FREQUENCY CHARACTERISTICS OF HYDRAULIC LINES

Yu. Ya. Komisarenko, P. Ya. Krauinsh, O. I. Vasenkov

(Presented by the Conference of Young Scientists)

In recent years, for the automation of various processes, hydroautomatic devices have found wide application, possessing a number of advantages [1, 2, 4, 5]. As a rule, some units of a hydraulic system undergo significant mutual displacements; to connect such units, high-pressure hoses (HPH) (GOST 6286–60) are used, whose elasticity substantially affects the characteristics of the machine as a whole [2, 6]. A number of studies [7, 1] have shown that the compliance coefficient \(k(p)\) depends on pressure. At the same time, the cited works do not propose an empirical dependence \(k(p)=f(p)\), do not show the degree of stability of \(k(p)\) for hoses (HPH) of one standard size, and do not provide a method for taking into account the variable compliance of HPH in an analytical study of dynamic characteristics.

Our aim is to develop a method for taking into account the nonlinear characteristics of HPH in the dynamic analysis of the hydraulic mechanisms of machines.

An HPH may be represented as a dynamic element whose input is the flow rate going into its deformation, and whose output is the pressure in the HPH. The relationship between the flow rate going into deformation of the line \(Q(t)\) and the pressure in the line \(p(t)\) can be written in the form [1, 2, 6]:

\[ Q(t)=k(p)W_0\frac{dp}{dt}, \tag{1} \]

where \(k(p)\) is the compliance coefficient of the line (a quantity inverse to the modulus of elasticity), in \(\text{cm}^2/\text{kg}\);

\(W_0\) is the initial volume of the line, in \(\text{cm}^3\).

We carried out a refinement of the dependence \(k(p)=f(p)\) in order to determine the stability of this dependence for one standard size of HPH on the setup shown in Fig. 1. A plunger 2, connected by means of string 6 to indicator 5, is placed in the bore of housing 3. The investigated HPH 4 with pressure sensor 1, filled with working fluid, was connected to cavity B. Air was carefully removed from the system. Fluid under variable pressure \(p\) was supplied to cavity A. Since the area of the plunger on the side of cavities A and B is the same, leakage of fluid from cavity B was eliminated. The dependence \(k(p)=f(p)\) was determined from the known expression:

\[ d\Delta W = k(p) W_0\, dp . \tag{2} \]

A study of a batch of RVD in accordance with GOST 6886—60 showed the stability of the dependence \(k(p)=f(p)\).

The dependence shown in Fig. 1 can be approximated by the following expression:

\[ k(p)=\left(\frac{60}{p+6}+2\right)\cdot 10^{-4}. \tag{3} \]

Thus, in order to increase the accuracy of dynamic calculations, it is necessary to take into account in expression (1) the variable compliance \(k(p)\) (3). To calculate the frequency characteristics we used the method of energy linearization [3, 4], which assumes equality of the works over a period in the nonlinear and linearized systems:

\[ A_{\mathrm{n}}=A_{\mathrm{l}}. \tag{4} \]

Fig. 1. Dependence of the compliance coefficient on pressure

The work performed by the nonlinear system over a period, taking (2) and (3) into account, can be calculated as:

\[ A_{\mathrm{n}}=\int_{0}^{p_a} p\cdot d\Delta W = W_0\left[ P_a^2+60p_a+360\ln\frac{6}{p_a+6} \right]\cdot 10^{-4}. \tag{5} \]

The work performed by the linearized system is

\[ A_{\mathrm{l}} = \int_{0}^{p_a} p\cdot d\Delta W = \int_{0}^{p_a} p\cdot W_0\beta\cdot dp = \frac{1}{2}p_a^2 W_0\beta, \tag{6} \]

where \(p_a\) is the amplitude value of the pressure in the system;
\(\beta\) is the coefficient of energy linearization, constant for the given \(p_a\).

