Generating spectrum-compatible time histories using eigenfunctions

In this article, the causes of drift in the velocity and the displacement time history are investigated. It is found that, in addition to numerical error, drift is caused by overdeterminacy in the constants of integration. Because there are six independent prescribed at-rest conditions (three initial and three terminal), the eigenfunctions from an eigenvalue problem, which is described by a sixth-order ordinary differential equation satisfying the six initial and terminal at-rest conditions, are chosen as a basis of expansion. The eigenfunctions form a dense and complete set and span a vector space. The eigenfunctions are used to expand an accelerogram. Drift-free consistent velocity and displacement time histories are then obtained, also in terms of the eigenfunctions, without direct integration and baseline correction. A method is also proposed to modify a real recorded accelerogram using the eigenfunctions to generate time histories compatible with the target response spectra without drift.