Summary on Kinematics:

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Summary on Kinematics Motion of Rigid Body can
be described by TRANSLATIONAL MOTION
of any one point ROTATIONAL MOTION
of entire body
?
Vector description
Component description
Note formulas are for ? positive in a CCW (right
hand rule) sense
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Example of General Motion (but with a constraint)
Ladder is leaning against the wall and is
observed to slip. Velocity at B is observed to
be VB(t) What is angular velocity ? ?
General relationship
L
And, what is A in this case? What is ? ?
Need to identify existence of a constraint
another point on the body where something is
known about the motion
B
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Ladder is leaning against the wall and is
observed to slip. Velocity at B is observed to
be VB(t) What is angular acceleration ? ?
General relationship
L
B
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Important constrained rigid body motion the
ROLLING WHEEL
Physical Picture
C
C
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Recall General Motion equations
G is one point where we know something about the
motion What other point do we know something
about the motion?
VG
R
G
Contact point C
VC0
(Ground is not moving think of the ground as a
really huge gear)
A?C B?G
?3?/2
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Wheel instantaneously pivots around contact point!
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Now we can understand the CABLE REEL
B
C
A moves to the right. What is the linear velocity
of the reel?
Strategy relate A to B, then B to C, then C to O
C to O
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What is the velocity of O ?
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Back to Power Transmission's primary job is to
allow an engine/motor to operate in its narrow
range of speeds near maximum power while
providing a wide range of output speeds. Use
system with multiple gears, like a Planetary
Gear system
Controlling input gear and output gear, can
achieve different ratios e.g. rotate Sun gear
(input), hold Ring gear fixed,
output to Planet gear
Ring gear
Sun gear
From left to right the ring gear, planet
carrier, and two sun gears
Planet gear
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Coupled Bodies the Slider Crank
Given VB, what is ?AO ? Or vice-versa?
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Two rigid bodies AB and AO, connected at O
For rigid body AB relate A to B
For rigid body AO relate A to O
Velocity of point A is unique, so two results
must be equal
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Velocity of point A is unique, so two results
must be equal
If we dont want ? to appear, use geometry to
eliminate it
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For determining motion from forces (dynamics), we
often need the acceleration relationships that
are constraints on the motion
Note that the accelerations depend on the angular
velocity, so understanding velocity relationships
remains important
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Rotate rigid body OB at constant rate ?OB
What is angular acceleration ?AC of rigid body AC
?
General strategy is similar Body OB Relate
motion of B to motion of O (fixed) Body AB
Relate motion of A to motion of B Body AC
Relate motion of A to motion of C Eliminate
motions of A and B, solve for angular motions of
all 3 bodies
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Dynamics of Rigid Bodies
What is the equivalent of F ma ? How does the
angular velocity change ? What are momentum,
kinetic energy, angular moment of a rigid body
? - Rigid body has both translation and rotation
What is the rotational analog of mass? i.e.
what is resistance to angular acceleration? ?
Moment of Inertia
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Dynamics of Rigid Bodies
Forces are applied at different locations on the
body Forces friction, normal reaction, gravity,
other applied forces
Forces causes the body to both translate and
rotate
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For now, just one quantity Angular momentum
Sum of angular momenta of masses mi at ri moving
at Vi
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Recall Rate of change of h is equal to total sum
of all moments acting on the body
Sum of the moments around the center of mass is
proportional to the angular acceleration
Coefficient of proportionality is the Moment of
Inertia IG
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Suppose point O is FIXED in space?
Fixed-Axis Rotational Motion