ME 459 Dynamics of Machinery
Review of Rectilinear (Straight Line) Motion
General Concepts
Position, Velocity, and Acceleration
A particle P has rectilinear motion when it moves in a straight line. As shown in the
figure, define the direction of motion as
ME 459 Dynamics of Machinery
Kinematics of a Point Moving on a Rigid Body Two Dimensions
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 d
ME 459 Dynamics of Machinery
Definitions of Inverse Kinematics and Dynamics
Inverse Kinematics
The study of the individual body kinematics required to produce a desired motion
of some point or body within a mechanical system.
e.g. the robot problem of cal
ME 459 Dynamics of Machinery
Definitions of Forward Kinematics and Dynamics
Forward Kinematics
The study of the kinematics of specific points within a mechanical
system given the kinematics of each of the individual bodies within the
system.
e.g. the robo
ME 459 Dynamics of Machinery
Newtons Law for Rigid Body Motion (2D)
General Plane Motion
The figure at the right depicts a rigid
body moving in two dimensions. The
motion is caused by a series of N forces
Fi (i = 1,K, N ) . Each force, in general, has
%
t
ME 459 Dynamics of Machinery
Center of Mass in Two Dimensions
Definition of Mass Center
The figure at the right depicts a rigid body in
two dimensions. The mass center of the body is
defined as that point where
rdm = 0
%
B
xdm = 0
B
ydm = 0
B
A more pr
ME 459 Dynamics of Machinery
Mass Moments of Inertia (2D)
Definition of Mass Moment of Inertia
The figure at the right depicts a rigid body in
two dimensions. The moment of inertia of the body
about the Z-axis is defined as
(
)
I Z = r 2dm = x 2 + y 2 dm
ME 459 Dynamics of Machinery
Angular Momentum of a Rigid Body (2D)
The figure at the right depicts a rigid body moving
in two dimensions. The "XY" axes are fixed, the "xy"
axes are fixed in (and move with) the body, and G is
the mass center of the body. T
ME 459 Dynamics of Machinery
Relative Kinematics of Two Points Fixed on a Rigid Body (2D)
Relative Velocity (2D)
Consider two points P and Q fixed on the
same rigid body B as shown in the figure at the
right. The position vector of point P relative to
a f
ME 459 Dynamics of Machinery
Rolling (without Slipping) in Two Dimensions
Rolling on a Fixed Surface
If a rigid body rolls (without slipping) on a fixed surface, the point that is in contact
with the surface has zero velocity. For example, consider a circ
ME 459 Dynamics of Machinery
Curvilinear Motion Review
General Concepts
Position, Velocity, and Acceleration
If a particle does not move in a straight line,
then its motion is said to be curvilinear. Given
r (t ) the position vector of a particle P, the
%
ME 459 Dynamics of Machinery
Curvilinear Motion Review Part 2
Normal and Tangential Components
Normal and tangential components refer to
components that are normal and tangential to
the path of P. These directions are defined by
the unit vectors en and et
ME 459 Dynamics of Machinery
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. For
example, consider the vector r P / Q
ME 459 Dynamics of Machinery
Curvilinear Motion Review Part 3
Radial and Transverse Components
Another way to describe the motion of P as it
moves along a curved path is to use radial and
transverse components. Here, we define the unit
vector er to point