Some Issues in Selecting Parameters of a Hydraulic Drive with Throttle Speed Control

Yu. Ya. Komisarenko, P. Ya. Krauinsh

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

Hydraulic drives are finding ever wider application in machine tools; for example, the drive of the main motion with throttle speed control is becoming widespread; a simplified diagram of it is shown in Fig. 1.

Operating experience with such systems shows that, in a number of cases, the systems prove to be inoperable because of intense self-oscillations. Studies conducted by I. Z. Zaichenko [1] on self-oscillations in hydraulic transmissions of machine tools showed that necessary conditions for the occurrence of oscillations are a falling characteristic of the resistance forces as a function of speed and the elasticity of the hydraulic connections.

Since the slope of the load characteristic \(\dfrac{\partial R}{\partial V}\) may vary under real conditions, in designing the drive the problem arises of selecting the drive parameters (i.e., the areas \(F\) and \(F_1\) of the cylinder, the value of the back pressure \(p_1\), etc.) so that the system has the greatest possible stability margin under the given conditions.

To solve this problem, let us turn to a linear mathematical model of the system with a throttle at the outlet. In choosing the equations we shall assume that the inlet pressure \(p\) is constant, neglect wave processes, and restrict ourselves to considering those transient processes for which the piston velocity \(V\) and the back pressure \(p_1\) remain positive.

Taking the adopted assumptions into account, we obtain a system of equations describing the transient process caused by a change in the load \(R(t; V)\):

\[ \begin{aligned} \frac{d\Delta V}{dt} &= \frac{\alpha}{m}\,\Delta V - \frac{F_1}{m}\,\Delta p_1 - \frac{1}{m}\,\Delta R_n, \\[6pt] \frac{d\Delta p_1}{dt} &= \frac{F}{W k(p)}\,\Delta V - \frac{1}{W k(p)}\,\Delta Q_{\mathrm{dr}}, \\[6pt] \Delta Q_{\mathrm{dr}} &= \varepsilon_{\mathrm{dr}}\,\Delta p_1 \end{aligned} \qquad\ldots \tag{1} \]

here

\[ \Delta V \equiv V(t)-V_0;\quad \Delta p_1 \equiv p_1(t)-p_{10};\quad \Delta R_n \equiv R_n(t)-R_{n0}; \]

\[ \Delta Q_{\mathrm{dr}} \equiv Q_{\mathrm{dr}}(t)-Q_{\mathrm{dr}0}; \]

where

\(V\) and \(p_1\) are, respectively, the piston velocity and the pressure drop across the throttle;

\(Q_{\mathrm{thr}}\) is the flow rate through the throttle;

\(m\) is the mass of the moving parts;

\(F_1\) is the working area of the piston on the back-pressure side;

Fig. 1. Diagram of a hydraulic drive with throttle speed control

\(W;\ k(p)\) are, respectively, the volume and the coefficient of compliance of the cavities under pressure \(p_1\);

\(R_{\mathrm{n}}\) is the external load, \(\alpha = -\dfrac{\partial R_{\mathrm{n}}}{\partial V}\) is the slope of the load characteristic;

\[ \varphi_{\mathrm{thr}}=\frac{\partial Q_{\mathrm{thr}}}{\partial p_1} \]

is the compliance of the throttle.

If the variables of system (1) are expressed in dimensionless coordinates and the system is solved with respect to the velocity \(V\), we obtain the equation:

\[ \frac{d^2 v}{dt^2} + \frac{1}{T_p} \left( 1-\frac{T_p}{T_v}\eta_v \right) \frac{dv}{dt} + \frac{1}{T_p T_v} (1-\eta_v)v = -\frac{1}{T_v}\frac{dR}{dt} - \frac{1}{T_v T_p}R \ldots \tag{2} \]

where

\[ v=\frac{\Delta V}{V_0}; \qquad R=\frac{\Delta R_{\mathrm{n}}}{R_{\mathrm{n0}}}; \qquad R_{\mathrm{n0}}=p_{1v}F_1\eta_v=\frac{x\varphi_{\mathrm{thr}}}{F_1^2}; \qquad p_{1v}=V_0F_1\frac{1}{\varphi_{\mathrm{thr}}}; \]

\[ T_v=\frac{m\varphi_{\mathrm{thr}}}{F_1^2}; \qquad T_p=\frac{Wk(p)}{\varphi_{\mathrm{thr}}}. \]

Thus, we have a second-order dynamic system; for its stability it is necessary and sufficient that all coefficients be positive, i.e.,

\[ \eta_v<1; \qquad \frac{T_p}{T_v}\eta_v<1. \]

