ME 555 Intermediate Dynamics
Simple Angular Motion
Simple Angular Velocity
The rigid body B shown in the diagram below rotates about the Z-axis. The
XYZ reference frame is a fixed frame, while the xyz reference frame is fixed in (and
rotates with) the bod
ME 555 Intermediate Dynamics
Orientation Angles of a Rigid Body in Three Dimensions
To describe the general orientation of a rigid body in
three dimensions, consider the rigid body shown in the
figure at the right. Here there are two reference frames
the
ME 555 Intermediate Dynamics
Rolling Constraints Line Contact
Cone Rolling on a Flat Plane
In earlier notes, we discussed
rolling (without slipping) where there
was a single contact point. Here we
want to consider rolling when there is
a line of contact b
ME 555 Intermediate Dynamics
Angular Velocity and Orientation Angles
When using angle sequences to describe the orientation of a rigid body, the
addition rule for angular velocities is used to find the angular velocity of the body.
Consider the case where
ME 555 Intermediate Dynamics: Thrust Bearing Example
(Reference: Kane and Levinson, Dynamics: Theory and Applications, McGraw-Hill, 1985.
Problem:
Given the thrust bearing in the diagram,
show that for pure rolling between the shaft S
and the bearing B, i
ME 555 Intermediate Dynamics
Rolling Constraints Point Contact
When a rigid body rolls (without slipping)
on a rigid surface, its motion is constrained
by the surface. Consider the body B rolling
on the rigid surface S as shown at the right.
Here, we have
ME 555 Intermediate Dynamics
Example: Slider on a Rotating Bar
Example: Slider on a Rotating Bar
Problem: Given the position coordinates ( x, , ) ,
find vP the velocity of P and a P the
%
%
acceleration of P.
Solution:
D : (e1 , e2 , e3 )
%
B : (er , e
ME 555 Intermediate Dynamics
Derivatives of a Vector in Two Different Reference Frames
("The Derivative Rule")
Motivation
It is often convenient to express vectors
in terms of local (or rotating) unit vector sets
(reference frames). For example, consider
ME 555 Intermediate Dynamics
Systems with Closed Kinematic Chains (Constraints)
The systems we have studied so far are "open chain" systems. The kinematic
descriptions start at a point and advance into the system to ever more remote
points. Many mechanica
ME 555 Intermediate Dynamics
Kinematics of a Point Moving on a Rigid Body
We now extend our kinematic analysis to
include systems where interconnected bodies
may rotate and translate relative to each other.
In this case, we have a need to describe the
kin
ME 555 Intermediate Dynamics
Relative Kinematics of Two Points Fixed on a Rigid Body
General Concept
Consider the three dimensional motion of a
rigid body B as shown in the diagram at the right.
The points P and Q represent two points that are
fixed in th
ME 555 Intermediate Dynamics
Summation Rule for Angular Velocities
Consider a rigid body B undergoing three dimensional motion as shown in the
diagram below. R and S represent two reference frames that are rotating relative to
each other. The angular velo
ME 5550 Intermediate Dynamics
Differential (Reference: Kane and Levinson, Dynamics: Theory and Applications, McGraw-Hill, 1985.)
Nomenclature
D : drive shaft
A : left axle (rotates relative to
casings C and F )
A : right axle (rotates relative to
casings