Reference frame for harmonic motion.

[rollcast]

If you a mass being accelerated by a force which is acting upon a spring attached to the mass it will exhibit harmonic motion. However unlike a fixed harmonic oscillator there is no explicit solution to the equation which describes the motion of the mass in a reference frame outside of the system.

However what if you took the reference frame and imagined a reference frame where the mass or spring was stationary would this allow you to calculate an explicit solution for the equation?

Thanks
AL

 

[Studiot]

Perhaps a diagram to make your intent more clear?

Why would the force cause SHM unless it is variable (oscillatory) or impulsive itself?

What other forces are acting?

 

[rollcast]

Here's a diagram which might make it a bit more clear as to the set up which I'm referring to.

I forgot to say that the force is stopped after a certain amount of time and the system s allowed to deccelerate freely.

http://www.flyforums.co.uk/members/james9118-albums-technical-picture8185-rigid-lever.jpg

That shows the velocity against time for the system.

http://www.flyforums.co.uk/members/james9118-albums-technical-picture8186-flexible-lever.jpg

The red curve shows the harmonic motion - there are better graphs but I haven't got excel on this laptop.

 

[Studiot]

I still can't see it.

Assume your system is in free space so there are no other forces acting.

Assume the force is applied at t=o as shown.

What causes the spring to extend? ie what opposes F?

 

[AlephZero]

I don't understand why you are saying "there is no explicit solution to the equations of motion". Have you studied ordinary differential equations yet? If you have, you should know how to solve this. And if you know about Laplace Transforms, you can almost write down the solution without doing any "plug and chug" algebra.

 

[rollcast]

I still can't see it.

Assume your system is in free space so there are no other forces acting.

Assume the force is applied at t=o as shown.

What causes the spring to extend? ie what opposes F?

The inertia of the mass on one end of the spring opposes the force being applied to the other end of the spring.

I don't understand why you are saying "there is no explicit solution to the equations of motion". Have you studied ordinary differential equations yet? If you have, you should know how to solve this. And if you know about Laplace Transforms, you can almost write down the solution without doing any "plug and chug" algebra.

I understand 1st and 2nd ODEs but I'm not familiar with the formula that is being used to describe the motion of the mass. A poster on another forum has put it into an excel doc. which calculates the solution to the ODE using a Runge Kutta method - which from what I can figure out about it means the formula has no explicit solution?

 

[Studiot]

Are you quite sure that your description of the system is correct?

With the system as currently described what happens depends upon how long F is applied for and if it can move its point of application, since it is the only external force acting in the system.

Runge Kutta methods are simply numerical methods which may be more convenient for the calculation of a differential equation. They were developed long before we could put complicated formulae into computers and have the answer in microseconds for many mesh points.
In particular their use does not mean the equation has no analytic solution.

However without some more boundary conditions the equation has no solution of any description.

What is this differential equation to which you refer?

It is usual to offer a system similar to yours, but with the spring connected to a fixed point at one end, the mass at the other, to demonstrate simple harmonic motion. Is this what you really mean?