Simple and driven harmonic motion

[Andrei0408]

Homework Statement

SUPERPOSITION OF TWO PERPENDICULAR SIMPLE
HARMONIC OSCILLATIONS : find the trajectory equation for a mass
point simoultaneously subjected at 2 SHO with the same frequency

DRIVEN HARMONIC MOTION : find the amplitude and the initial
phase for the steady solution and the resonance frequency and amplitude

In caps I wrote the name of the lecture, also how can I demonstrate the equation in the green box? I've also attached my attempt so far but I'm unsure how I should continue. Thank you!!

Relevant Equations

I've attached the equations that I believed to be relevant in the thread, I hope I was right

I know you can't solve it and just give it to me, I just want to know what I'm supposed to do, if you need any more information or clarification please let me know. Thank you for taking the time to help me!

 

[kuruman]

The statement of the problem tells you what to do. For the first part you need to "
find the trajectory equation for a mass point simultaneously subjected at 2 SHO with the same frequency". The first figure you posted has already started you along the way by giving you equations (1) and (2). You need to manipulate them so that you can get the trajectory that appears in the green box.

For the second part you need to write down the differential equation for the driven harmonic oscillator and find the amplitude assuming that the transients have died out. This is something that you can find in any intermediate level textbook on Classical Mechanics and on the web.

Your request is too general for me to be more specific. Start working, post your work and we will help you to the extent that we can.

 

[Andrei0408]

The statement of the problem tells you what to do. For the first part you need to "
find the trajectory equation for a mass point simultaneously subjected at 2 SHO with the same frequency". The first figure you posted has already started you along the way by giving you equations (1) and (2). You need to manipulate them so that you can get the trajectory that appears in the green box.

For the second part you need to write down the differential equation for the driven harmonic oscillator and find the amplitude assuming that the transients have died out. This is something that you can find in any intermediate level textbook on Classical Mechanics and on the web.

Your request is too general for me to be more specific. Start working, post your work and we will help you to the extent that we can.

Thank you! I've solved the first one but I'm still having some trouble regarding the second one, and I wrote the differential equation. I also know the solution x(t) but I don't fully understand how I can get to that solution. Could you help me a bit more?

 

[kuruman]

Thank you! I've solved the first one but I'm still having some trouble regarding the second one, and I wrote the differential equation. I also know the solution x(t) but I don't fully understand how I can get to that solution. Could you help me a bit more?

Like I said, you can find the derivation of the solution in textbooks or on the web. For example, http://farside.ph.utexas.edu/teaching/315/Waves/node13.html

 

[DaveE]

There is a really good set of lectures about SHOs here:


The recitation lectures are the good part IMO.