ABSTRACT
The spring constant Ks is defined by applying a static force f with the terminals short-circuited so that e = 0. Since d2xjdt2 = dxjdt = 0 for a static force, Eq. (5-10) gives Ks =fIx, which can be measured in a lab test by applying dead weights and measuring the deflection. The piezoelectric force constant C1 is found by clamping the crystal rigidly so that x = dxjdt = d2xjdt2 = 0, applying a known voltage e, and measuring the force produced by the crystal pushing against the clamp. Equation (5-10) then gives us C1 =fie. The electric circuit equations are
C de-C dx = i (5-11) dt q dt
conversion: positive-displacement (Fig. 5-8) and centrifugal (Fig. 5-9). While positive-displacement pumps take a variety of forms (piston, vane, gear, etc.), their overall characteristics are basically similar, and a general model adequate for system dynamic analysis can be formulated. Figure 5-8 shows a multiple-piston pump with a rotary mechanical input, which we shall use as a concrete example in developing the general model. As the input shaft is rotated, the individual pistons are sequentially forced in and out of their cylinders, drawing fluid from the input port and expelling it at the output. Valves (not shown) are properly sequenced with rotation so that each cylinder is alternately exposed to the inlet port and then the discharge. The outflows from each cylinder are summed at the discharge port, so that, while each individual cylinder flow rate is pulsating, the total pump flow rate is relatively smooth. Intuitively one would guess that smoothness would increase with the number of cylinders; commercial pumps with seven or nine cylinders are not unusual. While hydraulic motors will be discussed later in this chapter, it might be well to mention now that, just as in the de motor/generator, the hydraulic pump and motor are in essence the same machine. That is, in Fig. 5-8, if we force fluid under pressure through the machine, we will develop a mechanical torque at the shaft, which is the function of a motor.
