Download - Planar Kinematics of a Rigid Body
16 Planar Kinematics of a Rigid Body
16.1 Rigid-Body Motion
- The study of kinematics is a mathematical problem
- We assume homogenous bodies made of the same material.
- A body undergoes planer motion when all the particles of a rigid body move
along paths which are equidistant from a fixed plane.
- There are types of rigid body motion:
1. Translation:
If every line segment on the body remains parallel to its original direction
during the motion. → 2 types
- rectilinear translation
- curved translation
2. Rotation about a fixed axis
- When a rigid body rotates about a fixed
axis, all particles of the body, except those
which lie on the axis of rotation, move
along circular paths.
- The rotation axis may be located inside the
body or outside of the body.
- Pure rotation, if the fixed axis goes
through the centroid of the body
- General rotation, if the fixed axis does not
go through the centroid of the body.
3. General plane motion
- The body undergoes translation and rotation at the same time.
- In fact, general rotation is general plane motion.
- Each general plane motion may be momentarily considered as a general
rotation about a fixed axis.
- The general plane motion is completely specified if
1. The motions of two points in the body are known ( → relative-motion
analysis), or
2. The rotational motion of a line fixed in the body and the translation of
a point located on this line ( → absolute-motion analysis).
Example: Types of planar motion
16.2 Translation
Position:
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- The position vectors rB and rA are absolute, they are measured from the x, y axes.
- The position vector rB/A is relative and gives the position of B with respect to A.
rB/A is measured from the translating x', y' axes.
- The magnitude of rB/A is constant since the body is rigid.
- The direction of rB/A is constant since x', y' coordinate system does not rotate.
Velocity:
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The time derivative of rB/A is zero since rB/A is constant. We get then:
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Acceleration:
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Note: Equations (3) and (4) indicates that all points in a rigid body subjected to either
rectilinear or curvilinear translation move with the same velocity and acceleration.
→ A translating rigid body may be considered as a particle.
Summary:
Position: �� � �� � ��/� Velocity: �� � �� Acceleration: �� � ��
16.3 Rotation about a Fixed Axis
- When a rigid body is rotating about a fixed axis, all
particles of the body, except those which lie on the
axis of rotation, travels along circular paths.
- only lines or bodies undergo angular motion.
Angular Position:
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- θ is measured between a fixed reference line and r.
- θ is positive counterclockwise.
- Since motion is a bout a fixed axis, the direction of θ is
always along the axis.
- Is measured in degrees, radians, or revolutions. (π
=180°,1rev = 2π)
Angular Displacement:
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Angular Velocity:
The time rate of change in the angular position is called
angular velocity ω and is measured in rad/s.
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with the magnitude
( ��"�
�8
Angular Acceleration:
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�9
With the magnitude
, ��ω�
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- The direction of α is the same as that for ω. However, its sense of direction
depends on whether ω is increasing or decreasing.
- If ω is increasing, then α is called angular acceleration.
- If ω is decreasing, then α is called angular deceleration.
By eliminating dt from Eq. (8) and (10), we get:
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Example: Constant angular acceleration
Given:
- α = αc = constant
- initial conditions: ω(t = 0) = ω0, θ(t = 0) = θ0.
-
Find: ω(t), θ(t).
We get:
( � (0 � ,1 �12
" � "0 � (0 �12,1 . �13
(. � (0. � 2,1�" 2 "0 �14
Motion of Point P (Circular Motion):
Point P travels along a circular path.
Position:
- The position of P is defined by a position vector rP,
which extends from an arbitrary point lie on the axis
of rotation to P.
- Usually the vector r which is a special case of rP is
used.
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Velocity:
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Since r = const, it follows 36 � 0, so that we get
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With "6 � ( and 47 � # 8 45 we get from (17)
� � (# 8 345 � & 8 � �18
By circular motion 47 � 49, so that we get from (17)
� � 3(49 � :49 �19
→ The direction of v is tangent to the circular path.
Note:
- & 8 � ; � 8 &, however, & 8 � � 2� 8 &. - vθ is always perpendicular to r and �7 � & 8 �. - The relation v = ωr can only be used when the following three conditions are
satisfied:
1. r extends from a fixed point with zero velocity
2. ω and v have the same direction.
3. v is always perpendicular to r.
