Investigation on the Effects of Operating Parameters on the Transient Thermal Behavior of the Wet Clutch in Helicopters

Abstract

The aviation wet clutch, as an indispensable component in helicopters, is particularly vulnerable to performance deterioration due to temperature rises, especially in high-power-density and high-torque conditions. Consequently, a comprehensive thermal-fluid-dynamic model, coupled with a dynamic model considering the spline friction and split spring and a thermal model considering the heat transfer parameters in friction pair gaps, was proposed in this work. The effects of operating parameters on the transient thermal behaviors of friction discs were investigated. A rise in rotation speed from 2000 rpm to 2400 rpm facilitates a 10.1% increase in the maximum temperature of the friction discs. An increase in control oil pressure from 1.5 MPa to 1.9 MPa rises the maximum temperature of the friction disc by 19.4%. Moreover, increased lubrication oil flow not only depresses the maximum temperature of the friction disc by 14.5% but also significantly narrows the temperature gradient by 16.7% and improves the temperature field uniformity. Therefore, reasonably increasing lubricant oil flow and decreasing control oil pressure can effectively reduce temperature rises and improve the temperature field uniformity. These results contribute to designing and developing optimal control strategies to enhance the comprehensive performance of helicopter transmission.

1. Introduction

Wet multi-disc friction clutches, with their advantages of excellent high-power transmission capacity and other excellent properties, are extensively utilized across various sectors, including aviation, aerospace, and transportation. Compared with automotive friction clutches, the service conditions of the aviation friction clutch are more severe, with speeds exceeding 6000 rpm, power exceeding 1000 kW, power density exceeding 4 kW/kg, and torque transmission exceeding 1000 N·m [1]. As illustrated in Figure 1, there is usually a driving shaft, a driven shaft, a piston, multiple friction discs, several split springs, and other necessary parts in an aviation wet multi-disc clutch. The operation states of wet multi-disc clutches are controlled by employing an electro-hydraulic proportional valve to regulate the flow of pressurized oil entering the piston cylinder. There is a lot of frictional heat between the friction pair resulting from the excessive rotational speed during differential engagement, causing a sharp temperature rise at the friction interface and a significant deterioration in clutch performance.

Recently, many scholars have performed extensive studies concerning the temperature distribution characteristics of wet clutches, primarily concentrating on automotive friction clutches. Li et al. [3] proposed a thermal model considering heat flux, convection, and conduction heat transfer for friction components made of carbon fabric in wet clutches, and a parametric analysis of the temperature of friction components was performed using FEM in the engagement phase. Moreover, to explore the temperature dynamics of wet clutches in the clutch engagement phase, the three-dimensional FEM was employed to analyze the transient thermal behaviors for wet clutches with different open grooved friction discs in references [4,5,6,7,8,9,10,11]. Kong et al. [12] formulated a CFD model to dissect the fluid dynamic behaviors between friction pair gaps in wet clutches and concurrently studied the temperature distribution across contact surfaces during both the engagement and disengagement phases. Subsequently, Xiong et al. [13] employed coupled FEM to establish a thermal analysis model for a multi-disc clutch, and accordingly investigated the heat flux distribution coefficients and thermal behaviors in friction pair gaps during the sliding process in the long term, which was substantiated by a test bench for the multi-disc clutch. Li et al. [14] presented a thermoelastic FEM for the friction clutch in a heavy-duty vehicle to study the temperature distribution characteristics and the thermoelastic stress in the friction pair under different conditions. Yevtushenko et al. [15,16] proposed effective 2D and 3D friction clutch temperature field calculation models based on FEM to investigate the effects of carbon-containing additives in different friction material structures on the clutch temperature field and verified them using a bench experiment. Padmanabhan et al. [17] employed the commercial software ANSYS Workbench 18 to analyze the temperature field distribution in the failure region of automobile clutches during operation. Nevertheless, research data on the temperature field of aviation clutches is scarce. Tan et al. [18] explored the effect of oil groove structure on the temperature field of an aviation clutch with a maximum rotation speed of 2450 rpm using ANSYS Workbench 18. With the help of FLUENT 18.0, Wei et al. [19] investigated the temperature distribution of an aviation clutch under the condition of a maximum relative rotation speed of 3000 rpm.

Taking the heat transfer characteristics on the contact surface of the friction pair into consideration, Wu et al. [20] developed a 2D thermal model of wet clutches to explore the temperature dynamic characteristics in the unstable and stable sliding phases of clutch operation and validated the constructed model using corresponding bench experiments. Zheng et al. [21] formulated a comprehensive numerical heat transfer model that considered friction pair gaps to examine temperature rise regularities of wet clutches in successive shift processes and subsequently corroborated the simulation results employing SAE#2 bench equipment. Liu et al. [22] employed a resistance network to construct a real-time temperature prediction model and investigated thermal characteristics of friction components in automobile clutches. Xue et al. [23] constructed a numerical model to calculate the equivalent thermal resistance in wet clutches and investigated the temperature distribution across different friction interfaces. To explore the temperature distribution law of separator plates in wet clutches, Li et al. [24] utilized FEM to formulate a calculation model for capturing the thermal characteristics of separator plates and concurrently revealed the thermoelastic instability mechanism.

