FOLLOW-UP HYDROMECHANISM WITH TWO PUMPS AND FLOW FEEDBACK FOR AUTOMATION OF MACHINE TOOLS

I. A. Nemirovskii

(Presented by the scientific seminar of the Departments of Machine Tools and Metal Cutting and of Mechanical-Engineering Technology)

The principal qualitative indicator of follow-up hydromechanisms (FH), widely used for the automation of metal-cutting machine tools, is the accuracy with which the profile of the copier is reproduced, or the accuracy with which the program is executed. In the process of copying or program execution, depending on the velocity of motion and the force developed by the FH, various displacements of the follow-up spool from its neutral position are required. These displacements of the spool, approximately proportional to the velocity and force developed by the FH, are the cause of error in processing the input action. In particular, the need to displace the spool from its neutral position in order to develop forces overcoming static-friction forces is the cause of the appearance of a dead zone, which substantially reduces the accuracy of the FH. The greater the force overcome by the follow-up hydraulic system, the greater the displacement of the spool and, correspondingly, the error in copying or program execution.

Among the most accurate is the hydraulic copying mechanism with two pumps [1, 2]. However, this type of FH also has errors that depend on the forces developed by the FH.

Using the graphoanalytic synthesis method [3], the author has developed an FH with two pumps and a servo-spool that provides flow feedback. Investigation of an experimental model of the developed FH showed that it is distinguished by increased accuracy and practically has no dead zone.

The diagram of the new FH is shown in Fig. 1.

In the stationary hydraulic cylinder 1 there is a piston 2, whose rod 3 is connected by a rigid mechanical linkage 4 to the body 5 of the follow-up spool 6, spring-loaded by spring 7. The grooves in body 5 and the working edges of spool 6 form slots \(\delta_3\) and \(\delta_4\). The programmed action \(F(t)\) is transmitted to spool 6 through the feeler of lever 3. Cavity \(I\) of cylinder 1 is connected to pump \(H_1\), and cavity \(II\) to pump \(H_2\). On the drain from the cavities of the hydraulic cylinder there is installed a servo-spool 9, forming with body 10 the working slots \(\delta_1\) and \(\delta_2\). The drain cavities of body 10 are connected by lines 11 and 12 to the inlet openings of body 5 of the follow-up spool. Line 13 connects body 5 to the drain.

The follow-up hydromechanism shown in Fig. 1 operates as follows (for simplicity of exposition we consider a symmetrical system, in which pumps \(H_1\) and \(H_2\) have identical parameters, and the working areas of the piston in cavities \(I\) and \(II\) are equal to one another, i.e., \(F_1 = F_2 = F\)).

In the neutral position of spool 6, when \(\delta_3=\delta_4\), the pressures \(p_3\) and \(p_4\) will be equal to each other when the equality \(q_1=q_2\) is satisfied, which, with identical pumps \(H_1\) and \(H_2\) and closed safety valves \(p_{k1}\) and \(p_{k2}\), is possible in the case of a stationary piston 2. In this case the servo spool 9 is also in the neutral position, i.e., \(\delta_1=\delta_2\).

Let, under the action of the input signal \(F(t)\), the tracking spool 6 begin to move from left to right, so that \(\delta_4>\delta_3\). Since the resistance of the gap \(\delta_3\) has increased and \(\delta_4\) has decreased, then, for \(q_1=q_2\), the pressure \(p_3\) has become greater than \(p_4\). As a result, spool 9, on whose end faces

Fig. 1. Diagram of a follow-up hydromechanism with two pumps and flow feedback

the pressures \(p_3\) and \(p_4\) act, will begin to move from left to right. This will lead to a decrease in the gap \(\delta_1\) and an increase in the gap \(\delta_2\), which in turn will cause an increase in the pressure \(p_1\) and a decrease in \(p_2\). The displacement of spool 9 will continue until the force \(P_{\mathrm{dv}}=F(p_2-p_1)\), acting on the piston, overcomes the forces resisting the displacement of piston 2. The mechanism must be designed so that, at the maximum calculated load on rod 3, the pressure \(p_1\) remains less than the pressure \(p_{k1}\), the setting of the safety valve; that is, the hydraulic system must operate in a constant-flow regime.

