Hello, question about the energy of a variable length pendulum.
Suppose you have a pendulum in the standard sense where θ is the angle, and we let the length r be a function of time r = r(t). What is the energy of the pendulum?
So far, I have determined that kinetic energy is = (1/2)m(r*dθ/dt)^2 + (1/2)m*(dr/dt)^2
and the potential energy is = mgr(1-cosθ).
In my homework problem i need to come up with a suitable Lyapunov function to study the stability of the pendulum and the typical approach approach is to set the lyapunoc function V = E (energy).
But the problem is that this is a time varying lyapunov function, i.e. V = V(θ,t). And i have to satisfy a positive definite constraint on V that is V(0,t) = 0 for all t. the problem is that dr/dt term.
Is there another lyapunov function i can choose here? or am i misrepresenting the energy?
Thank you
Suppose you have a pendulum in the standard sense where θ is the angle, and we let the length r be a function of time r = r(t). What is the energy of the pendulum?
So far, I have determined that kinetic energy is = (1/2)m(r*dθ/dt)^2 + (1/2)m*(dr/dt)^2
and the potential energy is = mgr(1-cosθ).
In my homework problem i need to come up with a suitable Lyapunov function to study the stability of the pendulum and the typical approach approach is to set the lyapunoc function V = E (energy).
But the problem is that this is a time varying lyapunov function, i.e. V = V(θ,t). And i have to satisfy a positive definite constraint on V that is V(0,t) = 0 for all t. the problem is that dr/dt term.
Is there another lyapunov function i can choose here? or am i misrepresenting the energy?
Thank you
0 commentaires:
Enregistrer un commentaire