetic diagrams to sum components in any direction and sum moments about any point can be used, obtaining equations that can be solved for the desired unknowns.
If more than three unknowns are involved, a new system must be chosen, use kinematics, or use additional information in the problem statement to find additional equations.

constrained plane motion

Most engineering applications deal with rigid bodies that are moving under given constraints. For example, cranks must rotate about a fixed axis, wheels must roll without sliding, and connecting rods must describe certain prescribed motions. Definite relations exist between the components of the acceleration a of the mass center G of the body considered and its angular acceleration α. The corresponding motion is said to be a constrained motion.
Two other particular cases of constrained plane motion are of special interest: noncentroidal rotation of a rigid body and rolling motion of a disk or wheel.

noncentroidal rotation

The motion of a rigid body constrained to rotate about a fixed axis that does not pass through its mass center is called noncentroidal rotation. The mass center G of the body moves along a circle with a radius r centered at point O, where the axis of rotation intersects the plane of reference
The expressions for the tangential and normal components of the acceleration of G are shown in the figure. The equations define the kinematic relation between the motion of the mass center G and the motion of the body about G.
The moment of inertia of the rigid body about the fixed axis is expressed usingg the formula shown in the right figure.

rolling motion

Another important case of plane motion is the motion of a disk or wheel rolling on a plane surface. If the disk is constrained to roll without sliding, the acceleration a of its mass center G and its angular acceleration α are not independent.
Recall that the system of the inertial terms in plane motion reduces to a vector ma and a couple Iα. We find that, in the particular case of the rolling motion of a balanced disk, these terms reduce to a vector of magnitude mrα attached at G and to a couple with a magnitude of Iα.
When a disk rolls without sliding, there is no relative motion between the point of the disk in contact with the ground and the ground itself. Thus, as far as the computation of the friction force F is concerned, a rolling disk can be compared with a block at rest on a surface.
The three different cases of rolling motion are shown in the left figure.

classwork assignment #4

Assignment#4_BARTOLOME(DORB001).pdf

coursework 4

BARTOLOME-FAUSTO-COURSEWORK-4-DORB001 (1).pdf

reflection:

The module is a continuation of kinetics of rigid bodies (module 3) which discusses the relations between the forces acting on a rigid body, the shape and masss of the body, and the analysis of motion produced by the rigid body.I found out that most of the cases of motion of rigid body under this module, the results derived is limited to the plane motion, where all motion occurs in singe 2d reference plane and the rigid bodies considered consist only of plane rigid bodies and of bodies that are symmetrical with respect to a reference plane.

Additionally, I have learned that the rate of change of the angular momentum is equivalent to the product of moment of inerta times angular acceleration of the body. Also, I learned that the external forces acting on a rigid body are also equal to the vector ma (mass x acceleration) attached to the mass center and a couple of moment (moment of inerta x angular acceleration). Also, I have learned on how construct a free-body diagram and kinetic diagram in solving problems concerning plane motion of rigid bodies, by solving problems that are included in our coursework. Although it seemed difficult at first, understanding the principles of plane motion of rigid body, it helped me to solve variety of problems especially those involving translation, centroidal rotation, and unconstrained motion of rigid bodies.

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