Finding the Fourier Coefficients for mechanics homework

[\Theta]

Homework Statement

Find the Fourier Coefficients for the triangular wave equation shown in this picture:

Homework Equations

##f(t)= a_0 + \sum_{n=1}^\infty a_{n}cos(n{\omega}t) + \sum_{n=1}^\infty b_{n}sin(n{\omega}t)##
##a_0 = \frac{1}{\tau}\int_{-\tau/2}^{\tau/2} f(t) \, dt##
## \omega = \frac{2\pi}{\tau}##
##a_n = \frac{2}{\tau}\int_{-\tau/2}^{\tau/2}f(t)cos(n\omega t) \, dt##
##b_n = \frac{2}{\tau}\int_{-\tau/2}^{\tau/2}f(t)sin(n\omega t) \, dt = 0, \forall n##, because it's an even function

The Attempt at a Solution

## \tau = 2##
##a_0 = \frac{1}{2}\int_{-1}^{1} 1-|t| \, dt= \frac{1}{2}[\int_{-1}^{0} 1+t \, dt+\int_{0}^{1} 1-t \, dt]=\frac{1}{2} [\frac{1}{2}+\frac{1}{2}]=\frac{1}{2} .##

##a_n = \frac{2}{2}\int_{-1}^{1}f(t)cos(n\omega t) \, dt= \int_{-1}^{0}(1+t)cos(n\omega t) \, dt + \int_{0}^{1}(1-t)cos(n\omega t) \, dt ,##
Which, when computed through many tiresome intermediate steps, gives:
##a_n= \frac{2}{n^2\pi^2} ##, which differs from the answer in the textbook, ##a_n= \frac{4}{n^2\pi^2}, n=1,3,5,7,...; a_n=0, n=2,4,6,8,10,...##

So my question is, have I set up the problem correctly and made a computational mistake somewhere along the way in my integrals? Or am I missing something in the setup of the problem and missing the point entirely? It's a *** problem in our textbook(aka the most difficult level) and it's also a "computer problem", so I'm sure my prof won't expect me to do a problem this laborious on a test, but I'd still like to know where I went sour.

 

[DrClaude]

Homework Equations

##f(t)= a_0 + \sum_{n=1}^\infty a_{n}cos(n{\omega}t) + \sum_{n=1}^\infty b_{n}sin(n{\omega}t)##
##a_0 = \frac{1}{\tau}\int_{-\tau/2}^{\tau/2} f(t) \, dt##
## \omega = \frac{2\pi}{\tau}##
##a_n = \frac{2}{\tau}\int_{-\tau/2}^{\tau/2}f(t)cos(n\omega t) \, dt##
##b_n = \frac{2}{\tau}\int_{-\tau/2}^{\tau/2}f(t)sin(n\omega t) \, dt = 0, \forall n##, because it's an even function

Are you sure about these equation? If yes, what is ##\omega##?

 

[\Theta]

Are you sure about these equation? If yes, what is ##\omega##?

Well, these are the generic formula for the Fourier Series for a function. ##\omega = \frac{2\pi}{\tau}=\frac{2\pi}{2}=\pi##, I guess I wasn't as explicit about it but I considered it trivial.

 

[DrClaude]

Well, these are the generic formula for the Fourier Series for a function. ##\omega = \frac{2\pi}{\tau}=\frac{2\pi}{2}=\pi##, I guess I wasn't as explicit about it but I considered it trivial.

There can be different conventions, so I just wanted to make sure that I understood.

##a_n = \frac{2}{2}\int_{-1}^{1}f(t)cos(n\omega t) \, dt= \int_{-1}^{0}(1+t)cos(n\omega t) \, dt + \int_{0}^{1}(1-t)cos(n\omega t) \, dt ,##
Which, when computed through many tiresome intermediate steps, gives:
##a_n= \frac{2}{n^2\pi^2} ##

You'll have to give the intermediate steps, because the first line is correct, but the last equality doesn't follow.

 

[\Theta]

There can be different conventions, so I just wanted to make sure that I understood.
You'll have to give the intermediate steps, because the first line is correct, but the last equality doesn't follow.

Oh, I see what you were asking. Sorry.

##a_n = \frac{2}{2}\int_{-1}^{1}f(t)cos(n\omega t) \, dt= \int_{-1}^{0}(1+t)cos(n\omega t) \, dt + \int_{0}^{1}(1-t)cos(n\omega t) \, dt ,##
##=\int_{-1}^{0}cos(n\omega t) \, dt +\int_{-1}^{0}tcos(n\omega t) \, dt +\int_{0}^{1}cos(n\omega t) \, dt -\int_{0}^{1}tcos(n\omega t) \, dt ##
##= \int_{-1}^{1}cos(n\omega t) \, dt +\int_{-1}^{0}tcos(n\omega t) \, dt -\int_{0}^{1}tcos(n\omega t) \, dt##
##= \frac{sin(n{\pi}t)}{n\pi}|_{-1}^1+\int_{-1}^{0}tcos(n\omega t) \, dt -\int_{0}^{1}tcos(n\omega t) \, dt##
##= 0 +[\frac{tsin(n{\pi}t)}{n\pi}+\frac{cos(n{\pi}t)}{n^2\pi^2}|_{-1}^0-[\frac{tsin(n{\pi}t)}{n\pi}+\frac{cos(n{\pi}t)}{n^2\pi^2}|_{0}^1##
##=0 +[0+\frac{cos(n{\pi}t)}{n^2\pi^2}|_{-1}^0-[0+\frac{cos(n{\pi}t)}{n^2\pi^2}|_{0}^1##
##= 0 + \frac{1}{n^2\pi^2}+\frac{1}{n^2\pi^2}-\frac{1}{n^2\pi^2}+\frac{1}{n^2\pi^2}##
##=\frac{2}{n^2\pi^2}##

BTW, thanks a bunch for helping out DrClaude.

