Kinematics of Rigid Bodies in Three Dimensions

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Chapter 18

• Kinematics of Rigid Bodies in Three Dimensions

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18.1 Introduction

• The fundamental relations developed for the plane
  motion of rigid bodies may also be applied to the
  general motion of three dimensional bodies.

• The current chapter is concerned with evaluation
  of the angular momentum and its rate of change
  for three dimensional motion and application to
  effective forces, the impulse-momentum and the
  work-energy principles.

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18.2 Rigid Body Angular Momentum in Three
Dimensions

• Angular momentum of a body about its mass center,

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18.2 Rigid Body Angular Momentum in Three
Dimensions
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18.2 Rigid Body Angular Momentum in Three
Dimensions

• Transformation of into is
  characterized by the inertia tensor for the body,

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z
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18.2 Rigid Body Angular Momentum in Three
Dimensions
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18.2 Rigid Body Angular Momentum in Three
Dimensions
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18.3 Principle of Impulse and Momentum

• The free-body diagram equation is used to develop
  component and moment equations.

• For bodies rotating about a fixed point,
  eliminate the impulse of the reactions at O by
  writing equation for moments of momenta and
  impulses about O.

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18.4 Kinetic Energy

• Kinetic energy of particles forming rigid body,

• With these results, the principles of work and
  energy and conservation of energy may be applied
  to the three-dimensional motion of a rigid body.

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18.4 Kinetic Energy
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Sample Problem 18.1

• SOLUTION
• Apply the principle of impulse and momentum.
  Since the initial momenta is zero, the system of
  impulses must be equivalent to the final system
  of momenta.

• Assume that the supporting cables remain taut
  such that the vertical velocity and the rotation
  about an axis normal to the plate is zero.

• Rectangular plate of mass m that is suspended
  from two wires is hit at D in a direction
  perpendicular to the plate.
• Immediately after the impact, determine
• the velocity of the mass center G,
• the angular velocity of the plate.

• Principle of impulse and momentum yields two
  equations for linear momentum and two equations
  for angular momentum.

• Solve for the two horizontal components of the
  linear and angular velocity vectors.

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Sample Problem 18.1

• SOLUTION
• Apply the principle of impulse and momentum.
  Since the initial momenta is zero, the system of
  impulses must be equivalent to the final system
  of momenta.

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Sample Problem 18.1
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Sample Problem 18.1
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Sample Problem 18.2

• SOLUTION
• The disk rotates about the vertical axis through
  O as well as about OG. Combine the rotation
  components for the angular velocity of the disk.

• A homogeneous disk of mass m ismounted on an
  axle OG of negligible mass. The disk rotates
  counter-clockwise at the rate ?1 about OG.
• Determine
• the angular velocity of the disk
• its angular momentum about O,
• its kinetic energy
• the vector and couple at G equivalent to the
  momenta of the particles of the disk.

• Compute the angular momentum of the disk using
  principle axes of inertia and noting that O is a
  fixed point.

• The kinetic energy is computed from the angular
  velocity and moments of inertia.

• The vector and couple at G are also computed from
  the angular velocity and moments of inertia.

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Sample Problem 18.2
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Sample Problem 18.2
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Sample Problem 18.2
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18.5 Motion of a Rigid Body in Three Dimensions

• Transformation of into is
  independent of the system of coordinate axes.

• Convenient to use body fixed axes Gxyz where
  moments and products of inertia are not time
  dependent.

• Define rate of change of change of with
  respect to the rotating frame,

Then,
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18.6 Eulers Eqs of Motion DAlemberts
Principle

• With and Gxyz chosen to correspond
  to the principal axes of inertia,

Eulers Equations

• System of external forces are equivalent to the
  vector and couple,

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18.7 Motion About a Fixed Point or a Fixed Axis

• For a rigid body rotation around a fixed Z
  axis(?z ?, ?x?z0),

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18.8 Rotation About a Fixed Axis

• If symmetrical with respect to the xy plane,
  IxzIyz0 and,

• A rotating shaft requires both static
  and dynamic balancing to avoid
  excessive vibration and bearing reactions.

