Handout #18 Inertia tensor Tops and Free bodies using Euler equations

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Presentation transcript:

Handout #18 Inertia tensor Tops and Free bodies using Euler equations Inertia tensor for a continuous body Kinetic energy from inertia tensor. Tops and Free bodies using Euler equations Precession Lamina theorem Free-body wobble :02

Inertia Tensor For continuous body I is symettric and also I is hermitian (Equal to the conjugate of the transpose).

Lamina Theorem :60

L 18-1 Angular Momentum and Kinetic Energy A square plate of side L and mass M is rotated about a diagonal. In the coordinate system with the origin at lower left corner of the square, the inertia tensor is? What are the eigenvalues and eigenvectors for this square plate? L Matrix |4 3 0| ML^2 / 12 *|3 4 0| |0 0 8| T=(ML^2)*(w-0^2)/12 L=ML^2w-0/12sqrt(2) * (1 1 0) Eigenvectors w=w/root 2 (1 1 0), (1 –1 0), w(0 0 1) I= (ML^2/12)[4 –3 0 -3 4 0 0 0 8] :02

Angular Momentum and Kinetic Energy We derived the moment of inertia tensor from the fundamental definitions of L, by working out the double cross-product Do the same for T (kinetic energy) :02

L 18-2 Angular Momentum and Kinetic Energy A complex arbitrary system is subject to multi-axis rotation. The inertia tensor is A 3-axis rotation is applied Eigenvalue is 2 for vector 5.0 8.2 3.0 Energy is 203 :02

Symmetrical top Euler equation :02

Precession Ignore in limit :02 Three principal axes. Do demo. Show how things go other direction if spin disk backwards. Gyro 0) Balance it 1) Precession --- with w reversed 2) Remove weight – rap the gyro 3) Now rap the gyro with a weight on it. :02

Euler’s equations for symmetrical bodies Note even for non-laminar symmetrical tops AND even for W-3 is general. Constant may not be 1 for non-laminar :60

Euler’s equations for symmetrical bodies 3-axis balancing! Plates in air Precession frequency=rotation frequency for symmetrical lamina :60

Euler’s equations for symmetrical bodies Free body – L fixed (space frame) Body frame (w-3 fixed, L and w precess around it). :60

L18-3 – Chandler Wobble The earth is an ovoid thinner at the poles than the equator. For a general ovoid, For Earth, what are W1-dotdot=-(w-3)^2)(2/5M-eps-b)^2 / (2/5M-b)^2=(eps/b)^2*(w-3)^2-> w-p=20/6400 * w-3 :60

Handout #18 windup :02