## 2 A Refined Friction Model in the Spool-Sleeve Pair
* Frictional force in the spool pair: the literature [24] indicates that friction in this type of pair is fluid friction due to the presence of an oil film between the spool and the sleeve and can be calculated using the following formula [14]: \[F_{t}=\pi\cdot d_{t}\cdot\frac{l}{\eta}\cdot\mu\cdot\frac{dx_{su}}{dt},\] (12)  
Figure 2: The following forces (projection of forces) act on the spool of a directional control valve equipped with proportional electromagnets: \(F_{s}\)—the force generated by the stiffness of springs centring the spool, \(F_{t}\)—the frictional force in the spool pair, \(F_{d}\)—the hydrodynamic force, \(F_{b}\)—the spool inertia force and \(F_{M}\)—the force generated by control elements (e.g., proportional solenoids). where \(d_{t}\) and \(l\) are the piston diameter and length (m) and \(h\) is the thickness of the gap in the spool pair (m). According to [14], Coulomb friction can be disregarded where the spool pair has adequate lubrication. This simplification can lead to large discrepancies among both experimental results and results based on this model for higher spool shifting speeds [25]. Based on tribological considerations [26; 27], a more detailed model of friction in the form of mixed friction can be assumed to exist in the spool-sleeve pair. The forces acting on the spool moving in the sleeve can be calculated using the following formulas--model 1:  
\[m_{c}\frac{d^{2}x_{su}}{dt^{2}}+\pi d_{t}\frac{l}{h}\mu\bigg{(}\frac{dx_{su}}{ dt}-\frac{dw}{dt}\bigg{)}+c_{sz}(x_{su}-w)+k_{t}(d_{k0}-a_{k0})\delta\bigg{(} \frac{dx_{su}}{dt}-\frac{dw}{dt}\bigg{)}=0, \tag{13}\]  
\[k_{t}=C_{k}u_{s}, \tag{14}\]  
where \(w\) is the amplitude of external excitation (m), \(\mu\) is the dynamic viscosity of the fluid (Ns/m\({}^{2}\)), \(d_{k0}\) and \(a_{k0}\) are the dimensions of the microwedge (m), \(k_{t}\) is the coefficient of proportionality between the frictional force and the value of contact strain (N/m), \(C_{k}\) is the contact stiffness of the surface (N/m) and \(\mu_{s}\) is the coefficient of static friction (-). The first term of Equation (13) describes the inertia of the spool movement, the mass of the associated fluid and \(1/3\) of the mass of the springs. The second term describes the fluid frictional force that occurs during the movement of the spool and proportional to its relative speed, while the third term describes the stiffness force from the centring springs, and the fourth term describes the static friction force. The second and last terms of Equation (13) therefore describe the mixed frictional force, which can be written as follows:  
\[F_{Tm}=\pi d_{t}\frac{l}{h}\mu\frac{dx_{su}}{dt}+k_{t}(d_{k0}-a_{k0}), \tag{15}\]  
The term \(\delta\) corresponds to the Dirac delta function, which takes the value of \(1\) when the relative velocity of the spool and sleeve is \(0\); it otherwise takes the value of \(0\). This function can be approximated as:  
\[\delta\bigg{(}\frac{dx_{su}}{dt}-\frac{dw}{dt}\bigg{)}\approx\frac{1}{1+10^{ \theta}\cdot\bigg{(}\frac{dx_{su}}{dt}-\frac{dw}{dt}\bigg{)}^{t}}, \tag{16}\]  
The value of exponent \(\theta\) should be as large as possible (\(\theta>>1\)) and the exponent \(t\) should take a large value and be an even number (\(t>>1\) and \(t=2\times k_{p}\), where \(k_{p}\) is a positive integer). These requirements are due to the fact that according to the literature [10], the denominator of expression (16) should tend towards infinity when the relative velocity of the spool and sleeve is other than \(0\). If there are seals in the spool pair, this must also be taken into consideration in model (13).