Chapter 16 Planar Kinematics of a Rigid Bodyw3.gazi.edu.tr/~ksoyluk/resimler/IM 224 Ch16.pdf · The...

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Chapter 16

Planar Kinematics of a Rigid Body

1. Rigid-Body Motion

2. Translation

3. Rotation about a Fixed Axis

4. Absolute Motion Analysis

5. Relative-Motion Analysis: Velocity

6. Instantaneous Center of Zero Velocity

7. Relative-Motion Analysis: Acceleration

8. Relative-Motion Analysis Using Rotating Axis


16.1 Planar Rigid-Body Motion

When all the particles of a rigid body move along

paths which are equidistant from a fixed plane,

the body is undergoing planar motion.

3 types of rigid body planar motion:

1) Translation

This type of motion occurs when every line

segment on the body remains parallel to its

original orientation throughout the motion.



1) Translation

Rectilinear translation - when two particles paths

of motion are along equidistant straight lines

Curvilinear translation - when paths of motion are

along curved lines are equidistant



2) Rotation about a fixed axis

When a rigid body rotates about a fixed axis, all

the particles of the body, except on the axis of

rotation, move along circular paths.

3) General Plane Motion

When a body is subjected to general plane

motion, there is a combination of translation and

rotation.


16.2 Translation

Position

By vector addition,

Velocity

Since magnitude of rB/A is

constant due to rigid body,

Acceleration

Time derivative of velocity equation yields

The above two equations indicate that all points in a rigid

body subjected to either rectilinear or curvilinear translation

move with the same velocity and acceleration.

ABAB /rrr

AB vv

AB aa


16.3 Rotation about a Fixed Axis

When a body rotates about a fixed axis, any point P located

in the body travels along a circular path.

Angular Motion

Only lines or bodies undergo angular motion

Angular Position

Angular position of r is defined by θ

Angular Displacement

Change in the angular position

measured as a differential dθ



Angular Velocity

Time rate of change in the angular position

Since dθ occurs during an instant of time dt,

Angular Acceleration

Time rate of change of the

angular velocity

Magnitude of vector is

dt

d ( +)

( +) 2

2

dt

d

dt

d



Angular Acceleration

Differential relation between the angular

acceleration, angular velocity and angular

displacement is

Constant Angular Acceleration

dd ( +)

)(2

2

1

020

2

200

0

c

c

c

tt

t

( +)

( +)

( +)



Motion of Point P

Point P travels along a circular path of radius r

and center at point O

Position

P is defined by the position vector r

Velocity

The velocity of P is

By using the cross product,

rv

prv



Acceleration

Acceleration of P can be expressed as

For acceleration in vector form,

Magnitude of acceleration is

rara nt

2 ;

rraaa2 nt

22tn aaa


16.4 Absolute Motion Analysis

A body subjected to general plane motion

undergoes a simultaneous translation and

rotation

One way to define these motions is to use a

rectilinear position coordinate s to locate the

point along its path and an angular position

coordinate θ to specify the orientation of the line

By direct application of the time-differential

equations v = ds/dt, a = dv/dt, ω = dθ/dt, α =

dω/dt, the motion of the point and the angular

motion of the line can be related.


The large window is opened using a hydraulic

cylinder AB. If the cylinder extends at a constant rate

of 0.5 m/s, determine the angular velocity and

angular acceleration of the window at the instant θ =

30°

Example 16.5


Solution

Position Coordinate Equation

The angular motion of the window can be obtained

using the coordinate θ and motion along the

hydraulic cylinder is defined using a coordinate s.

When θ = 30°,

Example 16.5

cos45

cos)1)(2(2)1()2(

2

222

s

s

(1)

m239.1s


Solution

Time Derivatives

Taking the time derivatives of Eq. 1,

Since vs = 0.5 m/s, then at θ = 30°

Example 16.5

)(sin2)(

)sin(402

svs

dt

d

dt

dss

(2)

rad/s620.0


Solution

Time Derivatives

Taking the time derivatives of Eq. 2 yields,

Since as = dvs/dt = 0, then

Example 16.5

)(sin2)(cos2

)(sin2)(cos2

22

ss

ss

sav

dt

d

dt

d

dt

dvsv

dt

ds

2

22

rad/s415.0

30sin2)620.0(30cos20)5.0(


16.5 Relative-Motion Analysis: Velocity

The general plane motion of a rigid body can be

a combination of translation and rotation

We use a relative-motion analysis involving two

sets of coordinate axes

x, y coordinate system has a known motion

The axes of this coordinate

system will only be allowed

to translate with respect to

the fixed frame



Position

Position vector rA specifies the location of the

“base point” A, and the rB/A locates point B with

respect to point A

By vector addition, the position of B is

Displacement

A and B undergo

displacements drA and drB

ABAB /rrr



Displacement

Due to the rotation about A, drB/A = rB/A dθ, and

the displacement of B is

ABAB ddd /rrr

due to rotation about A

due to translation about A

due to translation and rotation



Velocity

To determine the relationship between the

velocities of points A and B,

dt

d

dt

d

dt

d ABAB /rrr



Velocity

vB/A has a magnitude of vB/A = ωrB/A and a

direction which is perpendicular to rB/A

ABAB /vvv



Velocity

vB is determined by considering the entire body

to translate with a velocity of vA, and rotate about

A with an angular velocity ω

Vector addition of these two effects, applied to B,

yields vB

vB/A represents the effect of circular motion,

about A. It can be expressed by the cross

product

ABAB /rvv



Velocity

Point A on link AB must move along a horizontal

path, whereas point B moves on a circular path

The wheel rolls without slipping

where point A can be

selected at the ground

Chapter 16 Planar Kinematics of a Rigid Bodyw3.gazi.edu.tr/~ksoyluk/resimler/IM 224 Ch16.pdf · The...

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