SolSec 4.7 - Problems and Solutions for Section 4.7 (4.68...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Problems and Solutions for Section 4.7 (4.68 through 4.71) 4.68 Use Lagrange's equation to derive the equations of motion of the lathe of Fig. 4.21 for the undamped case. Solution: Let the generalized coordinates be 3 2 1 and , θ . The kinetic energy is 2 3 3 2 2 2 2 1 1 2 1 2 1 2 1 & & & J J J T + + = The potential energy is Uk k =− () +− 1 2 1 2 12 2 2 23 2 2 θθ There is a nonconservative moment M(t) on inertia 3. The Lagrangian is LTU J J J k k =−= + + −− 1 2 1 2 1 2 1 2 1 2 11 2 22 2 33 2 1 2 2 2 ˙˙˙ θθθ Calculate the derivatives from Eq. (4.136): = = = = = = + L J d dt L J L J d dt L J L J d dt L J L kk L ˙ ˙ ˙ ˙˙ ˙ ˙ ˙ ˙ ˙ ˙ 1 1 2 2 3 3 1 + + 2 1 2 2 3 k k L Using Eq. (4.136) yields t M k k J k k k k J k k J = + = + + = + 3 2 2 2 3 3 3 2 2 2 1 1 1 2 2 2 2 1 1 1 1 0 0 & & & & & & In matrix form this yields J J J kkk k M t 1 2 3 112 2 00 0 0 0 0 + −+ =
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
4.69 Use Lagrange's equations to rederive the equations of motion for the automobile of Ex. 4.8.2 illustrated in Figure 4.21 for the case 0 2 1 = = c c . Solution: Let the generalized coordinates be x and θ . The kinetic energy is 2 2 2 1 2 1 θ & & J x m T + = The potential energy is (ignoring gravity) () 2 2 2 2 1 1 2 1 2 1 I x k I x k U + + =
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 6

SolSec 4.7 - Problems and Solutions for Section 4.7 (4.68...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online