The digital modeling of HSCS is the key basis for studying the characteristics of the HSCS. According to the ratio relationship between the comprehensive value of the surface roughness of the friction plate and the oil film thickness \(\delta\), the friction state of the main and driven friction plates during the working process is divided into four types: Fluid lubrication, mixed friction, boundary friction and static friction.
German scholar Stribeck tested the friction of rolling bearing and sliding bearing, and established the relationship between speed \({v}\), normal load \({G}\), dynamic viscosity of lubricating oil \({{\mu}}\) and friction coefficient \({f}\), which is called Stribeck curve[14]. As shown in Fig. 1, this paper will explain the change of friction state of HSCS in working process.
Comprehensive value of surface roughness of friction pair
$$R={\left( {R_{1}^{2}+R_{2}^{2}} \right)^{1/2}}$$
1
Where \({R_1}\) and \({R_2}\)are the maximum height \({R_y}\) of the surface profile of the two dual friction plates.
(1) Liquid lubrication: The ratio \(\lambda\) of oil film thickness to surface roughness is about \([3,5]\), and the surface of the two friction plates is completely separated by the continuous oil film. The high speed difference operation condition of HSCS belongs to this mode, and the output torque is small under this condition.
(2) Mixed friction: The ratio of oil film thickness to the comprehensive value of surface roughness \(\lambda\) is small(about 3 or less), part of the surface of the two dual friction plates is separated by the oil film, and part of the contact between the asperities occurs, and the friction plate under the speed difference operation condition of HSCS is in this mode.
(3) Boundary friction: The ratio of oil film thickness to surface roughness \(\lambda\) is small(about 0.4-1), the contact between the asperities on the friction surface increases, and the friction surface has only a thin boundary oil film (about 0.1um below). The friction coefficient increases sharply. The HSCS belongs to this working condition when the speed difference is small, and the output speed fluctuates greatly, which is an unstable working area.
(4) Static friction: The oil film thickness is zero, the active and passive friction plate is pressed into one, the relative motion speed is zero, which is the synchronous transmission condition of the hydro-viscous speed control clutch.
The friction state of the hydro-viscous clutch is closely related to the oil film thickness. The relationship between the oil film thickness and the force is shown as Fig. 2
The HSCS adopts the mechanical structure of oil cylinder driving piston to realize the control of oil film thickness. The force of oil film in the clearance of piston extrusion friction pair is as follows.
(1)Force of cylinder pressure on piston \({F_\mathcal{1}}\). (2)Force generated by the disc spring between each pair of friction pairs \({F_{i2}}\). (3)Friction force produced by the sealing ring when the piston moves\({F_3}\), its direction is opposite to the direction of piston movement. (4)Friction force produced by the external spline when the piston moves \({F_{i4}}\), the direction of which is opposite to the direction of piston movement.
When the piston squeezes the oil film in the clearance of the friction pair, the oil film will have a reaction force on the piston, which is called the oil film bearing capacity, as follows:
(1) The oil groove on the friction plate produces a dynamic pressure \({F_{i5}}\) bearing capacity when it moves relative to the piston, and its value is positive. (2) The static pressure \({F_{i6}}\) carrying capacity generated when the lubricating oil passes through the gap between the stationary friction plates is positive.
