The use of hydraulic transmission lines in automatic control, liquid-propellant rocket, and other systems requires accurate knowledge of their dynamic response. In this paper the effects of fluid viscosity and compressibility are included to derive transfer functions relating the pressure and velocity variables at the two cross sections of a line. The results of theoretical analysis are compared with experimental data obtained from frequency-response tests. The analysis includes the significant effect on the dynamic response caused by the natural frequency of vibration of the line in the longitudinal direction. It is shown that in small-diameter lines the viscous effects cannot be neglected. The theoretical analysis may be used to improve the performance of systems incorporating hydraulic networks.

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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.