# Investigation of dynamic characteristics of three-linear flow regulator

### Abstract

### Keywords

#### For citation:

Khramov B.A., Gusev A.V. Investigation of dynamic characteristics of three-linear flow regulator. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2019;(1):91-97.
https://doi.org/10.38013/2542-0542-2019-1-91-97

## Introduction

Three-linear spool-type flow regulators (hereinafter – SFR) are widely used in throttle controlled hydraulic drives to limit the volumetric flow rate supplied to the actuators in a wide load range. Depending on the hydraulic drive intended use and the SFR loading conditions, different stability and response time requirements may be specified for SFR. In most cases, if SFR is used in hoisting machine hydraulic drives, no special requirements to the response time are specified, however, the regulator operation stability must be ensured in this case.

The SFR in question is applied in a highcapacity hydraulic drive not requiring fast response time. During operation of the drive with the installed SFR, in some cases abnormal noise was observed, caused by oscillations of the SFR movable elements. The acoustic noise level and pressure fluctuations observed in this case enable speaking of resonance phenomena in the hydraulic system and SFR. Unstable operation of SFR can be due to the resonance phenomena resulting from:

- erroneously selected geometry and physical characteristics of the SFR elements (spring rigidity, throttle orifice closure conditions in the spool and sleeve, hydraulic damping, etc.);
- fluctuations of volumetric flow rate and discharge line pressure caused by pump operation;
- load fluctuations on the hydraulic drive actuator;
- action on the actuator – spool valve – of internal perturbing forces (flow turbulence, cavitation, conditions of friction in the spool and sleeve, etc.).

This article provides the results of the work developing study [1] on the search of the zones of SFR resistance to pressure fluctuations with the spools of different geometry. Dynamic model description Fig. 1 shows the analytical diagram of the SFR operation dynamic model. The following assumptions were made during its building:

- hydrodynamic forces and friction forces in the spool and sleeve affecting the spool are small compared to other forces;
- the fluid is incompressible.

**Fig. 1.** Analytical diagram of the three-linear SFR dynamic model

As the mathematical description of D1 and D2 dampers, two models were considered [2]:

1) quadratic – for the opening in the thin wall:

2) linear – for the long crevice flow:

The SFR stability study was conducted for three geometry options of the spool shown in Fig. 2 [1].

**Fig. 2.** Spool geometry options under consideration: a – the first option; b – the second option; c – the third option

Dependence of the throttle orifice local drag coefficient in the spool and sleeve on the spool position А_{1}(х) for the first option:

for the third option:

in general form:

А_{1}( х) = k_{i}x ^{-2},

where ζ - flow coefficient for calculations of ζ = 2 [2];

ρ - working fluid density for calculations of р= 860 kg/m^{3};

α, - spool groove angle, for the second option α_{2} = 30°, for the third option - α_{3} = 45°; i - option number.

The SFR dynamic mathematical model corresponding to the analytical diagram described is a system of differential algebraic equations:

where m – spool weight;

- spool acceleration;

Р_{п1}, Р_{п2} - pressure in spool piston underside cavities;

Q_{п}_{1}, Q_{п}_{2} - volumetric flow from spool piston underside cavities caused by spool motion;

Ад_{1}, Ад_{2}, Вд_{1}, Вд_{2} - drag coefficients of the quadratic and linear damping throttles. In the original SFR design the linear damping throttle Д2 is not provided; however, A_{п}_{2} and B_{п}_{2} coefficients are introduced in the mathematical model to evaluate the impact of potential installation thereof;

х - spool speed.

## Model linearization

Systems (1) and (2) are the systems of non-linear differential equations. To investigate the stability they must be linearized. In this work a small-deflection method was applied for linearization of the system of equations.

Let us introduce the conventions: р_{2}, р - input variables; х - output variable; р_{2.0}, р_{0}, х_{0} - variable values in the linearization point; р'_{2}, р', х' - small deflections in the area of the linearization point. Further expressions are written as applicable to the system of equations (1).

