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Step response and weight-lift test

The simplest, but only moderately accurate, calibration method is to observe the response of the system to a step input. It can be generated by switching on or off a current through the calibration coil, or by applying or removing a constant mechanical force on the seismic mass, usually by lifting a weight. Horizontal sensors used to be calibrated with a V-shaped thread attached at one end to the mass, to a fixed point at the other end, and to the test weight at half length. The thread was then burned off for a soft release.

The step-response experiment can be used both for a relative and an absolute calibration; when applicable, it is probably the simplest method for the latter. Using a known test weight w and knowing the seismic mass M, we also know the test signal: it is a step in acceleration whose magnitude is w/M times gravity (times a geometry factor when a thread is used). In case of a rotational pendulum, a correction factor must be applied when the force does not act at the center of gravity. The method has lost its former importance because the seismic mass of modern seismometers is not easily accessible, and the correction factor for rotational motion is rarely supplied by the manufacturers.

Interestingly, in the case of a simple electromagnetic seismometer with linear motion and a known mass, not even a calibration coil or the insertion of a test mass are required for an absolute calibration. A simple experiment where a step current is sent through the signal coil of the undamped sensor can supply all parameters of interest: the generator constant E, the free period, and the mechanical damping. The method is described in chapter 4 of the old MSOP [Willmore, P. L. (ed.) 1979]. See also section 7.3 for an alternative method.

In the context of relative calibration, the step-response method is still useful as a quick and intuitive test, and has the advantage that it can be evaluated by hand. Software like PREPROC or CALEX covers the step response as well (section 9). Fig. 26 shows the characteristic step responses of second-order high-pass, band-pass, and low-pass filters with $1/\sqrt{2}$ of critical damping. The amplitude responses of these systems were shown in the left column of Fig. 6. Each response is a strongly damped oscillation around its asymptotic value. With the specified damping, the systems are Butterworth filters, and the amplitude decays to $e^{-\pi}$ or 4.3% within one halfwave. The ratio of two subsequent amplitudes of opposite polarity is known as the overshoot ratio. It can be evaluated for the numerical damping h: when xi and xi+n are two amplitudes n periods apart, with integer or half-integer n, then

\begin{displaymath}\frac{1}{h^2}=1+\left(\frac{2 \pi n}{\ln x_{i} - \ln x_{i+n}} \right)^2
\end{displaymath} (39)

The free period can in principle also be determined from the impulse or step response of the damped system but should preferably be measured without electrical damping so that more oscillations can be observed. A system with the free period T0 and numerical damping h oscillates with the period $T_0 / \sqrt{1-h^2}$ and the overshoot ratio $e^{- \pi h / \sqrt{1-h^2}}$.


  
Figure 26: Normalized step responses of second-order high-pass, band-pass and low-pass filters. The transfer functions are the same as in Fig. 6, left column. Compare also Fig. 4 which shows slightly smoothed impulse responses.
\includegraphics[width=\textwidth]{Fig/pulses.eps}


next up previous contents
Next: Calibration with arbitrary signals Up: Calibration Previous: Calibration with sinewaves
Erhard Wielandt
2002-11-08