Harmonic oscillator and fourier series

[The Oak]

Hello,
Attached are two problems I can not solve, thanks for the help.

The Attempt at a Solution

For the first question, I understand that I need insert A1coswt+A2sinwt into the homogenous equation , but don't know what's then ..

But I'm pretty much lost on both of em :(

 

[vela]

You should really make the effort to use the template and post the relevant information in the thread itself rather than expecting others to download a file just to read the problem.

Here's the text of the first problem:

Consider a damped driven harmonic oscillator, for which $\beta=\omega_0/4$ and the driving force is given by $F=F_0 \cos \omega t$. Our goal is to find the general solution, which is the sum of the complementary and particular solutions:

(a) Write down the equation of motion.

(b) Write down the complementary (transient, homogenous) solution. (No specific initial conditions are given so your answer will contain two arbitrary coefficients.)

(c) For the particular solution, try the following function: (this is equivalent to equation 3.55 but with a different form)

$$x_p(t) = B_1 \cos \omega t + B_2 \sin \omega t$$

Substitute this trial solution into your equation of motion, rearrange, and equate the coefficients to find expressions for B₁ and B₂.

(d) Now write down the expression for the general solution.
​

Start with part (a). What is the equation of motion for a forced, damped harmonic oscillator?

 

[The Oak]

oops sry I thought it's more convienet in this way

I got the Fourier series question solved.

Ok for a damped harmonic oscillator the equation of motion is

mx^.. + cx^. +kx = F0cos(wt)

is that right?

 

[vela]

It's right, except you don't say what c is.

 

[paranormal]

Hello,

I am happy to help you with your questions about the harmonic oscillator and Fourier series.

To solve the first problem, you are correct in using the general solution A1coswt + A2sinwt for the homogenous equation. From there, you can use the initial conditions given to solve for the constants A1 and A2. This will give you the particular solution for the equation.

For the second problem, you will need to use the Fourier series to represent the given function as an infinite sum of sine and cosine terms. This involves calculating the coefficients a0, an, and bn using the given formula. Once you have these coefficients, you can use them to write the function as a sum of sine and cosine terms.

I hope this helps and if you have any further questions, please don't hesitate to ask. Keep working at it and you will be able to solve these problems successfully. Best of luck!