When the HSCS is in a fluid friction state, the dynamic equation of the first friction plate is as follows
$${F_1} - {F_3} - {F_{14}} - {F_{15}} - {F_{16}}={m_1}{\mathop \delta \limits^{{\cdot \cdot }} _1}+{\xi _1}{\mathop \delta \limits^{\cdot } _1}+{K_1}\left( {{\delta _1} - {\delta _2}} \right)$$
2
The dynamic equation of the second friction plate is as follows
$${K_1}\left( {{\delta _1} - {\delta _2}} \right) - {F_{24}}+{F_{15}}+{F_{16}} - {F_{25}} - {F_{26}}={m_2}{\mathop \delta \limits^{{\cdot \cdot }} _2}+{\xi _2}{\mathop \delta \limits^{\cdot } _2}+{K_2}\left( {{\delta _2} - {\delta _3}} \right)$$
3
The dynamic equation of the third friction plate is as follows
$${K_2}\left( {{\delta _2} - {\delta _3}} \right) - {F_{34}}+{F_{25}}+{F_{26}} - {F_{35}} - {F_{36}}={m_3}{\mathop \delta \limits^{{\cdot \cdot }} _3}+{\xi _3}{\mathop \delta \limits^{\cdot } _3}+{K_3}\left( {{\delta _3} - {\delta _4}} \right)$$
4
The dynamic equation of the (i-2)th friction plate is as follows
$$\begin{gathered} {K_{i - 3}}\left( {{\delta _{i - 3}} - {\delta _{i - 2}}} \right) - {F_{\left( {i - 2} \right)4}}+{F_{\left( {i - 3} \right)5}}+{F_{\left( {i - 3} \right)6}} - {F_{\left( {i - 2} \right)5}} - {F_{\left( {i - 2} \right)6}} \hfill \\ ={m_{\left( {i - 2} \right)}}{\mathop \delta \limits^{{\cdot \cdot }} _{\left( {i - 2} \right)}}+{\xi _{\left( {i - 2} \right)}}{\mathop \delta \limits^{\cdot } _{\left( {i - 2} \right)}}+{K_{i - 2}}\left( {{\delta _{i - 2}} - {\delta _{i - 1}}} \right) \hfill \\ \end{gathered}$$
5
The dynamic equation of the (i-1)th friction plate is as follows
$$\begin{gathered} {K_{i - 2}}\left( {{\delta _{i - 2}} - {\delta _{i - 1}}} \right) - {F_{\left( {i - 1} \right)4}}+{F_{\left( {i - 2} \right)5}}+{F_{\left( {i - 2} \right)6}} - {F_{\left( {i - 1} \right)5}} - {F_{\left( {i - 1} \right)6}} \hfill \\ ={m_{\left( {i - 1} \right)}}{\mathop \delta \limits^{{\cdot \cdot }} _{\left( {i - 1} \right)}}+{\xi _{\left( {i - 1} \right)}}{\mathop \delta \limits^{\cdot } _{\left( {i - 1} \right)}}+{K_{i - 1}}\left( {{\delta _{i - 1}} - {\delta _i}} \right) \hfill \\ \end{gathered}$$
6
For the above expression, \({F_\mathcal{1}}\) is the input, \({F_{i2}}\) and spring compression linear relationship(= 1,2,3,4,……,28), \({F_3}\) and \({F_{i4}}\) can be estimated from relevant test data. Therefore, in the digital test modeling, it is necessary to obtain the relationship between \({F_{i5}}\), \({F_{i6}}\) and parameters such as speed, oil film thickness and fluid static pressure. The specific treatment method will be described in detail in section 3.2.
After solving the above mathematical model, theoretically, the displacement of each friction plate \({\delta _1},{\delta _2},{\delta _3}\)…, ,according to the friction pair clearance initial value \({\delta _1},{\delta _2},{\delta _3}\), …༌so that each pair of friction oil film thickness \({\delta _{sj}}\). The oil film transmission torque of each pair of friction pairs can be expressed as
\(\begin{array}{*{20}{c}} {{\delta _{s1}}={\delta _{01}}+{\delta _2}-{\delta _1}} \\ {{\delta _{s2}}={\delta _{02}}+{\delta _3}-{\delta _2}} \\ \vdots \\ {{\delta _{s\left( {i-2} \right)}}={\delta _{0\left( {j-2} \right)}}+{\delta _{\left( {j-1} \right)}}-{\delta _{\left( {j-2} \right)}}} \\ {{\delta _{s\left( {i-1} \right)}}={\delta _{0\left( {j-1} \right)}}-{\delta _{\left( {j-1} \right)}}} \end{array}\) (7)
Where \({M_j}\) is the torque transmitted by the j-th pair of oil film (\({j}\)= 1,2,3,……,2, ),\(f( \cdot )\) represents the function, \(\mu\) is the fluid viscosity, \({\omega _1}\) is the input rotational speed, \({\omega _2}\) is the output speed, and \(\delta\) is the oil film thickness.
The total output torque of a HSC clutch can be expressed as
$$M=\sum {_{{i=1}}^{{i=28}}} {M_j}$$
9
Where M is the total output torque.
