CALCULATION OF PRESSURE REGULATORS OF MACHINE-TOOL HYDRAULIC SYSTEMS FOR SENSITIVITY AND RESPONSE SPEED

Yu. Ya. Komisarenko

(Presented by the scientific seminar of the Department of Mechanical-Engineering Technology)

The tremendous tasks set by the XXII Congress of the CPSU for creating the material and technical base of communism require the introduction into the national economy of new machines of high productivity and the maximum automation and mechanization of production processes.

The use of hydraulically powered machines considerably simplifies the automation of production processes, improves the quality of machines, and also makes it possible to reduce substantially their weight and overall dimensions.

As indicated in the Proceedings of the Third Conference on the Fundamental Problems of the Theory of Machines and Mechanisms, for the successful application of hydraulics in industry it is necessary to develop more broadly both the study of hydraulic systems and the creation of methods for their calculation.

The relevance of research in the field of industrial hydraulic drives has been substantiated in the works of both Soviet and foreign scientists.

In a typical hydraulic-drive system, as a rule, safety valves are used, protecting the system from pressure overloads, or else maintaining the pressure in the system at a specified level.

The basic requirement imposed on the quality of operation of a safety valve in a hydraulic system is the stability of the maintained pressure, minimal pressure fluctuations both under slow and sufficiently rapid changes in the speeds of motion of the working member, and the absence of significant pressure peaks when operating in the safety-valve mode.

The studies of I. Z. Zaichenko [6] and G. T. Grosschmidt [7; 8] showed that the parameters of pressure spool valves affect the operability of hydraulic systems as a whole.

At ENIKMASH, under the direction of I. B. Matveev, an analysis was carried out of the operation of a safety valve with an overflow spool. In these works, when deriving analytical dependences, a number of assumptions were made: the character of the fluid outflow through the working slots was taken to be linear; the true character of the friction forces was not taken into account; the assumption was made that the pressure at the inlet to the hydraulic system was constant; the actual characteristics of pumping stations were not taken into account, etc.

Studies carried out in the Hydroautomatics Laboratory of the Tomsk Polytechnic Institute showed a discrepancy between experimental data and the data obtained using the formulas proposed in the above-mentioned works.

The method of graphical-analytical calculation being developed at TPI makes it possible to take into account all essential nonlinearities quite simply and to obtain accurate results [3].

The present paper sets forth a method for graph-analytic calculation of the transient process arising in a relief valve when the operating mode of the system changes, and also presents a method for selecting the design parameters of the valve according to prescribed dynamic characteristics of the transient process in a hydraulic system.

Let us consider a relief valve with an overflow spool, Fig. 1, of type G52-13. The overflow spool has mass \(m = 0.123 \times 10^{-3}\ \mathrm{kg\,sec^2\,cm^{-1}}\), mushroom diameter \(D = 3.5\ \mathrm{cm}\), diameter of the working part of the spool \(d_3 = 1.6\ \mathrm{cm}\), and mushroom area \(F = 9.7\ \mathrm{cm^2}\). The spool is pressed downward by a spring having an initial compression \(l_3 = 0.1\ \mathrm{cm}\)

Fig. 1. Calculation diagram of a relief valve with an overflow spool

and stiffness \(c_3 = 18.4\ \mathrm{kg/cm}\). The cavity above the spool mushroom is connected to the pressure line through a throttle with flow cross-sectional area \(f_{\mathrm{d}} = 0.018\ \mathrm{cm^2}\).

At the extreme lower position of the spool, the working slot of the valve has an overlap \(l = 0.2\ \mathrm{cm}\).

The auxiliary valve, which shuts off the discharge of oil to drain from the upper cavity of the spool, is a ball of diameter \(d_1 = 1.2\ \mathrm{cm}\), pressed by a spring of stiffness \(c_{\mathrm{sh}} = 158\ \mathrm{kg/cm}\) and initial compression \(\lambda\) against a seat having diameter \(d = 0.8\ \mathrm{cm}\).

In the experimental setup, the valve protected a system of volume \(W = 10^3\ \mathrm{cm^3}\), to which a pump with delivery was connected

\(Q_{\mathrm{n}}=166\ \mathrm{cm^3/sec}\) at pressure \(p_{\mathrm{n}}=20\ \mathrm{kg/cm^2}\); the flow rate passing through throttle \(f_{\mathrm{d}}\) in this case was \(Q_{\mathrm{d}}=16\ \mathrm{cm^3/sec}\).

