Structure and Interpretation of Classical Mechanics
The two-bar linkage shown in figure 1.3 is constrained to move in the plane. It is composed of three small massive bodies interconnected by two massless rigid rods in a uniform gravitational field with vertical acceleration . The rods are pinned to the central body by a hinge that allows the linkage to fold. The system is arranged so that the hinge is completely free: the members can go through all configurations without collision. Formulate a Lagrangian that describes the system and find the Lagrange equations of motion. Use the computer to do this, because the equations are rather big.
(define ((F->C F) local) (->local (time local) (F local) (+ (((partial 0) F) local) (* (((partial 1) F) local) (velocity local))))) ; (define ((q->r l1 l2) local) (let ((q (coordinate local))) (let ((x2 (ref q 0)) (y2 (ref q 1)) (theta (ref q 2)) (phi (ref q 3))) (let ((x1 (+ x2 (* l1 (cos theta)))) (y1 (+ y2 (* l1 (sin theta)))) (x3 (+ x2 (* l2 (cos phi)))) (y3 (+ y2 (* l2 (sin phi))))) (up x1 y1 x2 y2 x3 y3))))) ; (show-expression ((q->r 'l_1 'l_2) (up 't (up 'x_2 'y_2 'theta 'phi) (up 'xdot_2 'ydot_2 'thetadot 'phidot)))) ; (define (KE m vx vy) (* 1/2 m (+ (square vx) (square vy)))) (define ((T-rect m1 m2 m3) local) (let ((q (coordinate local)) (v (velocity local))) (let ((x1dot (ref v 0)) (y1dot (ref v 1)) (x2dot (ref v 2)) (y2dot (ref v 3)) (x3dot (ref v 4)) (y3dot (ref v 5))) (+ (KE m1 x1dot y1dot) (KE m2 x2dot y2dot) (KE m3 x3dot y3dot))))) (show-expression ((T-rect 'm_1 'm_2 'm_3) (up 't (up 'x_1 'y_1 'x_2 'y_2 'x_3 'y_3) (up 'xdot_1 'ydot_1 'xdot_2 'ydot_2 'xdot_3 'ydot_3)))) ;
— Me@2021-03-12 05:37:27 PM
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