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Johnrobjr
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I did read some of the posts on the topic. However, I just want to check my qualitative understanding without employing too much mathematical dynamics. If anyone has some feedback, this would be much appreciated.
The System.
A spring hangs vertically. There is a mass attached to it.
Assumptions
The motion is restricted to act in the x-y plane. There is no friction or damping.
Initial Conditions
The mass can be perturbed from its resting position (the origin, -y axis parallel to the gravitational force).
My Understanding
There are three differential equations:
one for the mass moving along the y-axis.
And, one the mass moving along the x-axis.
Also, the is one for the energy of the system.
Most generally, the equations describing the motion in the plane are coupled, and a closed form solution is not possible. However, if the spring is stretched only so that it operates in the linear region, and the pendulum is displaced only so that it is operating in the linear region...then the equations are decoupled and a solution is possible.
My question is under those initial conditions where the equations are decoupled, will the system be stable, assuming no friction or damping? I think it will degenerate, but I am not sure why.
Also, if the system is operating so that the equations are coupled then many trajectories are possible (perhaps infinite). Is this true?
It the system operates so that the equations are coupled then are there periodic trajectories that remain stable or will they always evolve to others with time?
The System.
A spring hangs vertically. There is a mass attached to it.
Assumptions
The motion is restricted to act in the x-y plane. There is no friction or damping.
Initial Conditions
The mass can be perturbed from its resting position (the origin, -y axis parallel to the gravitational force).
My Understanding
There are three differential equations:
one for the mass moving along the y-axis.
And, one the mass moving along the x-axis.
Also, the is one for the energy of the system.
Most generally, the equations describing the motion in the plane are coupled, and a closed form solution is not possible. However, if the spring is stretched only so that it operates in the linear region, and the pendulum is displaced only so that it is operating in the linear region...then the equations are decoupled and a solution is possible.
My question is under those initial conditions where the equations are decoupled, will the system be stable, assuming no friction or damping? I think it will degenerate, but I am not sure why.
Also, if the system is operating so that the equations are coupled then many trajectories are possible (perhaps infinite). Is this true?
It the system operates so that the equations are coupled then are there periodic trajectories that remain stable or will they always evolve to others with time?