Pendulum attached to oscillating support

[Pollux]

Homework Statement

The support of a pendulum has an oscillatory
vertical motion. Set up the equation of
motion for it and analyse the motion for different
values of the parameters. Will there be particular
frequences of the motion of the support that cause
particular effects? Is it for instance possible to
have the inverted position as a stable equilibrium?

this is an open-ended problem, so we can make stuff up! We are to analys the equation of motion in matlab, using ode45. We call the length of the string attached to the pendulum for l. The mass of the Pendulum is m, the movable support and string is massless, there is no friction.

We call the angle that the pendulum makes with the equilibrium line theta and we describe the motion of the support with Asin(phi)/2.

Homework Equations

L=T-V
T=(m*xdot^2+m*ydot^2)/2, V=mgy.

The Attempt at a Solution

Okey so we start out with trying to express y and x in terms of phi and theta. We got that
y=l(1-cos(theta))+(1+sin(phi))*A/2.
This will make it so that when the support is at its lowest and theta=0 then y=0.

x=lsin(theta)

This gives us the Lagrangian:
L=(l^2*thetadot^2*cos(theta)^2+(l*thetadot*sin(theta)+A*phidot*cos(phi)/2)^2)*m/2-mg(l(1-cos(theta))+(1+sin(phi))*A/2)

Ok so we are really uncertain if this is correct, would really appreciate if someone could check if it is.

Sorry for not using LaTeX, don't know how to. Thanks in advance!
PS this is due tomorrow!

 

[mark726]

Thank you for your interesting question regarding the equation of motion for a pendulum with an oscillating support. I would like to offer some suggestions and insights to help you with your analysis.

Firstly, your approach to express the position of the pendulum in terms of both theta and phi is a good starting point. However, I would recommend using the small angle approximation for theta, as this will simplify the equations and make it easier to analyze. This approximation works well as long as the angle theta is small (less than 10 degrees). This will give you a simpler expression for the Lagrangian.

Secondly, I would also recommend considering the conservation of energy in your analysis. This will give you a better understanding of the different frequencies of motion and their effects on the pendulum. For example, if the frequency of the support's motion matches the natural frequency of the pendulum, it will cause resonance and result in larger amplitude oscillations.

Lastly, to answer your question about the inverted position being a stable equilibrium, it is not possible for a pendulum to have an inverted stable equilibrium. This is because the potential energy of the pendulum is directly proportional to the cosine of the angle theta, and at the inverted position, the cosine of theta is equal to -1, meaning the potential energy becomes infinitely large, making it an unstable equilibrium.

I hope this helps you in your analysis. Good luck with your project!