If, for any reason, the pressure in the system rises above the pressure for which the spring of the auxiliary valve is set, then the latter will rise by an amount \(z\); oil from the upper cavity of the spool will flow under pressure \(p_1\) through the gap \(h_{\mathrm{sh}}\) between the ball valve and the seat; the spool will rise upward until a gap opens through which part of the pump delivery will drain into the tank. If the pressure falls below the setting pressure, then the ball valve will close, oil through the damper \(f_{\mathrm{d}}\) will fill the upper cavity of the spool, and the spool, under the action of spring \(c_3\), will descend, closing the working gap.

The operation of the valve can be analyzed on the basis of the following equations: continuity, outflow of the working fluid through the throttle orifice, the auxiliary valve and the overflow spool, and the equilibrium equations of the spool and the auxiliary valve

\[ Q_{\mathrm{n}}=Q_w+Q_{\mathrm{d}}+Q_h+Q_3, \tag{1} \]

where

\[ Q_w=\frac{dp}{dt}\cdot\frac{W}{E_{\mathrm{m}}}; \]

\[ Q_{\mathrm{d}}=\mu f_{\mathrm{d}}\sqrt{\frac{2g}{\gamma}(p-p_1)}; \tag{2} \]

\[ Q_h=\frac{dh}{dt}\cdot F;\qquad Q_{\mathrm{sh}}=\mu \pi d_{\mathrm{sh}}\sqrt{\frac{2g}{\gamma}(p_1-p_{\mathrm{sl}})}; \tag{3} \]

\[ Q_3=\mu \pi d_3(h-l)\sqrt{\frac{2g}{\gamma}(p-p_{\mathrm{sl}})}; \tag{4} \]

\[ p-p_1=\frac{1}{F}\left(m\frac{d^2h}{dt^2}+R_1+mg+c_3(l+h)\right); \tag{5} \]

\[ (\lambda+z)c_{\mathrm{sh}}=(p_1-p_{\mathrm{sl}})\frac{\pi d^2}{4}, \tag{6} \]

where \(g\) is the acceleration of gravity, \(g=981\ \mathrm{cm/sec^2}\),
\(\gamma\) is the specific weight of the oil, \(\gamma=0.9\cdot10^{-3}\ \mathrm{kg/cm^3}\),
\(E_{\mathrm{m}}\) is the bulk modulus of elasticity of the oil, \(E_{\mathrm{m}}=1.3\cdot10^4\ \mathrm{kg/cm^2}\),
\(p_{\mathrm{sl}}\) is the drain pressure, \(\mathrm{kg/cm^2}\),
\(\mu\) is the discharge coefficient.

Let us consider the first phase of the transient process in the valve (section \(a\)—\(a\) in Fig. 4). In this section, as a result of the increase in oil pressure above the spool setting pressure, the spool begins to rise upward. Obviously, the process of pressure rise will end when the valve rises by the amount of overlap \(l\) plus \(h_0\).

\[ h_0=\frac{Q_{\mathrm{n}}-Q_{\mathrm{d}}} {\mu\pi d_3\sqrt{\dfrac{2g}{\gamma}p_{\mathrm{n}}}}. \]

Thus, in order to determine the maximum increase of pressure in the system, it is necessary to analyze the process of valve operation during the period in which the spool rises by the amount \(l+h_0\). Since the magni-

for \(h_0\) amounts to no more than \(4\div 5\%\) of \(l\), it may be assumed that during this phase of the transient process the working slot of the overflow spool remains closed, i.e. \(Q_3=0\).

We shall also neglect the mass of the moving elements of the valve, on the basis of the recommendations of [6], and also on the basis of experiments carried out in the Laboratory of Hydraulic Automation.

We shall regard the forces resisting the motion of the spool as constant; the drain pressure \(p_{\text{dr}}\) shall be taken equal to 0.