According to study [3], if we decompose a non-linear function F (u, y, t) into Taylor’s series about the values u_{0}, y_{0} and omit the addends containing deflections u' and у' to the power higher than one, we obtain:

The first equation of the system (1), considering equations for dampers Д1, Д2 and flow rates Q_{п1} and Q_{п}_{2} will look as follows:

Then after linearization

Because the linearization point determines the equilibrium position of the system, then, , and the sum of forces acting on the spool in the linearization point is also equal to zero, then

Then the equation will be as follows

The second equation of the system (1), the equation of flow rates, considering expressions for Q_{1}, Q_{2} and Q_{п}_{1} will be as follows:

where

Then after application of equation (5)

Because the linearization point determines the system equilibrium position, then

Let us write down the linearized equation

Having expressed р′ from the equation, we obtain

Based on equation (7), expression (6) will be as follows

or

Having standardized equation (8), we obtain

System (4) may be linearized in a similar way. Then the system of equations (4) can be written as the following equation

Equations (9) and (10) differ from each other only by the constants at variable x′ ( a_{2} and a_{22} ), consequently, with the admissions made, the quadratic dampers described as the local drag, according to equation (1), have no effect on the system stability.

Considering type similarity of equations (9) and (10), let us then omit the designation of coefficient a_{22} and substitute it for a_{2}.

To analyze the stability of the systems described using equations (9) and (10), the Nyquist frequency criterion was used.

Let us designate: s – complex variable in the image space, X(s) – image function corresponding to the original function x(t), P_{2}(s) – image function corresponding to the original function P_{2}(t). After transition to the image space and standardization, equations (9) and (10) will be as follows:

Fig. 3 (a) shows the block diagram of a negative feedback SFR with the feedback ratio K_{о.}_{с}.

**Fig. 3.** Block diagrams of closed system and respective unity feedback system: a – closed block diagram of SFR; b – respective open block diagram of the unity feedback SFR

Fig. 3 (b) shows the SFR block diagram scaled to the unity feedback view. The transfer function of such a system is as follows:

## Stability study

The SFR stability was studied for its operation conditions with the following parameters Р_{2.0} = (1, 5, 12, 20) MPa, р_{1.0} = 0 Pa, A_{2} = 7,2 х х 1013 kg/m^{7}, В_{д1} = 0, Вд_{2} = 1,379 · 10^{12}kg/(m^{4} ·s). The linearization points (р_{2.0}, р_{0}, х_{0}), for which the system stability was determined, were calculated using the static model of SFR operation [1]:

According to the Nyquist criterion of stability, the closed system is stable if its equivalent open system is stable and the amplitude-phasefrequency characteristic (APFC) thereof does not

envelop point (-1,j_{0}) [4].

The conducted analysis showed stability of the open system with the transfer function (12) on Hurwitz criterion.

The results of build-up of the APFC of a system with transfer function (12) are provided in Fig. 4.

The curves in Fig. 4 indicate that the APFC of the open system 12 envelops point (-1, j_{0}) in none of the considered cases, which means that the system (11) is stable. Moreover, the largest stability margin both in terms of amplitude and in terms of phase have APFC systems with Case 1 geometry and a linear damper.

## Conclusion

The conducted SFR stability analysis enables concluding that with the assumptions made:

- SFR of the design in question retains stability with unsteady pressure performance at load р
_{2}and at the inlet р; - none of the considered spool geometry options cause unstable operation of SFR.

The results also showed that in fact the SFR performance stability is not affected by the dampers with the mathematical description similar to that of the local drag (orifice, opening in the thin wall). Herewith, the dampers described by the linear dependence of pressure on flow rate (slot opening) significantly affect the SFR stability margin.

Based on the study results, it is possible to conclude that neither input pressure or flow rate fluctuations, nor SFR geometry cause unstable operation thereof and the reasons should be sought in the internal factors – effect of internal hydrodynamic forces and friction conditions in the spool and sleeve.

During preparation and development of the study methodology the following references [5][6][7] were also used.

### About the Authors

**B. A. Khramov**Russian Federation

**A. V. Gusev**Russian Federation

### Review

#### For citation:

Khramov B.A., Gusev A.V. Investigation of dynamic characteristics of three-linear flow regulator. *Journal of «Almaz – Antey» Air and Space Defence Corporation*. 2019;(1):91-97.
https://doi.org/10.38013/2542-0542-2019-1-91-97