Problem 4

Periodic input to multi-flywheel system. Return to the three flywheel system from Home-

work 6

bi

Use the same parameter values as the analysis in Homework 6. The motor moment, s, is the

periodic signal shown below:

2

bs

Lbs

Here are some general facts established from prior homework (see Problem 3, Homework 3):

● Because u is bounded and the system is asymptotically stable, fly, fly and fly are bounded.

Because u is periodic all dependent variables are periodic/nThese points provide a priori justification for using Fourier series to analyze the dependent vari

ables.

1. Compute the Fourier series coefficients for u.

2. Write the Fourier series representations for f₁, fly and fly. Hint: use the Fourier series repre-

sentation of the input and also the transfer functions/frequency response functions determined

in Homework 6.

3. Graph, in a single figure, the periodic responses off, f and fly and the input u by computing

a partial sum of the Fourier series expression over the index range &-0,1,2,...,1000.

Use the time grid t-[-2:0.0001:2];

4. Note how 1₂ appears "smoother than fh, and fly is smoother than f₂. How can this be

rigorously explained? Hint: Graph the frequency response magnitudes associated with fh,

fly and fly on the interval 0.01 Hz to 100 Hz and note how the magnitudes decrease at the

frequencies represented in the Fourier series.