An appraisal of numerical integration methods as applied to strong-motion data
Authors: Neumann Frank 
Published: May 2021

Accuracy of displacement curves computed from accelerograph records by numerical double integration.—The following results were obtained from Coast and Geodetic Survey accelerograph records in shaking-table tests made at the Massachusetts Institute of Technology:

Accelerograph performance.—The preceding paragraphs show that the pivot type of accelerometer now in use is satisfactory from the engineering viewpoint and that wave forms in terms of displacement can be satisfactorily computed for all but the longer-period waves. In transferring from the quadrifilar type of pendulum suspension to the pivot type to obtain a sturdier and more readily adjustable instrument, some sacrifice was made in accuracy of performance, but it is not serious. Although the pivot suspensions embody the highest quality of workmanship, they nevertheless undergo (when recording an earthquake) a certain amount of minute shifting, and this is greatly amplified in the double-integration process. This necessitates a high standard of servicing, and some adjusting in the mathematical treatment.

The present drum speed of 1 cm/sec. seems satisfactory enough for the present. Any expected increase in the accuracy of computed displacements through opening up the time scale would, at the present time, be nullified by errors resulting from pendulum instability. A more immediate advantage would be greater ease in disentangling overlapping curves and extrapolating those which go off the sheet entirely. Reduction of accelerometer sensitivity solves this problem, which in practice is serious. Errors due to imperfections in the uniformity of the paper speed are of secondary importance.

A test with one accelerometer recording a 45° component of the true table motion indicated that accelerographs correctly record the components of an impressed motion according to theoretical expectations, but obviously within the limits of normal instrumental performance.

Numerical integration.—The shaking-table tests prove the validity of the basis on which axis adjustments are made when one is double-integrating an accelerograph record to obtain displacement. All shaking-table motions were computed from the recorded acceleration (or seismograph) records without advance knowledge of the table motion, and no preliminary tests were made to investigate possible sources of error. They demonstrated that even permanent displacements can be detected under favorable conditions; but with most accelerograph records this is problematical.

In the accelerometer tests a systematic error was found to be due to heat distortion of the accelerogram in the lantern enlargement process. After the tests, a specially designed mechanical enlarging apparatus eliminated this and incorporated many other practical advantages.

With respect to the more complex type of shaking-table accelerograph record, it was found that a time increment five times larger than the 1/30 second actually used would have given practically the same result in computation of the shaking-table displacement. This means that the time employed on the summation processes could safely have been reduced to one-fifth that required for the smaller increment. Caution is necessary, however, if the velocity curve is to be used for period investigations or other special purposes, as the increment must be small enough to outline correctly all important waves. Time increments between 0.07 and 0.15 second would appear to serve satisfactorily for active types of accelerograms.

The effect of omitting the first two terms of the fundamental equation of pendulum motion was determined for a complex type of shaking-table motion and was found to be rather insignificant. Current practice assumes that an accelerometer registers true acceleration for very rapid motions as well as for the slower ones, but there are limitations. The effect would be even less if the accelerometer pendulum period should be shortened, a step which would also effect a desirable decrease in sensitivity.

The time required to process accelerograms is not prohibitive. The actual summation processes require less time than enlarging and scaling the acceleration curves and constructing the computed curves, but a considerable amount of additional work is usually involved because of adjustments and recomputations made necessary by accelerometer-pendulum zero shifts.

Displacement with a torsion-pendulum analyzer.—An actual earthquake accelerograph record was used to test the practicability of determining displacement by making an experimental torsion pendulum simulate the response of a long-period seismograph pendulum. A comparison between the pendulum curve and the displacement computed by double-integrating the accelerograph record revealed a difference which was only half the smallest displacement error found in the M.I.T. shaking-table tests. Pendulum results, however, are subject to some uncertainty at the beginning of the motion, because acceleration records lose a certain amount of the initial ground motion in getting started. They “smooth out” rather than correct the effects of unstable accelerometer pendulums. The torsion pendulum, nevertheless, is well suited to play an important part in the practical solution of seismological as well as engineering problems.