chapter 20: 3-d kinematics of a rigid body
TRANSCRIPT
Chapter 20: 3-D Kinematics of a Rigid Body
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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.
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• 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
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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
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Rotational About a Fixed Point
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.
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Rotational About a Fixed Point
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
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Rotational About a Fixed Point
• 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
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Rotational About a Fixed Point
• 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
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Rotational About a Fixed Point
• 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
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Rotational About a Fixed Point
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
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• 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’
Rotational About a Fixed Point
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Rotational About a Fixed Point
• 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
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Rotational About a Fixed Point
• 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
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Rotational About a Fixed Point
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
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Rotational About a Fixed Point
• 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
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Rotational About a Fixed Point
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.
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Rotational About a Fixed Point
• 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.
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Rotational About a Fixed Point
• 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
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Rotational About a Fixed Point
• 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
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Rotational About a Fixed Point
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
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Rotational About a Fixed Point
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
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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
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The Time Derivative of a Vector Measured From
Either a Fixed or Translating-Rotating System
• 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
.
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The Time Derivative of a Vector Measured From
Either a Fixed or Translating-Rotating System
• 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.
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The Time Derivative of a Vector Measured From
Either a Fixed or Translating-Rotating System
• 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
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The Time Derivative of a Vector Measured From
Either a Fixed or Translating-Rotating System
• 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
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The Time Derivative of a Vector Measured From
Either a Fixed or Translating-Rotating System
• 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
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The Time Derivative of a Vector Measured From
Either a Fixed or Translating-Rotating System
• 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
.
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The Time Derivative of a Vector Measured From
Either a Fixed or Translating-Rotating System
• 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
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The Time Derivative of a Vector Measured From
Either a Fixed or Translating-Rotating System
• 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
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The Time Derivative of a Vector Measured From
Either a Fixed or Translating-Rotating System
• 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.
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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
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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
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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
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General Motion
and
• These two equations are identical to those
describing the general plane motion of a rigid
body
)( //
/
ABABAB
ABAB
rraa
rvv
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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
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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
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Relative-Motion Analysis Using Translation and
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.
.
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Relative-Motion Analysis Using Translation and
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
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Relative-Motion Analysis Using Translation and
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
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Relative-Motion Analysis Using Translation and
Rotating Axes
• Here (vB/A)xyz is the relative velocity of B w.r.t
A measured from x, y, z. Thus,
xyzABABAB )( // vrvv
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Relative-Motion Analysis Using Translation and
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
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Relative-Motion Analysis Using Translation and
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
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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
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PROCEDURE FOR ANALYSIS
• 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
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PROCEDURE FOR ANALYSIS
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
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PROCEDURE FOR ANALYSIS
• 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
.
. .
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PROCEDURE FOR ANALYSIS
• 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.
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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.
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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
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CHAPTER REVIEW
• Once ω and α are known, then the velocity
and acceleration of point P can be
determined from
r
rra
v
)(
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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
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CHAPTER REVIEW
xyzABxyzAB
ABABAB
)()(2
)(
//
//
av
rraa
xyzABABAB )( // vrvv
• The relationship are
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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
. .
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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