STUDY OF THE DYNAMIC PROPERTIES OF AN ELECTROHYDRAULIC SERVO DRIVE

E. G. FRANK

(Presented at the city scientific conference of young scientists and specialists)

Electrohydraulic servo drives are finding increasingly wide application in automatic-control technology, owing to the well-known advantages of electrical control and hydraulic actuating elements [2]. Therefore, the study of the dynamic properties of electrohydraulic converters and drives as a whole, as well as ways of improving them, is a timely problem.

In the present work, possibilities are considered for improving the dynamic properties of the G68—13 electrohydraulic converter and of a servo drive with a rotary hydraulic motor and position feedback, on the basis of an analysis of a mathematical model of the drive dynamics and experimental investigations.

Figure 1 gives the schematic of a drive with a G68—13 spool valve, the operating principle of which is known [1, 3]. It need only be said that the MN-7 analog computer (AVM) is used here to introduce a number of correcting connections with respect to the system variables and their derivatives.

1. Dynamic properties of the G68—13 converter

The simultaneous solution of the equation of motion of the moving elements of the electromechanical converter (EMC), the equation of motion of the four-edged spool valve, and the continuity equation for the control line, taking into account the masses of the moving elements, dissipative forces, fluid compressibility, and compliance of the pipelines, after linearization by expansion in a Taylor series, gives a fourth-order linear differential equation, which is the mathematical model of the dynamics of the G68—13 converter:

\[ T_4 \frac{d^4 y}{dt^4} + T_3 \frac{d^3 y}{dt^3} + T_2 \frac{d^2 y}{dt^2} + T_1 \frac{dy}{dt} + y = Ik, \tag{1} \]

where \(y\) is the coordinate of the four-edged spool valve;
\(I\) is the current supplied to the EMC.

\[ T_1 = \frac{\rho_{\mathrm{н}} f_{\mathrm{и0}}}{a} + \frac{2\left(c_{\mathrm{и}} + k_F P_{y0}\right) k_{py} W_y f_{\mathrm{и0}} P_{y0}}{Q_y a} + \frac{2\left(c_{\mathrm{и}} + k_F P_{y0}\right) F_3^2 f_{\mathrm{и0}} P_{y0}}{Q_y \left(c_3 + \beta_3\right) a} + \]
\[ + \frac{\left(c_{\mathrm{и}} + k_F P_{y0}\right)\rho_3 f_{\mathrm{и0}}}{\left(c_3 + \beta_3\right)a} + \frac{2 \rho_3 F_{\mathrm{и0}} P_{y0} k_{\mathrm{и}}}{\left(c_3 + \beta_3\right)a}; \tag{2} \]

Fig. 1. Schematic diagram of an electrohydraulic tracking drive with slide valve Г68-13.

\[ \begin{aligned} T_2={}&\frac{m_i f_{i0}}{a} +\frac{2\rho_i k_{py} W_y f_{i0} P_{y0}}{Q_y a} +\frac{2\rho_i F_z^{2} f_{i0} P_{y0}}{Q_y(c_z+\beta_z)a} \\ &+\frac{\rho_i\beta_z f_{i0}}{(c_z+\beta_z)a} +\frac{2(c_i+k_F P_{y0})k_{py} W_y f_{i0}\beta_z P_{y0}}{Q_y(c_z+\beta_z)a}; \end{aligned} \tag{3} \]

\[ \begin{aligned} T_3={}&\frac{2m_i k_{py} W_y f_{i0}P_{y0}}{Q_y a} +\frac{2m_i F_z^{2} f_{i0}P_{y0}}{Q_y(c_z+\beta_z)a} \\ &+\frac{m_i\beta_z f_{i0}}{(c_z+\beta_z)a} +\frac{2\rho_i\beta_z k_{py} W_y f_{i0}P_{y0}}{Q_y(c_z+\beta_z)a}; \end{aligned} \tag{4} \]

\[ T_4=\frac{2m_i\beta_z k_{py} W_y f_{i0}P_{y0}}{Q_y(c_z+\beta_z)a}; \tag{5} \]

\[ k=\frac{k_1 k_i 2P_{y0}F_z}{(c_z+\beta_z)a}. \tag{6} \]

Here

\[ a=(c_i+k_F P_{y0})f_{i0}+2P_{y0}F_{i0}k_i, \]

\(c_i\) is the stiffness of the EMP spring, \(c_i=0.0864\ \mathrm{kg/cm}\);

\(m_i\) is the mass of the moving elements of the EMP, \(m_i=1.53\cdot10^{-5}\ \mathrm{kg\,sec^2/cm}\);

\(k_{py}\) and \(W_y\) are, respectively, the compliance coefficient and the volume of the control line, \(k_{py}=2.66\cdot10^{-1}\ \mathrm{cm^2/kg}\),
\(W_y=150\ \mathrm{cm^3}\);

