ON THE PRESSURE AT THE INLET OF A HYDRAULIC TRACING SYSTEM WITH A DIFFERENTIAL CYLINDER

I. A. NEMIROVSKII

(Presented by the Department of Mechanical-Engineering Technology)

The diagram of the system under consideration, which is widely used for the automation of metal-cutting machine tools, is shown in Fig. 1.

Fig. 1. Diagram of a hydraulic tracing system with a differential cylinder and a single-edge spool valve:
1 — spool-valve slide, 2 — spool-valve body, 3 — machine-tool table, 4 — cylinder, 5 — piston, 6 — throttle, 7 — pump, 8 — safety-and-overflow valve, 9 — lever of the tracing probe.

The pump, supplying the system with working fluid, develops at the system inlet a pressure \(p\). In this case the system consumes a flow rate \(Q_c\).

In analytical studies of such systems it is assumed that the pressure at the system inlet \(p = \mathrm{const}\) [1, 2, and others], i.e., that part of the flow rate

\(Q_{\mathrm{n}}\), developed by the pump, is discharged through the overflow valve. The characteristics of the system obtained under this assumption do not make it possible to analyze the copying process in regimes occurring with the overflow valve closed. It is therefore of interest to estimate the magnitude of the interval of copying regimes in which the pressure at the system inlet is variable, and the dependence of the magnitude of this interval on the copying-motion speed \(V\) developed by the system, the nominal pump delivery \(Q_{\mathrm{nom}}\), and the setting of the overflow valve \(p_k\).

If leakage in the hydraulic cylinder and the resistance of the pipelines and channels of the copying device are neglected, then, at a steady liquid temperature in the system and with sharp working edges of the spool, the following relations may be written for the system under consideration [6]:

\[ \frac{h}{h_0} = \frac{ \frac{Q_c}{Q_0}-\frac{VF_2(1-k)}{Q_0} }{ \sqrt{ \frac{P_{\mathrm{dv}}}{p_0 F_2 k}\cdot \frac{1}{1-k} + \left[ \frac{Q_c}{Q_0}+\frac{VF_2 k}{Q_0} \right]^2 } }; \tag{1} \]

\[ \frac{p}{p_0} = \left[ \frac{VF_2 k}{Q_0}+\frac{Q_c}{Q_0} \right]^2 + \frac{P_{\mathrm{dv}}}{p_0 F_2 k}\frac{1}{1-k}, \tag{2} \]

where \(h\) is the spool opening (Fig. 1);

\(F_2\) is the working area of the piston in chamber II;

\(k\) is the ratio of the working areas of the piston,

\[ k=\frac{F_1}{F_2}; \]

\(h_0, p_0, Q_0\) are, respectively, the spool opening, the inlet pressure, and the flow rate consumed by the system at \(V=0\) and \(P_{\mathrm{dv}}=0\).

Denoting

\[ \frac{P_{\mathrm{dv}}}{p_0F_2k}\frac{1}{1-k}=r, \qquad \frac{VF_2k}{Q_0}=s, \]

\[ \frac{h}{h_0}=j, \qquad \frac{Q_c}{Q_0}=q, \qquad \frac{p}{p_0}=n, \]

we write equations (1) and (2) in the form

\[ j= \frac{ q-s\frac{1-k}{k} }{ \sqrt{r+(q+s)^2} }, \tag{1-a} \]

\[ n=[s+q]^2+r. \tag{2-a} \]

Figure 2 shows the characteristic curves of the system, constructed from equations (1-a) and (2-a) for \(k=0.5\). The characteristic of the pumping station \(MN_1Z_1\) is also plotted there. The segment \(MN_1\) corresponds to a closed overflow valve. The dependence of the pump flow rate \(Q_{\mathrm{n}}\) on the pressure \(p\) is linear for most pumps [3].

The slope of the line \(MN_1\) is determined by the leakage coefficient in the pump and the closed overflow valve. The segment \(N_1Z_1\) corresponds to copying regimes with the overflow valve open, under the assumption that the pressure maintained by the valve is approximately constant. The operating points of the system are determined by the intersection of the characteristic curves of the system with the characteristic of the pumping station.

Considering Fig. 2, the following conclusions may be drawn:

Fig. 2. Characteristic curves of a hydraulic tracing system with a differential cylinder and a single-edge spool valve, and the characteristic of the pumping station.

