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

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.

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): r ij = distance between i & j = c ij = const (1) N particles, n p = # pairs = (½)N(N-1) = # eqtns like (1)  Naively: s = # degrees of freedom = 3N - n p However, this is NOT valid because –All eqtns like (1) are not independent of each other! –ALSO: n p = (½)N(N-1) > 3N (if N  7) n p >> 3N (N >>1)

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

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): r 12 = c 12 = const, r 23 = c 23 = const, r 13 = c 13 = const  s = 6

See figure: Can also sees = 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 r 12 = c 12. With 2 points determined, point 3 needs only 1 coord, since r 13 = c 13 & r 23 = c 23 constrain its location.  s = = 6

 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:

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.

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 cosθ 11, cosθ 12, cosθ 13, with angles as shown in the figure.

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

Relns between i, j, k  unit vectors along x, y, z & i´, j´, k´  unit vectors along x´, y´, z´: i´ = cosθ 11 i+ cosθ 12 j+ cosθ 13 k j´ = cosθ 21 i+ cosθ 22 j+ cosθ 23 k k´= cosθ 31 i+ cosθ 32 j+ cosθ 33 k 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´) = cosθ 11 x+ cosθ 12 y+ cosθ 13 z y´ = (r  j´) = cosθ 21 x+ cosθ 22 y+ cosθ 23 z z´ = (r  k´) = cosθ 31 x+ cosθ 32 y+ cosθ 33 z Inverse relns are similar.

Relations between components of arbitrary vector G in the 2 systems: We had x´ = (r  i´) = cosθ 11 x+ cosθ 12 y+ cosθ 13 z y´ = (r  j´) = cosθ 21 x+ cosθ 22 y+ cosθ 23 z z´ = (r  k´) = cosθ 31 x+ cosθ 32 y+ cosθ 33 z Procedure to get these  procedure to get components of G: G x´ = (G  i´) = cosθ 11 G x + cosθ 12 G y + cosθ 13 G z G y´ = (G  j´) = cosθ 21 G x + cosθ 22 G y + cosθ 23 G z G z´ = (G  k´) = cosθ 31 G x + cosθ 32 G y + cosθ 33 G z Inverse relations are similar.

Primed axes are fixed in body:  9 direction cosines cosθ 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.

Relns between different cosθ 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´ =cosθ 11 i+ cosθ 12 j+ cosθ 13 k j´=cosθ 21 i+cosθ 22 j+ cosθ 23 k, k´ =cosθ 31 i+ cosθ 32 j+ cosθ 33 k with above dot products gives relns between cosθ ij : ∑ cosθ m´ cosθ m = 0 (m  m´, sum = 1,2,3) and: ∑ cos 2 θ m = 1 ( sum = 1,2,3) These  Orthogonality Relations between direction cosines

Use the Kronecker delta δ m,m´  0 (m  m´), δ m,m´  1 (m = m´), Orthogonality relations become: ∑ cosθ m´ cosθ m = δ m,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!