Engine Test Bed Fourier Analysis

[ajd-brown]

I have done a test on a 4 piston test engine which is expected to exhibit torsional resonance at 800RPM and a vertical translational resonance at 1200RPM.

The data we gathered from the test bed machine was as follows:

Theta | Signal
0 | -5
60 | -1
120 | 7
180 | 4
240 | 6
300 | -2
360 | -5

We are assuming that the accelerations are periodic with a period of 2∏ (hence 0 and 360 = -5)

I am trying to decompose the Signal using Fourier series of sin and cos, I just have no idea as how to use the data gathered into do this, could someone please point me in the right direction?

Regards,

Anthony

 

[Baluncore]

Your data is in the time domain. You must convert it to the frequency domain.
You have 6 data items over one revolution. You can only extract 6 Fourier terms.
For each frequency there will be a cosine term, (complex real), and a sine term, (complex imaginary).
To identify the coefficients you must correlate your data with the functions. Correlation is multiplication.
So for a frequency of f, tabulate your data and the function Cos(f*theta).
Multiply each data value by it's corresponding function value and accumulate the products.
Divide the sum of products by 6, the number of data items.
That gives you the cosine coefficient for frequency f.
Repeat the process with the sine function to get the sine coefficient for the same frequency.
Those cosine and sine coefficients make a complex number, convert it from rectangular to polar to get amplitude and phase.

 

[Baluncore]

Here is a worked example, probably with some errors to keep you awake.
Any further questions welcome.

 Frequency = 0   ( DC component )
theta f*theta  data  cos(f*theta) sin(f*theta)  d*cos    d*sin
   0      0   -5.00    1.000000     0.000000   -5.0000  -0.0000
  30      0   -1.00    1.000000     0.000000   -1.0000  -0.0000
  60      0    7.00    1.000000     0.000000    7.0000   0.0000
  90      0    4.00    1.000000     0.000000    4.0000   0.0000
 120      0    6.00    1.000000     0.000000    6.0000   0.0000
 150      0   -2.00    1.000000     0.000000   -2.0000  -0.0000
                                       Total    9.0000   0.0000
                        Fourier coefficients    1.5000   0.0000
                        Amplitude =   1.5000 at   0.00 deg

 Frequency = 1
theta f*theta  data  cos(f*theta) sin(f*theta)  d*cos    d*sin
   0      0   -5.00    1.000000     0.000000   -5.0000  -0.0000
  30     30   -1.00    0.500000     0.866025   -0.5000  -0.8660
  60     60    7.00   -0.500000     0.866025   -3.5000   6.0622
  90     90    4.00   -1.000000     0.000000   -4.0000   0.0000
 120    120    6.00   -0.500000    -0.866025   -3.0000  -5.1962
 150    150   -2.00    0.500000    -0.866025   -1.0000   1.7321
                                       Total  -15.5000   1.7321
                        Fourier coefficients   -2.5833   0.2887
                        Amplitude =   2.5994 at 173.62 deg

 Frequency = 2
theta f*theta  data  cos(f*theta) sin(f*theta)  d*cos    d*sin
   0      0   -5.00    1.000000     0.000000   -5.0000  -0.0000
  30     60   -1.00   -0.500000     0.866025    0.5000  -0.8660
  60    120    7.00   -0.500000    -0.866025   -3.5000  -6.0622
  90    180    4.00    1.000000    -0.000000    4.0000  -0.0000
 120    240    6.00   -0.500000     0.866025   -3.0000   5.1962
 150    300   -2.00   -0.500000    -0.866025    1.0000   1.7321
                                       Total   -8.5833   0.2887
                        Fourier coefficients   -1.4306   0.0481
                        Amplitude =   1.4314 at 178.07 deg

 

[ajd-brown]

Thats great thanks for your help, I now understand the last few steps of generating the coefficients which I was missing before! I did notice a few mistakes as you said.

Are these last fews steps anything to do with the Riemann sum or is that something else entirely?

 

[Baluncore]

The discrete time data you sampled, from a continuous function in the time domain, can be seen to represent an area bounded by a curve. The Riemann sum is the DC component only.

Once transformed into the frequency domain, only Fourier coefficients for discrete integer frequency harmonics are present. With the exception of the zero frequency DC component, the area of all other Fourier terms over one revolution is zero since all terms represent sinusoids.

The Fourier transform does not analyse the area so much as the way area is distributed over one full rotation. Your 4 cylinder engine had a firing order that would have set up torsional vibrations along the shaft. It should be apparent that analysis over one rotation is OK for 2 stroke engines, but for four stroke, the vibration analysis should be over two full rotations of the crank shaft since the crank rotates at twice the rate of the cam shaft.