Chapter 20: 3-D Kinematics of a Rigid Body

Chapter 20: 3-D Kinematics of a Rigid Body

©2007 Pearson Education South Asia Pte Ltd

Chapter Objectives

• To analyze the kinematics of a body subjected to

rotation about a fixed axis and general plane motion.

• To provide a relative-motion analysis of a rigid body

using translating and rotating axes.


• Rotation About a Fixed Point

• The Time Derivative of a Vector Measured from

Either a Fixed or Translating- Rotating System

• General Motion

• Relative-Motion Analysis Using Translating and

Rotating Axes

Chapter Outline


Rotational About a Fixed Point

• When a rigid body rotates about a fixed point, the distance r from the point to a particle P located on the body is the same for any position of the body

• Thus, the path of motion for the particle lines on the surface of a sphere having a radius r and centered at the fixed point

• Since motion along this path occurs only from a series of rotations made during a finite time interval, we will first develop a familiarity with some of the properties of rotational displacements



Euler’s Theorem

• Euler’s Theorem states that two “component”

rotations about different axes passing

through a point are equivalent to a single

resultant rotation about an axis passing

through the point

• If more than two rotations are applied, they

can be combined into pairs, and each pair

can be further reduced to combine into one

rotation.



Finite Rotations

• if component rotations used in Euler’s

theorem are finite, it is important that the order in which they are applied be

maintained

• This is because finite rotations do not obey

the law of vector addition, and hence they

cannot be classified as vector quantities



• Consider the 2 finite rotations θ1 + θ2 applied to

block. Each rotation has a magnitude of 90° and

a direction defined by the right-hand rule, as

indicated by the arrow

• The resultant orientation of the block is shown at

the right



• When these two rotations are applied in the

order θ2 + θ1 ,as shown, the resultant

position of the block is not the same as in the

previous diagram



• Consequently, finite rotations do not obey the

commutative law of addition (θ1 + θ2 ≠ θ2 + θ1 ),

and therefore they cannot be classified as vectors

• If smaller, yet finite rotations had been used to

illustrate this point, e.g., 10° instead of 90°, the

resultant orientation of the block after each

combination of rotations would also be different;

however, in this case, the difference is only a

small amount



Infinitesimal Rotations

• When defining the angular motions of the

body subjected to three-dimensional motion,

only rotations which are infinitesimally small will be considered

• Such rotations may be classified as vectors, since they can be added vectorially in any manner


• Consider the rigid body itself to

be a sphere which is allowed to

rotate about its central fixed

point O

• If we impose two infinitesimal

rotations dθ1 + dθ2 on the body,

it is seen that point P moves

along the path dθ1 x r + dθ2 x r

and ends up at P’




• Had the two successive rotations occurred in

the order dθ2 + dθ1, then the resultant

displacements of P would have been dθ2 x r

+ dθ1 x r

• Since the vector cross product obeys the

distributive law, by comparison (dθ1 + dθ2) x r

= (dθ2 + dθ1) x r



• Here infinitesimal rotations dθ are vectors,

since these quantities have both a magnitude

and direction for which the order of (vector)

addition is not important, i.e., dθ1 + dθ2 = dθ2

+ dθ1

• Furthermore, the two “component” rotations

dθ1 + dθ2 are equivalent to a single resultant

rotation dθ = dθ1 + dθ2 , a consequence of

Euler’s theorem



Angular Velocity

• If the body is subjected to an angular rotation dθ

about a fixed point, the angular velocity of the

body is defined by the time derivative,

• The line specifying the direction of ω, which is

collinear with dθ is referred to as the

instantaneous axis of rotation



• In general, this axis changes

direction during each instant of

time

• Since dθ is a vector quantity, so

too is ω, and it follows from

vector addition that if the body is

subjected to two component

angular motions, and

• The resultant angular velocity is

ω = ω1 + ω2

22 11



Angular Acceleration

• The body’s angular acceleration is

determined from time derivative of angular

velocity,

• For motion about a fixed point, α must

account for a change in both the magnitude

and direction of ω, so that, in general, α is

not directed along the instantaneous axis of

rotation.



• As the direction of the instantaneous axis of rotation (or the line of action of ω) changes in space, the locus of points defined by the axis generates a fixed space cone.



