Title: Chap' 13
1Chap. 13 Angular Kinematics
2Angular Kinematics of Human Movement
Goals
- Rectilinear and Curvilinear Motions
- Relationships among Angular Kinematic Variables
- Angular Kinematic Quantities with their Units of
Measure - Relationships between Angular Kinematic
Quantities and Linear Kinematic - Quantities
- Equations for Angular Kinematics
Questions 1. Why is a driver longer than
9-iron? 2. Hands up the handle of the bat
in bunt 3. Angular motion of discus or
hammer
313.1 Polar Coordinates
P
O
413.2 Angular Position or Displacement
- Angular Distance
- The sum of all angular changes undergone by a
rotating body - Scalar representation
- Angular Displacement
- The difference in the initial and final
positions of the moving body - Change in angular position
- Vector representation
Angular distance
Angular displacement
513.2 Angular Position or Displacement (continued)
- Angular Displacement
- Counter-clockwise is positive.
1 rad
radius
-135o
radius ( arc length)
radius
135o
613.3 Angular Velocity 13.4 Angular Acceleration
713.6 Definitions of Basic Concepts
Angular Velocity
Angular Acceleration
813.6 Definitions of Basic Concepts
- Damped Oscillation
- Damping effect (resistance)
Damped oscillations
9Example 13.1 Shoulder abduction
13.6 Definitions of Basic Concepts
- Shoulder abduction in the frontal plane
- Assume that time it takes for the arm to cover
the angles between OA and OB, OB and OA, OA and
OC, and OC and OA are approximately equal. - Derive expressions for the angular displacement,
velocity and acceleration of the arm. - Take the period of angular motion of the arm to
be 3s and the angle to be 80
Figure 13.1(a) Shoulder abduction
10Example 13.1 Shoulder abduction
Assume
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?
?
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Equations and can alternatively be
written as
amplitude of the angular velocity
Figure 13.1(b) Graph of function
amplitude of the angular acceleration
11Example 13.1 Shoulder abduction
ltSolutiongt
Figure 13.12 Angular position, velocity and
acceleration vs. time.
1213.7 Rotational Motion About a Fixed Axis
- n and t are the normal (radial) and tangential
directions at point P. - For circular motion, the velocity vector has only
a tangential component. - For a circular motion, the acceleration vector
can have both tangential and normal components.
1313.8 Relationships Between Linear Angular
Quantities
Recall that For a circular motion, radius is
constant.
13.9 Uniform Circular Motion
Angular velocity constant ? Angular
acceleration zero
1413.10 Rotational Motion with Constant Acceleration
- The equations for rotational motion about a
fixed axis with - constant angular acceleration
constant angular acceleration
the initial angular position and velocity of
the object at time
1513.11 Relative Motion
Relative Angle vs. Absolute Angle
- (1) Relative angle Angle at a joint formed
between the longitudinal axes of - adjacent body segments
- Should be consistently measured on the same side
of a given joint - The straight, fully extended position at a joint
is regarded as 0 - degrees.
- (2) Absolute angle Angular orientation of a
body segment w.r.t. a fixed reference line - Should be consistently measured in the same
direction from a single - reference - either horizontal or vertical.
Measuring angles
Caliper, Filmed Images, Videotapes, Computer
measurements
1613.11 Relative Motion
Relationship Between Angular Motions and Linear
Motions
B2
B2
B1
B1
B1
r
r
r
r
r
q
r
A2
A2
A1
A1
A2
A1
1713.11 Relative Motion
Relationship Between Angular Motions and Linear
Motions
sA sA/B sB vA vA/B vB aA aA/B aB
sA, sB Linear displacement of A, B sA/B
Relative displacement of A w.r.t. B vA, vB
Linear velocity of A, B vA/B Relative
velocity of A w.r.t. B aA, aB Linear
acceleration of A, B aA/B Relative
acceleration of A w.r.t. B
18Relationship Between Angular Motions and Linear
Motions
y
P
Y
r
o
For vector mechanics,
r
x
R
O
X
19For vector mechanics,
201
2
3
Centripetal acceleration (normal component)
1
Tangential acceleration (tangential component)
2
Coriolis acceleration
3
21Centripetal Motions
y
j
x
i
Note the sign of v and a !
22Example 13.3
- Consider the motion described in Fig. 13.22. A
person (B) riding on a vehicle that is moving
toward the right by a constant speed of 2m/s
throws a ball straight up into the air with an
initial speed of 10m/s. - Describe the motion of the ball as observed by a
stationary person (A) in the time interval
between when the ball is first released and when
it reaches its maximum elevation.
Figure 13.3(a) Relative to the xy frame, the ball
is undergoing a translational motion in the y
direction.
23Example 13.3 ltSolutiongt
The three step to solve the problem.
First, let x and y represent a coordinate frame
moving with the vehicle. The speed of the ball
in the y direction between the instant of release
and peak elevation can be determined from
Assume This equation is valid in the time
interval between t0 and
Person B observed that the ball has no motion in
direction. Therefore, the velocity
of the ball relative to person B
24Example 13.3 ltSolutiongt
Next, let and represent a coordinate
frame fixed to the ground. With respect to the
frame, or with respect to the stationary person
A, the vehicle is moving in the positive
direction with a constant speed of vx0 2m/s
(Figure 13.3(b)).
Figure 13.3(b) Relative to the XY frame, the
vehicle is undergoing a translational motion in
the X direction with constant velocity.
Therefore
25Example 13.3 ltSolutiongt
- Finally, to determine the velocity of the ball
relative to person A, we have to add velocity
and together - Or by substituting the known parameters
-
26Example 13.3 ltSolutiongt
- For example, 0.5s after the ball is released, the
ball has a velocity - That is, according to person A or relative to the
XY coordinate frame, the ball is moving to the
right with a speed of 2m/s and upward with a
speed of 5m/s(Fig 13.3(c)). At this instant, the
magnitude of the net velocity of the ball is
Figure 13.3(c) Relative to the XY frame, the ball
moves both in the X and Y directions.
2713.12 Linkage Systems
- Linkage system
- - Double pendulum
- consists of two bars hinged together and to
the ground. - Multi-link systems
- Degree of Freedom (DOF)
Double pendulum
28Example 13.4 Double Pendulum
Figure 13.4(a)
29Example 13.4 Double Pendulum
a) Motion of B observed from A
A
B
B
Figure 13.4(a)
30Example 13.4 Double Pendulum
Coordinate Transformation(1)
31Example 13.4 Double Pendulum
Coordinate Transformation(2)
32Example 13.4 Double Pendulum
Coordinate Transformation (3)
Transformation matrix
33Example 13.4 Double Pendulum
a) Motion of B observed from A
B
34Example 13.4 Double Pendulum
a) Motion of B observed from A
B
Figure 13.4(c)
Since 0.3m, 30,
35Example 13.4 Double Pendulum
b) Motion of C observed from B
36Example 13.4 Double Pendulum
b) Motion of C observed from B
Since 0.3m, 45, 4rad/s
Figure 13.4(e)
37Example 13.4 Double Pendulum
c) Motion of C observed from A
The magnitudes
38Instant Center of Rotation
Instant Center of Rotation (1)
The center of rotation at a given joint angle, or
at a given instant in time during dynamic movement
39Instant Center of Rotation (2)
C instant center of velocity
Component of vB in BC direction is zero.
Component of vA in AC direction is zero.