\(Q_y\) is the flow rate of the control pump, \(Q_y=53.6\ \mathrm{cm^3/sec}\);

\(P_{y0}\) is the pressure in the control line at the neutral position of the spool, \(P_{y0}=7.5\ \mathrm{kg/cm^2}\);

\(\rho_i\) and \(\beta_z\) are the coefficients of dissipative forces in the EMP and in the spool, respectively, \(\rho_i=1.9\cdot10^{-3}\ \mathrm{kg\,sec/cm}\), \(\beta_z=6\cdot10^{-3}\ \mathrm{kg\,sec/cm}\);

\(f_{i0}\) is the area of the working opening of the needle valve in the neutral position, \(f_{i0}=0.0185\ \mathrm{cm^2}\);

\(F_{i0}\) is the cross-sectional area of the EMP needle cone on which the pressure \(P_{y0}\) acts in the neutral position,
\(F_{i0}=0.00885\ \mathrm{cm^2}\);

\(k_i\) is the proportionality coefficient of the increment in the area of the working opening of the needle valve with respect to the needle coordinate,
\(k_i=0.167\ \mathrm{cm}\);

\(k_F\) is the proportionality coefficient of the increment in the cross-sectional area of the needle cone, on which the control pressure acts, with respect to the needle coordinate, \(k_F=0.145\ \mathrm{cm}\);

\(F_z\) is the area of the spool end face, \(F_z=2.07\ \mathrm{cm^2}\);

\(c_z\) is the stiffness of the spool spring, \(c_z=22\ \mathrm{kg/cm}\);

\(\beta_z\) is the coefficient accounting for the hydrodynamic forces on the spool, \(\beta_z=30\ \mathrm{kg/cm}\);

\(k_1\) is the proportionality coefficient of the current \(J\) in the coil to the axial force developed by the EMP, \(k_1=0.57\ \mathrm{kg/a}\).

To investigate the dynamics of the G68-13 transducer by the frequency method, we write the expressions for the logarithmic amplitude \(A\) and phase \(\Theta\) frequency characteristics (LAFCh and FPhCh) for equation (1) [5]:

\[ A=20\lg k-20\lg\sqrt{(1-T_2\omega^2+T_4\omega^4)^2+(T_1\omega-T_3\omega^3)^2}, \tag{7} \]

\[ \Theta=-\operatorname{arc\,tg}\frac{T_1\omega-T_3\omega^3}{1-T_2\omega^2+T_4\omega^4}. \tag{8} \]

Here \(\omega\) is the frequency of the input action, in \(\mathrm{rad/sec}\).

The numerical solution of expressions (7) and (8), with substitution of the parameters of the G68–13 converter into expressions (2–6) for the time constants, gave good agreement with the experimental frequency characteristics over a wide range of variation of a number of parameters

\[ (Q_y;\ k_{py};\ W_y;\ \beta_3;\ c_i\ \text{and others}). \]

Analysis of the frequency characteristics (7) and (8) showed that improvement of the response speed of the G68–13 converter is promoted by:

1. An increase in the numerical value of the parameters \(c_i,\ Q_y,\ c_3\), as well as of the needle cone angle \(\alpha\) (an increase in \(k_i,\ k_F,\) and \(F_{i0}\)).
2. A decrease in \(m_i,\ F_3,\) and \(P_{y0}\).
3. Introduction of negative feedback on the spool position.

The stability of the G68–13 converter was analyzed using the Routh–Hurwitz algebraic stability criterion [5], transformed into a form convenient for graphical interpretation:

\[ \frac{T_1T_2}{T_3} > \frac{1}{1-\dfrac{T_1T_4}{T_2T_3}} . \tag{9} \]

Figure 2 shows the influence of the main parameters of the G68–13 converter on the stability region in the plane

\[ \frac{T_1T_2}{T_3} - \frac{T_1T_4}{T_2T_3}. \]

Fig. 2. Influence of the parameters of the G68 converter on its stability.

(The direction of the arrows corresponds to an increase in the values of the parameters; point 2 corresponds to the parameter values of converter G68-13). The large stability margin of converter G68-13 (point 2 is far from stability boundary 1) makes it possible to change a number of parameters in order to improve the frequency response. Thus, increasing \(c_{\mathrm{и}}, Q_{\mathrm{у}}, c_3, \alpha\) and decreasing \(m_{\mathrm{и}}\), as well as introducing feedback with respect to the spool position in the actual converter, increased the pass band (speed of response) by 30% without a substantial loss of stability.

2. Dynamic properties of the drive as a whole

The mathematical model of the drive, under a number of assumptions [3], can be represented by the linear differential equation [3]

\[ a_4 \frac{d^4 \varphi}{d t^4} + a_3 \frac{d^3 \varphi}{d t^3} + a_2 \frac{d^2 \varphi}{d t^2} + a_1 \frac{d \varphi}{d t} + \varphi = I \cdot k_1 . \tag{10} \]

Here \(\varphi\) is the coordinate of the hydraulic-motor shaft;

\(a_1\)–\(a_4\) are time constants depending on the constant parameters of the drive;

\(k_1\) is the static gain coefficient, depending mainly on the coefficient of the main feedback \(k_{\mathrm{oc}}\).