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Sample Problem 18.3

• SOLUTION
• Evaluate the system of effective forces by
  reducing them to a vector attached at G and
  couple

• Expressing that the system of external forces is
  equivalent to the system of effective forces,
  write vector expressions for the sum of moments
  about A and the summation of forces.

Rod AB with weight W 40 lb is pinned at A to a
vertical axle which rotates with constant angular
velocity ? 15 rad/s. The rod position is
maintained by a horizontal wire BC. Determine
the tension in the wire and the reaction at A.

• Solve for the wire tension and the reactions at A.

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Sample Problem 18.3

• SOLUTION
• Evaluate the system of effective forces by
  reducing them to a vector attached at G
  and couple

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Sample Problem 18.3
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Prob. 18.65

• A homogeneous 8 lb disk is mounted on the
  horizontal shaft AB. The plane of the disk forms
  a 20? angle with the yz plane as shown.
• Knowing that the shaft rotates with a constant
  angular velocity ? of magnitude 10 rad/sec,
  determine the dynamic reactions at points A and B.

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Prob. 18.78

• The essential structure of a certain type of
  aircraft turn indicator is shown.
• Each spring has a constant of 40 lb/ft, and the
  7-oz. uniform disk of 2 in. radius spins at the
  rate of 10,000 rpm.
• The springs are stretched and exert equal
  vertical forces on yoke AB when the airplane is
  traveling in a straight path.
• Determine the angle through which the yoke will
  rotate when the pilot executes a horizontal turn
  of 2250 ft radius to the right at a speed of 500
  mi/hr. Indicate whether point A will move up or
  down.

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Start of Class
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18.9 Motion of a Gyroscope. Eulerian Angles

• A gyroscope consists of a rotor with its mass
  center fixed in space but which can spin freely
  about its geometric axis and assume any
  orientation.

• From a reference position with gimbals and a
  reference diameter of the rotor aligned, the
  gyroscope may be brought to any orientation
  through a succession of three steps
• rotation of outer gimbal through f about AA,
• rotation of inner gimbal through ? about BB
• rotation of the rotor through ? about CC.

• j, q, and y are called the Eulerian Angles and

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18.9 Motion of a Gyroscope. Eulerian Angles
z'
y'

• Equation of motion,

x'
let I be the moment of inertia about the z
axis let I ' be the moment of inertia about the x
y axes
align fixed x'y'z' coordinate system as shown
attach xyz coord. system to orange frame (inner
gimbal)
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18.9 Motion of a Gyroscope. Eulerian Angles

• Equation of motion,

z'
y'
x'
align fixed x'y'z' coordinate system as shown
attach xyz coord. system to orange frame (inner
gimbal)
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18.10 Steady Precession of a Gyroscope
?z
Steady precession,
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18.10 Steady Precession of a Gyroscope
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18.11 Motion of an Axisymmetrical Body Under No
Force

• Define the Z axis to be aligned with and z
  in a rotating axes system along the axis of
  symmetry. The x axis is chosen to lie in the Zz
  plane.

• ? constant and body is in steady precession.

x axis lies in Zz plane (although it doesnt look
like it)
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18.11 Motion of an Axisymmetrical Body Under No
Force
Two cases of motion of an axisymmetrical body
which under no force which involve no precession
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18.11 Motion of an Axisymmetrical Body Under No
Force
The motion of a body about a fixed point (or its
mass center) can be represented by the motion of
a body cone rolling on a space cone. In the case
of steady precession the two cones are circular.

• I lt I. Case of an elongated body. g lt q and
  the vector w lies inside the angle ZGz. The
  space cone and body cone are tangent externally
  the spin and precession are both counterclockwise
  from the positive z axis. The precession is said
  to be direct.

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Prob. 18.123

• A coin is tossed into the air. It is observed to
  spin at a rate of 600 rpm about the axis GC and
  to precess about the axis GD.
• Knowing that GC forms an angle of 15? with GD,
  determine
• the angle that the velocity vector of the
  coin forms with GD
• the rate of precession of the coin about GD

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Prob. 18.123
The total angular velocity of the coin can be
written as
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Prob. 18.123
The angle ß is the solution to part (a). From
slide 40,
?
ß
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Prob. 18.123
Recall that
?
ß
?x
?z
From slide 40,
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End of Class