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Correct the answer 1.2.2-2 #1

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4 changes: 2 additions & 2 deletions 01_introduction-motivation-and-background/QuizWeek1
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Expand Up @@ -49,7 +49,7 @@
d) derivada segunda de theta = -(g/l · sin theta) -(b/m · derivada primera de theta) -> ESTA

2. Assume a nonlinear system has total energy η=(ml²/2)·θ²+mgl(theta−1)² and that the system is dissipative such that the total energy decreases to zero. What do you expect the final angle θ to be when the system comes to a rest? You can assume that there are perturbations that move the system off any unstable fixed points. (Note: your answer should be numeric)
a) 0
a) 1

3. Assume that the nonlinear system in Question 2 with total energy η=(ml²/2)·θ²+mgl(theta−1)² is actually the model of some robot arm you are building and wish to control so that θ goes to the answer of Question 2. If the mass is doubled and the length is quadrupled, how will the answer to Question 2 change?
a) It will go up
Expand All @@ -74,4 +74,4 @@
It turns out that the nonlinear solutions also oscillate but we need more advanced methods to show this mathematically.
2. Linearize the equations of motion of an undamped pendulum with equation of motion second derivate θ=−g·sin(⁡θ)/l about the point θ=π, derivate θ=0.
Sol: -g/l
You should get a Jacobian [0 1; g/l 0]. This matrix has one positive and one negative eigenvalue. Since the linearized dynamics are hyperbolic we can predict the nonlinear stability behavior from that of the linearized dynamics (which our eigenvalue analysis indeed shows to be unstable).
You should get a Jacobian [0 1; g/l 0]. This matrix has one positive and one negative eigenvalue. Since the linearized dynamics are hyperbolic we can predict the nonlinear stability behavior from that of the linearized dynamics (which our eigenvalue analysis indeed shows to be unstable).