# Lomb-Scargle Periodogram The Lomb-Scargle Periodogram is a least-square method which is useful for calculating the Fourier transform of a time history with gaps or an uneven sampling rate.
Reference Paper

This function has been added to the vibrationdata GUI package

Python script & Utility:
lomb_scargle.py
tompy.py

– Tom Irvine

# Inertial Navigation System Vibration Ring Laser Gyros

Acronyms:

IMU – inertial measurement unit
INS – inertial navigation system
RLG – ring laser gyro

* * *

An INS uses the output from an IMU, and combines the information on acceleration and rotation with initial information about position, velocity and attitude. It then delivers a navigation solution with every new measurement.

This process, called mechanization, is the summation of acceleration and attitude rate over time to produce position, velocity and attitude.   The mathematics require coordinate transformation and integration.

An IMU is typically composed of the following components:

• Three accelerometers
• Three gyroscopes
• Digital signal processing hardware/software
• Power conditioning
• Communication hardware/software
• An enclosure

Three accelerometers are mounted at right angles to each other, so that acceleration can be measured independently in three axes: X, Y and Z. Three gyroscopes are also at right angles to each other, so the angular rate can be measured around each of the orthogonal axes.

The gyroscopes were traditionally spinning wheel devices.  Nowadays, there are MEMS, fiber optic and ring laser gyros.

Vibration environments can adversely affect the accuracy of the IMU data.   Some of the potential issues are: aliasing, stability, bias drift, saturation, linearity, random walk and latency.

An IMU may be mounted via isolators.  As an example, the Space Integrated GPS/INS (SIGI) Inertial Sensor Assembly is isolated with a natural frequency of 55 Hz and 13.5% damping, equivalent to Q=3.7.

The purpose of the IMU is to measure rigid-body motion.  But the sensors also record the vehicle’s  elastic body vibration.  The control algorithms must be designed accordingly. Also, any isolation method must not be allowed to degrade the IMU accuracy.

* * *

The Nyquist frequency is equal to one-half the sampling rate.

Shannon’s sampling theorem states that a sampled time signal must not contain components at frequencies above the Nyquist frequency. Otherwise, an aliasing error will occur.

* * *

Here are some papers:

Notes on sample rate and aliasing:aliasing_notes.pdf

Inertial Navigation System Dither Sound & Vibration Test: INS_dither.pdf

Sound File:  dither.mp3

– Tom Irvine

# Correcting Acceleration, Velocity & Displacement Time Histories

There is a need in certain analysis problems to correct an acceleration signal so that its integrated velocity and double integrated displacement each oscillates about its respective zero baseline.  This may require using high pass filtering, trend removal and tapering throughout the integration process.  Trial-and-error is required to select the optimum combination of steps.

The steps are needed in part because the initial velocity and displacement are undefined.  Also, the acceleration may have a spurious offset or trend for the case of measured data.

Furthermore, the resulting displacement should be such that it can be recovered if it is then double differentiated to acceleration and the acceleration is then double integrated back to displacement.  This requirement is for rigor.  It yields a consistent set of acceleration, velocity and displacement time histories where each oscillates about its respective zero baseline.

This feature is now included in:

Matlab script: Vibrationdata Signal Analysis Package

Time History > Integrate or Differentiate > Correct Acceleration, Velocity, Displacement

– Tom Irvine