• If the change in this axis is viewed

with respect to the rotating body,

the locus of the axis generates a

body cone

• At any given instant, these cones

are tangent along the

instantaneous axis of rotation, and

when the body is in motion, the

body cone appears to roll either on

the inside or the outside surface of

the fixed space cone



• Provided the paths defined by the open ends

of the cones are described by the head of the

ω vector, α must act tangent to these paths

at any given instant, since the time rate of

change of ω is equal to α



Velocity

• Once ω is specified, the velocity of any point P

on a body rotating about a fixed point can be

determined using the same methods for a body

rotating about a fixed axis

• Hence, by cross product,

• r defines the position of P measure from the fixed

point O

rv



Acceleration

• If ω and α are known at a given instant, the

acceleration of any point P on the body can be

obtained by time differentiation of previous

equation, which yields,

• The equation defines the acceleration of a point

located on a body subjected to rotation about a

fixed axis

)( rra


The Time Derivative of a Vector Measured From

Either a Fixed or Translating-Rotating System

• In many types of problems involving the motion of the body about a fixed point, the angular velocity ω is specified in terms of its component angular motions

• For example, the disk as shown spins about the horizontal y axis at ωs while it rotates or precesses about the vertical z axis at ωp




• Therefore its resultant angular velocity is ω =

ωs + ωp

• If the angular acceleration α = ω of such a

body is to be determined, it is sometimes

easier to compute the time derivative of ω by

using a coordinate system which has a

rotation defined by one or more of the

components of ω

.




• For this reason, and for other uses later, an

equation will presently be derived that relates

the time derivative of any vector A defined

from a translating-rotating reference to its

time derivative defined from a fixed

reference.




• Consider the x, y, z axes of the

moving frame reference to have

an angular velocity Ω which is

measured from the fixed X, Y, Z

axes

• It will be convenient to express

vector A in terms of its i, j, k

components, which defined the

directions of the moving axes




• Hence,

• The time derivative of A must account for the

change in both the vector’s magnitude and

direction

• However, if this derivative is taken with respect

to the moving frame of reference, only a change

in the magnitudes of the components of A must

be accounted for

kjiA zyx AAA




• Since the directions of the components do

not change with respect to the moving

reference. Hence,

• When the time derivative of A is taken with respect to the fixed frame of reference, the

directions of i, j, k change only on account of

the rotation, Ω, of the axes and not their

translation

kjiA zyxxyz AAA




• Hence in general,

• i = di/dt represents only a change in the

direction of i with respect to time, since i has

a fixed magnitude of 1 unit

• The change, di is tangent to the path as i

moves due to the rotation Ω

kjikjiA zyxzyx AAAAAA

.




• Accounting for both the magnitude and

direction of di, we can define i using the

cross product, i = Ω x i

• Therefore we can obtain

kkjjii

.

AAA xyz




• The equation states that the time derivative of

any vector A as observed from the fixed X, Y, Z

frame of reference is equal to the time rate of

change of A as observed from x, y, z translating-

rotating frame of reference

• Ω x A is the change of A caused by the rotation

of the of the x, y, z frame

• It should always be used whenever Ω produces a

change in the direction of A as seen from the X,

Y, Z reference




• If this change does not occur, i.e., Ω = 0,

then the time rate of change of A as

observed from both coordinate system will be

the same.


General Motion

• In this section, a translating coordinate system will be used to define relative motion

• Shown in figure is a rigid body subjected to

general motion in three dimensions for which

the angular velocity is ω and the angular

acceleration is α


General Motion

• If point A has a known motion of vA and aA, the

motion of any other point B may be determined

by using this relative motion analysis

• If the origin of the translating coordinate system

x, y, z (Ω = 0) is located at the “base point” A,

then, at the instant shown, the motion of the body

may be regarded as the sum of an instantaneous

translation of the body having a motion of vA and

aA and a rotation of body about an instantaneous

axis passing through the base point


General Motion

• Since the body is rigid, the motion of point B

measured by an observer located at A is the

same as motion of the body about a fixed point.

• This relative motion occurs about the

instantaneous axis of rotation is defined by vB/A =

ω x rB/A , aB/A = α x rB/A + ω x (ω x rB/A)

• For translating axes the relative motions are

related to absolute motions by vB = vA + vB/A and

aB = aA + aB/A so that the absolute velocity and

acceleration of point B can be determined from


General Motion

and

• These two equations are identical to those

describing the general plane motion of a rigid

body

)( //

/

ABABAB

ABAB

rraa

rvv


General Motion

• However, there will be difficulty in application

for three-dimensional motion, because α

measures the change in both the magnitude

and direction of ω

• This is because for general plane motion, α

and ω are always parallel or perpendicular to

the plane of motion, and therefore α

measures only a change in the magnitude of

ω


Relative-Motion Analysis Using Translation and

Rotating Axes

• The locations of points A and B are specified

relative to the X, Y, and Z frame of reference

by position vectors rA and rB.