After \(P_{\mathrm{dr}}\) becomes greater than the resistance forces, piston 2 will begin to move from left to right, which will cause a corresponding decrease in the flow rate \(q_1\) and an increase in \(q_2\):

\[ q_1 = Q_{\mathrm{н}1} - F V, \]

\[ q_2 = Q_{\mathrm{н}2} + F V, \tag{1} \]

where \(Q_{\mathrm{н}1}\) and \(Q_{\mathrm{н}2}\) are, respectively, the flow rates of pumps \(H_1\) and \(H_2\); \(v\) is the velocity of the follow-up motion of piston 2.

Spool 9 will cease its motion when \(p_3 = p_4\), i.e., when the relation

\[ \frac{q_1}{q_2} = \frac{\delta_3}{\delta_4} \tag{2} \]

is satisfied.

From equalities (1) and (2) it is evident that the velocity of the follow-up motion will be proportional to the displacement of follow-up spool 6 from the neutral position, i.e.,

\[ V = \frac{Q_{\mathrm{н}}}{F}\cdot \frac{\delta_3 - \delta_4}{\delta_3 + \delta_4}. \tag{3} \]

The motion of piston 2 with rod 3 is transmitted through the rigid mechanical linkage 4 to the body 5 of the follow-up spool. Thus, negative position feedback is realized.

If, in the process of piston 2 following the input action \(F(t)\), the force being overcome changes, this is compensated by a corresponding automatic displacement of servo spool 9. Suppose, for example, that the force overcome by the piston moving from left to right increases. Then pressure \(p_1\) will increase, which will cause an increase in flow rate \(q_1\) through slot \(\delta_1\) and a corresponding decrease in the velocity of piston \(V\). Flow rate \(q_2\), on the contrary, will decrease. An increase in \(q_1\) and a decrease in \(q_2\) at unchanged \(\delta_3\) and \(\delta_4\) will lead to pressure \(p_3\) becoming greater than \(p_4\); spool 9 will begin to move from left to right, decreasing slot \(\delta_1\) and increasing \(\delta_2\). This will lead to an increase in pressure \(p_1\) and a decrease in \(p_3\), i.e., to an increase in the driving force \(P_{\mathrm{dr}} = F(p_2 - p_1)\).

After the relation

\[ \frac{q_1}{q_2} = \frac{\delta_3}{\delta_4}, \]

has been restored, i.e., the value of the follow-up velocity \(V\) corresponding to the ratio of \(\delta_3\) and \(\delta_4\) has been restored, pressures \(p_3\) and \(p_4\) will become equal, and the motion of spool 9 will cease.

The system operates analogously when follow-up spool 6 is displaced in the other direction. Let us consider the operation of the system when the cylinder and piston are fixed immovably, i.e., when the forces resisting piston displacement exceed the maximum design force developed by the hydraulic system. In this case, with the slightest displacement of spool 6 from the neutral position, the equality of pressures \(p_3\) and \(p_4\) will be disturbed and servo spool 9 will begin to move toward the lower pressure. Suppose spool 9 moves from left to right. Pressure \(p_1\) will increase and \(p_2\) will decrease. Since the piston is immobile and no decrease in flow rate \(q_1\) due to an increase in the volume of cavity \(I\) occurs, the motion of spool 9 and the corresponding increase in pressure \(p_1\) will continue until the value of \(p_1\) reaches the setting pressure of valve \(p_{\mathrm{к}1}\). The valve will begin to bypass part of the flow rate \(Q_{\mathrm{н}1}\). Flow rate \(q_1\) will decrease to the value corresponding to the given ratio-

the ratio of the working-slot dimensions \(\frac{\sigma_3}{\sigma_4}\), the pressures \(p_3\) and \(p_4\) will become equal, and the servo spool 9 will stop. The greatest pressure \(p_{\mathrm{k}}\) will act on piston 2 from the left, and a small pressure \(p_2\) will act from the right, equal to the sum of the pressure drops in slots \(\delta_2\) and \(\delta_4\) (since, during the motion of spool 9, slot \(\delta_1\) decreased and slot \(\delta_2\) increased, the pressure drop across \(\delta_2\) is small).

