# Dynamics of rigid bodies

Rigid Body Dynamics Two-Dimensional Rigid Body Dynamics For two-dimensional rigid body dynamics problems, the body experiences motion in one plane, due to forces acting in that plane. A general rigid body subjected to arbitrary forces in two dimensions is shown below. The full set of scalar equations describing the motion of the body are: IG is the rotational inertia of the rigid body about an axis passing through the center of mass G, and pointing in the z-direction out of the page Note that, if the rigid body were rotating about a fixed point O, the final moment equation would retain the same form if we were to choose point O instead of point G. Rigid Body Dynamics Two-Dimensional Rigid Body Dynamics For two-dimensional rigid body dynamics problems, the body experiences motion in one plane, due to forces acting in that plane.

A general rigid body subjected to arbitrary forces in two dimensions is shown below.

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The full set of scalar equations describing the motion of the body are: IG is the rotational inertia of the rigid body about an axis passing through the center of mass G, and pointing in the z-direction out of the page Note that, if the rigid body were rotating about a fixed point O, the final moment equation would retain the same form if we were to choose point O instead of point G.

So, the equation would become: The figure below illustrates this situation. Where the point O is a fixed point, attached to ground.

A specific example of this would be a pendulum swinging about a fixed point. Here are some examples of problems solved using two-dimensional rigid body dynamics equations: Trebuchet Physics Three-Dimensional Rigid Body Dynamics For three-dimensional rigid body dynamics problems, the body experiences motion in all three dimensions, due to forces acting in all three dimensions.

This is the most general case for a rigid body. A general rigid body subjected to arbitrary forces in three dimensions is shown below. The first three of the six scalar equations describing the motion of the body are force equations. For example, a force acting along the Z-axis is resolved into its components along the xyz axes in the above three equations.

This can generally be done using trigonometry. However, it is not necessary to resolve the quantities along the xyz axis. For the above three force equations, one can resolve the quantities along the XYZ axes instead.

To solve three-dimensional rigid body dynamics problems it is necessary to calculate six inertia terms for the rigid body, corresponding to the extra complexity of the three dimensional system. To do this, it is necessary to define a local xyz axes which lies within the rigid body and is attached to it as shown in the figure aboveso that it moves with the body.

The six inertia terms are calculated with respect to xyz and depend on the orientation of xyz relative to the rigid body. So, a different orientation of xyz relative to the rigid body will result in different inertia terms.

The reason that xyz is said to "move with the body" is because the inertia terms will not change with time as the body moves. So you only need to calculate the inertia terms once, at the initial position of the rigid body, and you are done. This has the advantage of keeping the mathematics as simple as possible.

An added benefit of having xyz move with the rigid body is when simulating the motion of the body, over time. We can track the orientation of the body by tracking the orientation of xyz since they move together.

For two-dimensional rigid body dynamics problems there is only one inertia term to consider, and it is IG, as given above.

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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.

Therefore, we don't need to define an axes xyz that is attached to the rigid body, and has a certain orientation relative to it like we do in three-dimensional problems. 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.

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. To calculate these components, one must first determine the angular velocity vector of the rigid body with respect to the global XYZ axes, and then resolve this vector along the x, y, z directions to find the components wx, wy, wz.

This is often done using trigonometry. IGx is the rotational inertia of the rigid body about the x-axis, passing through the center of mass G IGy is the rotational inertia of the rigid body about the y-axis, passing through the center of mass G IGz is the rotational inertia of the rigid body about the z-axis, passing through the center of mass G IGxy is the product of inertia xy of the rigid body, relative to xyz IGyz is the product of inertia yz of the rigid body, relative to xyz IGzx is the product of inertia zx of the rigid body, relative to xyz The six inertia terms are evaluated as follows, using integration: The orientation of xyz relative to the rigid body can be chosen such that This orientation is defined as the principal direction of xyz.

With this simplification, the moment equations become:Engineering Systems in Motion: Dynamics of Particles and Bodies in 2D Motion from Georgia Institute of Technology. This course is an introduction to the study of bodies in motion as applied to engineering systems and structures.

We will study. The dynamics of the rigid body consists of the study of the effects of external forces and couples on the variation of its six degrees of freedom.

The trajectory of any point in the body, used as reference point, gives the variation of three of these degrees of freedom. Dynamics is the branch of mechanics which deals with the study of bodies in motion. Branches of Dynamics Dynamics is divided into two branches called kinematics and kinetics.

Rigid Body Simulation David Baraff Robotics Institute Carnegie Mellon University Introduction This portion of the course notes deals with the problem of rigid body dynamics.

## EN4 Notes: Kinematics of Rigid Bodies

To help get you started simulating rigid body motion, we’ve provided code fragments that implement most of . 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. Rigid-Body Dynamics Below are selected topics from rigid-body dynamics, a subtopic of classical mechanics involving the use of Newton's laws of motion to solve for the motion of rigid bodies moving in 1D, 2D, or 3D space.

B We may think of a rigid body as a distributed mass, that is, a mass that has length, area, and/or volume rather than occupying only a single point in space.

Rigid Body Dynamics