# Experimental Methods for Measuring the Viscous Friction Coefficient in Hydraulic Spool Valves

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Tested Component

_{LS}) of the hydraulic actuators with the pump delivery pressure (p

_{S}). In particular, the analyzed FC is mounted on an axial piston pump, and it has the function of regulating the swash plate angle.

_{S}and p

_{LS}and on the force of an adjustable spring. The pump displacement is a function of the equilibrium of two actuators 1 and 2 with different diameters. The smaller actuator 2 is always connected to the delivery pressure, while the pressure acting on the larger actuator 1 is modulated by the FC. When the FC is not regulating (control in saturation), the actuator 1 is connected to the atmospheric pressure, and the actuator 2 holds the maximum pump displacement. On the contrary, when the valve spool is in equilibrium in an intermediate position, the pressure in the piston 1 is increased, its exerted force is able to balance the force of the piston 2, and the pump displacement is reduced. The condition of equilibrium of the spool implies that the delivery pressure p

_{S}is maintained equal to the load sensing pressure p

_{LS}incremented by a contribution given the adjustable spring, typically around 20 bar. In this way the FC maintains a fixed differential pressure across the main control valve, represented in the scheme by a variable restrictor CV, so that the flow rate delivered to the actuator depends only on the flow area of CV and not on the load pressure p

_{LS}.

_{LS}represents the feedback signal, it features poor behavior in damping high inertial loads [15]. Hence a possible risk is the instability that is also influenced by the viscous friction force of the FC. It is desirable that a simulation model of the hydraulic circuit could be able to predict the instability, but if the real damping coefficient of the spool of the FC is unknown, the reliability of the model for such a study could be questionable. For this reason, the spool of a load sensing pressure compensator represents a good example of application of the proposed methods for the measurement of the viscous friction.

## 3. First Method

#### 3.1. Theoretical Analysis

_{1}is generated (i.e., a sinusoidal signal), while the spring chamber is connected at atmospheric pressure. As a consequence, the mass will oscillate, with the same frequency of the input signal, but with amplitude and phase delay that are influenced by the viscous friction between the mass and the housing. However, the measurement of the position x of the mass by means of an LVDT will alter the dynamics of the system, above all if the size of the system is small. The solution is to measure both pressures p

_{1}and p

_{2}since their amplitude ratio and phase shift are also influenced by the viscous friction of the mass.

- -
- mechanical natural frequency:$${\mathsf{\omega}}_{\mathrm{m}}=\sqrt{\frac{\mathrm{k}}{\mathrm{m}}}$$
- -
- mechanical damping ratio:$${\mathsf{\xi}}_{\mathrm{m}}=\frac{\mathrm{c}}{2\sqrt{\mathrm{mk}}}$$
- -
- hydraulic natural frequency:$${\mathsf{\omega}}_{\mathrm{h}}=\sqrt{\frac{1}{\mathrm{LC}}}$$
- -
- hydraulic damping ratio:$${\mathsf{\xi}}_{\mathrm{h}}=\frac{\mathrm{R}}{2}\sqrt{\frac{\mathrm{C}}{\mathrm{L}}}$$

_{1}and p

_{2}(more details are available in reference [16]):

_{2}has two natural frequencies, since the denominator is of the 4th order. Such frequencies can be calculated analytically only if the dissipative terms c and R are neglected; however, the main influence of such terms is on the damping and not on the frequency values that therefore can be expressed by Equation (17):

_{2}has also two complex conjugate zeros coincident with the poles of the function F

_{1}.

_{2}can be used for determining c.

#### 3.2. Numerical Case

_{2}is calculated using the same parameters of the pipe and of the fluid used in the experimental tests described in Section 3.3. It must be noted that the spring stiffness was deliberately reduced with respect to the original control valve in order to lower the value of the first frequency given by Equation (17). In Table 1 the calculated quantities are listed.

_{1}< ω

_{m}< ω

_{h}< ω

_{2}. The transfer function F

_{2}(s) is plotted in Figure 3 in terms of magnitude and phase shift.

_{1}–ω

_{m}is highly influenced by the viscous friction coefficient c, as shown in Figure 4, therefore, it is enough to test the system up to a maximum frequency of the order of ω

_{m}. It can be convenient to use a low stiffness spring; in this way it is possible to limit the maximum frequency of the input pressure p

_{1}with two advantages:

- It is easier to generate the sinusoidal signal with a reasonably high amplitude;
- If the mechanical frequency ω
_{m}is far away from the hydraulic frequency ω_{h}, the transfer function around the lower natural frequency ω_{1}is mainly influenced by the mechanical system mass–spring and not by the dynamics of the pipe; therefore, any uncertainty in the simulation of the hydraulic system has a negligible influence on the evaluation of the friction coefficient of the mass.

