ME 470 01 B Finding the Fourier series expansion of a periodic function

Fourier Series Example: Triangle Wave (Periodic Ramp) and Fourier Coefficients This video walks through a complete Fourier series expansion for a periodic triangle-wave style signal. You will learn how to (1) identify the smallest repeating period, (2) write the function over one period, (3) compute a0, an, and bn, and (4) assemble the final Fourier series and interpret what the coefficients mean for frequency content. Key takeaways Find the period and the fundamental frequency omega = 2*pi/T Define x(t) over a single period, then extend it by periodic repetition Compute a0 (DC term), then compute an and bn using the standard integrals Use integration by parts (or an integral table shortcut) for t*cos( ) and t*sin( ) Simplify trig terms using integer-multiple identities Recognize when the series becomes sine-only (an = 0) and what that implies Final result shown in the lesson (for the triangle-wave style example) x(t) = 5/2 - (5/pi) * sum from n = 1 to infinity of (1/n) * sin(n*(2*pi/3)*t) Course page for notes, examples, and the full video set https://sites.google.com/view/mechani... Related videos from this channel (good next steps) Procedure for Fourier series expansion (analytic function) -    • ME 470 01 A Procedure for Fourier Series e...   Fourier series of a periodic function -    • ME 470 01 B Finding the Fourier series exp...   Fourier Example 1 in MATLAB -    • ME 470 01 B1 Fourier Example 1 in MATLAB   Fourier series of a periodic piecewise function -    • ME 470 01 C Finding the Fourier series exp...   Fourier Example 2 in MATLAB -    • ME 470 01 C1 Fourier Example 2 in MATLAB   Discrete Fourier series -    • ME 470 01 D Discrete Fourier Series   DFT small-data example -    • ME 470 01 E DFT Small Data Example   Spectral analysis and PSD -    • ME 470 01 F Spectral Analysis and PSD   PSD example -    • ME 470 01 G PSD Example   DFT with differentiation and integration -    • ME 470 01 H DFT with Differentiation and I...   Transmissibility and vibration isolator (Bode diagram) -    • ME 357 14 D1 Transmissibility, Vibration I...   #MechanicalVibrations #FourierSeries #SignalProcessing #FrequencyAnalysis #PowerSpectralDensity #VibrationAnalysis Timestamps 0:00 Problem setup and what the Fourier series represents 0:32 Step 1: identify the period and fundamental frequency 1:32 Step 2: define x(t) over one period (triangle-wave style ramp) 3:20 Step 3: compute a0 (DC term) 6:55 Step 4: set up an integral and why it is tougher than a0 9:20 Using an integral table or integration by parts for t*cos( ) 13:05 Simplifying trig terms and showing an = 0 17:05 Set up bn and evaluate the definite integral 20:40 Simplify bn and interpret the 1/n amplitude trend 22:10 Assemble the full Fourier series 23:25 Convergence and what more terms do to the approximation