## 2 A Refined Friction Model in the Spool-Sleeve Pair  
Below, the results of theoretical work to describe the nature of interaction in the basic structural node of a spool valve, namely, the spool-sleeve pair, are presented. This will enable a more detailed description of the transmission pathways of external mechanical oscillations to the directional control valve spool, which in turn can be used to minimise the transmission of these oscillations without reducing the static and dynamic parameters of the valve. A mathematical model of spool movement is presented using the following general simplifying assumptions [21]: minor influences are disregarded; the tested system does not cause any changes to its environment; distributed parameters are replaced by lumped parameters; simple linear (linearised) relationships exist among the physical variables that describe cause and effect; the physical parameters do not change over time (they are not functions of time); and uncertainty and noise are disregarded. Based on these principles, the following detailed simplifying assumptions were used in the mathematical model of the valve: the effect of the elasticity of the directional control valve body and spool was disregarded and the directional control valve was assumed to be completely rigidly mounted to a tight, oscillating substrate; the mass of oil contacting the directional control valve spool, compared to the mass of the spool, is small enough to be disregarded in the analysis; lumped parameters were used, because distributed-parameter systems must be described by partial differential equations, which are generally very difficult to solve; the dimensions of all gaps in the spool pair remain unchanged; the physical properties of the process fluid remain unchanged; and the kinematic excitation acting on the directional control valve body is harmonic. In addition, in order to simplify the mathematical description, it was assumed that the spool is in the neutral position (there is no fluid flow through the directional control valve), it is not loaded by forces caused by static pressure and there are no resistive forces caused by seal friction. When analysing the theoretical model of a spool valve, the balance of forces acting on the spool can be considered as a starting point (Figure 2). The literature [14] divides forces into transverse and longitudinal, depending on their direction. These forces determine the resistance of the spool movement that must be overcome when shifting the flow direction. Lateral forces caused by the movement of the spool relative to the sleeve have no direct effect, but by acting perpendicularly to the axis of the spool they determine the frictional forces (and, as a result, the resistive forces) on the surface of the spool; these forces have been examined, for example, in [22]. On the other hand, the values of longitudinal loads are often much greater than the values of inertia and the frictional forces acting on the spool; they are therefore decisive when determining the force necessary to shift the valve spool [23]. The forces acting on a moving spool are as follows:  
* Spool inertia: \[F_{b}=m_{c}\frac{d^{2}x_{su}}{dt^{2}},\] (10) where \(m_{c}\) is the mass of the moving control elements of a piston spool and its associated liquid column (kg) and \(x_{su}\) is the spool displacement (m). * Stiffness of centring springs: \[F_{s}=c_{sz}\cdot x_{su},\] (11) where \(c_{sz}\) is the equivalent stiffness of centring springs (N/m).