Visible labels in the figure include:

\[ \frac{q_c}{q_0}, \qquad \frac{q_n}{q_0}, \qquad \frac{p}{p_0} \]

\[ z=\frac{P d^2}{q_2^{\,2}\kappa}\,\frac{1}{1-\kappa}, \qquad S=\frac{\mu k f_2 \kappa}{Q_0}. \]

Points and markings shown on the plot include \(O\), \(z\), \(z_1\), \(C\), \(M_1\), \(M_2\), \(N\), and \(N_2\), with curve labels of the form \(S=\cdots\), \(z=\cdots\).

1. In the hydraulic copying system under consideration there is always a part of the operating modes corresponding to the segment \(MN_1\), i.e., occurring at variable pressure.

2. Depending on the specific parameters of the hydraulic copying system, the pump delivery, and the setting of the safety valve, one may obtain one or another magnitude of the linear segment of the static characteristic of the system (i.e., the segment corresponding to

\[ \frac{p}{p_0}=\mathrm{const}). \]

Comparing \(MNZ\) and \(MN_1Z_1\), we see that the higher the setting pressure of the safety valve (all other factors being unchanged), the greater the part of the modes that occurs at variable pressure.

1. Comparing the operation of the system with a pumping station having the characteristic \(MN_1Z_1\) and a station with the characteristic \(M_2N_2Z_1\), we see that the greater the pump delivery (all other factors being unchanged), the greater the part of the copying modes that occurs with the valve open, i.e., at \(p=\mathrm{const}\); but for any pump delivery there exist copying modes that occur at variable pressure at the system inlet.

2. In analyzing the dynamic properties of the system, in addition to such nonlinearities of the static characteristic as the dead zone, saturation, etc., it is necessary to take into account the break in the characteristic caused by the transition from modes occurring at \(p=\mathrm{const}\) to modes corresponding to the segment of the characteristic \(MN_1\).

In order to estimate what part of the copying modes of the system corresponds to variable pressure at the inlet, introduce the coefficient \(\varepsilon\)

\[ \varepsilon=\frac{|s'_{\max}|+s_{\mathrm{i}}}{|s'_{\max}|+|s''_{\max}|}, \tag{3} \]

where

\[ s'_{\max}=\frac{V'_{\max}F_2k}{Q_0}, \]

\(V'_{\max}\) is the maximum value of the copying speed developed by the system in the negative direction (Fig. 1) at \(P_{\mathrm{dv}}=0\);

\[ s_{\mathrm{i}}=\frac{V_{\mathrm{i}}F_2k}{Q_0}; \]

\(V_{\mathrm{i}}\) is the copying speed corresponding to the break point of the pumping-station characteristic (point \(N\) in Fig. 2) at \(P_{\mathrm{dv}}=0\);

\[ s''_{\max}=\frac{V''_{\max}F_2k}{Q_0}; \]

\(V''_{\max}\) is the maximum copying speed developed by the system in the positive direction (Fig. 1) at \(P_{\mathrm{dv}}=0\).

The value \(s'_{\max}\) can be determined from formula (1–a), assuming that \(r=0\) and \(j\to\infty\).

Then \(s'_{\max}=-q'\); taking into account that, for \(r=0\) and \(j=\infty\), the quantity \(n\) is also close to zero, we determine

\[ q'=\frac{Q_{\mathrm{nom}}}{Q_0}\frac{1}{\eta}, \]

where \(Q_{\mathrm{nom}}\) is the flow rate developed by the pump at the maximum working pressure \(p_{\max}\) permissible for the given pump;

\(\eta\) is the volumetric efficiency of the pump at \(p_{\max}\).

The final expression for \(s'_{\max}\) is

\[ s'_{\max}=-\frac{Q_{\text{nom}}}{Q_0}\frac{1}{\eta}. \tag{4} \]

We obtain the expression for \(s_i\) from formula (2—a), assuming that

\[ r=0,\quad \text{and}\quad n=n_i=\frac{p_k}{p_0}, \]

\[ s_i=\sqrt{n_i}-q_i. \tag{5} \]

In order to determine \(q_i\), corresponding to the break point of the characteristic of the pumping station, we shall use the dependence of the dimensionless flow rate developed by the pump, \(\dfrac{Q_{\text{n}}}{Q_0}\), on the likewise dimensionless pressure \(\dfrac{p}{p_0}\) [6]:

\[ \frac{Q_{\text{n}}}{Q_0} = \frac{Q_{\text{nom}}}{Q_0} \left[ \frac{1}{\eta} - \frac{p}{p_0}\frac{p_0}{p_{\max}} \left( \frac{1}{\eta}-1 \right) \right]. \tag{6} \]

