Question

Figured it out

Answer 1

@Michele_Laino say equation of motion is something like \[\dot{\theta} + \frac{ \dot l }{ l } \dot \theta + \frac{ \cos \theta }{ l } \dot{y}-\frac{ \sin \theta }{ l }\dot x=0\] the theta dot, y dot, and x dot should be all double dots, this is just a random equation of motion I found and was hoping you could help me understand if I need to find small frequency of oscillations how I would go about doing so @Michele_Laino

Answer 2

I know we can expand it right, but I'm not exactly sure what that means, if you can help explain it

Answer 3

in order to get the little oscillations, we have to write the kinetic energy, and the potential energy of our system in quadratic form

Answer 4

namely we have to write:
\[\large V\left( {{q_1},{q_2},...,{q_L}} \right) = \frac{1}{2}\sum\limits_{i,j} {{B_{ij}}{q_i}{q_j}} \]

Answer 5

where:
\[\large {B_{ij}} = {\left. {\frac{{{\partial ^2}V}}{{\partial {q_i}\partial {q_j}}}} \right|_{{P_0}}}\]

Answer 6

and P_0, is a point of stable equilibrium, furthermore, L is the number of degree of freedom of our system

Answer 7

similarly for the kinetic energy KE

Answer 8

Oh I was thinking we need this\[\cos \theta \approx 1 - \frac{ \theta^2 }{ 2 }~~~~~\sin \theta \approx \theta\]

Answer 9

So wait, we just set up our equation of motion in quadratic form? OR Lagrange/ Hamiltonian

Answer 10

it should be better if we solve an exercise

Answer 11

Yes, um let me see if I can find a problem

Answer 12

I have found one exercise of mine

Answer 13

Ok that works

Answer 14

here is the text:
A system is composed by three identical pendulums, with the same mass and with the same length. Those masses are connecte each other by two springs, whose constant is K. At the equilibrium all three pendulums are vertical and the springs are in a rest position

Answer 15

here is the drawing: