STUDY BY THE METHOD OF MATHEMATICAL MODELING OF THE DYNAMICS OF AN ELECTROHYDRAULIC SERVO DRIVE WITH A G68-1 SPOOL VALVE

E. G. Frank

(Presented by the scientific seminar of the Department of Mining Machines and Mine Transport)

The study of drive dynamics as a linear system is not always acceptable, since it does not make it possible to take into account the substantial nonlinearities inherent in hydraulic drives. In addition, allowance for the main parameters affecting the dynamics of an electrohydraulic drive leads to differential equations of high order, which also complicates analysis by ordinary mathematical methods.

In the present work, differential equations are given for an electrohydraulic servo drive, taking into account the compressibility of the fluid in the main lines, the hydrodynamic force on the spool valve, the nonlinear friction characteristic in the load, and the masses of the moving elements of the system; an electronic model for studying these equations is considered; and modeling results are presented and compared with experiment.

Figure 1 shows the layout of the experimental drive. The G68-13 spool valve, whose operating principle is known [4], controls the motion of the MG-153a hydraulic motor (position 1). On shaft 6, connected to the hydraulic-motor shaft through strain-gauge dynamometer 2, replaceable disks 5 and the disk of friction brake 3 are mounted. Load pump 7 is provided to create a constant load torque.

The power pumping station included two NP-26 pumps, and an NS-3 pumping station was used in the control system.

Electrical control signals to the input of the G68-13 spool valve were supplied from an NGPK-3M low-frequency generator.

Potentiometer 4 was used as a feedback sensor for the angular coordinate of the hydraulic-motor shaft.

The summation of the control signal and the feedback signal was performed by a UPT direct-current amplifier.

In the experimental studies, the following quantities were recorded: the control signal \(I_t\), the feedback signal \(\varphi K_0\), proportional to the angular coordinate \(\varphi\) of shaft 6, the control pressure \(P_y\), the coordinate \(y\) of the spool valve, the inlet pressure \(P_0\), the pressures \(P_1\) and \(P_2\) supplied to the hydraulic motor, and the driving torque \(M_d\) developed by the hydraulic motor.

Assuming the hydraulic losses in the lines and local resistances to be insignificant, the flow coefficients to be constant, and also neglecting wave processes owing to the small length of the lines,

Fig. 1. Schematic diagram of the experimental electrohydraulic drive with a G68-1 spool valve.

... pipelines, the dynamic processes in the drive considered here can be described by the following system of equations:

\[ m\frac{d^{2}x}{dt^{2}}+\rho_{\mathrm{u}}\frac{dx}{dt}+(C+\beta)x+\Delta P_{y}F_{\mathrm{u}}^{0}=Ik; \tag{1} \]

\[ \mu_{\mathrm{u}}(f_{\mathrm{u}}^{0}-\Delta f_{\mathrm{u}}) \sqrt{\frac{2}{\rho}}\sqrt{P_{y}^{0}+\Delta P_{y}} +F_{3}\frac{dy}{dt} +k_{p}W\frac{dP_{y}}{dt}=Q_{y}; \tag{2} \]

\[ \rho_{3}\frac{dy}{dt}+c_{3}y+\beta_{3}\frac{d\varphi}{dt} =\Delta P_{y}F_{3}; \tag{3} \]

\[ (y^{0}+y)\mu b\sqrt{\frac{2}{\rho}}\sqrt{P_{0}-P_{1}} -\sigma_{1}P_{1} -\sigma_{1-2}(P_{1}-P_{2}) - \]

\[ -k_{p_{1}}W_{1}\frac{dP_{1}}{dt} =q_{0}\frac{d\varphi}{dt}; \tag{4} \]

\[ (y^{0}+y)\mu b\sqrt{\frac{2}{\rho}}\sqrt{P_{2}-P_{\mathrm{T}}} +\sigma_{2}P_{2} -\sigma_{1-2}(P_{1}-P_{2}) - \]

\[ -k_{p_{2}}W_{2}\frac{dP_{2}}{dt} =q_{0}\frac{d\varphi}{dt}; \tag{5} \]

\[ I_{M}\frac{d^{2}\varphi}{dt^{2}}+M_{\mathrm{n}}+F\left(\frac{d\varphi}{dt}\right) =\psi(P_{1}-P_{2}); \tag{6} \]

\[ I_{t}-\varphi k_{0}=I, \tag{7} \]

where

\(x\) — needle coordinate;
\(m\) — mass of the needle;
\(\rho_{\mathrm{u}}\) — coefficient of viscous friction of the needle;
\(C\) — stiffness of the needle spring;
\(\beta\) — coefficient accounting for the increment of the axial force on the needle, equal to \(P_{y}^{0}\Delta F_{\mathrm{u}}\);
\(\Delta F_{\mathrm{u}}\) — increment of the cross-sectional area of the conical portion of the needle;
\(P_{y}^{0}\) — pressure in the control line with the coil de-energized;
\(\Delta P_{y}\) — pressure increment in the control line;
\(F_{\mathrm{u}}^{0}\) — cross-sectional area of the conical portion of the needle on which the pressure \(P_{y}^{0}\) acts in the mean position of the spool;
\(I\) — current in the needle coil;
\(k\) — coefficient of proportionality between the current \(I\) and the axial force developed by the coil;
\(\mu_{\mathrm{u}}\) — flow coefficient through the working port of the control line;
\(f_{\mathrm{u}}^{0}\) — area of the working port of the control line with the coil de-energized;
\(\Delta f_{\mathrm{u}}\) — increment of the area of the working port of the control line;

\[ \Delta f_{\mathrm{u}}=k_{\mathrm{u}}x;\qquad k_{\mathrm{u}}= \frac{2\sqrt{\pi F_{\mathrm{u}}^{0}}} {\sin \frac{\alpha}{2}\left(1+\operatorname{ctg}^{2}\frac{\alpha}{2}\right)}; \]

\(\alpha\) — needle-cone angle;
\(\rho\) — density of the working fluid;
\(Q_y\) — flow rate of the control pump;
\(F_3\) — area of the spool end face;
\(k_p\) — compliance coefficient of the control line;
\(W\) — volume of the control line;
\(\rho_3\) — coefficient of viscous friction of the spool;
\(y\) — spool coordinate;
\(c_3\) — stiffness of the spool spring;
\(\beta_3\) — coefficient of the axial hydrodynamic force acting on the spool;
\(\varphi\) — angular coordinate of the hydraulic-motor shaft;
\(q_0\) — displacement of the hydraulic motor;
\(y^0\) — axial opening of the spool in the neutral position;
\(\mu\) — coefficient of flow through the spool ports;
\(b\) — length of the edge of the spool working ports;
\(P_1\) and \(P_2\) — pressures supplied to the hydraulic motor;
\(\sigma_1\) and \(\sigma_2\) — leakage coefficients from the corresponding cavities of the hydraulic motor;
\(\sigma_{1-2}\) — leakage coefficient in the distribution mechanism of the hydraulic motor;
\(k_{p1}\) and \(k_{p2}\) — compliance coefficients of the corresponding cavities of the hydraulic motor;
\(W_1\) and \(W_2\) — volumes of the corresponding cavities of the hydraulic motor;
\(I_M\) — moment of inertia of the rotating parts, reduced to the hydraulic-motor shaft;
\(M_n\) — constant load torque;
\(F\!\left(\dfrac{d\varphi}{dt}\right)\) — relay friction function;
\(\psi\) — coefficient of proportionality between the pressure drop across the hydraulic motor and the torque developed;
\(I_t\) — control signal;
\(k_0\) — feedback gain.

After simple transformations and the generally accepted linearization of the radical expressions, we write equations (1)—(7) in a form convenient for modeling:

\[ \frac{d^2 y}{dt^2} = a_1 x - a_2 \frac{dy}{dt} - a_3 y - a_4 \frac{d^2 \varphi}{dt^2} - a_5 \frac{d\varphi}{dt}; \tag{8} \]

\[ \frac{d^2 x}{dt^2} = b_1 I_t - b_2 \frac{dx}{dt} - b_3 x - b_4 \frac{dy}{dt} - b_5 y - b_6 \frac{d\varphi}{dt} - b_7 \varphi; \tag{9} \]

\[ \frac{dU_1}{dt} = c_1 - c_2 \frac{d\varphi}{dt} + c_3 y - c_4 U_1 - c_5 yU_1; \tag{10} \]

\[ \frac{dU_2}{dt} = d_1 - d_2 \frac{d\varphi}{dt} + d_3 y - d_4 U_2 - d_5 yU_2; \tag{11} \]

\[ \frac{d^2 \varphi}{dt^2} = \frac{\psi}{I_M} U_1 + \frac{\psi}{I_M} U_2 - \frac{M_n}{I_M} - \frac{F\!\left(\dfrac{d\varphi}{dt}\right)}{I_M}. \tag{12} \]

Here

\[ a_1 = \frac{k_i F_3 Q_y}{f_n^0 \rho_3 k_p W}; \qquad a_2 = \frac{c_3}{\rho_3} + \frac{Q_y}{2P_y^0 k_p W} + \frac{F_3^2}{\rho_3 k_p W}; \]

\[ a_3 = \frac{\varepsilon_3 Q_y}{2P_y^0 \rho_3 k_p W}; \qquad a_4 = \frac{\beta_3}{\rho_3}; \qquad a_5 = \frac{\beta_3 Q_y}{2P_y^0 \rho_3 k_p W}; \]

\[ b_1 = \frac{k}{m}; \qquad b_2 = \frac{\beta_{\mathrm{и}}}{m}; \qquad b_3 = \frac{c+\beta}{m}; \qquad b_4 = \frac{\beta_3 F_{\mathrm{и}}^0}{mF_3}; \]

\[ b_5 = \frac{c_3 F_{\mathrm{и}}^0}{mF_3}; \qquad b_6 = \frac{\beta_3 F_{\mathrm{и}}^0}{mF_3}; \qquad b_7 = \frac{k_0 k}{m}; \]

\[ c_1 = \frac{Q_1 - \sigma_1 P_1^0}{k_{P1} W_1}; \qquad c_2 = \frac{q_0}{k_{P1} W_1}; \qquad c_3 = \frac{Q_1}{y^0 k_{P1} W_1}; \]

\[ c_4 = \frac{Q_1}{2(P_0^0 - P_1^0) k_{P1} W_1} + \frac{\sigma_1 + \sigma_{1-2}}{k_{P1} W_1}; \qquad c_5 = \frac{Q_1}{2y^0(P_0^0 - P_1^0) k_{P1} W_1}; \]

\[ d_1 = \frac{Q_4 + \sigma_2 P_0^2}{k_{P2} W_2}; \qquad d_2 = \frac{q_0}{k_{P2} W_2}; \qquad d_3 = \frac{Q_4}{y^0 k_{P2} W_2}; \]

\[ d_4 = \frac{Q_4}{2(P_2^0 - P_T) k_{P2} W_2} + \frac{\sigma_1 + \sigma_{1-2}}{k_{P2} W_2}; \]

\[ d_5 = \frac{Q_4}{2y^0(P_2^0 - P_T) k_{P2} W_2}. \]

\(P_T\) is the drain pressure before the spool land;

\(U_1\) and \(U_2\) are pressure increments in the corresponding cavities of the hydraulic motor;

\(P_1^0\) and \(P_2^0\) are the pressures in the cavities of the hydraulic motor at the neutral position of the spool;

\(Q_1\) and \(Q_4\) are the fluid flow rates through the corresponding working ports at the neutral position of the spool.

Figure 2 presents a block diagram for modeling the system of equations (8)—(12). In setting up this diagram on the EMU-10 electronic model, multiplier blocks and a nonlinearity block were used. The obtained modeling diagram makes it possible to trace the interrelations among the drive elements and to investigate qualitatively and quantitatively the influence of their parameters and the nature of the loads on the dynamic characteristics of the drive.

On the basis of the block diagram considered, a study was carried out of the system of differential equations with substitution of the parameters and coefficients of the actual drive.

Figure 3 presents the characteristics of the transient process for one combination of drive parameters, obtained experimentally (solid lines) and on the model (dashed lines). The good agreement of the transient process on the experimental and mathematical models shows that, in the mathematical description, the principal processes occurring during operation of the system were correctly taken into account, and it permits the results of investigating the mathematical model to be used for improving the actual drive. In the modeling, oscillograms were obtained characterizing the influence of the feedback coefficient \(k_0\), the constant load torque \(M_{\mathrm{н}}\), the friction torque in the load

\[ F\left(\frac{d\varphi}{dt}\right), \]

the moment of inertia of the masses \(I_M\), the stiffness of the needle spring \(c\), the coefficient of hydrodynamic forces on the spool \(\beta_3\), the compressibility of the fluid in the control plane \(k_P\), and also in the cavities of the hydraulic motor \(k_{P1}\)

and \(k_{p2}\) and the mass of the needle \(m\) on the character of the transient process. The possibility was investigated of improving the dynamic characteristics by introducing additional feedbacks with respect to the pressure increment \(U_1\) in one of the cavities of the hydraulic motor, with respect to the pressure drop \((U_1+U_2)\) across the hydraulic motor, with respect to the angular velocity \(\dfrac{d\varphi}{dt}\) of the hydraulic-motor shaft, and with respect to the coordinate \(y\) of the spool.

Fig. 2. Block diagram for modeling an electrohydraulic servo drive

Figure 4 gives examples of oscillograms from which one can see the influence of the friction torque in the load \(F\left(\dfrac{d\varphi}{dt}\right)\), of the moment of inertia \(I_m\), of the compressibility of the fluid \(k_{p1}\) and \(k_{p2}\), and of the stiffness of the needle spring \(c\) on the character of the transient process of variation of the angle of the hydraulic-motor shaft.

The influence of the functional dependence of the friction torque on the rotational speed of the hydraulic-motor shaft on the stability of the drive was shown. The stability region of a drive with a friction torque independent of the rotational speed of the hydraulic-motor shaft ...

Fig. 3. Characteristics of the transient process of the drive: 1 — angular coordinate of the hydraulic-motor shaft \(\varphi\); 2 — spool coordinate \(y\); 3 — pressure in one of the hydraulic-motor chambers \(p_2\).

Fig. 4. Influence of certain drive parameters on the characteristics of the transient process:
a — friction torque \(F\left(\dfrac{d\varphi}{dt}\right)\); b — needle-spring stiffness \(c\); c — mass moment of inertia \(I_M\); d — compressibility of the fluid in the hydraulic-motor chambers \(K_{p1}; K_{p2}\).

of the hydraulic motor (relay function), is considerably greater than with a nonlinear descending segment of the friction-versus-velocity function modeled by means of a piecewise-linear approximation.

This indicates the need to take into account the real dependence of the friction forces on the velocity of the actuating elements in the study and design of such systems.

A study of the influence of additional feedbacks showed that the greatest improvement in the nature of the transient process is produced by negative feedback with respect to the pressure drop \((U_1 - U_2)\) at the hydraulic motor and positive feedback with respect to the spool coordinate \(y\).

REFERENCES

1. L. V. Balkind. Dynamic properties of an electrohydraulic drive with feedback on the velocity of the actuating mechanism. Collection of articles Pneumo- and Hydroautomatics, ed. by Prof. M. A. Aizerman, Nauka, 1964.

2. A. M. Banshchik, L. I. Radovskii, B. G. Turbin. Derivation of differential equations and study, by the method of mathematical modeling, of the dynamic characteristics of an electrohydraulic follow-up drive. Collection of articles Pneumo- and Hydroautomatics, ed. by Prof. M. A. Aizerman, Nauka, 1964.

3. P. K. Zauervald, E. M. Khaimovich. Analysis of the results of studying a high-speed follow-up system. Collection of articles Hydraulic Drive and Hydropneumoautomatics, No. 3, Tekhnika, Kiev, 1968.

4. G. I. Kamenetskii. Follow-up spool valves for machine tools with program control. Machines and Tools, No. 5, 1961.

5. B. Ya. Kogan. Electronic modeling devices and their application to the study of automatic control systems. Fizmatgiz, Moscow, 1963.