Chap' 13

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Chap. 13 Angular Kinematics
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Angular 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
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13.1 Polar Coordinates
P
O
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13.2 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
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13.2 Angular Position or Displacement (continued)

• Angular Displacement
• Counter-clockwise is positive.

1 rad
radius
-135o
radius ( arc length)
radius
135o
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13.3 Angular Velocity 13.4 Angular Acceleration
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13.6 Definitions of Basic Concepts

• Free Oscillation

• Angular
• Displacement

Angular Velocity
Angular Acceleration
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13.6 Definitions of Basic Concepts

• Damped Oscillation
• Damping effect (resistance)

Damped oscillations
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Example 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
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Example 13.1 Shoulder abduction

• ltSolutiongt

Assume
?
?
?
?
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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
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Example 13.1 Shoulder abduction
ltSolutiongt
Figure 13.12 Angular position, velocity and
acceleration vs. time.
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13.7 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.

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13.8 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
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13.10 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
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13.11 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
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13.11 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
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13.11 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
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Relationship Between Angular Motions and Linear
Motions
y
P
Y
r
o
For vector mechanics,
r
x
R
O
X
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For vector mechanics,
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1
2
3
Centripetal acceleration (normal component)
1
Tangential acceleration (tangential component)
2
Coriolis acceleration
3
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Centripetal Motions
y
j
x
i
Note the sign of v and a !
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Example 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.
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Example 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
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Example 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
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Example 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

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Example 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.
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13.12 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
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Example 13.4 Double Pendulum
Figure 13.4(a)
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Example 13.4 Double Pendulum
a) Motion of B observed from A
A
B
B
Figure 13.4(a)
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Example 13.4 Double Pendulum
Coordinate Transformation(1)
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Example 13.4 Double Pendulum
Coordinate Transformation(2)
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Example 13.4 Double Pendulum
Coordinate Transformation (3)
Transformation matrix
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Example 13.4 Double Pendulum
a) Motion of B observed from A
B
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Example 13.4 Double Pendulum
a) Motion of B observed from A
B
Figure 13.4(c)
Since 0.3m, 30,
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Example 13.4 Double Pendulum
b) Motion of C observed from B
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Example 13.4 Double Pendulum
b) Motion of C observed from B

Since 0.3m, 45, 4rad/s
Figure 13.4(e)
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Example 13.4 Double Pendulum
c) Motion of C observed from A

The magnitudes
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Instant 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
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Instant 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.