The stability of the drive is investigated in [3]. We shall consider only the possibility of improving its dynamic properties by introducing additional correcting feedbacks.

Let us introduce feedback with respect to the derivatives of the error (mismatch signal) of regulation

\[ I = \pm k_1 \frac{d\left(I_t - \varphi k_{\mathrm{oc}}\right)}{d t} \pm k_2 \frac{d^2\left(I_t - \varphi k_{\mathrm{oc}}\right)}{d t^2} \pm k_3 \frac{d^3\left(I_t - \varphi k_{\mathrm{oc}}\right)}{d t^3} \pm \ldots \tag{11} \]

Here \(I_t\) is the input action;

\(k_1, k_2\) are the coefficients of the additional feedbacks.

Then equation (10) will take the form

\[ b_4 \frac{d^4 \varphi}{d t^4} + b_3 \frac{d^3 \varphi}{d t^3} + b^2 \frac{d^2 \varphi}{d t^2} + b_1 \frac{d \varphi}{d t} + \varphi = \]

\[ = I k_2 \pm \frac{d I_t}{d t} k_3 \pm \frac{d^2 I_t}{d t^2} k_4 \pm \frac{d^3 I_t}{d t^3} k_5 \pm \frac{d^4 I_t}{d t^4} k_6 . \tag{12} \]

Here \(b_1\)–\(b_4\) are the new time constants.

Thus, the introduction of feedback (11) makes it possible to change the constant coefficients in the left-hand side of equation (10), which determine the stability and the form of the logarithmic amplitude and phase-frequency characteristics, and, in addition, makes it possible to use the known advantages of regulation with respect to derivatives of the control signal [5, 6]. It should be noted that the introduction of feedback (11) is equivalent to the introduction of feedbacks with respect to the four derivatives of the hydraulic-motor shaft coordinate \(\varphi\) and to the introduction of four derivatives of the control signal \(I_t\). Moreover, because of the difficulty of obtaining high orders of derivatives in real systems, in a number of cases it is possible to replace a high order of one system variable by a lower order of another variable. Thus, in hydraulic systems with an inertial load, feedback with respect to the second derivative \(\dfrac{d^2 \varphi}{d t^2}\) of the coordinate of the actuating motor is equivalent to feedback with respect to the pressure drop (load) \(\Delta P\) at the hydraulic engine-

body [2]; feedback with respect to $\dfrac{d\varphi}{dt}$ is equivalent to feedback with respect to the spool coordinate $y$, etc.

To verify the proposed method for correcting the dynamic characteristics of the drive, an experiment was carried out on an actual drive (Fig. 1) together with the MN-7 analog computer, which made it possible to obtain a number of derivatives of the drive variables $y$, $\Delta P$, $\varphi$, and $I$. For this purpose the nonelectrical variables $y$, $\varphi$, and $\Delta P$ were converted into electrical ones by means of inductive, potentiometric, and strain-gauge sensors, respectively. The derivatives on the MN-7 were reproduced by known methods for implicit functions [4].

Figure 3 shows the experimental logarithmic amplitude characteristics and phase-frequency characteristics without additional feedbacks (dashed lines) and with positive

Fig. 3. Experimental logarithmic amplitude characteristics and phase-frequency characteristics of an electrohydraulic servo drive

feedbacks with respect to the first two derivatives of the mismatch signal $I = I_t - \varphi K_{oc}$ and negative feedback with respect to the spool coordinate $y$ (solid lines). It is evident from Fig. 3 that these feedbacks substantially increased the phase margin (by approximately $90^\circ$ over the entire frequency range).

Experimental studies carried out for a large range of feedback magnitudes and combinations have shown that the use of analog-computer elements to correct the dynamic properties of electrohydraulic servo systems is an effective means.

REFERENCES

1. G. I. Kamenetskii. Tracking spools for machine tools with numerical control. Stanki i instrumenty, No. 5, 1961.

2. I. M. Krassov. Hydraulic Elements in Control Systems. “Mashinostroenie,” Moscow, 1967.

3. I. A. Nemirovskii, E. G. Frank. Experimental study of periodic motion with stops of a servo hydraulic drive. Izv. vuzov, “Mashinostroenie,” No. 5, Moscow, 1969.

4. G. Korn, T. Korn. Electronic Analog and Analog-Digital Computing Machines. “Mir” Publishing House, Moscow, 1967.

5. E. P. Popov. Automatic Regulation and Control. “Nauka” Publishing House, Moscow, 1966.

6. P. Hammond. Feedback Theory and Its Applications. Fizmatgiz, Moscow, 1961.