If one uses the diagram in the coordinates

\[ A=\frac{\alpha\varphi_{\mathrm{thr}}}{F_1^2}; \qquad B= \frac{\alpha Wk(p)}{\varphi_{\mathrm{thr}}m_{\mathrm{thr}}} \]

(Fig. 2), then the stability region will be the square bounded—

... by the coordinate axes and the straight lines \(A=1\), \(B=1\). On this diagram one can find the region of aperiodic stability, which corresponds to the presence of real roots of equation (2); for this the following condition must be satisfied:

\[ \frac{1}{4}\left[\frac{1}{T_{\mathrm{p}}}\left(1-\frac{T_{\mathrm{p}}}{T_v}\eta_v\right)\right]^2 -\frac{1}{T_{\mathrm{p}}T_v}(1-\eta_v)>0\ldots \tag{3} \]

The boundary of aperiodic stability is determined if the left-hand side of (3) is set equal to zero. Carrying out the necessary transformations, we obtain the equation of the boundary of aperiodic stability:

\[ A=\frac{4B}{(B+1)^2}. \]

This boundary (line \(G\) in Fig. 2) divides the stability region into two: I—the region of aperiodic stability; II—the region of oscillatory stability.

If the system is assigned certain parameters, then a definite point on the \(A\)—\(B\) diagram will correspond to this system,

Fig. 2. Stability diagram of the system: I—the region of aperiodic stability; II—the region of oscillatory stability; \(G\)—the boundary of aperiodic stability

let us call this point the representative point*). When any system parameter changes, the representative point describes a trajectory showing the influence of this parameter on the stability of the system.

Thus, from Fig. 2, where such trajectories are plotted, it is evident that an increase in the mass \(m\) and in the cylinder area \(F_1\) increases stability, while an increase in the compliance of the system \(W_k(p)\) and an increase in the steepness \(\alpha\) of the load characteristic decrease stability. It is also evident here that the system may be unstable both when the compliance of the throttle \(\Phi_{\mathrm{dr}}\) is too small and when its value is too large—

* Not to be confused with the concept of a “representative point” used in constructing the phase portrait of a system.

tions. Physically this can be explained by the fact that the maximum ability of the throttle to dissipate energy occurs at a definite value of the compliance.

Considering the parameters \(m\), \(F_1\), and \(W_K(p)\) as given, the maximum possible stability can be achieved by varying the throttle compliance \(\varphi_{\mathrm{dr}}\) so that the representative point is equally distant from both boundaries of the stability region.

Mathematically this is expressed by the equality \(A = B\), i.e. \(\eta_v = \eta_v \dfrac{T_p}{T_v}\), or \(T_p = T_v\), whence we find the optimal value of the throttle compliance:

\[ \varphi_{\mathrm{dr}}^{*} = F_1 \sqrt{\frac{\overline{W_K(p)}}{m}} . \qquad \ldots \tag{4} \]

When the actual value of the compliance deviates from the optimal value \(\varphi_{\mathrm{dr}}^{*}\), the system will approach the stability boundary.

If the stability margin is understood as the ratio \(\beta = \dfrac{\alpha_{\max}}{\alpha_0}\), where \(\alpha_0\) is the nominal value of the slope of the load characteristic, and \(\alpha_{\max}\) is its value at which the system loses stability, then the dependence of the stability margin on the throttle compliance \(\varphi_{\mathrm{dr}}\) may be represented by the graph shown in Fig. 3. In the same figure it is evident that the region of aperiodic stability corresponds to approximately \(1/4\) of the range of permissible variation of the throttle compliance; the stability margin in this region is also considerably lower than the optimum.

Fig. 3. Dependence of the system stability margin on throttle compliance. The graph is plotted for the following parameters: \(m = 1\ \mathrm{kg\,sec^2/cm}\); \(l = 100\ \mathrm{cm}\); \(F = 75\ \mathrm{kg/cm}\); \(P_{10} = 35\ \mathrm{kg/cm^2}\); \(V_0 = 60\ \mathrm{cm^3/sec}\); \(R_0 = 7190\ \mathrm{cm}\); \(W_K(p) = 2\ \mathrm{cm^3/kg}\); \(\alpha = 5\ \mathrm{kg\,sec/cm}\); \(\varphi_{\mathrm{dr}}^{*} = 50\ \mathrm{cm/sec\,kg}\).

References

1. I. Z. Zaichenko. Self-Oscillations in Hydraulic Transmissions of Metallurgical Machines. Moscow, Mashgiz, 1958.

2. V. I. Prokofiev. Calculation of forced oscillations of a hydraulic drive. Mechanical Engineering Bulletin, No. 12, 1967.