 

[\Theta]

Oh, I see what you were asking. Sorry.

##a_n = \frac{2}{2}\int_{-1}^{1}f(t)cos(n\omega t) \, dt= \int_{-1}^{0}(1+t)cos(n\omega t) \, dt + \int_{0}^{1}(1-t)cos(n\omega t) \, dt ,##
##=\int_{-1}^{0}cos(n\omega t) \, dt +\int_{-1}^{0}tcos(n\omega t) \, dt +\int_{0}^{1}cos(n\omega t) \, dt -\int_{0}^{1}tcos(n\omega t) \, dt ##
##= \int_{-1}^{1}cos(n\omega t) \, dt +\int_{-1}^{0}tcos(n\omega t) \, dt -\int_{0}^{1}tcos(n\omega t) \, dt##
##= \frac{sin(n{\pi}t)}{n\pi}|_{-1}^1+\int_{-1}^{0}tcos(n\omega t) \, dt -\int_{0}^{1}tcos(n\omega t) \, dt##
##= 0 +[\frac{tsin(n{\pi}t)}{n\pi}+\frac{cos(n{\pi}t)}{n^2\pi^2}|_{-1}^0-[\frac{tsin(n{\pi}t)}{n\pi}+\frac{cos(n{\pi}t)}{n^2\pi^2}|_{0}^1##
##=0 +[0+\frac{cos(n{\pi}t)}{n^2\pi^2}|_{-1}^0-[0+\frac{cos(n{\pi}t)}{n^2\pi^2}|_{0}^1##
##= 0 + \frac{1}{n^2\pi^2}+\frac{1}{n^2\pi^2}-\frac{1}{n^2\pi^2}+\frac{1}{n^2\pi^2}##
##=\frac{2}{n^2\pi^2}##

BTW, thanks a bunch for helping out DrClaude.

I think I see where I went wrong
Looks like it was just a silly case of the negative signs cancelling each other.
Should be:
##=0 +[0+\frac{cos(n{\pi}t)}{n^2\pi^2}|_{-1}^0-[0+\frac{cos(n{\pi}t)}{n^2\pi^2}|_{0}^1##
##=\frac{cos(n\pi)}{n^2\pi^2}+\frac{cos(n\pi)}{n^2\pi^2}+\frac{cos(n\pi)}{n^2\pi^2}+\frac{cos(n\pi)}{n^2\pi^2} ##
##= \frac{4cos(n\pi)}{n^2\pi^2}##
##=\frac{4(-1)^n}{n^2\pi^2} ##

 

[DrClaude]

##=0 +[0+\frac{cos(n{\pi}t)}{n^2\pi^2}|_{-1}^0-[0+\frac{cos(n{\pi}t)}{n^2\pi^2}|_{0}^1##
##=\frac{cos(n\pi)}{n^2\pi^2}+\frac{cos(n\pi)}{n^2\pi^2}+\frac{cos(n\pi)}{n^2\pi^2}+\frac{cos(n\pi)}{n^2\pi^2} ##

Getting closer, but that second line doesn't follow from the first.

 

[\Theta]

Getting closer, but that second line doesn't follow from the first.

##=0 +[0+\frac{cos(n{\pi}t)}{n^2\pi^2}|_{-1}^0-[0+\frac{cos(n{\pi}t)}{n^2\pi^2}|_{0}^1##
##=\frac{cos(0)}{n^2\pi^2} - \frac{cos(-n\pi)}{n^2\pi^2}-\frac{cos(n\pi)}{n^2\pi^2}+\frac{cos(0)}{n^2\pi^2}##
##=\frac{1}{n^2\pi^2} - \frac{cos(-n\pi)}{n^2\pi^2}-\frac{cos(n\pi)}{n^2\pi^2}+\frac{1}{n^2\pi^2} ##
##=\frac{1}{n^2\pi^2} - \frac{cos(n\pi)}{n^2\pi^2}-\frac{cos(n\pi)}{n^2\pi^2}+\frac{1}{n^2\pi^2}##
##=\frac{1}{n^2\pi^2} +\frac{1}{n^2\pi^2}+\frac{1}{n^2\pi^2}+\frac{1}{n^2\pi^2}## when n is odd
and ##=\frac{1}{n^2\pi^2} - \frac{1}{n^2\pi^2}-\frac{1}{n^2\pi^2}+\frac{1}{n^2\pi^2}=0## when n is even
So the Fourier series for the function would be:
##f(t)= a_0 + \sum_{n=1}^\infty a_{n}cos(n{\omega}t) + \sum_{n=1}^\infty b_{n}sin(n{\omega}t)##
##=\frac{1}{2}+\sum_{n=1}^\infty \frac{4}{(2n-1)^2\pi^2}cos((2n-1){\pi}t)##

That looks right. DrClaude, I definitely owe you one.

 

[\Theta]

Case closed.