Taking in formula (6) that \(\dfrac{p}{p_0}=n_i\), we obtain from (5) and (6)

\[ s_i = \sqrt{n_i} - \frac{Q_{\text{nom}}}{Q_0} \left[ \frac{1}{\eta} - n_i\frac{p_0}{p_{\max}} \left( \frac{1}{\eta}-1 \right) \right]. \tag{7} \]

In order to determine \(s''_{\max}\) in formula (2—a), we assume that
\(r=0;\ n=n_i;\ s=q\) [6].
Then

\[ s''_{\max}=\frac{1}{2}\sqrt{n_i}. \tag{8} \]

From equations (3), (4), (7), and (8) it follows that

\[ \varepsilon = \frac{ 1+\sqrt{n_i}\dfrac{p_0}{p_{\max}}\left(\dfrac{1}{\eta}-1\right) }{ 0.5+\dfrac{Q_{\text{nom}}}{Q_0}\dfrac{1}{\eta}\dfrac{1}{\sqrt{n_i}} }. \tag{9} \]

If leakage in the pump is neglected, then \(\eta=1\), and

\[ \varepsilon = \frac{1}{ 0.5+\dfrac{Q_{\text{nom}}}{Q_0}\dfrac{1}{\sqrt{n_i}} }. \tag{9—a} \]

Let us denote

\[ q_{\text{n}}=\frac{Q_{\text{n}}}{Q_{\text{nom}}} \quad \text{and} \quad \psi=\frac{Q'_0}{Q_0} = \frac{f''_{\text{dr}}}{f_{\text{dr}}} \sqrt{\frac{p'_0}{p_0}}, \]

where \(f_{\text{dr}}\) is the cross-sectional area of the throttle opening 6 (Fig. 1).
Then equation (9—a) can be written in the form

\[ \varepsilon = \frac{1}{ 0.5+\dfrac{q_{\text{n}}}{\psi}\cdot\dfrac{1}{\sqrt{n_i}} }. \tag{10} \]

Taking \(n_i=1;\ \dfrac{p'_0}{p_0}=1\), we construct the dependence of \(\varepsilon\) on \(q_{\text{n}}\) for differ-

…i.e., for different \(\dfrac{f'_{\mathrm{dr}}}{f_{\mathrm{dr}}}\). As can be seen from Fig. 3, the range of the copying mode in which the pressure at the inlet to the tracking hydraulic system is variable decreases with increasing \(q_{\mathrm{n}}\) and decreasing \(f_{\mathrm{dr}}\). At the same time, this range is sufficiently large.

Fig. 3. Dependence of the magnitude of the interval of copying modes with variable pressure at the system inlet, \(\varepsilon\), on the relative pump delivery \(q_{\mathrm{n}}\) and the working area of the throttle orifice \(f_{\mathrm{dr}}\).

Therefore, the calculation of the static characteristics of the hydraulic system must be carried out for all modes, not excluding modes with variable pressure at the system inlet.

A method for such calculations for piston hydraulic motors was developed by L. S. Bron and E. A. Tsukhanova [4], and for tracking hydraulic systems it is discussed in the author’s works [5, 6].

REFERENCES

1. Korobochkin B. L. Hydraulic tracking systems of copying machine tools and the fundamentals of their design. Collection “Automation in Mechanical Engineering,” Mashgiz, 1957.

2. Khokhlov V. A. Speed characteristics of hydraulic actuating mechanisms with spool control, “Automation and Telemechanics,” vol. 16, No. 5, 1955.

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

4. Bron L. S., Tsukhanova E. A. Investigation of the hydraulic drive of transport devices of automatic lines. New Developments in Pneumatic-Hydraulic Automation. Edited by Doctor of Technical Sciences M. A. Aizerman, Academy of Sciences of the USSR, 1962.

5. Nemirovskii I. A. Calculation of the static characteristic of a single-coordinate hydraulic copying system with variable inlet pressure. Proceedings of Higher Educational Institutions, Mechanical Engineering, No. 8, 1962.

6. Nemirovskii I. A. Graphical-analytical method for calculating the accuracy of single-coordinate hydraulic copying systems for the automation of lathes. TsITEIN, No. M—62—15/3, Moscow, 1962.