SolSec 4.7

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

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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 + −+ =

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

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