The value of the amplitude pressure \(p_a\) can be determined, for a sinusoidal input signal (1), as follows:

\[ Q(t)=Q_0\sin\omega t, \]

over a half-period a volume of liquid \(\Delta W\) is pumped into the system, equal to

\[ \Delta W = 2\int_{0}^{\pi/2\omega} Q_0\cdot \sin\omega t\cdot dt = \frac{2Q_0}{\omega}. \tag{7} \]

The given change in volume \(\Delta W\), created by the flow rate \(Q(t)\), corresponds to an increase in pressure up to \(p_a\) according to the following relation:

\[ \frac{2Q_0}{\omega}=\int_0^{p_a} d\Delta W . \tag{8} \]

After transformations, taking (2) and (3) into account, expression (8) takes the following form:

\[ \ln \frac{p_a+6}{6}+\frac{p_a}{30} = \frac{Q_0\cdot 10^4}{30W_0\omega}. \tag{9} \]

Relation (9) expresses the connection between \(p_a\), \(Q_0\), and \(\omega\), i.e., the dependence \(p_a=f(Q_0\omega)\), needed for determining the coefficient of energy linearization.

Fig. 2. Frequency characteristics of hydraulic lines.
Curve 1 — calculation by formulas 11, 9, 10.
Curve 2 — calculation by formula 12.
Curve 3 — experimental data [1]

From expressions (5) and (6), taking (4) into account, the coefficient of energy linearization \(\beta\) can be found:

\[ \beta = 2\cdot 10^{-4} \left[ \frac{360}{p_a^2}\cdot \ln \frac{6}{p_a+6} + \frac{60}{p_a} + 1 \right]. \tag{10} \]

Using relations (1), (9), (10), one can construct the logarithmic amplitude frequency characteristic (LAFCh) of high-pressure hoses according to GOST 6886—60. As an example, in Fig. 2 LAFCh are constructed:

curve 1 —

\[ \left. L(\omega) \right|_{\beta=f(Q_0,\omega)} = 20\lg |W(j\omega)| = 20\lg \frac{1}{W_0\beta\omega}, \]

curve 2 —

\[ \left. L(\omega) \right|_{k(p)=\mathrm{const}.} = 20\lg |W(j\omega)| = 20\lg \frac{1}{W_0 k(p)}\cdot \frac{1}{\omega}, \]

curve 3—experimental data from [1] for the following quantities:

\(Q_0 = 13.2\ \text{cm}^3/\text{s}\)—amplitude value of the input signal,

\(W_0 = 100\ \text{cm}^3\)—initial volume of the line,

\(k(p)=\text{const}=3.5\cdot10^{-4}\ \text{cm}^2/\text{kg}\)—the recommended constant value of compliance for this pressure range [2, 5, 6].

As can be seen from Fig. 2, the logarithmic amplitude–frequency characteristic (LAFCh) of the linearized element (curve 1) differs substantially from the LAFCh of the linear element \((k(p)=\text{const})\) (curve 2). The cutoff of \(L(\omega)\) occurs considerably earlier than in the linear model. The dependence presented agrees well with the experimental data given in [1] (curve 3). This makes it possible to adopt the method set forth as a basis for the dynamic calculation of control lines of hydroautomatic systems made in the form of RVD. Evidently, the application of this calculation method may be considered promising when local small gas inclusions are taken into account in the system, when the elastic modulus of the system (or the reciprocal quantity—the compliance of the system) becomes a function of pressure [1, 2, 5, 6].

As for determining the phase shift \(\psi(\omega)\), theoretical investigations lie beyond the scope of the present work; apparently, it also differs from the phase shift of the linear model. Analysis of the obtained curves (Figs. 1 and 2) makes it possible to conclude that it is unjustified to treat an RVD as a first-order linear element (an integrating element).

REFERENCES

1. I. A. Nemirovskii. A grapho-analytical method for calculating hydraulic drives. “Mashinostroenie,” Moscow, 1965.

2. V. A. Leshchenko. Hydraulic servodrives for the automation of machine tools. Mashgiz, Moscow, 1962.

3. L. S. Goldfarb. Theory of Automatic Control, Parts I and III, MEI, Moscow, 1965.

4. N. Guyon. Studies of hydraulic systems. Mashgiz, Moscow, 1964.

5. T. M. Bashta. Hydraulic systems of aircraft. Mashgiz, Moscow, 1966.

6. V. A. Khokhlov. Electrohydraulic servodrive. “Nauka” Publishing House, Moscow, 1964.

7. I. A. Nemirovskii et al. A graphical method for investigating nonlinear processes in the hydraulic systems of machine tools. “Stanki i instrument,” No. 9, 1965.

8. Theory of Automatic Control, ed. by Ketumik, Moscow, “Vysshaya shkola,” 1968.