Lecture 30: Illustrations using the quadruple pendulum

1. DRAFT Lecture 30: Illustrations using the quadruple pendulum Develop the governing equations Linearize (and learn a new, more rational, way to do this) The linear problem Diagonalization

2. The quadruple pendulum We’ll have eight simple coordinates — the locations of the bobs, yi, zi Four degrees of freedom four generalized coordinates qi We’ll eventually have a state vector

3. How do we get there? Start with the Euler-Lagrange process Constraints

4. Define the generalized coordinates The differential equations are These are all second order equations, and we want to go to first order equations for some state Q1 is the only nonzero generalized force, equal to t, supposed to be small enough to keep the angles small

5. The first four equations are trivial we’ve seen their like before We can get the second four equations from the EL equations Solve them for These depend only on the angles, which are

6. We can write these equations in the form These equations are aggressively nonlinear It’s time to learn a nice linearization technique We can treat all the angles as small at the same time by letting

7. The second derivatives we just found are all then functions of e We can write a Taylor series in e for each of them, and we just keep the first term The calculation is quite messy. I’ll look at this in Mathematica shortly.

8. We get at the small angle assumption by rewriting these as The Taylor series for any one of these looks like The first (zeroth order) term is zero for all of them, so we need only retain the first term

9. We aren’t going to do it this way. We have Mathematica to do it for us I’ll go through that later For now we have the last four state equations, from which we can cancel the e

10. When we cancel the dummy small parameter we have the second four linearized state equations (without forcing) Because the forcing is small, b will be evaluated ate = 0 which gives us a simple b

11. Now let’s build A

12. The upper left hand block is empty

13. The upper right hand block is the 4 x 4 unit matrix

14. We get the lower left hand block from the linearized EL equations

15. We get the lower left hand block from the linearized EL equations

16. We get the lower left hand block from the linearized EL equations

17. We get the lower left hand block from the linearized EL equations

18. The lower right hand block is empty

19. The eigenvalues of this matrix are ±3.06514j, ±2.12993j, ±1.32327j, ±0.567993j The corresponding eigenvectors are (and their complex conjugates)

20. We can experiment with the diagonalizing transformation We have the eigenvector equations and the question before us is the nature of the right hand side

21. After some manipulation we have Each eigenvector is excited by a torque at the upper pivot: the system is controllable

22. What about the output? Every component of z will contribute to the output if and we can check that for this system for all possible outputs

24. Every one of these passes the test. The first four are real The second four are imaginary A sample of each We can look at all of them in Mathematica

25. Let’s see what we can say about all of this in simulation I can look at unforced motion (free oscillations) and see how the eigenfunctions and eigenfrequencies work I can look at forced motion as well Start with free motion

26. If I start the pendulum in one of its eigenvector shapes then I ought to get response at that eigenvalue if the magnitude isn’t too big

31. Let’s also look at forced motion

32. Slow forcing, 0.2 x lowest eigenfrequency (four periods)

33. Intermediate forcing, unity (eight periods)

34. Rapid forcing, 5 x highest frequency (16 periods)

35. Rapid forcing, 5 x highest frequency (50 periods)