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Lecture 22’

Lecture 22’. We didn’t do very well. I’m particularly disturbed by problem 2 so I want to look at that one in some detail. If we have time I’ll look at three and four as well. Find the steady state horizontal motion of a 1 kg mass attached to a vertical wall

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Lecture 22’

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  1. Lecture 22’

  2. We didn’t do very well

  3. I’m particularly disturbed by problem 2 so I want to look at that one in some detail If we have time I’ll look at three and four as well

  4. Find the steady state horizontal motion of a 1 kg mass attached to a vertical wall by a 16 N/m spring and a 1 Ns/m damper if the wall oscillates in the horizontal direction at a frequency of 4 rad/sec with a peak to peak amplitude of 2 mm

  5. Find the steady state horizontal motion of a 1 kg mass attached to a vertical wall by a 16 N/m spring and a 1 Ns/m damper if the wall oscillates in the horizontal direction at a frequency of 4 rad/sec with a peak to peak amplitude of 2 mm No homogeneous solution No need to think about finding natural frequencies No need to do anything about exp(st)

  6. Find the steady state horizontal motion of a 1 kg mass attached to a vertical wall by a 16 N/m spring and a 1 Ns/m damper if the wall oscillates in the horizontal direction at a frequency of 4 rad/sec with a peak to peak amplitude of 2 mm gravity is irrelevant

  7. Find the steady state horizontal motion of a 1 kg mass attached to a vertical wall by a 16 N/m spring and a 1 Ns/m damper if the wall oscillates in the horizontal direction at a frequency of 4 rad/sec with a peak to peak amplitude of 2 mm Numbers for when we get to the analysis

  8. Find the steady state horizontal motion of a 1 kg mass attached to a vertical wall by a 16 N/m spring and a 1 Ns/m damper if the wall oscillates in the horizontal direction at a frequency of 4 rad/sec with a peak to peak amplitude of 2 mm It’s attached to the world in such a way that the horizontal motion is possible

  9. Find the steady state horizontal motion of a 1 kg mass attached to a vertical wall by a 16 N/m spring and a 1 Ns/m damper if the wall oscillates in the horizontal direction at a frequency of 4 rad/sec with a peak to peak amplitude of 2 mm There is no external forcing This is a “ground motion” problem

  10. Find the steady state horizontal motion of a 1 kg mass attached to a vertical wall by a 16 N/m spring and a 1 Ns/m damper if the wall oscillates in the horizontal direction at a frequency of 4 rad/sec with a peak to peak amplitude of 2 mm wf= 4 yW = 1 sin(4 t) OR yW = 1 cos(4 t) or anything else that is harmonic with frequency 4

  11. Draw a picture m c k yW = sin(4t)

  12. Now we need equations of motion and it doesn’t matter how you find them I think FBD is easier for this problem, but I’ll do both

  13. The Lagrangian which we can see is the same thing, as it must be

  14. The Lagrangian What people missed

  15. So how do we solve the problem?? Let’s look at the sine choice (one person did cosine and got it pretty much right)

  16. the equation

  17. QUESTIONS?

  18. I’d also like to take a look at problem 3 the only 2DOF problem

  19. Find the general homogeneous solution for the motion of two carts, one of mass 1.5 kg and the other of 3 kg mass if they are connected by a 100 N/m spring. The carts are free to roll on the ground.

  20. Find the general homogeneous solution for the motion of two carts, one of mass 1.5 kg and the other of 3 kg mass if they are connected by a 100 N/m spring. The carts are free to roll on the ground. No particular solution we are going to have to do the exp(st) procedure

  21. Find the general homogeneous solution for the motion of two carts, one of mass 1.5 kg and the other of 3 kg mass if they are connected by a 100 N/m spring. The carts are free to roll on the ground. Almost certainly two degrees of freedom

  22. Find the general homogeneous solution for the motion of two carts, one of mass 1.5 kg and the other of 3 kg mass if they are connected by a 100 N/m spring. The carts are free to roll on the ground. Data

  23. Find the general homogeneous solution for the motion of two carts, one of mass 1.5 kg and the other of 3 kg mass if they are connected by a 100 N/m spring. The carts are free to roll on the ground. If there is a zero frequency you are going to need to use it

  24. Draw a picture y1 y2 k m1 m2

  25. I really like the Euler-Lagrange method for this one

  26. dots become s and we can go to a matrix form

  27. The determinant has to vanish Expand

  28. The modal vectors for this come from the “eigenvectors” For s = 0

  29. For s = 10j So the general solution is

  30. QUESTIONS?

  31. Find the motion of a 1 m long inverted simple pendulum with a 0.5 kg bob if the bob is connected to a fixed vertical wall by a 54.905 N/m spring k. Suppose the system to start from rest with an initial angle of 5° from the vertical. Suppose that the small angle approximation is valid.

  32. Find the motion of a 1 m long inverted simple pendulum with a 0.5 kg bob if the bob is connected to a fixed vertical wall by a 54.905 N/m spring k. Suppose the system to start from rest with an initial angle of 5° from the vertical. Suppose that the small angle approximation is valid. means upsidedown means the rod is massless

  33. Find the motion of a 1 m long inverted simple pendulum with a 0.5 kg bob if the bob is connected to a fixed vertical wall by a 54.905 N/m spring k. Suppose the system to start from rest with an initial angle of 5° from the vertical. Suppose that the small angle approximation is valid. one object, one degree of freedom

  34. Find the motion of a 1 m long inverted simple pendulum with a 0.5 kg bob if the bob is connected to a fixed vertical wall by a 54.905 N/m spring k. Suppose the system to start from rest with an initial angle of 5° from the vertical. Suppose that the small angle approximation is valid. initial condition on velocity is zero

  35. Find the motion of a 1 m long inverted simple pendulum with a 0.5 kg bob if the bob is connected to a fixed vertical wall by a 54.905 N/m spring k. Suppose the system to start from rest with an initial angle of 5° from the vertical. Suppose that the small angle approximation is valid. means that you can linearize the equations and is a hint that the angle might be the best variable

  36. The rest of the information is just data, so it’s time to draw a picture k m1 (y1, z1) q1 z y

  37. Again I like the Lagrangian method constraints Don’t linearize here unless you are very clever

  38. Linearize Natural frequency

  39. initial conditions

  40. QUESTIONS?

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