Carrying out the necessary transformations and linearization of the system of equations (1; 2; 3; 4; 5; 6) and taking into account that \(Q_3=0\), we obtain

\[ \tau-\frac{h_t\cdot F}{TQ}=1-l^{-\tau}, \tag{7} \]

\[ \frac{\Delta p\cdot k_{\text{sh}}}{Q}=1-l^{-\tau}, \tag{7a} \]

where \(h_t\) is the current coordinate of the spool, cm.

\[ T=\frac{W}{E_m\cdot k_{\text{sh}}},\quad \text{sec};\qquad \tau=\frac{t}{T}; \]

\[ Q=Q_{\text{n}}-Q_{\text{dr}},\quad \text{cm}^3/\text{sec}; \]

\[ \Delta p=p-p_{\text{n}},\quad \text{kg}/\text{cm}^2; \]

\[ k_{\text{sh}}=\mu\frac{\pi^2 d^3}{4c_{\text{sh}}}\cos\alpha\sqrt{\frac{2g}{\gamma}\cdot p_{\text{n}}},\quad \frac{\text{cm}^5}{\text{kg}\cdot\text{sec}}. \]

If in equation (7a) we substitute \(h_t=l+h_0\), then we shall find the maximum pressure rise in the system.

In order to determine the value of \(k_{\text{sh}}\) from the permissible pressure rise \(\Delta p_{\text{perm}}\) in the system, let us turn to graphical constructions.

In Fig. 2, in the coordinates \(\tau\) and \(\dfrac{\Delta p\cdot k_{\text{sh}}}{Q}\), the exponential is shown; to the left along the abscissa axis a scale \(\Delta p\) is plotted.

We set, approximately, a value of \(k_{\text{sh}}\); calculate the quantity

\[ \frac{hFk_{\text{sh}}E_m}{W} \]

and lay off to the right along the abscissa axis the segment

\[ oa=\frac{hFk_{\text{sh}}E_m m_x}{W}; \]

from point \(a\) we draw the ray \(a-a\) at the angle

\[ \beta=\arc\tg\frac{m_y}{m_x} \]

to the abscissa axis until it intersects the exponential \(l\).

From point \(b\) we draw a straight line parallel to the abscissa axis until it intersects the ray \(OS\), passing at the angle

\[ \alpha=\arc\tg\frac{Qm_p}{k_{\text{sh}}\cdot m_y} \]

to the ordinate axis (point \(d\)).

The perpendicular dropped from point \(d\) to the abscissa axis cuts off the segment \(ol\), which, on the scale \(m_p\), will express the maximum pressure rise in the system, i.e.

\[ \Delta p=\frac{ol}{m_p}. \]

In fact, the right-hand side of equation (7) is an exponential, while the left-hand side is a straight line cutting off on the abscissa axis a segment \(\dfrac{Fh}{TQ}\) with slope \(k=1\).

Obviously, the point of intersection \(b\) (Fig. 2) of these two lines determines the value \(\tau\) satisfying equation (7). At the same time, the ordinate of point \(b\) is

\[ y_b=(1-l^{-\tau})m_y, \]

and the segment

\[ cd=y_b\cdot \operatorname{tg}\alpha=(1-l^{-\tau})m_y\cdot \operatorname{tg}\alpha . \tag{8} \]

Fig. 2. Nomogram for determining the pressure rise in the system.

Taking into account that, by the construction condition,

\[ \operatorname{tg}\alpha=\frac{Qm_p}{k_{\mathrm{sh}}\cdot m_y} \tag{9} \]

and finding from equation (7a) \((1-e^{-\tau})\)

\[ (1-l^{-\tau})=\frac{\Delta p\cdot k_{\mathrm{sh}}}{Q}, \tag{10} \]

we substitute (10) and (9) into (8):

\[ cd=\frac{\Delta p\cdot k_{\mathrm{sh}}}{Q}\,m_y\, \frac{Qm_p}{k_{\mathrm{sh}}\cdot m_y} =\Delta p\cdot m_p, \]

which was required to be proved.

Thus, by the method of successive approximations one can find the coefficient \(k_{\mathrm{sh}}\) at which the pressure rise in the system caused by a sharp increase in the flow rate through the valve will not exceed the allowable value.

