Automatic control systems  Examples
Examples
E x a m p l e 1. Control system of a hydraulic drive on the basis of typical linear links of ACS.
On Fig. 2 the hydraulic drive circuit with a control system described by a set of typical linear links of automatic control is presented.
Fig. 2. Control system of a hydraulic drive on the basis of typical linear links of ACS
Object of control – hydraulic system 1 with the closed contour of circulation of a working fluid. The control system 2 consists of the electro hydraulic amplifier (EHA) 3, a computer 4, the block of comparison 5 and the general feedback 6.
The electro hydraulic amplifier 3 is presented as consecutive connection of typical linear dynamic links: a conservative link 7 (the electro mechanical converter), the 2nd order aperiodic link 8 (a spool of the hydraulic amplifier) and an inertial integrating link 9 (the piston of the hydraulic amplifier). In a feedback 6 there is an intensifying link. The block of comparison 5 is the summer. Concrete realization of a computer 4 depends on algorithm of transformations carried out by it. Designations on Fig. 2:– pressures in pipelines of basic contour of the hydraulic system;an angular speed and an angle of turn of the hydraulic motor shaft; a signal on an input of a control system; a control signal on an input of the electro hydraulic amplifier.
E x a m p l e 2. Simulation of a control system by a hydraulic drive.
Let's consider a problem of simulation of dynamics of a hydraulic drive of the testing stand with an electro mechanical control system (Fig. 3).
Fig. 3. The circuit diagram of a hydraulic drive of the testing stand with an electro mechanical control system
The hydraulic drive of the stand consists of the hydraulic cylinder (7–8–14), moving the lever mechanism (1423) with cargo in a mass m, having a rigidity c and damping properties h and has been described by the 2nd order aperiodic link; directional spool valve (4–5–10–11); pipelines (1–2) and (11–12); hoses of high pressure (3–4), 5–6) and (9–10); throttles (2–3), (6–7), (8–9) and (12–13); electro mechanical control system, consisting from 15 linear dynamic links: amplifiers (3115), (2526), (3233), (3218), (2527), (2528), (7824), an inertial differentiating link (22–16), the 1st order aperiodic link (19–21), adders (15–17–16), (20–22–21), (27–19–18), (24–20–28), (293130), (26–32–17); and two nonlinear links: electro mechanical converter z (I) of the following form:
where z is a spool position, I – a current;
and the entrance signal generator as function of movement of the hydraulic cylinder piston relatively the average position described by the following nonlinear function:
where rectangular impulse (see Fig. 4).
Thus, the system consists of 29 elements connected in 36 nodes, and is described by the 40th order system of algebraicdifferential equations.
Basic parameters of hydraulic system:
hydraulic cylinder: the piston diameter 100 mm, the rod diameter 45 mm, the piston stroke 240 mm;
pipelines and hoses of high pressure: diameter 22 mm, lengths: (1–2) – 6 m, (3–4) – 5.34 m, (5–6) –0.69 m, (9–10) – 0.64 m, (11–12) – 11.3 m;
throttles: sectional areas: (2–3) – 3.8 sm², (6–7) – 3.14 sm², (8–9) – 3.14 sm², (12–13) – 1.17 sm², resistance coefficients: (2–3) –5.54, (6–7) –3.28, (8–9) –3.28 и (12–13) –11.38;
directional spool valve: diameter of conditional passage 20 mm, a spool stroke 1.8 mm, flow coefficient of spool channels 0.62; geometric characteristics of spool channels, expressing dependence of sectional areas of channels on spool position z , is presented on Fig. 3c.
working fluid: density 850 kg/m^{3}, kinetic viscosity 2^{.}10^{5} m^{2}/s, the volumetric elasticity module 1765 MPa;
mechanism: mass m = 19600 kg, rigidity c = 6.86^{.}10^{6} N/m, coefficient of viscous friction h = 0.
Force on the rod R_{c} = c (x_{14} – x_{23}). Dependence of the flow Q_{1} in node 1 (Fig. 3b) considers the static characteristic of pressure relief valve not shown on the circuit diagram. Coefficients of equations of dynamic links are specified Fig. 3а.
Transient processes in a hydraulic drive of the testing stand, received as a result of mathematical simulation, are presented on Fig. 4.
Fig. 4. Dynamics of a hydraulic drive of the testing stand with an electro mechanical control system
Similar calculations and researches executed on a design stage, allow to find an optimum variant of a control system for concrete object of control, for example, for a hydraulic drive.