It follows from this that the given follow-up hydraulic system has a relay-type force characteristic, i.e., the dead zone is equal to zero. The high hydraulic stiffness of the hydraulic system is combined with dynamic stability. Servo spool 9 is damped by the working slots of the follow-up spool 6, since during its motion it displaces an additional volume of liquid through one of the slots of spool 6.

Thus, if the force overcome by the system does not exceed the maximum design value, the speed of the follow-up motion will depend only on the ratio \(\frac{\sigma_3}{\sigma_4}\), while changes in forces will be compensated by the servo spool; i.e., the system has only a velocity error, and the magnitude of the overcome force does not affect the copying accuracy.

In practice, owing to friction forces in servo spool 9 and leakage in the pumps, there will be a very small dead zone. Thus, in an experimental model of the hydraulic copying system the size of the dead zone was \(\pm 0.0015\) mm.

Let us consider the statics of the FS, taking into account that in the general case \(Q_{\mathrm{n}1} \ne Q_{\mathrm{n}2}\) and \(F_1 \ne F_2\). Denote \(\frac{F_1}{F_2}=k\) and \(\frac{Q_{\mathrm{n}1}}{Q_{\mathrm{n}2}}=k'\).

The flow rates through the control slots \(\delta_3\) and \(\delta_4\) of the follow-up spool 6 are respectively equal to:

\[ \left. \begin{aligned} q_1 &= \pi d_3 \mu_3 \delta_3 \sqrt{\frac{2g}{\gamma}} \sqrt{p_3-p_{\mathrm{sl}3}},\\ q_2 &= \pi d_4 \mu_4 \delta_4 \sqrt{\frac{2g}{\gamma}} \sqrt{p_4-p_{\mathrm{sl}4}}, \end{aligned} \right\} \tag{4} \]

where \(d_3\) and \(d_4\) are the diameters of the working lands of the follow-up spool;
\(\mu_3\) and \(\mu_4\) are the flow coefficients through slots \(\delta_3\) and \(\delta_4\);
\(\gamma\) is the specific weight of the working fluid;
\(p_3\) and \(p_4\) are the pressures under the end faces of the servo spool;
\(p_{\mathrm{sl}3}\) and \(p_{\mathrm{sl}4}\) are the pressures at the drain from slots \(\delta_3\) and \(\delta_4\).

Usually \(p_{\mathrm{sl}}\approx 0\). Therefore, denoting

\[ \pi d_3 \mu_3 \sqrt{\frac{2g}{\gamma}}=a_3;\qquad \pi d_4 \mu_4 \sqrt{\frac{2g}{\gamma}}=a_4, \]

we obtain

\[ \left. \begin{aligned} q_1 &= a_3 \delta_3 \sqrt{p_3},\\ q_2 &= a_4 \delta_4 \sqrt{p_4}. \end{aligned} \right\} \tag{5} \]

From equations (1) and (5), after transformations we obtain

\[ V=\frac{Q_{\mathrm{n}2}}{F_2}\cdot \frac{a_3\delta_3-k'a_4\delta_4}{a_3\delta_3+ka_4\delta_4}. \]

For a symmetric spool \(d_3=d_4;\ \mu_3=\mu_4\), and consequently \(\alpha_3=\alpha_4\). Then

\[ V=\frac{Q_{\mathrm{H}2}}{F_2}\cdot \frac{\delta_3-\kappa'\delta_4}{\delta_3+\kappa\delta_4}. \tag{6} \]