From the equation

\[ k_{\mathrm{sh}}=\mu\frac{\pi^2 d^3}{4c_{\mathrm{sh}}}\cos\alpha \sqrt{\frac{2g}{\gamma}\cdot p_{\mathrm{n}}}. \]

Determining the values \(d\), \(c_{\mathrm{sh}}\), and \(\cos\alpha\) from the found \(k_{\mathrm{sh}}\) presents no difficulty:

\[ d = (0.5 \div 0.8)\, d_{\mathrm{э}}, \quad \alpha = 42 \div 47^\circ \]
(from an analysis of existing types of standard equipment)
\[ d_1 = \frac{d}{\cos \alpha}, \]

\[ c_{\mathrm{ш}} = \frac{\pi^2 d^3}{4 k_{\mathrm{ш}}}\cdot \cos \alpha \cdot \sqrt{\frac{2q}{\gamma}}\, p_{\mathrm{н}} . \]

Figure 2 shows an example of determining the quantity \(\Delta p\) for the standard design of a valve of type G 52-13.

Experimental verification of the results obtained was carried out on a special test stand, the schematic of which is given in Fig. 3 [4].

Fig. 3. Schematic of the experimental installation.

1 — oil tank; 2 — vane pump LF-1,
\(Q = 11.5\ \text{l/min}\); 3 — plate filter; 4 — safety valve G52-13; 5 — impact mechanism; 6 — flat spool; 7 — throttle for setting the initial pressure; 8 — strain-gauge pressure transducer with natural-oscillation frequency \(f = 10\ \text{kHz}\).

Fig. 4. Oscillogram of the transient process in the system whose schematic is given in Fig. 3.

1 — time marker; 2 — indicator of actuation of the flat spool; 3 — zero line; 4 — curve of pressure variation in the system; 5 — maximum pressure in the system, determined from the graph (Fig. 2); 6 — maximum pressure in the system, calculated by the formula given by ENIKMASH [5].

Instantaneous shutoff of the pipeline was carried out by a special impact mechanism with a flat spool. The beginning and end of the motion of the flat spool are marked on the oscillogram (line 2).

The original system of equations (1; 2; 3; 4; 5; 6) can be used to construct the entire curve of the transient process, which makes it possible to judge the duration and oscillatory nature of the transient process.

For this purpose, a calculation method [3] was developed using constructions proposed by A. V. Basharin [11] for automatic control systems.

The advantages of this method consist in the simplicity of the construction, the accuracy of the solution, and the possibility of taking into account such nonlinearities as cannot be allowed for in analytical methods (friction forces, the reaction of the fluid jet on the spool valve, etc.).

Thus, the proposed calculation method makes it possible:

a) to determine the increase in pressure in the system that arises when the flow rate through the valve changes;

b) to calculate the valve parameters that ensure the required transient process;

c) to calculate the curve of the transient process, on the basis of which the quality of regulation can be judged.

The accuracy and simplicity of the method make it possible to recommend it for practical engineering calculations and research work.

LITERATURE

1. Theory of automatic machines and hydropneumatic drive. Proceedings of the Third Conference on Fundamental Problems in the Theory of Machines and Mechanisms. Mashgiz, 1963.

2. Hydraulic and pneumatic power control systems. Edited by J. Blackburn et al. Foreign Literature Publishing House, Moscow, 1962.

3. I. A. Nemirovskii. Study of the technological possibilities of hydroautomatic systems of metal-cutting machine tools. Abstracts of reports of the First Scientific Session of Higher Educational Institutions of Western Siberia. Tomsk, 1963.

4. I. A. Nemirovskii, Yu. Ya. Komisarenko. Dynamics of pump stations of hydroautomatic systems of machine tools. Proceedings of TPI, 1964.

5. ENIKMASH. Report on topic K 58-8. Voronezh, 1962.

6. I. Z. Zaichenko. Self-oscillations in hydraulic transmissions of metal-cutting machine tools. Mashgiz, 1962.

7. G. T. Grossshmidt. Investigation of the operation of certain overflow valves of machine-tool hydraulic drives under steady-state conditions. Tallinn, 1959.

8. G. T. Grossshmidt. Calculation of the dynamic characteristics of a typical machine-tool hydraulic drive with throttling speed control. Tallinn, 1958.

9. E. M. Khaimovich. Hydraulic drives and hydroautomatics of machine tools. Mashgiz, 1959.

10. V. V. Ermakov. Fundamentals of hydraulic-drive calculation. Mashgiz, 1951.

11. A. V. Basharin. Calculation of dynamics and synthesis of nonlinear control systems. Gosenergoizdat, 1960.