INFLUENCE OF NONLINEAR ELASTIC CHARACTERISTICS OF HYDRAULIC SYSTEM CAVITIES ON TRANSIENT PROCESSES

I. A. Nemirovskii, O. I. Vasenkov, P. Ya. Krauinsh, E. G. Frank

(Presented by the scientific seminar of the Faculty of Automatic Systems)

The cavities of a hydraulic system, filled with working fluid under pressure, possess elastic properties that affect the dynamics of the entire system. Allowance for these elastic properties is necessary in studies of fast transient processes.

It should be noted that until recently it was considered possible to regard the working fluids in hydraulic systems as incompressible, and the cavities as absolutely rigid.

At present, as a rule, both the compressibility of the fluid and the elasticity of the cavities are taken into account; however, the elastic characteristic of a cavity filled with fluid is considered to be a constant quantity independent of the parameters of the dynamic process.

Studies have shown that these characteristics are not constant [1; 2; 3].

In the general case, the following elastic elements may be included in some line of a hydraulic system:

1. Working fluid. The compressibility of the working fluid, characterized by the coefficient $\beta$, decreases with increasing pressure. When undissolved air is present in the working fluid, the dependence of $\beta$ on the pressure $p$ is very significant and depends on the quantity of undissolved air, the exponent $n$ of the polytropic process occurring in the gas inclusions, and the character of mass exchange between the liquid and gas phases during the dynamic process.

2. Cavities formed by metal pipelines, hydraulic-motor chambers, and hydraulic apparatus have a constant compliance $K(p)_{\mathrm{m}}$, but the magnitude of their compliance, determined experimentally, is, as a rule, greater than the calculated value of $K(p)_{\mathrm{m}}$. This can be explained by deformations of joints, seals, threaded connections, etc., which are not taken into account in calculations.

3. Flexible high-pressure hoses possess a pressure-dependent nonlinear compliance, which can be estimated by the coefficient $K(p)_{\mathrm{sh}}$, determined experimentally by static loading of the hose with pressure.

In dynamics, the complex cord-rubber-metal structure of the hose exhibits specific properties:

a) depending on the rate of change of the volume of the hose cavity (the rate of change of deformation of the hose walls), the magnitude of the hose compliance changes.

As a first approximation, this property of the hose can be estimated by the quantity \(\gamma \left[\dfrac{dW}{dt}\right]^x\), where \(W\) is the hose volume, and \(\gamma\) and \(x\) are coefficients determined experimentally.

The authors’ experiments showed that \(\gamma\) and \(x\) may be taken as constant in each particular interval of rates of pressure change \(\dfrac{dp}{dt}\);

b) individual standard sizes of flexible hoses have clearly expressed hysteresis properties.

Thus, the line of the hydraulic system, including all the elastic elements listed above, can be estimated by the total compliance coefficient:

\[ K(p)_{\Sigma}=\frac{1}{W_{\Sigma}}\left[\beta W_{\mathrm{l}}+K(p)_{\mathrm{m}}W_{\mathrm{m}}+K(p)_{\mathrm{h}}W_{\mathrm{h}}\right], \tag{1} \]

where \(W_{\Sigma}\), \(W_{\mathrm{m}}\), \(W_{\mathrm{h}}\), and \(W_{\mathrm{l}}\) are the volumes of the entire line, the cavities in the metal parts, the hoses, and the volume of liquid located in the line, respectively. If it is necessary to take into account cavities in metal parts that differ in configuration and housing material, then

\[ K(p)_{\mathrm{m}}W_{\mathrm{m}}=\sum K(p)_{\mathrm{m}i}W_{\mathrm{m}i}. \]

The coefficient \(K(p)_{\Sigma}\) makes it possible to take into account the entire line in the dynamics in the form of the term

\[ q_{\mathrm{def}}=K(p)_{\Sigma}W_{\Sigma}\frac{dp}{dt}. \tag{2} \]

Since \(K(p)_{\Sigma}\) depends on the parameters of the transient process, i.e., is a nonconstant quantity, it is convenient to calculate the dynamics of the hydraulic system by approximate graphical-analytical methods [2].

