Next, consider the motion of a rigid body over the interval as shown, with arbitrary point taken as reference.
To determine how forces and other sliding vectors must be added, let us start by the simplest case, when the forces to be added have the same line of action.
If a body has two or three planes of symmetry, the principal directions will be aligned with these planes. For the general case where we have an arbitrary orientation of xyz within the rigid bodythe last three equations describing the motion of the rigid body are moment torque equations. So, the equation would become: For example, if Dynamics of rigid bodies assumes planar motion, the six equations reduce to the two-dimensional dynamics equations given above.
We will apply some of the results we have derived for transformation from an inertial frame to a rotating frame. Therefore, the composition of the former three angles has to be equal to only one rotation, whose axis was complicated to calculate until matrices were developed.
For every rigid body a principal direction exists. To satisfy this, the particles that comprise a rigid body must move in concert, making the kinematics almost trivial.
For a rigid body, the angular momentum L is the product of the moment of inertia and the angular velocity: In two-dimensions, to find the angular acceleration you simply differentiate the magnitude of the angular velocity, with respect to time.
While the translational motion is described by the velocity and acceleration of the reference point, the rotational motion is characterized by the unique angular velocity and angular acceleration associated with the body.
To illustrate, see the figure below. You just replace the subscript G with the subscript O, and everything else stays the same.
The rotations were described by orthogonal matrices referred to as rotation matrices or direction cosine matrices. Angular velocity and angular acceleration of a rigid body. Whenever it was necessary to add forces in chapter 4we assumed that they could be moved freely and added together as free vectors, using the parallelogram rule.
Combining the first and last equation in this example leads to Solution: Motion in which every line in the body remains parallel to its original position.
The six inertia terms are calculated with respect to xyz and depend on the orientation of xyz relative to the rigid body. Any plane motion that is neither a pure rotation nor a translation falls into this class.
The direction of the velocity is tangent to the path of the point of rotation. By comparison with expressions derived previously: To illustrate, see the figure below.
Similarly, the positive directions of the individual x, y, z axes are also in the directions shown in the first figure. The high speeds in motorcycle races imply larger tilt angles. A rigid body is called chiral if its mirror image is different in that sense, i.
A general rigid body subjected to arbitrary forces in two dimensions is shown below. But for the moment equations, the reason is rather complicated, but basically it comes down to how they are derived, which is discussed here.
With this simplification, the moment equations become: Objects deform elastically, but these deformation are negligible for a wide range of problems. If we could apply the upward force in a point in the same vertical line as the center of gravity C, the chair would go straight up without rotating.
The study of particle and rigid body dynamics is a fundamental part of curricula for students pursuing graduate degrees in areas involving dynamics and control of systems. These include physics, robotics, nonlinear dynamics, aerospace, celestial mechanics and automotive engineering, among others.
Rigid Body Dynamics November 15, 1 Non-inertial frames of reference So far we have formulated classical mechanics in inertial frames of reference, i.e., those vector bases in.
thus, a rigid body in space is a dynamic system with six degrees of freedom. The translational motion of a rigid body in space was treated in Part II. This chapter is concerned with the rotational motion of a rigid vehicle with or without the influence of gravitational and other external forces.
Rigid Body Dynamics. In video game physics, we want to animate objects on screen and give them realistic physical behavior. This is achieved with physics-based procedural animation, which is animation produced by numerical computations applied to the theoretical laws of physics. We shall analyze the motion of systems of particles and rigid bodies that are undergoing translational and rotational motion about a fixed direction.
Because the body is translating, the axis of rotation is no longer fixed in space. We shall describe the motion by a translation of the center of mass and a rotation about the center of mass. Chapter 21 Rigid Body Dynamics: Rotation and Translation about a Fixed Axis Introduction patience to the establishment of the laws of rotation of the solid bodies.
For a rigid body rotating about the center of mass with.Dynamics of rigid bodies