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The conversion between acceleration, velocity, and displacement is based on one of the fundamental laws in Newtonian physics, represented by the following equations:

Velocity is the first derivative of displacement with respect to time. Acceleration is the first derivative of velocity and the second derivative of displacement with respect to time. Therefore, given acceleration, perform a single integration with respect to time to compute the velocity or perform a double integration with respect to time to compute the displacement.

When representing the acceleration of a point by a simple sinusoid, the velocity and the displacement of the point are represented by the following equations:

a = Asin(t)

Note  The initial condition is set arbitrarily to zero in Equations A and B.

The amplitude of the velocity is inversely proportional to the frequency of the vibration. The amplitude of the displacement is inversely proportional to the square of the frequency of the vibration. Furthermore, the phase of the velocity lags the acceleration by 90°. The phase of the displacement lags the acceleration by 180°. The following front panel illustrates the relationship between acceleration, velocity, and displacement.

The integration of a sinusoid is known in closed form. Integration of an arbitrary waveform typically requires a numerical approach. You can use several numerical integration schemes to evaluate an integral in the time domain.

In the frequency domain, you can define any arbitrary band-limited waveform as a sum of sinusoids. Because the amplitude and phase relationships are known for sinusoids, you can carry out the integration in the frequency domain.