Let us consider several questions associated with taking into account the dependence of the elastic properties of the hose on the rate of deformation of its cavity, i.e., with allowance for the quantity \(\gamma \left[\dfrac{dW_{\mathrm{h}}}{dt}\right]^x\).

Represent the increment \(\Delta p\) of pressure in the system as the sum

\[ \Delta p=\Delta p_y+\Delta p_c, \]

where \(\Delta p_y\) and \(\Delta p_c\) are components depending on the elastic forces of the flexible hose, i.e., proportional to \(\dfrac{1}{K(p)_{\mathrm{h}}}\), and on forces proportional to \(\left[\dfrac{dW}{dt}\right]^x\), respectively.

Then

\[ \frac{dp}{dt}=\frac{dp_y}{dt}+\frac{dp_c}{dt}. \tag{3} \]

In turn:

\[ \frac{dp_y}{dt}=\frac{Q_{\mathrm{h}}}{W_0K(p)_{\mathrm{h}}}\ldots \tag{4} \]

\[ \frac{dp_c}{dt}=\gamma\left[\frac{dQ_{\mathrm{h}}}{dt}\right]^x, \tag{5} \]

where \(Q_{\mathrm{h}}\) is the flow rate caused by the change in the hose volume.

Relations (3), (4), and (5) make it possible to find the values of $\gamma$ and $\varkappa$ from experimental data. For a finite time interval $\Delta t$ we have

\[ \gamma = \left[\frac{\Delta t}{\Delta Q_{\mathrm{sh}}}\right]^{\varkappa} \left[\frac{\Delta p}{\Delta t} - \frac{Q_{\mathrm{sh,avg}}}{W_0 K'(p)} \right]. \tag{6} \]

From oscillograms of the dynamic process, the following are determined:

\[ \Delta t = t_2 - t_1; \qquad \Delta Q_{\mathrm{sh}} = Q_{\mathrm{sh}2} - Q_{\mathrm{sh}1}; \qquad Q_{\mathrm{sh,avg}} = \frac{Q_{\mathrm{sh}1} + Q_{\mathrm{sh}2}}{2}; \]

\[ \Delta p \ \text{and}\ K'(p)_{\mathrm{sh}}\ \text{for}\ p + \frac{\Delta p}{2}. \]

Then, by the method of least squares, the results obtained are processed taking into account that $\gamma = \mathrm{const}$ and $\varkappa = \varkappa\!\left(\dfrac{dp}{dt}\right)$.

The investigations performed by the authors make it possible to conclude that calculation of transient processes in hydraulic systems without taking into account the elastic-dynamic characteristics of all elastic elements of the system leads to considerable errors (up to 60–70% in pressure amplitude and up to 30–40% in pressure phase).

In particular intervals of values of $\dfrac{dp}{dt}$, mean constant values of $\gamma$ and $\varkappa$ may be used. Calculation of the dynamics of a hydraulic system without taking into account the dependence of the elastic characteristics of the hose on the rate of deformation may be carried out when $\dfrac{dp}{dt} < 100\ \mathrm{kg}/\mathrm{cm}^2\!\cdot\!\mathrm{s}$. When $\dfrac{dp}{dt} > (100 \div 150)\ \mathrm{kg}/\mathrm{cm}^2\!\cdot\!\mathrm{s}$, the quantity

\[ \frac{dp_c}{dt} = \gamma \cdot \left[\frac{dQ_{\mathrm{sh}}}{dt}\right]^{\varkappa} \]

should be taken into account.

REFERENCES

1. T. M. Bashta. Aircraft Hydraulic Drives. Mashgiz, 1959.
2. I. A. Nemirovsky. A Graph-Analytical Method for Calculating Hydraulic Drives. Mashinostroenie Publishing House, 1968.
3. V. N. Prokofiev, I. A. Luzanova. Elastic characteristics of a hydraulic drive. Izv. vuzov, Mashinostroenie, No. 7, 1966.