# Thread: Fourier serier on a engiineering pulse.

1. ## Fourier serier on a engiineering pulse.

Morning All

I have a question I am completely stuck on.

f(t) =3t+3, -1<t<0
=t+3, 0<t<1
=6t-2t, 1<t<3

Make a sketch of this pulse.

Time is measured in milliseconds and amplitude of vibration is in microns.
The diagnostic machinery repeats this pulse every 4 milliseconds. Carry out a Fourier decomposition of the resulting waveform and find the amplitude at the fundamental frequency and at the next six higher harmonics.

Make a plot of this waveform in the frequency domain, showing amplitude against frequency in Hertz.

The engineering company want to ensure that there are no vibrations with an amplitude of more than 0.1 millimetres in the frequency band between 600 Hz and 800Hz.

This is the question. Any help on where I should start. I understand this is a sawtooth wave, and need to apply the equation to get A0, An, B0 but I don’t get how to apply this to a wave with more than one function.

Any Help

Regards

2. ## Re: Fourier serier on a engiineering pulse.

do you mean 6t-2 for 1<t<3 ?

otherwise it's just 4t

3. ## Re: Fourier serier on a engiineering pulse.

sorry i did mean 6t-2.

regards

4. ## Re: Fourier series on a engineering pulse.

Can anyone point in where to start?

Regards

James

5. ## Re: Fourier serier on a engiineering pulse.

The problem says "Carry out a Fourier decomposition". Do you know what that is?

6. ## Re: Fourier serier on a engiineering pulse.

It a fourier series. I know you use the formulas for a sawtooth wave. What i can't figure out is how the three functions at different time intervals fit into the formula.

7. ## Re: Fourier serier on a engiineering pulse.

You know that $\displaystyle \int_a^b g(x)\,\mathrm dx = \int_a^c g(x)\,\mathrm dx + \int_c^b g(x)\,\mathrm dx$?

This means that you can break the interval of integration into subintervals. Does that help?

8. ## Re: Fourier serier on a engiineering pulse.

ok so i've attempted with the help on youtube. Can someone have a look and see if i'm anywhere close.

i'm assume its an odd function even though when i plot it. it doesn't look like it is. Hope that bit is correct. If so the A0 & An = 0
f(t)=A/T *t = 16/4t
bn 4/L int(between 3 & -1) f(t)sin(Nπt) .dt
4/2 int 16/4tsin(Nπt)dt
8(int)tsin(Nπt)dt

can you (int)sin(ax)dx = 1/a^2(sin(ax)-axcos(ax)

therefore i get
8/(N^2*π^2) ((sin((Nπt)-Nπcos(Nπt))

can somebody have a look and see if i'm way off here and please point out where am i going wrong

regards
James

9. ## Re: Fourier serier on a engiineering pulse.

Originally Posted by jblakes
i'm assume its an odd function even though when i plot it. it doesn't look like it is. Hope that bit is correct.
It's not. The fact that $\displaystyle \lim_{t \to 0} f(t)$ exists and is non-zero tells you that.

$\displaystyle \int_{-1}^3 f(t) \cos{\left((t-1)\tfrac{\pi}{2}\right)} \, \mathrm dt = \int_{-1}^0 f(t) \cos{\left((t-1)\tfrac{\pi}{2}\right)} \, \mathrm dt + \int_0^1 f(t) \cos{\left((t-1)\tfrac{\pi}{2}\right)} \, \mathrm dt + \int_{1}^3 f(t) \cos{\left((t-1)\tfrac{\pi}{2}\right)} \, \mathrm dt$

If you prefer, you could work with the interval $\displaystyle (0,4)$ by using $\displaystyle f(t)=f(t-4)$ to find the values in the interval $\displaystyle (3,4)$. It may be slightly easier.

10. ## Re: Fourier serier on a engiineering pulse.

so i'm guessing it neither even or odd. I did think that. i assume i need to work out both A0, An, &Bn? I'm struggling to find an example of one that has both. Or looking at you response it is just an even function?

regards

11. ## Re: Fourier serier on a engiineering pulse.

Originally Posted by jblakes
so i'm guessing it neither even or odd. I did think that. i assume i need to work out both A0, An, &Bn? I'm struggling to find an example of one that has both. Or looking at you response it is just an even function?

regards
$c_n = \dfrac 1 T \displaystyle{\int_T}~f(t)e^{j \frac{n 2 \pi t}{T}}~dt$

where any interval of length $T$ is satisfactory for integration.

Here

$c_n = \dfrac 1 4 \displaystyle{\int_T}~f(t)e^{j \frac{n \pi t}{2}}~dt=$

$\dfrac 1 4 \displaystyle{\int_{-1}^0}~(3t+3)e^{j \frac{n \pi t}{2}}~dt + \dfrac 1 4 \displaystyle{\int_0^1}~(t+3)e^{j \frac{n \pi t}{2}}~dt _ + \dfrac 1 4 \displaystyle{\int_1^3}~(6t+2)e^{j \frac{n \pi t}{2}}~dt$

none of these are particularly hard to integrate.

12. ## Re: Fourier serier on a engiineering pulse.

No, it's not even either. You will also need
$\displaystyle \int_{-1}^3 f(t) \sin{\left((t-1)\tfrac{\pi}{2}\right)} \, \mathrm dt = \int_{-1}^0 f(t) \sin{\left((t-1)\tfrac{\pi}{2}\right)} \, \mathrm dt + \int_0^1 f(t) \sin{\left((t-1)\tfrac{\pi}{2}\right)} \, \mathrm dt + \int_{1}^3 f(t) \sin{\left((t-1)\tfrac{\pi}{2}\right)} \, \mathrm dt$

I rather thought you might be able to work that out for yourself.

Edit: Romsek is correct, you can replace $\displaystyle (t-1)$ with $\displaystyle t$ in those integrals.

13. ## Re: Fourier serier on a engiineering pulse.

i did think that Archie. Thanks for you reply, and you too Romsek.

((3t^2/2)-(6cos(pit/2))/pi |-1&0 + (t^2/2)-((6cos(pit/2)/pi)|0&1 + 2cos(t)+3t^2 |1&3

does that look right for An values? Just need to stick values in.

14. ## Re: Fourier serier on a engiineering pulse.

Is that correct for the Bn?

((3t^2)-3sin(t))|-1&0 +(t^2-3sin(t))|0&1+ (2sin(t)+3t^2) |1&3

regards

15. ## Re: Fourier serier on a engiineering pulse.

I don't think i have intergrated these correctly.
intergration between -1&0 3t+3(sin(t-1(pi/2))

Comes to

3/2π(sin(1)-sin(2)+cos(1) is what i get on symbolab. Can I ask if that is correct?

regards

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