In the neutral position of the spool and the servo spool, when \(P_{\mathrm{dv}}=0\) and \(V=0\), we have

\[ \delta_1=\delta_{1_0}; \qquad \delta_2=\delta_{2_0}; \]

\[ \delta_3=\delta_{3_0}; \qquad \delta_4=\delta_{4_0}; \qquad p_1=p_{1_0}; \qquad p_2=p_{2_0}. \]

Let us denote the displacement of the follow-up spool from the neutral position by \(h\), and let \(\delta_{3_0}+\delta_{4_0}=h_0\); then

\[ \left. \begin{aligned} \delta_3&=\delta_{3_0}-h,\\ \delta_4&=\delta_{4_0}+h. \end{aligned} \right\} \tag{7} \]

Taking (7) into account, equation (6) assumes the form:

\[ V=\frac{Q_{\mathrm{H}2}}{F_2}\cdot \frac{h_0-\delta_{4_0}(1+\kappa')-h(1+\kappa')} {h_0-\delta_{4_0}(1-\kappa)-h(1-\kappa)}. \tag{8} \]

When \(V=0\), the value \(h=0\). In this case, from equation (8) we obtain

\[ \delta_{4_0}=\frac{h_0}{1+\kappa'}. \tag{9} \]

Introduce the dimensionless parameters

\[ s=\frac{VF_2}{Q_{\mathrm{H}2}} \quad \text{and} \quad j=\frac{h}{h_0}. \]

Then the expression for the dimensionless velocity \(s\) is written in the form

\[ s=-\frac{j(1+\kappa')} {\dfrac{\kappa+\kappa'}{1+\kappa'}-j(1-\kappa)}. \tag{10} \]

The system velocity gain \(c\) is equal to

\[ c=\frac{\partial s}{\partial j} =-\frac{\kappa+\kappa'} {\left[\dfrac{\kappa+\kappa'}{1+\kappa'}-j(1-\kappa)\right]^2}. \tag{11} \]

Figure 2 gives the velocity characteristic of the considered follow-up system for \(\kappa=1;\ \kappa'=2\), calculated from equation (10). As follows from equation (10) and Fig. 2, for \(\kappa=1\) there is a linear relationship between \(s\) and \(j\), i.e., the velocity characteristic of the follow-up system is linear. The quantity \(\kappa'=\dfrac{Q_{\mathrm{H}1}}{Q_{\mathrm{H}2}}\) represents the ratio \(\dfrac{|+s_{\max}|}{|-s_{\max}|}\). If \(\kappa\ne1\), then the velocity characteristic of the system is nonlinear, and the direction of curvature of the characteristic depends on whether the coefficient \(\kappa\) is less than or greater than unity. Examples of nonlinear velocity characteristics for two combinations of \(\kappa\) and \(\kappa'\) are given in Fig. 3.

Thus, if \(\kappa=\kappa'=1\), then the velocity characteristic \(s(j)\) is linear and \(|+s_{\max}|=|-s_{\max}|\); for \(\kappa=1\) and \(\kappa'\ne1\), \(s(j)\) is linear, but \(|+s_{\max}|\ne|-s_{\max}|\); for \(\kappa\ne1\), \(s(j)\) is nonlinear, and, if \(\kappa'=\kappa\), then \(|+s_{\max}|=|-s_{\max}|\), whereas in the case \(\kappa'\ne\kappa\), \(|+s_{\max}|\ne|-s_{\max}|\).

The system velocity gain at \(\kappa = 1\) is equal to
\(c_{\kappa=1}=-(1+\kappa')\) and does not depend on \(j\).

For \(\kappa \ne 1\), according to (11), \(c\) depends on \(j\). In this case, systems with different values of \(\kappa\) and \(\kappa'\) are conveniently compared by the value \(c_0\), calculated from equation (11) at \(j=0\). The character of the variation of \(c_0\) as a function of \(\kappa\) and \(\kappa'\) for \(j=0\) is shown in Fig. 4.

