Separate even and odd parts of a mechanical signal
Any function on a symmetric interval can be separated into its even part f_e(x) = [f(x) + f(−x)]/2 and odd part f_o(x) = [f(x) − f(−x)]/2. The even part contributes the constant and cosine terms, while the odd part contributes sine terms. This is useful for a signal containing both symmetric displacement and an antisymmetric component.
Worked practice example
Separate f(x) = 2 + x + cos(2x) on −π ≤ x ≤ π, then find its mean and the contribution to b₁ from x.
| Step | Working |
|---|---|
| Reflect the function | f(−x) = 2 − x + cos(2x). |
| Even and odd parts | f_e = 2 + cos(2x), and f_o = x. |
| Mean | The average of x and cos(2x) is zero, so the mean is 2 and a₀ = 4. |
| First sine coefficient | b₁ = (2/π)∫ from 0 to π x sin x dx = 2. |
Check your method
The signal as a whole is neither even nor odd, so it is incorrect to discard every sine or every cosine term. The known cosine term has a₂ = 1. The ramp contributes bₙ = 2(−1)ⁿ⁺¹/n. At the joined endpoints, the series represents the midpoint of the periodic jump.
Try it yourself
Separate f(x) = x² + 3x − 4 into even and odd parts.
Show the answer and method
The even part is x² − 4 and the odd part is 3x. Expand the former using cosine coefficients and the latter using sine coefficients, on the stated symmetric interval.
This written practice example was prepared for Jemshah revision. It is additional practice, not an official examination question or a transcript of the video.