To perform FFT (Fast Fourier Transform) analysis on vibration sensor data in MATLAB

To perform FFT (Fast Fourier Transform) analysis on vibration sensor data in MATLAB, follow this structured guide. FFT helps convert time-domain vibration signals into the frequency domain, revealing dominant frequencies and their amplitudes, which is critical for diagnosing machinery faults (e.g., imbalance, misalignment, bearing defects).

Step 1: Load and Preprocess the Data

Import Time-Domain Data

Assume your vibration data is stored in a CSV or text file with two columns: time (s) and amplitude (e.g., acceleration in m/s²).

% Load data
data = readtable('vibration_data.csv');
time = data.Time;          % Time vector
amplitude = data.Amplitude;% Vibration signal

Check Sampling Rate

Calculate the sampling frequency ( $f_{s}$ ):

Fs = 1 / (time(2) - time(1)); % Sampling frequency (Hz)
N = length(amplitude);         % Number of data points

Step 2: Preprocess the Signal

Detrend the Data

Remove DC offset or linear trends:

amplitude = detrend(amplitude);

Apply a Window Function

Reduce spectral leakage using a Hanning window:

window = hann(N);
amplitude_windowed = amplitude .* window;

Step 3: Perform FFT

Compute the FFT

Y = fft(amplitude_windowed);     % Compute FFT
P2 = abs(Y/N);                   % Two-sided spectrum
P1 = P2(1:N/2+1);               % Single-sided spectrum
P1(2:end-1) = 2*P1(2:end-1);    % Adjust for energy in single-sided

Define Frequency Axis

f = Fs * (0:(N/2)) / N; % Frequency vector (Hz)

Step 4: Plot the Frequency Spectrum

figure;
plot(f, P1);
title('Single-Sided Amplitude Spectrum');
xlabel('Frequency (Hz)');
ylabel('Amplitude');
grid on;

Step 5: Identify Dominant Frequencies

Find Peaks

Use MATLAB’s findpeaks to detect significant frequencies:

[peaks, locs] = findpeaks(P1, f, 'MinPeakHeight', 0.1*max(P1), 'MinPeakDistance', 1);

Display Peaks

hold on;
plot(locs, peaks, 'ro');
legend('Spectrum', 'Dominant Frequencies');

Step 6: Analyze Results

Peak Frequencies: Compare dominant frequencies ( $f_{peak}$ ) with known fault frequencies (e.g., rotational speed harmonics).
Amplitude: Higher amplitudes at specific frequencies indicate resonance or defects.

Example Workflow for Simulated Data

% Simulate a vibration signal with 10 Hz and 30 Hz components
Fs = 1000;            % Sampling frequency (Hz)
t = 0:1/Fs:1-1/Fs;    % Time vector (1 second)
f1 = 10;              % Frequency 1 (Hz)
f2 = 30;              % Frequency 2 (Hz)
amplitude = 0.7*sin(2*pi*f1*t) + 1.2*sin(2*pi*f2*t) + 0.1*randn(size(t));

% Perform FFT
Y = fft(amplitude);
P2 = abs(Y/length(t));
P1 = P2(1:length(t)/2+1);
P1(2:end-1) = 2*P1(2:end-1);
f = Fs*(0:(length(t)/2))/length(t);

% Plot
plot(f, P1);
xlabel('Frequency (Hz)');
ylabel('Amplitude');

Key Considerations

Aliasing: Ensure $f_{s} > 2 \times f_{max}$ (Nyquist theorem).
Zero-Padding: Use NFFT = 2^nextpow2(N) to improve frequency resolution.

Averaging: For noisy data, use pwelch for power spectral density (PSD) estimation:

[pxx, f] = pwelch(amplitude, hann(512), 256, 512, Fs);
plot(f, 10*log10(pxx)); % Plot in dB

Log Scale: Visualize small amplitudes with a logarithmic y-axis:
matlab

Copy
```
semilogy(f, P1);
```

Troubleshooting

Low Amplitude Peaks: Check signal-to-noise ratio (SNR) and apply filtering (e.g., lowpass or bandpass).
Frequency Resolution: Increase data acquisition time ( $T = N / f_{s}$ ).
Non-Stationary Signals: Use Short-Time FFT (spectrogram) for time-frequency analysis:
matlab

Copy
```
spectrogram(amplitude, hann(256), 128, 256, Fs, 'yaxis');
```

By following these steps, you can effectively analyze vibration sensor data to diagnose mechanical issues or validate system dynamics.