Let us examine the force characteristic of the SG under consideration, i.e., the dependence of the displacement \(h\) of the spool from the neutral position on the force developed at \(V=0\). If, because of their small magnitude, we neglect the friction forces in servospool 9, as well as the reaction forces of the flowing working fluid, then in statics the equality \(p_3=p_4\) always holds.

\[ S=-\frac{j(1+\kappa')}{1-\frac{1-\kappa}{1+\kappa'}\,j(1-\kappa)} \]

Fig. 2. Velocity characteristic of an SG with two pumps and flow feedback for \(\kappa=1,\ \kappa'=2.0\)

\[ S=-\frac{j(1+\kappa')}{1-\frac{1-\kappa}{1+\kappa'}\,j(1-\kappa)} \]

\[ S=-\frac{j(1+\alpha\kappa)}{1+\frac{1-\kappa}{1+\alpha\kappa}\,j(1-\kappa)} \]

Fig. 3. Nonlinear velocity characteristics of the SG: for \(\kappa=0.5,\ \kappa'=1.0\); for \(\kappa=2.0,\ \kappa'=2.0\)

Let the follow-up spool be shifted from the neutral position by \(+h\). Then, at \(V=0\), the pressure in cavity 1 will rise to \(p_1=p_{1\kappa}\), and part of the flow delivered by pump \(H_1\) will drain through the valve into the tank.

Pump \(H_2\) will develop a pressure \(p_2<p_{2\kappa}\), and therefore the flow \(Q_{H_2}\) will pass through gap \(\delta_2\) of the servospool and gap \(\delta_4\) of the follow-up spool.

Then from (5) it follows that

\[ p_3=p_4=\frac{q_2^2}{\alpha_4^2\delta_4^2} =\frac{Q_{\mathrm{H}_2}^2}{\alpha_4^2\delta_4^2}. \tag{12} \]

From (5) and (12) it follows that the flow rate through the slots \(\delta_1\) and \(\delta_3\) is equal to

\[ q_1=Q_{\mathrm{H}_2}\frac{\delta_3}{\delta_4}. \]

Hence

\[ \begin{aligned} \delta_1 &=\frac{q_1}{\alpha_1\sqrt{p_1-p_3}} \\[4pt] &=Q_{\mathrm{H}_2}\frac{\delta_3}{\delta_4}\cdot \frac{1}{\alpha_1\sqrt{p_1-p_3}} \\[4pt] &=\frac{Q_{\mathrm{H}_2}\dfrac{\delta_3}{\delta_4}} {\alpha_1\sqrt{\,p_{1\mathrm{k}}-\dfrac{Q_{\mathrm{H}_2}^2}{\alpha_4^2\delta_4^2}\,}} . \end{aligned} \tag{13} \]

Fig. 4. Dependence of the gain coefficient of the hydraulic follow-up system with respect to velocity on \(\kappa\) and \(\kappa'\).

In turn,

\[ \delta_2=u_0-\delta_1 =u_0- \frac{Q_{\mathrm{H}_2}\dfrac{\delta_3}{\delta_4}} {\alpha_1\sqrt{\,p_{1\mathrm{k}}-\dfrac{Q_{\mathrm{H}_2}^2}{\alpha_4^2\delta_4^2}\,}}, \tag{14} \]

where

\[ u_0=\delta_{10}+\delta_{20}=\delta_1+\delta_2. \]

The pressure drop across the slot \(\delta_2\) of the servo valve is equal to

\[ p_2-p_4 =\frac{q_2^2}{\alpha_2^2\delta_2^2} = \left[ \frac{Q_{\mathrm{H}_2}} {\alpha_2\left( u_0- \dfrac{Q_{\mathrm{H}_2}\dfrac{\delta_3}{\delta_4}} {\alpha_1\sqrt{\,p_{1\mathrm{k}}-\dfrac{Q_{\mathrm{H}_2}^2}{\alpha_4^2\delta_4^2}\,}} \right)} \right]^2 . \tag{15} \]

