Simplification of Hydraulic Line Dynamics by Use of Infinite Products

Abstract

In many hydraulic control and other systems the effect of fluid carrying lines is an important factor in system dynamics. Following electrical transmission line technique a hydraulic line between two cross sections is characterized by a four-terminal network with pressure and flow the interacting variables. Use of this four-terminal network in a variety of system problems leads to transfer functions relating pairs of variables in the system, where these transfer functions are transcendental. These transfer functions cause serious mathematical difficulties when employed for the computation of system transients. The standard mathematical technique of using power series expansions fails in that this yields instability in most applications where this instability does not actually occur. In this paper these difficulties are overcome by writing these functions as quotients of infinite products of linear factors. It is shown that it is necessary to keep only a few of these factors to compute transients accurately. The transfer functions are thus replaced by rational approximations. However, in contrast to the classical lumped constant approach to distributed systems the accuracy of the approximation can be seen from the factors directly, facilitating system analysis and synthesis. The technique applies to electrical transmission lines as well as hydraulic pipes. This method yields a technique for automatically smoothing stepwise transient responses obtained in water hammer studies. Good agreement has been obtained between theory and experiment on the four terminal hydraulic network approach. The paper covers the results of the experiments made in the United States to verify the theory.