APPLICATION OF THE BILINEAR Z-TRANSFORM METHOD TO GROUND MOTION STUDIES By J. RAY STAGNER AND GARY

A recursive relationship in the time domain is derived for the response of a single degree of freedom oscillator to an arbitrary digitized ground motion. In developing the relationship the oscillator's continuous frequency response characteristics are expressed in terms of an equivalent sampled function Z-plane representation using the bilinear Z-transform method. Oscillator response time histories and response spectra for prescribed ground motions are compared with exact solutions. Application of the presented method to instrument correction studies is discussed. INTRODUCTION The calculation of the response time history for a linear single degree of freedom oscillator to a prescribed ground motion time history has been discussed by other researchers (Benioff, 1934; Housner, 1941). Prior to the impact of the digital computer, emphasis was placed on methods of calculating which considered the ground motion to be a continuous function of time. However, now with the increased availability and speed of digital computers the response of a single degree of freedom oscillator to an arbitrary ground motion is usually calculated using digital techniques. Different authors have proposed methods for calculating digitized response time histories (Nigam and Jennings, 1969; Brady, 1966). In general, these methods represent both the forcing function and oscillator response as a digitized stepwise linear function in the time domain. The relative merit of each method is described in the original papers. In this technical paper we shall explicitly represent the oscillator's response characteristics in the frequency domain using a bilinear Z-transform. This frequency representation enables us to develop a time domain recursive relation for calculating structural response. Three basic steps are necessary to develop this relation. First, the standard continuous Laplace transform s-plane representation of the ground motion-oscillator displacement response relationship is derived. Second, this standard continouous Laplace s-plane representation is expressed in terms of an equivalent sampled function Z-plane frequency representation. Third, the parallelism between the Z-plane bilinear transform and finite difference operators is utilized in order to develop a recursive relation in the time domain for calculating oscillator response. The recursive relationship we derive, using this digitized frequency approach, expresses the oscillator's relative displacement in terms of the ground acceleration at the time of interest and the oscillator displacement and ground acceleration at the two previous instants in time. In addition to our method providing an alternate way of calculating oscillator response it also results in approximately a threefold savings in computer computation time when compared to Simpson's rule integration of Duhamel's integral. Application of the presented approach to instrument correction studies follows directly. We discuss briefly this problem and formulate the necessary reeursive equation for pre-processing the instrument record prior to response analysis. 8O9 ~10 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA THEORETICAL RESPONSE DEVELOPMENT The theoretical steps that are necessary in order to develop a recursive equation for calculating the response of a single degree of freedom oscillator to a prescribed ground motion time history are presented in this section. These steps are based in part on the bilinear Z-transform method (Golden and Kaiser, 1964). The differential equation of motion for a single degree of freedom oscillator subjeered to base motion is 2(t ) -+2BO~n2(t) + ~n2z(t) = -a( t ) (1) where 13 is the critical damping ratio, COn is the oscillator's undamped natural frequency of vibration and a(t ) is the ground acceleration. Taking the Laplace transform of equation (1) for 'the special case of zero initial conditions yields 2 [s ~ + 2~co,~s + ~on]X(S) ----A(S) (2) where s is the Laplace variable and A ( s ) and X ( s ) are the Laplace transforms of ground acceleration and relative displacement, respectively. Using equation (2) it follows that the s-plane transfer function relating the ground acceleration and relative displacement is _ x ( s ) _ 1/[s 2 -t2/~cons -t~on2]. (3) H ( s ) A(s) Equation (3) is a standard Laplace force-response relationship. While this equation is useful for some mathematical ground motions, in practice we usually represent the ground motion and corresponding oscillator response as sampled data points digitized at uniform time increments (data digitized at non-uniform time intervals can always be transformed into ~ uniformly sampled function by an interpolation process). In order to express equation (3) in a digitized form it is advantageous to introduce the concept of the Z-transform. However, while direct application of this transform me~hod preserves the amplitude characteristics of the s-plane function given by equation (3), it warps the frequency scale. In order to compensate for this effect, it is necessary to pre-warp the frequency characteristics of equation (3). With this in mind we define the new variable