Since \(P_{\mathrm{dv}}=F_2(p_2-p_1\kappa)\), taking (12) and (15) into account we obtain

\[ \left(\frac{P_{\mathrm{dv}}}{F_2}\right)_{h>0} = \left[ \frac{Q_{\mathrm{H}_2}} {\alpha_2\left( u_0- \dfrac{Q_{\mathrm{H}_2}\delta_3\delta_4} {\alpha_1\sqrt{\,p_{1\mathrm{k}}-\dfrac{Q_{\mathrm{H}_2}^2}{\alpha_4^2\delta_4^2}\,}} \right)} \right]^2 + \frac{Q_{\mathrm{H}_2}^2}{\alpha_4^2\delta_4^2} - p_{1\mathrm{k}}\kappa . \tag{16} \]

Analogously, one can obtain the expression for \(\dfrac{P_{\mathrm{dv}}}{F_2}\) for \(h<0\):

\[ \left(\frac{P_{\mathrm{dv}}}{F_2}\right)_{h<0} = p_{2\mathrm{k}} - \frac{Q_{\mathrm{H}_1}^2}{\alpha_3^2\delta_3^2}\,\kappa - \left[ \frac{Q_{\mathrm{H}_1}} {\alpha_1\left( u_0- \dfrac{Q_{\mathrm{H}_1}\delta_4/\delta_3} {\alpha_2\sqrt{\,p_{2\mathrm{k}}-\dfrac{Q_{\mathrm{H}_1}^2}{\alpha_3^2\delta_3^2}\,}} \right)} \right]^2 . \tag{17} \]

Figure 5 shows the force characteristics of the HS, obtained by calculation from equations (16) and (17) and experimentally on a test stand.

Curve 2 was obtained experimentally (the experimental points are denoted by circles), and curve 3 by calculation from relations (16) and (17) for an HS with the following parameters: $d_3 = 2.4\ \text{cm}$; $d_{\text{f.b.}} = 2.4\ \text{cm}$; $u_0 = 0.054\ \text{cm}$; $h_0 = 0.027\ \text{cm}$; $\mu_1 = \mu_2 = \mu_3 = \mu_4 = 0.57$; $Q_{\text{p}1} = 200\ \text{cm}^3/\text{s}$; $Q_{\text{p}2} = 340\ \text{cm}^3/\text{s}$; $\kappa = 0.6$; $F_2 = 95.2\ \text{cm}^2$. Figure 5 also gives the experimental curve 1 for an ordinary system with two pumps without a servospool. From a comparison of the force characteristics of the ordinary system and of the system with flow feedback, the substantial advantages of the new HS are evident.

Fig. 5. Force characteristics of the HS with flow feedback (curves 2 and 3) and without feedback (curve 1)

The investigation of the dynamic characteristics of the new HS was carried out on an experimental test stand, by the graphoanalytic method [3, 4], and by calculations of transient processes on the Minsk-1 digital computer.

Figure 6 shows samples of experimental amplitude-phase frequency characteristics of the new HS (curve I) and of the HS without flow feedback (curve II).

The characteristics were obtained for the HS whose parameters are given above. In addition, it should be indicated that the mass of the moving units of the HS was

\[ m = 0.2\ \frac{\text{kg}\cdot\text{s}^2}{\text{cm}}; \]

the total coefficient of compres-

of the compressibility of the fluid and the compliance of the main-line pipelines from pump \(H_1\) to the servovalve was equal to \(k_1 = 0.6 \dfrac{\text{cm}^5}{\text{kg}}\); the same for the main line from pump \(H_2\) to the servovalve was \(k_2 = 0.5 \dfrac{\text{cm}^5}{\text{kg}}\); the transmission ratio of the coding-lever mechanism was \(i = 0.35\).

From a comparison of curves \(I\) and \(II\) in Fig. 6, it may be concluded that the amplitude and phase errors of the SG with flow feedback are smaller than those of the conventional system with two pumps.

