Introduction

This experiment investigates how varying the surface reflection angle affects laser vibrometer measurements. A vibrometer is aligned at several angles relative to the motion of a shaker’s stinger — orthogonally (0º), and at ±20º, ±30º, and ±45º. The results are compared to assess the accuracy of angle-corrected reconstructions.

Protocol

  1. Baseline Measurement: The vibrometer is initially aligned orthogonally (0º) to the shaker stinger. This measurement serves as the ground truth reference.

  2. Angled Measurements: The measurement surface is tilted to ±20º, ±30º, and ±45º, and the corresponding velocity responses are recorded for each angle.

  3. Correction and Averaging: Opposing angle pairs are averaged and corrected using cosine projection to estimate the longitudinal component.

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Mathematical Background

1. Projection Correction

The vibrometer measures the velocity projected onto its axis. For a laser angled at θ, the measured velocity is:

\[V_{\text{meas}} = V_z \cdot \cos(\theta)\]

To recover the true longitudinal velocity:

\[V_z = \frac{V_{\text{meas}}}{\cos(\theta)}\]

2. Averaging Symmetrical Angles

Opposing measurements (e.g., +θ and -θ) are averaged and corrected:

\[V_z^{(\theta)} = \frac{V_{+\theta} + V_{-\theta}}{2 \cos(\theta)}\]

The full estimate uses all three angle pairs:

\[V_z^{\text{avg}} = \frac{1}{3} \left( V_z^{(20^\circ)} + V_z^{(30^\circ)} + V_z^{(45^\circ)} \right)\]

3. Relative Error in Decibels

To quantify the difference between an estimated velocity and the orthogonal reference, we use:

\[\mathrm{Error}_{\mathrm{dB}}(f) = 20 \log_{10} \left| \frac{ \hat{V}(f) }{ V_{\text{ref}}(f) } \right|\]

Where:

  • 0 dB means perfect match
  • Positive values indicate overestimation
  • Negative values indicate underestimation

Results

Orthogonal vs. Reconstructed Velocity

Comparison of the orthogonal velocity measurement and the averaged reconstruction from all corrected angle pairs:

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Individual Angle Contributions

This plot shows the corrected measurements from each angle pair individually:

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Relative Error with Respect to Orthogonal Measurement

The relative error (in decibels) quantifies how much each estimate deviates from the orthogonal reference:

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Conclusions

Here’s the two main conclusions:

  • Non-orthogonal angles can be corrected using cosine projection.
  • Averaging across pair angles improves the results, since it cancels the in-plain vibrations (x and y).

Code and Data

The full source code and experimental data used are available on GitHub PhD repository.