Chapter 4. Section 4.1: Independent Coordinates.

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Chapter 4 Rigid Body Kinematics

• Rigid Body ? A system of mass points subject to
  (holonomic) constraints that all distances
  between all pairs of points remain constant
  throughout the motion.
• Of course, an idealization!
• However, quite a useful concept!! 2 Chapters!
• Ch. 4 Kinematics Description of motion without
  discussing causes
• Very mathematical!
• Ch. 5 Dynamics Causes of motion - forces,
  torques.
• Especially interested in rigid body rotation.
• As part of this discussion, we will discuss
  fictitious (non-inertial) forces Centrifugal
  Coriolis

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Sect. 4.1 Independent Coordinates

• How many independent coordinates does it take to
  describe a rigid body?
• How many degrees of freedom are there?
• 6 indep coordinates or degrees of freedom
• 3 external coordinates to specify position of
  some reference point in body (usually CM) with
  respect to arbitrary origin.
• 3 internal coordinates to specify how the body is
  oriented with respect to the external coordinate
  axes.
• Here, we justify this.

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• Rigid body, N particles
• ? (At most) 3N degrees of freedom.
• However, constraints are that the distances
  between each particle pair are fixed.
• ? All constraints are of the form (for pair i,
  j)
• rij distance between i j cij
  const (1)
• N particles, np pairs (½)N(N-1)
  eqtns like (1)
• ? Naively s degrees of freedom 3N -
  np
• However, this is NOT valid because
• All eqtns like (1) are not independent of each
  other!
• ALSO np (½)N(N-1) gt 3N (if N ? 7)
• np gtgt 3N (N gtgt1)

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• To fix a point in a rigid body, it is not
  necessary to specify its distances from ALL other
  points in body. Its ONLY necessary to specify
  distances to any three non-collinear points
  (figure).

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• See figure
• ? If positions of 3
• particles (figure) are
• given, constraints fix
• positions of all N-3
• other particles. That
• is, we must have degrees of freedom s ? 9 (3
  particles,
• dimensions). However, the 3 reference points are
  not
• all independent, but are related by eqtns like
  (1)
• r12 c12 const, r23 c23 const, r13 c13
  const
• ? s 6

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• See figure
• Can also see s 6 in
• another way To
• establish position of
• one reference point,
• need 3 coords. Once
• point 1 is fixed, point 2 can be specified by
  only 2
• coords, since it is constrained to move on a
  sphere of
• radius r12 c12. With 2 points determined, point
  3
• needs only 1 coord, since r13 c13 r23 c23
• constrain its location. ? s 3 2 1 6

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• ? A rigid body in space needs 6 independent
  generalized coords to specify its configuration
  to treat its dynamics, no matter how many
  particles it contains.
• Also, of course, there may be additional
  constraints on the body which reduce the
  independent coordinates further.
• How are these 6 coordinates assigned?
  Configuration of rigid body is completely
  specified by locating a set of Cartesian axes
  FIXED IN THE RIGID BODY (primed axes ? body
  axes in figure) relative to an arbitrary set of
  Cartesian axes (unprimed axes ? space or lab
  frame or reference axes) fixed in external
  space.
• See figure

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• See figure
• 3 coords
• (of necessary 6)
• Specify origin of
• body (primed)
• axes in space
• (unprimed) axes
• system.
• 3 coords Specify orientation of primed axes
  relative to unprimed axes (actually to axes
  parallel to unprimed axes but sharing origin with
  primed axes). Now focus on 3 orientation coords.

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• There are many ways to specify the orientation of
  one Cartesian set of axes with respect to
  another with a common origin. Common procedure
• Specify the DIRECTION COSINES of the primed axes
  relative to the unprimed axes.
• See figure
• For example,
• orientation of x
• in x, y, z system
• is specified by
• cos?11, cos?12,
• cos?13, with angles as shown in the figure.

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• Notation i, j, k ? unit vectors along x, y, z.
• i, j, k ? unit vectors along x, y, z.
• ? Direction cosines (9 of them!) cos?11 ?
  cos(i?i) i?i i?i
• cos?12 ? cos(i?j) i?j j?i cos?13
  ? cos(i?k) i?k k?i
• cos?21 ? cos(j?i) j?i i?j cos?22
  ? cos(j?j) j? j j?j
• cos?23 ? cos(j?k) j?k k?j cos?31 ?
  cos(k?i) k?i i?k
• cos?32 ? cos(k?j) k?j j?k cos?33
  ? cos(k?k) k?k k?k Convention 1st index
  is primed, 2nd is unprimed

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• Relns between i, j, k ?
• unit vectors along x, y, z
• i, j, k ? unit vectors along
• x, y, z
• i cos?11i cos?12 j cos?13k
• j cos?21i cos?22 j cos?23k
• k cos?31i cos?32 j cos?33k Inverse relns
  are similar.
• ? Can express an arbitrary point in either coord
  system
• r xi yi zk xi yj zk
• Primed coords unprimed coords are related by
• x (r?i) cos?11x cos?12 y cos?13z
• y (r?j) cos?21x cos?22 y cos?23z
• z (r?k) cos?31x cos?32 y cos?33z
• Inverse relns are similar.

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• Relations between components of arbitrary vector
  G in the 2 systems
• We had x (r?i) cos?11x cos?12 y cos?13z
• y (r?j) cos?21x cos?22 y cos?23z
• z (r?k) cos?31x cos?32 y cos?33z
• Procedure to get these ? procedure to get
  components of G
• Gx (G?i) cos?11Gx cos?12Gy
  cos?13Gz
• Gy (G?j) cos?21Gx cos?22Gy
  cos?23Gz
• Gz (G?k) cos?31Gx cos?32Gy
  cos?33Gz
• Inverse relations are similar.

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• Primed axes are fixed
• in body ? 9 direction
• cosines cos?ij will be
• functions of time as the
• body rotates.
• ? Can view direction cosines as generalized
  coordinates describing the orientation of the
  body. However, they cannot be independent! There
  are 9 of them to describe the orientation of
  rigid body, we need only 3 coordinates.

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• Relns between different cos?ij
• Obtained using orthogonality of
• unit vectors in both coord sets
• i?j j?k k?i 0, i?i j?j k?k
  1
• i?j j?k k?i 0, i?i j?j
  k?k 1
• ? Combining i cos?11i cos?12 j cos?13k
• jcos?21icos?22 j cos?23k, k cos?31i
  cos?32 j cos?33k
• with above dot products gives relns between
  cos?ij
• ??cos??m cos??m 0 (m ? m, sum ?
  1,2,3)
• and ??cos2??m 1 (sum ? 1,2,3)
• These ? Orthogonality Relations between direction
  cosines

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• Use the Kronecker delta
• dm,m ? 0 (m ? m), dm,m ? 1 (m m),
• Orthogonality relations become
• ??cos??m cos??m dm,m (sum ? 1,2,3)
• 6 orthogonality relns between 9 direction cosines
• ? 3 indep coords. ? Using direction cosines as
  generalized coordinates to set up Lagrangian is
  not possible. Instead choose some set of 3
  independent functions of the direction cosines.
  There is no unique choice for this set. A common
  set
• ? The Euler Angles. Described
  later.
• Relations we just derived are, however, very
  useful. Can use them to derive many theorems
  about properties of, rigid body motion. We do
  this next!