Acceleration:
� � <3= 2 3"6.>45 � <3"= � 236"6 >47 �20
Since r = const, it follows 36 � 3= � 0, so that we get
� � 3"=47 2 3"6 .45 �21 With
"6 � (, "= � ,, � � 345 , * � ,#, 47 � # 8 45 �22
We get:
� � * 8 � 2 (.� �23
In circular motion we have:
�9 � �7 � * 8 � �24
�@ � 2�5 � (.� �25
Note: All points in a rigid body rotate with the same ω and the same α. However, each
point in a plane perpendicular to the rotation axis, has its own different velocity and
acceleration since its position r, from the rotation axis, is different from the positions of
the other points. (: � (3, A9 � ,3, A@ � :. 3⁄ ).
Summary:
Rotation about a fixed axis:
( ��"�
, ��ω�
��."� .
,�" � (�(
Constant angular acceleration (α = αc = constant):
( � (0 � ,1 " � "0 � (0 �12,1 . (. � (0
. � 2,1�" 2 "0
Circular Motion of a Particle:
� � 345 � � & 8 � � � * 8 � 2 (.�
16.4 Absolute Motion Analysis
- A body subjected to general plane motion undergoes a simultaneous translation
and rotation.
- This motion can be completely specified by knowing both the angular motion of a
line fixed in the body and the rectilinear motion of a point on this line.
- By direct application of the relations:
: ��C�
A ��:�
( ��"�
, ��ω�
, �26
the motion of the point and the angular motion of the line can then be related.
Procedure for analysis:
- Locate point P using a position coordinate s.
- Measure from a fixed reference line the angular position θ of a line lying in the
body and passing through point P.
- From the dimensions of the body, relate s to θ, s = f(θ), using geometry and/or
trigonometry.
- Take the first time derivative of s = f(θ) to get a relation between v and ω.
- Take the second time derivative of s = f(θ) to get a relation between a and α.
- Use the chain rule when taking the derivatives.
16.5 Relative-Motion Analysis: Velocity
- The general plane motion of a rigid body
can be described as a combination of
translation and rotation.
- To view these “ components” motions
separately we will use a relative motion
analysis involving two sets of coordinate
axes.
- Fixed coordinate system x, y (absolute).
- Translating coordinate system x', y' (relative).
- The x', y' system is pin-connected to the body at the base point A.
- The axes of the x', y' coordinate system translate with respect to the fixed
system but not rotate with the body.
- The base point A has generally a known motion.
- The position vectors rA and rB are absolute and measured from the fixed x, y axes.
- The relative position vector rB/A is measured from the translating x', y' system and
gives the relative position of B with respect to A.
- Since the body is rigid, 3�/� � D��/�D � EFGC . - The relative motion analysis with translating axes can only be used to study the
motion of points
- on the same body
- on bodies which are pin-connected, or
- on bodies in contact without slipping between them.
→ For the other cases use translating and rotating axes.
Position:
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Displacement:
- During dt, points A and B undergo
displacements drA and drB.
- drA is pure translation
- drB consists of translation and rotation.
- The entire body first translates by an amount
drA, and then rotates about A by an amount
dθ. → point B undergoes a relative
displacement with drB/A = rB/Adθ.
-
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drB due to translation and rotation.
drA due to translation of A.
drB/A due to rotation about A.
Velocity:
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� I�KI9
�I�J/K
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With drB/A = rB/Adθ, we get
�3�/��
� 3�/��"�
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- vB and vA are measured from the fixed x, y axes and represent the absolute
velocityies of A and B, respectively.
- vB/A is the relative velocity of B with respect to A as measured by an observer
fixed to the translating x', y' axes.
- Since the body is rigid, the magnitude rB/A remains constant. Therefore, the
observer fixed to the translating x', y' axes sees point B move along a circular path
with radius rB/A and angular velocity ω.
→ vB/A is perpendicular to rB/A.
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→ The relative motion is circular, the magnitude is vB/A = ω rB/A and the
direction is perpendicular to rB/A.
From Equations (30) and (32) we get:
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- When two points have the same path of motion but located at two pin-connected
bodies or in contact (without slipping) with each other, then these points have the
same velocity and the same acceleration.
The choice of the base point A:
Summary:
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