FROM PARTICLE TO RIGID BODY From Particle to

FROM PARTICLE TO RIGID BODY

From Particle to Rigid Body This lecture concentrates on a body that cannot be treated as a single particle but as a combination of a large number of particles. The lecture discusses the kinematics of a rigid body in 3 -D space After this lecture, the student should be able to: • Appreciate the concept of rigid bodies and extended rigid bodies • Define the mathematical condition for the rigidity of a body • Define the rotational matrix between two frames of reference

Particle vs. Rigid Body Combination of many particles (size/shape has to be considered) Particle (size/shape not important) X A B The size/shape has to be considered, for example, because the length between point “A” and point “B” may affect the kinematics solution. For a particle, size/shape is not important as it is treated as a point. Rigid implies that body cannot be deformed.

Extended Rigid Body 2 Axis <a> Body 1 Body 3 Point “P” The 3 rigid bodies are linked: Body 1 and body 2 joined together by body 3 (pin). As body 2 rotates about axis <a>, any point on the axis (e. g. point “P”) remains fixed relative to all 3 bodies. We can consider this axis as an extension of any of the 3 rigid bodies. In general, we can associate a space outside as an extension of a rigid body if no point of that space undergoes motion relative to the body.

Spatial Configuration Consider the following rigid body and the inertia reference frame {X, Y, Z} with origin located at point “O”: To study the movement of the rigid body w. r. t. the inertia frame, we use another set of orthonormal frame attached to the body. For example, frame {e 1, e 2, e 3} is attached to the body with origin fixed at “A” Z-axis X “O” A X-axis B Y-axis Note that points”O” and “A” are at the same point at time t=0. As the body moves, the points are not the same

Spatial Configuration Consider the body has moved to a new location after time t=1 sec. To define the new position of the body w. r. t. frame {a}, we can use vector (from “O” to “A”) Obviously, position is not enough to describe the motion as the orientation of the body has changed (orientation is not a problem in particle kinematics) Z-axis X B A X “O” A X-axis B Y-axis

Position and Translation • Translation can be defined by the displacement between the origins of frame {b} =(e 1, e 2, e 3) and frame {a}=(X, Y, Z). Translation will not change the orientation between the frames: Z-axis X A X “O” A X-axis B Y-axis B

Orientation To study the effects of the orientation, we note that at time t=0: is in the positive X-axis: is in the positive Y-axis: is in the positive Z-axis: At time t=1 sec. : Z-axis X B A X “O” A X-axis B Y-axis We can use to define the orientation of the body

Orientation and Rotation Orientation: The changes in the orientation of the can be viewed as a result of rotations: e. g. 90° rotation about Zaxis Z-axis X X “O” A X-axis B Y-axis

General Motion of a Rigid Body General Motion = Translation + Rotation The analysis so far revealed that the motion of a rigid body can be defined w. r. t. the inertia reference frame {X, Y, Z} using another orthonormal frame attached to the body. The rigid body motion has two parts: • Translation: defined by the vector from the origin of frame {X, Y, Z} to the origin of frame • Orientation: The changes in orientation of frame can be defined by rotations w. r. t. frame {X, Y, Z}

Translation of a Rigid Body Consider the following translation of a rigid body: Z-axis X A X “O” A B B Y-axis X-axis Notice that all the points in the rigid body must have the same velocity and the same acceleration at any time instance

Rotation of a Rigid Body Unlike translation, rotation is defined using an angular coordinate like . The angular velocity and angular acceleration are respectively defined as Vector direction Rotation, angular velocity and angular acceleration are vectors: they have both magnitudes and directions (the direction is defined using the right-hand rule) Direction of rotation

Rigidity Condition The example on the motion of a rigid body also revealed the rigidity conditions: C Z-axis X C B A X “O” A B Y-axis X-axis The distance between points “A” and “B” on the body remains unchanged after the body has moved. Similarly the CAB also remains unchanged after the motion. The preservation of these properties are called rigidity conditions.

Rotational Tensor The changes in orientation can be defined by the rotational tensor. Let us review the case of pure rotation: The rotational tensor is defined as Z-axis C X “O” A B Y-axis X-axis is in the positive X-axis: is in the positive Y-axis: is in the positive Z-axis: There is no rotation!

Rotational Tensor What is the rotational tensor? Z-axis The rotational tensor is C X “O” A B Y-axis X-axis is in the positive Y-axis: is in the negative X-axis: is in the positive Z-axis:

Rotational Tensor Example X What is the rotational tensor for the following configuration? Z-axis The rotational tensor is B C “O” A Y-axis is in the positive X-axis: X-axis is in the positive Z-axis: is in the negative Y-axis:

Rotational Motion Consider two points “A” to “B” on the rigid body (the distance “AB” is 2) and the rotational tensor: At time t=0: At time t=1: Z-axis C X X “O” A X-axis B Y-axis

Rotational Motion Notice that: R(t 1) times vector (AB) before rotation = Vector (AB) after rotation

Rotational Motion In general, given two arbitrary point “P” and “Q” on the rigid body and the rotational tensor at time “t”: The equation describes how changes its orientation in space as a result of a rotation defined by R(t) Note that if R(t) is the identity matrix, then This means that either: • No rotation takes place • The rotation returns the body to its original position • The rotation is such that every line connecting 2 arbitrary points of the body remains parallel to itself

Example: Rotational Motion Given before rotation is Find the vector on the rigid body after rotation.

Rotation between two configurations Let us again examine the rotation between the following two configurations using a new orthonormal frame: At time t=t 0: Z-axis Let X “O” A X-axis B Y-axis

Rotation between two configurations At time t=t 1: Z-axis X “O” A X-axis B Y-axis

Rotation between two configurations At time t=t 1: Note that R(t 1) times Frame (t 0) before rotation = Frame (t 1) after rotation

Rotation between two configurations At time t=t 2: Z-axis X B “O” A X-axis Y-axis

Rotation between two configurations At time t=t 2: Again, note that R(t 2) times Frame (t 0) before rotation = Frame (t 2) after rotation

Rotation between two configurations At time t=t 1 and t=t 2:

Rotation between two configurations At time t=t 1 and t=t 2:

Rotation between two configurations In general, the concept can be extended to any arbitrary orthonormal set attached to the rigid body, i. e. using Therefore: The above shows that if two configurations F(t 2) and F(t 1) are known relative to F(t 0), then we can find the configuration of F(t 2) relative to F(t 1) using R, where

Summary This lecture concentrates on a body that cannot be treated as a single particle but as a combination of a large number of particles. The lecture discusses the finite motion of a rigid body in 3 -D space. The following were covered: • The concept of rigid bodies and extended rigid bodies • The mathematical condition for the rigidity of a body • The rotational matrix between two frames of reference