In this report, the external load of the HSC clutch is the fan, and the fan torque is proportional to the square of the fan speed.
$${M_o}=f\left( {\omega _{2}^{2}} \right)$$
10
Where \(f( \cdot )\) denotes the function.
Ignoring the elastic effect of the coupling, the dynamic equation of the HSC clutch driving fan system is
$$M - {M_0} - \Delta M=J\frac{{d{\omega _2}}}{{dt}}+B{\omega _2}$$
11
Where \({J}\) is the total moment of inertia of the system, and \({B}\) is the system damping.
Under stable or constant speed conditions, the following relationship exists in the HSCS driving fan system
$$M={M_o}+{\text{{\Delta}}}M$$
12
Where \({\text{{\Delta}}}M\) is the torque loss.
When the external control pressure is the larger, the local area of the friction pair begins to contact, and the friction plate is in a mixed friction or boundary friction state. Because the friction disc surface is processed with oil groove, the transmission torque of the friction pair can be decomposed into two parts, one part is the transmission torque of the friction plate contact part, and the other part is the oil film shear transmission torque. The i-th pair mechanical contact part of the friction pair transmits torque as follows
$$M_{i}^{\iota }=\frac{2}{3}\pi fk\left( {r_{2}^{3} - r_{1}^{3}} \right){p_{wi}}$$
13
Where is the contact friction coefficient of the friction pair, is the proportion of the area without oil groove on the surface of the friction pair, and \({p_{wi}}\) is the contact pressure of the i-th pair friction pair.
$${p_{wi}}=\frac{{{F_i}}}{{k\pi \left( {r_{2}^{2} - r_{1}^{2}} \right)}}$$
14
Where \({F_i}\) represents the external normal force on each pair of friction pairs, which is related to the output pressure of the proportional pressure reducing valve and the area of the control cylinder.
Substitution Eq. (14) into Eq. (13)
$${M^{\prime}_i}=\frac{2}{3}Ff\frac{{\left( {r_{2}^{3} - r_{1}^{3}} \right)}}{{\left( {r_{2}^{2} - r_{1}^{2}} \right)}}$$
15
The oil film shear transmission torque of the oil groove part in the i-th pair of the friction pair is
$${M^{\prime\prime}_i}=\frac{1}{2}n\pi \mu \left( {{\omega _1} - {\omega _2}} \right)\left( {1 - k} \right)\frac{1}{h}\left( {r_{2}^{4} - r_{1}^{4}} \right)$$
16
Where represents the depth of the oil groove of the friction plate.
The total torque that can be transmitted by the i-th pair friction pair of HSCS in the contact state is
$$\begin{gathered} {M_{ic}}=M_{i}^{\iota }+M_{i}^{{\iota \iota }} \\ =\frac{2}{3}{F_i}f\frac{{\left( {r_{2}^{3} - r_{1}^{3}} \right)}}{{\left( {r_{2}^{2} - r_{1}^{2}} \right)}}+\frac{1}{{2h}}n\pi \mu \left( {{\omega _1} - {\omega _2}} \right)\left( {1 - k} \right)\left( {r_{2}^{4} - r_{1}^{4}} \right) \\ \end{gathered}$$
17
The total output torque \({M_c}\) of the hydro-viscous clutch under contact condition is
$${M_c}=\sum {_{{i=1}}^{{i=28}}} {M_{{\text{ic}}}}$$
18
The transient acceleration process of the fan driven by the HSCS can be obtained by joint solution. In Eq. (17), \({F_i}\), and are input variables, \({F_i}\) is related to the output of the proportional pressure reducing valve, is related to the contact surface material, surface condition and relative motion speed of the friction pair, and is related to the structure of the friction plate.
The modeling process of the entire digital test model is shown in Fig. 3
(1) By controlling the piston to form the input drive of the axial position of the friction plate, the axial position of the friction plate is input into the axial dynamic calculation function and the rotational dynamic calculation function at the same time.
(2) Calculate the torque to obtain the output speed, and transfer the speed value to the axial dynamic calculation function, combined with the axial position to obtain the real-time axial force, and feedback to the axial force calculation input to obtain the real-time position balance value.
(3) The calculated real-time position balance value is input into the rotational force calculation function and the axial force calculation function again until the stable output value of the balance is calculated by the model.