• The base point A represents the origin of the

x, y, z coordinate system, which is translating

and rotating with respect to X, Y, Z



Rotating Axes

• At the instant considered, the velocity and

acceleration of point A are vA and aA,

respectively, and the angular velocity and

angular acceleration of the x, y, z axes are Ω

and Ω = dΩ/dt, respectively

• All these vectors are measured w.r.t the X, Y,

Z frame of reference, although they may be

expressed in Cartesian component form

along either set of axes.

.



Rotating Axes

Position

• If the position of “B w.r.t A” is specified by the

relative-position vector rB/A, then by vector

addition,

ABAB /rrr



Rotating Axes

Velocity

• The velocity of point B measured from X, Y,

Z is determined by taking the time derivative

of relative position equation,

• rB/A is measured between two points in a

rotating reference. Hence,

ABAB /rrr

ABxyzABABxyzABAB ///// )()( rvrrr



Rotating Axes

• Here (vB/A)xyz is the relative velocity of B w.r.t

A measured from x, y, z. Thus,

xyzABABAB )( // vrvv



Rotating Axes

Acceleration

• The acceleration of point B measured from

X, Y, Z is determined by taking time

derivative of relative velocity equation, which

yields,

• Where

xyzABABABABdt

d)( /// vrrvv

xyzABxyzAB

xyzABxyzABxyzABdt

d

)()(

)()()(

//

///

va

vvv



Rotating Axes

• Here (aB/A)xyz is the relative acceleration of B

w.r.t A measured from x, y, z. Thus,

xyzABxyzAB

ABABAB

)()(2

)(

//

//

av

rraa


PROCEDURE FOR ANALYSIS

Coordinate Axes

• Select the location and orientation of the X,

Y, Z and x, y, z coordinate axes

• Most often solutions are easily obtained if at

the instant considered:

1) The origins are coincident

2) The axes are collinear

3) The axes are parallel



• If several components of angular velocity are

involved in a problem, the calculations will be

reduced if the x, y, z axes are selected such

that only one component of angular velocity

is observed in this frame (Ωxyz) and the frame

rotates with Ω defined by the other

components of angular velocity



Kinematics Equations

• After the origin of the moving reference, A, is

defined and the moving point B is specified,

the equations should be written in symbolic

form as

xyzABxyzAB

ABABAB

xyzABABAB

)()(2

)(

)(

//

//

//

av

rraa

vrvv



• If rA and Ω appear to change direction when

observed from the fixed X, Y, Z reference

use a set of primed reference axes, x’, y’, z’ having Ω’ = Ω and to

determine Ωxyz and the relative motion

(vB/A)xyz and (aB/A)xyz

• After the final form of Ω, vA, aA, Ωxyz, (vB/A)xyz

and (aB/A)xyz are obtained, numerical problem

data may be substituted and the kinematic

terms evaluated

AAA xyz

.

. .



• The components of all these vectors may be

selected either along the X, Y, Z axes or

along x, y, z. The choice is arbitrary, provided

a consistent set of unit vectors is used.


CHAPTER REVIEW

Rotation About a Fixed Point

• When a body rotates about a fixed point O, then

points on the body follow a path that lies on the

surface of a sphere

• Infinitesimal rotations are vector quantities,

whereas finite rotations are not

• Since the angular acceleration is a time rate of

change in the angular velocity, then we must

account for both the magnitude and direction

changes of ω when finding it derivative.


CHAPTER REVIEW

• To do this, the angular velocity is often

specified in terms of its component motions,

such that some of these components will

remain constant relative to rotating x, y, z

axes

• If this is the case, then the time derivative

relative to the fixed axis can be determined,

that is AAA xyz


CHAPTER REVIEW

• Once ω and α are known, then the velocity

and acceleration of point P can be

determined from

r

rra

v

)(


CHAPTER REVIEW

Relative Motion Analysis Using Translating and Rotating Axes

• The motion of two points A and B on a body,

a series of connected bodies, or located on

two different paths, can be related using a

relative motion analysis with rotating and

translating axes at A


CHAPTER REVIEW

xyzABxyzAB

ABABAB

)()(2

)(

//

//

av

rraa

xyzABABAB )( // vrvv

• The relationship are


CHAPTER REVIEW

• When applying these equations, it is

important to account for both the magnitude

and directional changes of rA, (rB/A)xyz, Ω, and

Ωxyz when taking their time derivatives to find

vA, aA, (vB/A)xyz, (aB/A)xyz, and their Ω, and Ωxyz

• To do this properly, one must use

AAA xyz

. .


CHAPTER REVIEW

General Motion

• If the body undergoes general motion, then

the motion of a point B in the body can be

related to the motion of another point A using

a relative motion analysis, along with

translating axes at A

• The relationships are

)( //

/

ABABAB

ABAB

rraa

rvv

Documents

Chapter 20: 3-D Kinematics of a Rigid Body