At the same time, the nonrigidity of the system, i.e., the large values of the coefficients \(k_1\) and \(k_2\), did not make it possible in this case to realize fully all the advantages of the new SG.

A theoretical study of the dynamics of the new SG was carried out taking into account the discontinuous characteristic of friction forces, the compliance of pipelines and the compressibility of the fluid, and leakage in the pump [5].

In doing so, the following assumptions were adopted: the length of the pipelines is small, i.e., their resistance, as well as wave processes in them, may be neglected; the working edges of the spools and sleeves are sharp and perpendicular to the axes of the spools; the friction forces in the servovalve and the reaction of the fluid flow are insignificant; the mass of the servovalve is negligibly small; there are no leaks in the hydraulic cylinder and spools.

Fig. 6. Amplitude-phase frequency characteristics of an SG with flow feedback (curve \(I\)) and without feedback (curve \(II\)); \(\omega\) in cps

In this case the dynamics of the SG is described by the following system of nonlinear differential equations, which are given here without derivation:

\[ \left. \begin{aligned} \kappa_1 \frac{dp_1}{dt} &= Q_{H_1} - \sigma_1 p_1 + F_2 \frac{dx}{dt} - (u_{1_0} + u)\alpha_1 \sqrt{p_1 - p_t} \\ \kappa_2 \frac{dp_2}{dt} &= Q_{H_2} - \sigma_2 p_2 - F_2 \frac{dx}{dt} - (u_{2_0} - u)\alpha_2 \sqrt{p_2 - p_t}, \\ m \frac{d^2x}{dt^2} &= F_2 p_2 - F_2 k p_1 + F_{\mathrm{fr}}\!\left(\frac{dx}{dt}\right), \\ f \frac{du}{dt} &= [h_{3_0} - y(t) - i x]\alpha_3 \sqrt{p_t} - (u_{1_0} + u)\alpha_1 \sqrt{p_1 - p_t}, \end{aligned} \right\} \tag{18} \]

where

\[ p_t = p_3 = p_4 = \left[ \frac{ Q_{H_2} - \sigma_2 p_2 - F_2 \dfrac{dx}{dt} - \kappa_2 \dfrac{dp_2}{dt} - f \dfrac{du}{dt} }{ [h_{4_0} + y(t) + i x]\alpha_4 } \right]^2, \]

\(\sigma_1\) and \(\sigma_2\) are, respectively, the leakage coefficients in pumps \(H_1\) and \(H_2\);
\(u_{1_0}\) and \(u_{2_0}\) are the openings of the servovalve slots at \(P_{\mathrm{dv}} = 0\), \(V = 0\);
\(u\) is the displacement of the servovalve from the neutral position;
\(F_{\mathrm{fr}}\!\left(\dfrac{dx}{dt}\right)\) is the nonlinear dependence on velocity of the friction forces in the guides and seals of the SG.
\(f\) is the area of the end faces of the servovalve;
\(y(t)\) is the input function.

In calculating transient processes in the SG on a digital computer, the interrelation between the forces acting on the moving units of the mechanism, which occurs in the real process of the system’s processing of the input signal \(y(t)\), was taken into account; that is, the calculation program was compiled in such a way that the following condition was satisfied:

I) for

\[ \frac{dx}{dt} = 0: \]

a)

\[ \frac{d^2x}{dt^2} = 0, \quad \text{if } \left|F_2 p_2 - F_2 k p_1\right| < \left|F_{\mathrm{fr}}\right|; \]

b) \(F_{\mathrm{fr}} > 0\), if \(\left|F_2 p_2 - F_2 k p_1\right| > \left|F_{\mathrm{fr}}\right|\) and \((F_2 p_2 - F_2 k p_1) < 0\);

c) \(F_{\mathrm{fr}} < 0\), if \(\left|F_2 p_2 - F_2 k p_1\right| > \left|F_{\mathrm{fr}}\right|\) and \((F_2 p_2 - F_2 k p_1) > 0\).

