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Chapter 4: Rigid Body Kinematics

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!

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Chapter 4: Rigid Body Kinematics

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  1. 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

  2. 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 coordinatesto specify position of some reference point in body (usually CM) with respect to arbitrary origin. • 3 internal coordinatesto specify how the body is oriented with respect to the external coordinate axes. • Here, wejustify this.

  3. 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) Nparticles,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) > 3N(if N  7) np >> 3N (N >>1)

  4. To fix a point in a rigid body, it is not necessary to specify its distances from ALLother points in body. It’s ONLYnecessary to specify distances to any three non-collinear points (figure).

  5. 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

  6. See figure: Can also see s = 6 in another way: To establish position of one reference point, need3 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

  7.  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:

  8. 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.

  9. 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 cosθ11, cosθ12, cosθ13, with angles as shown in the figure.

  10. Notation:i, j, k  unit vectors along x, y, z. i´, j´, k´  unit vectors along x´, y´, z´.  Direction cosines (9 of them!): cosθ11  cos(i´i) = i´i =ii´ cosθ12  cos(i´j) = i´j =ji´ cosθ13  cos(i´k) = i´k = ki´ cosθ21  cos(j´i) = j´i = ij´cosθ22  cos(j´j) = j´ j = jj´ cosθ23  cos(j´k) = j´k = kj´cosθ31  cos(k´i) = k´i = ik´ cosθ32  cos(k´j) = k´j = jk´cosθ33  cos(k´k) =k´k = kk´Convention: 1st index is primed, 2nd is unprimed

  11. Relns betweeni, j, k  unit vectors along x, y, z & i´, j´, k´  unit vectors along x´, y´, z´: i´ = cosθ11i+ cosθ12 j+ cosθ13k j´ = cosθ21i+ cosθ22 j+ cosθ23k k´= cosθ31i+ cosθ32 j+ cosθ33k Inverse relns are similar.  Can express an arbitrary point in either coord system: r = xi + yi + zk = x´i´ + y´j´ + z´k´ • Primed coords & unprimed coords are related by: x´ = (ri´) = cosθ11x+ cosθ12 y+ cosθ13z y´ = (rj´) = cosθ21x+ cosθ22 y+ cosθ23z z´ = (rk´) = cosθ31x+ cosθ32 y+ cosθ33z Inverse relns are similar.

  12. Relations between components of arbitrary vector G in the 2 systems: • We had x´ = (ri´) = cosθ11x+ cosθ12 y+ cosθ13z y´ = (rj´) = cosθ21x+ cosθ22 y+ cosθ23z z´ = (rk´) = cosθ31x+ cosθ32 y+ cosθ33z • Procedure to get these  procedure to get components of G: Gx´ = (Gi´) = cosθ11Gx + cosθ12Gy+ cosθ13Gz Gy´ = (Gj´) = cosθ21Gx+ cosθ22Gy+ cosθ23Gz Gz´ = (Gk´) = cosθ31Gx + cosθ32Gy + cosθ33Gz Inverse relations are similar.

  13. Primed axes are fixed in body:  9 direction cosinescosθijwill 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.

  14. Relns between different cosθij 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´ =cosθ11i+ cosθ12 j+ cosθ13k j´=cosθ21i+cosθ22 j+ cosθ23k, k´ =cosθ31i+ cosθ32 j+ cosθ33k with above dot products gives relns between cosθij: ∑cosθm´cosθm = 0 (m  m´, sum  = 1,2,3) and: ∑cos2θm= 1 (sum  = 1,2,3) These  Orthogonality Relationsbetween direction cosines

  15. Use the Kronecker delta δm,m´  0 (m  m´), δm,m´  1 (m = m´), • Orthogonality relations become: ∑cosθm´cosθm = δm,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!

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