# Dynamics of rigid bodies

Charles Stark Draper is shown below with a gyroscope used for the inertial guidance systems. Relativity Rotational Motion of a Rigid Body Rotational motion is more complicated than linear motion, and only the motion of rigid bodies will be considered here.

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.

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. The kinematics equations for rotational motion at constant angular acceleration are Consider a wheel rolling without slipping in a straight line.

The values of these three rotations are called Euler angles. The first object might be a heavy ring supported by struts on an axle like a flywheel. Notice that the resultant force is also in the same direction as the original forces and with magnitude equal to the sum of the magnitudes of andsince.

Let and be the positions of particles A and B in the fixed reference. To do this, it is necessary to define a local xyz axes which lies within the rigid body Dynamics of rigid bodies is attached to it as shown in the figure aboveso that it moves with the body.

We conclude that to predict the effect of a force it is necessary to know its magnitude, direction, sense and also its line of action, which is the straight line passing through the point where the force is applied, in the direction of the force.

This has the advantage of keeping the mathematics as simple as possible. This choice of sign convention is important when using the rigid body dynamics equations given here especially the moment equations.

Differentiating the relation above with this in mind, These hold for arbitrary lines attached to the body, implying in turn that the body can be associated with a unique angular velocitydefined as, for an arbitrary line attached to the body.

A force produces different effects if it is moved to a different line of action, even if its magnitude, direction and sense are kept equal.

While the field of particle and rigid body dynamics has not evolved significantly over the past seven decades, neither have approaches to teaching this complex subject. Geometry[ edit ] Two rigid bodies are said to be different not copies if there is no proper rotation from one to the other. But for the moment equations, the reason is rather complicated, but basically it comes down to how they are derived, which is discussed here.

The three force equations, and three moment equations shown here for three-dimensional rigid body dynamics problems "fully describe" all possible rigid body motion.

As rigid bodies are viewed as collections of particles, this may appear an insurmountable task, requiring a description of the motion of each particle. The six inertia terms are calculated with respect to xyz and depend on the orientation of xyz relative to the rigid body. This is because, for planar problems where motion is in one planeIG would be independent of the orientation of xyz relative to the rigid body. These kind of vectors, with a specific line of action, are called sliding vectors.

Rotation matrix With the introduction of matrices the Euler theorems were rewritten. They are equivalent to rotation matrices and rotation vectors. With respect to rotation vectors, they can be more easily converted to and from matrices. Note that the positive directions of the individual X, Y, Z axes are in the directions shown in the first figure.

For every rigid body a principal direction exists. 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. However, it is not necessary to resolve the quantities along the xyz axis.

In the following, we will restrict attention to the planar motion of rigid bodies. Observe that, as the moving frame does not rotate, basis vectors i and j do not change in time. Clearly, it is a good idea to choose the orientation of xyz so that it lies in the principal direction.

The positions, velocities and accelerations determined in this way are referred to as absolute. A general rigid body subjected to arbitrary forces in two dimensions is shown below. You just replace the subscript G with the subscript O, and everything else stays the same. An added benefit of having xyz move with the rigid body is when simulating the motion of the body, over time. However, for three-dimensional rigid body dynamics problems these vectors might be pointing in different directions, as shown below.

Provides detailed, easy-to-understand explanations of concepts and mathematical derivations Includes numerous flipped-classroom exercises carefully designed to help students comprehend the material covered without actually solving the problem for them Features an extensive chapter on electromechanical modelling of systems involving particle and rigid body motion Provides examples from the state-of-the-art research on sensing, actuation, and energy harvesting mechanisms Offers access to a companion website featuring additional exercises, worked problems, diagrams and a solutions manual Ideal as a textbook for classes in dynamics and controls courses, Dynamics of Particles and Rigid Bodies: The second object could have its mass close to the central axis.

All points of the body have the same velocity and same acceleration. In order to do this, observe that an arbitrary rigid body motion falls into one of the three categories: The law of conservation of angular momentum can be stated that the angular momentum of a system of objects is conserved if there is not external net torque acting on the system.

For these problems IG can be calculated with respect to any orientation of the rigid body, and it will always be the same, since the problem is planar. 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. We will study the dynamics of particle motion and bodies in rigid planar (2D) motion. This will consist of both the kinematics and kinetics of motion. Kinematics deals with the geometrical aspects of motion describing position, velocity, and acceleration, all as a function of time. In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with its axes fixed to the body and parallel to the body's principal axes of degisiktatlar.com general form is: ˙ + ×.

where M is the applied torques, I is the inertia matrix, and ω is the angular. 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. Chapter 9.

Dynamics of Rigid Bodies (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. ) Notes on Notation.

Dynamics of rigid bodies
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Rigid body dynamics - Wikipedia