Hilbert Envelope and Cross-Correlation for Diagnosing Bearing Defects in Simulated Robot Motors

Project info

• Field of studyRobotics and Mechatronics
• LevelDoctoral dissertation (S3)
• Method typeSignal Processing
• Estimated build time± 7 working days
• Toolshilbert, xcorr (Signal Processing Toolbox)
• PriceUS$38 - US$231

Research goal

Apply the Hilbert transform to the bearing vibration envelope to identify the bearing defect frequency amid machine background noise through simulation.

Variables

Example result

• 1.007Envelope mean
• 0.7Modulation depth (est.)
• 0.533Modulation depth (true)
• 55Estimated delay (ms)
• 0.971Peak correlation
• 87.995Carrier frequency (Hz)

The Hilbert envelope of Bearing vibration envelope gives a modulation depth of 0.700 (true 0.533); cross-correlation with Bearing defect frequency peaks at 0.971 for a 55 ms delay, and the carrier is estimated at 88.0 Hz.

Computed from illustrative synthetic data, not from a real study.

About Envelope and Cross-Correlation (Hilbert)

The Hilbert transform produces the analytic signal to obtain the envelope (instantaneous amplitude), and cross-correlation measures the delay between two signals. It is used in vibration analysis, modulated signals, and sensor delay estimation.

Steps

1. 1Prepare the signals and sampling frequency
2. 2Compute the analytic signal with hilbert
3. 3Take the envelope from its magnitude
4. 4Compute the cross-correlation of two signals with xcorr
5. 5Find the lag at the correlation peak to estimate delay
6. 6Plot the envelope and the correlation function

Assumptions to check

• The signal is narrowband so the envelope is meaningful
• The delay between signals is constant over the record
• Noise is not so large that it hides the correlation peak

Reading and reporting the results

Report the envelope (for example peak value or trend) and the estimated delay in seconds with its error against the true value if known. State the delay resolution of 1/fs.

Example MATLAB code

A short starting script with this project's variable names. The delivered version is adapted to your data and fully commented.

rng(5);
fs = 1000;
bearing_defect_frequency = (0:1/fs:2-1/fs)';
env_true = 1 + 0.5*sin(2*pi*2*bearing_defect_frequency);
bearing_vibration_envelope = env_true .* sin(2*pi*80*bearing_defect_frequency);          % amplitude-modulated signal

analytic = hilbert(bearing_vibration_envelope);
env = abs(analytic);

delay_s = 0.037;
d = round(delay_s * fs);
y2 = [zeros(d,1); bearing_vibration_envelope(1:end-d)] + 0.3*randn(size(bearing_vibration_envelope));
[c, lags] = xcorr(y2, bearing_vibration_envelope + 0.3*randn(size(bearing_vibration_envelope)), 200, 'coeff');
[cmax, imax] = max(c);
fprintf('Estimated delay: %.3f s (true %.3f s), peak corr %.2f\n', lags(imax)/fs, delay_s, cmax);
fprintf('Envelope RMSE: %.3f\n', sqrt(mean((env(100:end-100) - env_true(100:end-100)).^2)));

figure;
subplot(2,1,1);
plot(bearing_defect_frequency, bearing_vibration_envelope, bearing_defect_frequency, env, 'r', 'LineWidth', 1.2);
legend('Signal', 'Envelope'); xlabel('Time (s)');
subplot(2,1,2);
plot(lags/fs, c); xlabel('Lag (s)'); ylabel('Correlation');

Packages and prices

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• Payment: 50% down payment before work starts, balance after the deliverables are handed over
• Tax: Prices exclude 11% VAT
• Data and academic integrity: Results are computed from your own research data; you write and stand behind your manuscript, we help with the analysis and your understanding of it
• Revisions: Up to 2 revisions are included in the price (Package 3: 3 rounds)

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