II) for

\[ \frac{dx}{dt} \ne 0: \]

a) \(F_{\mathrm{fr}} > 0\), if

\[ \frac{dx}{dt} < 0; \]

b) \(F_{\mathrm{fr}} < 0\), if

\[ \frac{dx}{dt} > 0. \]

In calculations of the dynamics by the graphoanalytic method, the basic principles of which are set forth in work [4], the nonlinear dependence of the compliance coefficient \(\kappa\) on pressure was additionally taken into account, as well as the complex character of the outflow of the working fluid through small open and covered control slots of the valves.

Figure 7 presents the results of calculations of the transient process in the new SG by the graphoanalytic method (curve 2), on the digital computer (curve 3), and also the experimental curve 1.

The main parameters of the system were given on p. 158, and the values of \(m\), \(\kappa_1\), and \(\kappa_2\) are:

\[ m = 0.163 \frac{\mathrm{kG}\,\mathrm{s}^2}{\mathrm{cm}^2}; \qquad \kappa_1 = 0.10 \frac{\mathrm{cm}^5}{\mathrm{kG}} \qquad \kappa_2 = 0.04 \frac{\mathrm{cm}^5}{\mathrm{kG}} . \]

As follows from Fig. 7, the accuracy of the graphical-analytical calculation is somewhat better than the results of the calculation on the digital computer, although in the graphical-analytical calculation the entire duration of the transient process was divided into 30 intervals, whereas in the calculation on the digital computer it was divided into 400. This is explained by the fact that, in the graphical-analytical calculation, additional essential factors determining the nonlinearity of the process were taken into account. Taking these factors into account in the digital-computer calculation considerably complicated the computation program.

The results of investigations of a follow-up hydromechanism (FH) with flow feedback make it possible to draw the following conclusions:

1. The follow-up hydromechanism under consideration has high static accuracy, is sufficiently stable, and possesses better dynamic characteristics than a conventional system with two pumps without flow feedback.

2. By varying the ratio \(k\) of the working piston areas in the cylinder cavities and the ratio \(k'\) of the pump delivery rates, any required velocity characteristic of the FH can be obtained.

Fig. 7. Characteristics of the transient process in an FH with flow feedback:
1 — experimental curve \(x(t)\); 2 — curve \(x(t)\), obtained by graphical-analytical calculation; 3 — curve \(x(t)\), obtained by calculation on the Minsk-1 digital computer.

1. When this hydromechanism is used with open negative feedback, it can ensure high stability of the piston velocity, which is especially valuable in moving the heavy units of large machine tools at low speeds, in the drives of broaching, planing, and slotting machines.

As the motor in the hydraulic system under consideration, not only a hydraulic cylinder but also a rotary hydraulic motor may be used.

Further investigations will help to establish the conditions for the dynamic stability of the new FH, as well as schemes for practical application of the FH in drive systems and in machine-tool automation.

REFERENCES

1. A. F. Domrachev. Experimental investigation of hydraulic copying systems at increased speeds of follow-up motion. Mechanization and Automation in Mechanical Engineering. Moscow—Kiev, Mashgiz, 1959.

1. V. A. Leshchenko. Hydraulic Follow-up Drives for the Automation of Machine Tools. Moscow, Mashgiz, 1962.

2. I. A. Nemirovskii. Graphoanalytic analysis and synthesis of single-coordinate follow-up hydraulic systems of machine tools. Academy of Sciences of the USSR. Fourth Conference on the Automation of Mechanical-Engineering Processes. Abstracts of Papers. Nauka Publishing House, Moscow, 1964.

3. I. A. Nemirovskii et al. Increasing the Productivity of Large Machine Tools. Mashinostroenie Publishing House, Moscow, 1965.

4. I. A. Nemirovskii. Investigation of the possibility of applying the graphoanalytic method to the analysis of nonlinear processes and the synthesis of follow-up hydromechanisms. Dissertation for the degree of Candidate of Technical Sciences, Tomsk, 1965.