13
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.

Planar Kinematics of a Rigid Body

Embed Size (px)

DESCRIPTION

Dynamics, a whole chapter on planar kinematics of a rigid body with illustrations and drawings.

Citation preview

Page 1: 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.

Page 2: Planar Kinematics of a Rigid 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

Page 3: Planar Kinematics of a Rigid Body

16.2 Translation

Position:

�� � �� � ��/� �1

- 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:

�� �����

�����

����/��

�2

The time derivative of rB/A is zero since rB/A is constant. We get then:

�� � �� �3

Acceleration:

�� � �� �4

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: �� � ��

Page 4: Planar Kinematics of a Rigid Body

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:

! � "# �5

- 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:

�! � �"# �6

Angular Velocity:

The time rate of change in the angular position is called

angular velocity ω and is measured in rad/s.

& ��!�

�7

with the magnitude

( ��"�

�8

Angular Acceleration:

* ��&�

�9

With the magnitude

, ��ω�

��."� .

�10

Page 5: Planar Kinematics of a Rigid Body

- 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:

,�" � (�( �11

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.

� � 345 �15

Page 6: Planar Kinematics of a Rigid Body

Velocity:

� � 3645 � 3"647 �16

Since r = const, it follows 36 � 0, so that we get

� � 3"647 �17

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

Page 7: Planar Kinematics of a Rigid Body

"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 (.�

Page 8: Planar Kinematics of a Rigid Body

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.

Page 9: Planar Kinematics of a Rigid Body

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:

�� � �� � ��/� �27

Page 10: Planar Kinematics of a Rigid Body

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.

-

��� � ��� � ���/� �28

drB due to translation and rotation.

drA due to translation of A.

drB/A due to rotation about A.

Velocity:

�� � I�JI9

� I�KI9

�I�J/K

I9 �29

�� � �� � ��/� �30

Page 11: Planar Kinematics of a Rigid Body

With drB/A = rB/Adθ, we get

�3�/��

� 3�/��"�

� (3�/� � :�/� �31

- 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.

L ��/� � &8 ��/� �32

→ 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:

�� � �� �& 8 ��/� �33

- 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:

Page 12: Planar Kinematics of a Rigid Body

Summary:

�� � �� � ��/� �� � �� �& 8 ��/�

Page 13: Planar Kinematics of a Rigid Body