Additional Problem (HW 10)

Transcription

1 1 Housekeeping - Three more lectures left including today: Nov. 20 st, Nov. 27 th, Dec. 4 th - Final Eam on Dec. 11 th at 4:30p (Eploratory Planetary 206)

2 2 Additional Problem (HW 10) z h y O Choose origin to be at bottom of bowl 2 2, h y y y 1 T m y z V mg 2 y 2 y 1 L m y z mg y y ,, 0,0,0 y z yz,,,, 0,0, y z

3 3 Additional Problem (HW 10) z h y Since, we have and O 2 2, h y y y 2 2, 2 2 z h y y y z y y y z y y y y y y, y 0,0 2 Near, z O( ) and 0 0 z O( ) 2 4 O( ) So, keeping terms up to 2, we have effectively a 2D system, 1 T m y V mg 2 y 2 y

4 4 Additional Problem (HW 10) z h y O, y 0,0 2 2, h y y y Near our equilibrium,, y, and,, 1 T m y T ij m 0 y 0 m y y 2 2 V mg y y V ij 2mg mg mg 2mg

5 5 Additional Problem (HW 10) z h y O 2 2, h y y y The Characteristic equation for the eigenvalues is: V T 2 det 2 m 2g 2 g 2 0 g 2 g g 2 g 2 2 g g

6 6 Additional Problem (HW 10) z h y O 2 2, h y y y 3g g with a a 1 1 2m m 1 y ata 1 T

7 7 Normal Modes (Eigenmodes) need both eigenvalues & Normalized eigenvectors Normal Frequency (Eigenfrequency) need just eigenvalues

8 Review from Last Lecture 8

9 9 How to describe a rigid body? Rigid Body - a system of point particles fied in space i r ij j subject to a holonomic constraint: r ij c ij for all i, j pairs A rigid body needs only si independent generalized coordinates to specify its configurations.

10 10 How to describe a rigid body? A better way to think about this i 1 d 1 d 3 d 2 2 ANY point i within the rigid body can be located by the distances from three fied reference points! d, d, d So, we only need to specify the coordinates of the three reference points; then ALL points in the 6 dofs rigid body are fied by the constraint equations, r ij c ij

11 11 Fied and Body Aes for a rigid body We will use two sets of coordinates: - 1 set of eternal fied coordinates (unprimed) - 1 set of internal body coordinates (primed) z ' z, yz, ', y', z' (As the name implied, the body aes are attached to the rigid body.) y ' 3 coords to specify the origin o of the body aes. o ' o y 3 coords to specify the orientation of the body aes wrt to the translated fied system (dotted aes).

12 12 Euler s and Chasles's Theorems Useful general principle for the analysis of rigid bodies Euler s Theorem: A general displacement of a rigid body with 1 pt fied in space is equivalent to a rotation about some ais. Chasles s Theorem: A general displacement = a translation + a rotation Therefore, for the discussion of rigid body motion, a rotation is an important operation/coordinate transformation to consider.

13 13 Rotation - The point P is fied and the prime frame rotates counter-clockwise ' 2 2 P This is the passive point of view. (convention: + counterclockwise) 1 ' 1 - One could equally consider the 2 transformation (rotation) as taking the point P (or vector) and rotating it by in the clockwise direction in the same frame. P P ' This is the active point of view. (convention: + clockwise) 1 Either way, the math is the same!

14 14 Rotation - A rotation (passive) about one of the coordinate aes is simple: Just put a 1 in the diagonal corresponding to that ais and squeeze the 2D rotation matri into the rest of the entries. i.e., to rotate about the 3 ais: 2 i.e., to rotate about the ais: cos sin 0 sin cos cos 0 sin sin 0 cos

15 15 Orientation of the Body Aes As for a general rotation, it is not as simple but we can build it up from these basic rotation around coordinate aes Most often used: Euler s angles but similar to (roll, pitch, yaw) The Euler s Angles,, consisting of a particular sequence of 3 rotations (D, C, B) along three principle aes: ' ' D C B

16 Euler s Angles D C B ' ' 16 - Then the general rotation from the fied aes to the body ', y', z ' aes is given by the product (note the specific order): A(,, ) B( ) C( ) D( ),, yz,, (The sequence of rotated angles are called the Euler s angles.) A cos cos cossinsin cos sin cos cossin sinsin sin cos cossincos sin sin cos coscos sin cos sinsin sin cos cos ' body A( fied)

17 Infinitesimal Rotation 17

18 18 Infinitesimal Rotations - In general, an infinitesimal transformation is commutative and can be represented as: A Iε - An infinitesimal orthogonal transformation is a proper rotation with ε being anti-symmetric and it can generally be written in component form as, ε 0 d d d d 2 1 d d 0 1 d d 0 d d 1 d 0 d 1 where the three components can also be represented as a vector (pseudo-vector): dω d, d, d T dω d,0, d d an infinitesimal Euler s rotation T

19 19 Changes due to an Infinitesimal Rotation - Under an infinitesimal rotation, the change in the coordinates can be epressed as, - Writing this out eplicitly in components, we have, r' r dr Iε rr εr d1 0 d3 d21 dr d εr d 0 d d 3 d2 d1 0 3 d d d d d d d d d The result can be represented as the cross product between two vectors: dr rdω (passive view) with r 1, 2, 3 dr dωr (active view) dr dω d, d, d T T

20 20 Rate of Change of a Vector under Rotation The rate of change of a position vector in the body frame as measured in the fied frame under an infinitesimal rotation will be give by : z z ' P y ' r ' y R ' Since this discussion applies equally well to ANY vectors in the body frame, we can abstract this out in general as an operator: r ' dr' dr' ωr' dt dt fied d ω Ω dt d d ω dt dt fied body body