ME 012 Engineering Dynamics
Lecture 21
Instantaneous Center of Zero Velocity
(Chapter 16, Section 6)
Tuesday,
Apr. 09, 2013
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
REMAINING COURSE MATERIAL
Today’s
Lecture
Chapter 16: Planar Kinematics of a Rigid Body
• 16.1 Rigid-Body Motion
• 16.2 Translation
• 16.3 Rotation About a Fixed Axis
• 16.4 Absolute Motion Analysis
• 16.5 Relative Motion Analysis: Velocity
• 16.6 Instantaneous Center of Zero Velocity
• 16.7 Relative Motion Analysis: Acceleration
Chapter 17: Planar Kinematics of a Rigid Body: Force and Acceleration
• 17.1 Moment of Inertia
• 17.2 Planar Kinetic Equations of Motion
• 17.3 Equations of Motion: Translation
• 17.4 Equations of Motion: Rotation About a Fixed Axis
• 17.5 Equations of Motion: General Plane Motion
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
2
TODAY’S OBJECTIVE
• Locate the instantaneous center of zero velocity.
• Use the instantaneous center to determine the velocity of any point on a rigid
body in general plane motion.
In-Class Activities:
• Applications
• Location of the Instantaneous
Center
• Velocity Analysis
• Problem Solving
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
3
APPLICATIONS
The instantaneous center ( ) of zero
velocity for this bicycle wheel is at the
point in contact with ground. The
velocity direction at any point on the
rim is perpendicular to the line
connecting the point to the .
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
As the board slides down the wall (to
the left) it is subjected to general plane
motion (both translation and rotation).
Since the directions of the velocities of
ends
and
are known, the
is
located as shown.
4
16.6 Instantaneous Center of Zero Velocity
For any body undergoing planar motion, there always exists a point in the plane
of motion at which the velocity is instantaneously zero (if it were rigidly
connected to the body).
This point is called the instantaneous center of zero velocity, or
or may not lie on the body!
. The
may
If the location of this point can be determined, the velocity analysis can be
simplified because the body appears to rotate about this point at that instant.
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
5
16.6 Instantaneous Center of Zero Velocity
LOCATION OF THE INSTANTANEOUS CENTER
To locate the , we can use the fact that the velocity of a point on a body is always
perpendicular to the relative position vector from the
to the point. Several
possibilities exist.
First recall the relation between velocity and angular
velocity : =
Now, consider the case when velocity
on the body and the angular velocity
are known.
of a point
of the body
In this case, the
is located along the line drawn
perpendicular to
at , a distance / = /
from . Note that the lies up and to the right of
since
must cause a clockwise angular velocity
about the .
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
6
16.6 Instantaneous Center of Zero Velocity
LOCATION OF THE INSTANTANEOUS CENTER
(2nd Case)
A second case is when the lines of action
of two non-parallel velocities, vA and vB,
are known.
First, construct line segments from A and B
perpendicular to vA and vB. The point of
intersection of these two line segments
locates the IC of the body.
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
7
16.6 Instantaneous Center of Zero Velocity
LOCATION OF THE INSTANTANEOUS CENTER
(3rd Case)
A third case is when the magnitude and direction of two parallel velocities at
and are known.
Here the location of the IC is determined by proportional triangles. As a special case,
would be located at
note that if the body is translating only (vA = vB), then the
infinity. Then equals zero, as expected.
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
8
16.6 Instantaneous Center of Zero Velocity
LOCATION OF THE INSTANTANEOUS CENTER
(Graphically Finding the )
In each case show graphically how to locate the of zero velocity of link
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
.
9
16.6 Instantaneous Center of Zero Velocity
VELOCITY ANALYSIS
The velocity of any point on a body undergoing general plane motion can be
determined easily once the instantaneous center of zero velocity of the body is
located.
• The body seems to rotate about the
instant, as shown in this kinematic diagram
at any
• The magnitude of velocity of any arbitrary point is:
=
where
point.
is the radial distance from the
to the
• The velocity’s line of action is perpendicular to its
associated radial line.
• Note the velocity has a sense of direction which
tends to move the point in a manner consistent with
the angular rotation direction.
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
10
16.6 Instantaneous Center of Zero Velocity
Important Notes
• The point chosen as the for the body can be used only for an instant of time since
the body changes its position from on e instant to the next.
• The does not, in general, have zero acceleration and so should not be used for
accelerations of point in a body
•
•
•
•
Solving Problems Using the
First establish the location of the using one of the methods described.
The body is then imagined as “extended and pinned” at the such that, at the
instant considered, it rotates about this pin with its angular velocity .
The magnitude of velocity for an arbitrary point on the body can be determined
using the equation =
where is the radial line drawn from the to each
point.
The line of action of each velocity vector is perpendicular to its associated radial
line and the velocity has a sense of direction which tends to move the point in a
manner consistent with the angular rotation of the radial line.
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
11
EXAMPLE 1
The wheel ( = 150 mm) is rotating with an
angular velocity = 8 rad/s. Determine the
velocity of the collar at the instant = 30°,
and
= 60°
using the method of
instantaneous center of zero velocity. Let
= 500 mm.
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
12
EXAMPLE 1: Solution
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
13
EXAMPLE 2
The pinion gear ( = 0.3 ft) rolls on the gear racks. Rack
is moving to the right at speed
= 8 ft/s and rack
is moving to the left at speed
= 4ft/s. Using the
method of instantaneous center of zero velocity,
determine the angular velocity of the pinion gear and
the velocity of its center .
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
14
EXAMPLE 2: Solution
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
15
EXAMPLE 3
The mechanism used in a marine engine consists of
a single crank
( = 0.2 m) and two connecting
rods
( ! = 0.4 m) and " ( # = 0.4 m).
Determine the velocity of the piston at
the
instant the crank is in the position shown ( = 45°,
= 30°, and $ = 45°) and has an angular velocity
= 5 rad/s.
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
16
EXAMPLE 3: Solution
ME 012 Engineering Dynamics: Lecture 21
J. M. Meyers, Ph.D. ([email protected])
17