1 Class #17 Lagrange’s equations examples Pendulum with sliding support (T7-31) Double Atwood (7-27) Accelerating Pendulum (7-22)

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Presentation transcript:

1 Class #17 Lagrange’s equations examples Pendulum with sliding support (T7-31) Double Atwood (7-27) Accelerating Pendulum (7-22)

2 1) Write down T and U in any convenient coordinate system. 2) Write down constraint equations Reduce 3N or 5N degrees of freedom to smaller number. 3) Define the generalized coordinates Consistent with the physical constraints 4) Rewrite in terms of 5) Calculate 6) Plug into 7) Solve ODE’s 8) Substitute back original variables Lagrange’s Kitchen Mechanics “Cookbook” for Lagrangian Formalism

3 Pendulum with sliding support-I x1x1 L m1m1 x2x2 z2z2 m2m2 A pendulum with mass m 2 and length L is suspended from a block of mass m 1 resting on a frictionless plane. What is the frequency of small oscillations of the pendulum? What is the center of mass motion?

4 Pendulum with sliding support-II The cosine and sine terms “annihilate” to 1.

5 Pendulum with sliding support-III These terms cancel in the final equation of motion. Set the cosine term = 1 This is a product of small terms, it can be ignored

6 Pendulum with sliding support-IV

7 Pendulum with sliding support-V A pendulum with mass m 2 and length L is suspended from a block of mass m 1 resting on a frictionless plane. What is the frequency of small oscillations of the pendulum? x1x1 L m1m1 x2x2 z2z2 m2m2

8 Pendulum with sliding support-VI A pendulum with mass m 2 and length L is suspended from a block of mass m 1 resting on a frictionless plane. What is the center of mass motion? x1x1 L m1m1 x2x2 z2z2 m2m2

9 Class #17 Windup New variables can be introduced so long as add additional constraints Generalized coordinates do not need to be of same type (e.g. angle / position).