#### 3.3. Experimental Procedure

_{1}at the inlet of the pipe. The servovalve was fed (supply line) at constant pressure by a pressure reducing valve. A manual variable restrictor R1, connected with the return line, was mounted downstream the working port A of the servovalve, while port B was closed. The LS control was connected through the pipe to the junction J located between the servovalve and the manual restrictor. The spring chamber (LS signal) was connected directly to atmosphere, while port A was closed. Two GS XPM5 pressure transducers, each with a measuring range 0–50 absolute bar, were mounted at the two ends of the pipe. The oil temperature was measured by means of the sensors mounted in the inlet and return lines. The pressure in the junction J was modulated by the flow area of the manual restrictor and of the servovalve. In fact, once a suitable value of the flow area of the manual restrictor is set, if a sinusoidal input current is supplied to the servovalve, then an oscillating pressure is generated in the junction J. A photo of the hydraulic circuit is shown in Figure 6.

_{1}oscillate between 3 and 9 bar at 5 Hz. A control and data acquisition program was developed in the NI Labview

^{®}to perform the test. The software generates trends of sinusoidal signals with increasing frequencies, and at the end of each stage the amplitude and the phase shift of the ratio between the pressures p

_{2}and p

_{1}are calculated. For the present study, the frequency was increased from 10 Hz to 150 Hz with a logarithmic scale, and for each frequency value the Bode diagram was calculated on a total time interval of about 4 seconds. The sampling frequency used for acquiring the pressure signals was 10 kHz. The results obtained at 40 °C with ISO VG 46 oil in six different tests are shown in Figure 8 and Figure 9. Before each test, the warm-up procedure was performed. Very good repeatability in both magnitude and phase was observed. Moreover, in spite of the simplifications used for determining the transfer function F

_{2}plotted in Figure 2, the experimental function was very similar not only qualitatively, but also quantitatively.

#### 3.4. Evaluation of the Friction Coefficient

_{1}), while a fixed hydraulic capacity simulates the volume in the T junction where the transducer for measuring p

_{2}is mounted. The pipe is simulated with a distributed parameter RLC model with 12 internal nodes and frequency dependent friction with 5 states. It was checked that such a pipe model gives the same results, in terms of time response, of a CFD 1D model that cannot be used for a linear analysis.

^{2}were also indicated. It can be noticed that the friction coefficient can be considered proportional to the viscosity with a high degree of confidence; moreover, the straight line passes very close to the origin of the axes. Therefore, the results shown in Figure 14 demonstrate that the nature of the friction is manly viscous, since it tends to zero when the viscosity tends to zero.

## 4. Second Method

^{®}5064 was used together with the pressure transducers reported in Table 2; the sampling rate was set equal to 20 kHz.

#### Evaluation of the Friction Coefficient

## 5. Discussion

_{f}and the contact surface S, is given by Equation (20):

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 7.**Cross section of the LS control in the configuration for the tests (by-pass P–T through R2 closed).

**Figure 24.**Comparison between experimental and simulated Bode diagrams (30 bar, 1.2 mm); LVDT installed.

Variable | Value | Unit | Variable | Value | Unit |
---|---|---|---|---|---|

L | 2.182 · 10^{7} | kg/m^{4} | ω_{m} | 108 | Hz |

R | 1.935 · 10^{9} | Pa·s/m^{3} | ξ_{m} | 0.126 | - |

C | 5.660 · 10^{−15} | m^{3}/Pa | ω_{h} | 453 | Hz |

a_{0} | 3.731 · 10^{12} | - | ξ_{h} | 0.0155 | - |

a_{1} | 2.567 · 10^{9} | - | ω_{1} | 67 | Hz |

a_{2} | 2.140 · 10^{7} | - | ω_{2} | 733 | Hz |

a_{3} | 2.602 · 10^{2} | - |

Variable | Sensor | Main Features |
---|---|---|

p_P, p_I | Kistler 6005 | 0–1000 bar Bandwidth 140 kHz |

p_P, p_I | Danfoss 1250 | 0–400 bar ± 0.5%FS Bandwidth 1 kHz |

Amplifier | KISTLER^{®} 5064 | 2-channel |

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**MDPI and ACS Style**

Rundo, M.; Casoli, P.; Lettini, A.
Experimental Methods for Measuring the Viscous Friction Coefficient in Hydraulic Spool Valves. *Sustainability* **2021**, *13*, 7174.
https://doi.org/10.3390/su13137174

**AMA Style**

Rundo M, Casoli P, Lettini A.
Experimental Methods for Measuring the Viscous Friction Coefficient in Hydraulic Spool Valves. *Sustainability*. 2021; 13(13):7174.
https://doi.org/10.3390/su13137174

**Chicago/Turabian Style**

Rundo, Massimo, Paolo Casoli, and Antonio Lettini.
2021. "Experimental Methods for Measuring the Viscous Friction Coefficient in Hydraulic Spool Valves" *Sustainability* 13, no. 13: 7174.
https://doi.org/10.3390/su13137174