A survey of the literature reveals that the existing research has extensively employed commercial software to explore the temperature characteristics of wet friction clutches in automobiles. Nevertheless, there is a noticeable lack of studies on the transient thermal characteristics of high-power-density and high-torque aviation clutches in helicopters. Additionally, the existing research often disregards the time-varying characteristics of parameters such as the heat flow density, friction coefficient, and heat transfer coefficient, an oversight that, in turn, leads to inaccuracies in simulation results. Against this backdrop, the present study aims to explore the thermodynamic behavior of aviation wet clutches in the helicopter power-shifting phase. To this end, a comprehensive thermal-fluid-dynamic model, coupled with a dynamic model considering the spline friction and split spring and a thermal model considering the time-varying thermal parameters, is proposed. Synchronously, the effects of operating parameters on the transient thermal behavior of friction discs are analyzed. In Section 2, a thermal-fluid-dynamic model is developed. Section 3 analyzes the dynamic and thermal behaviors of friction components. In Section 4, a discussion on the effects of operating parameters on thermal behaviors is showcased. Finally, the results are synthesized, and corresponding conclusions are drawn in Section 5.

2. Dynamic and Heat Transfer Modeling

2.1. Axial Dynamic Model

In the clutch engagement and disengagement phases, the proportional valve regulates the flow of pressurized oil entering the piston cylinder to control the axial shift of the piston and friction elements. Subsequently, the elements are consecutively numbered, and corresponding axial forces are analyzed in Figure 2.

The position of the j-th component can be calculated by

$$x_j=x_{j+1}+h_j,j=0,1,2\dots n$$

(1)

where $h_j$ represents the gap in the operation state of the friction pair. n represents the number of the friction pair.

Based on the principles of Newton’s second law, the axial kinetic equation for the moving components of the aviation clutch was developed and is shown in Equation (2).

$$\begin{array}{c}\hfill \left[\begin{array}{c}{F}_{c0}+F_{v0}+F_{d0}-F_{app}\\ F_{c1}+F_{v1}+F_{f1}-F_{c0}-F_{v0}\\ F_{c2}+F_{v2}+F_{s2}+F_{d2}-F_{c1}-F_{v1}\\ ⋮\\ F_{c(n-1)}+F_{v(n-1)}+F_{f(n-1)}-F_{c(n-2)}-F_{v(n-2)}\\ F_{cn}+F_{vn}+F_{sn}+F_{dn}-F_{c(n-1)}-F_{v(n-1)}\end{array}\right]\\ \hfill +\left[\begin{array}{cccccc}{k}_r& 0& 0& \dots & 0& 0\\ 0& k_s& k_s& \dots & 0& 0\\ 0& 0& 0& \dots & 0& 0\\ & & & \ddots & & \\ 0& \dots & -k_s& -k_s& k_s& k_s\\ 0& \dots & 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{h}_o-h_0\\ h_o-h_1\\ h_o-h_2\\ ⋮\\ h_o-h_{n-1}\\ h_o-h_n\end{array}\right]=\left[\begin{array}{cccccc}{m}_p& & & & & \\ & m_f& & & & \\ & & m_s& & & \\ & & & \ddots & & \\ & & & & m_f& \\ & & & & & m_s\end{array}\right]\left[\begin{array}{c}{x}_0^{″}\\ x_1^{″}\\ x_2^{″}\\ ⋮\\ x_{n-1}^{″}\\ x_n^{″}\end{array}\right]\end{array}$$

(2)

where $m_p$, $m_f$, and $m_s$ denote the mass of the piston, the friction disc, and the steel disc, respectively. $F_{ci}$ and $F_{vi}$ denote the asperity bearing capacity and the oil film bearing capacity, respectively. $F_{si}$ and $F_{fi}$ denote the spline friction force in the friction disc and the steel disc, respectively. $F_{di}$ denotes the damping force of the steel disc. $F_{app}$ denotes the applied force of the piston. $k_r$ denotes the return spring stiffness. $k_s$ denotes the split spring stiffness. $F_{rk}$ and $F_{sk}$ are spring forces developed by the return spring and the split spring, respectively. $k_s$, $F_{rk}$, $F_{sk}$, $F_{di}$, and $F_{f\left(s\right)i}$ are calculated using Equation (3) [2].

$$\left\{\begin{array}{c}{k}_s={\displaystyle \frac{16kEb{\delta }^3{W_n}^4}{{\pi }^3{D}_M^3}}\hfill \\ F_{rk}=k_r(h_o-h_0)\hfill \\ F_{sk}=k_s\Delta h\hfill \\ F_{di}=c_s{x}_i^{\prime }\hfill \\ F_{f\left(s\right)i}=\mathrm{sign}\left(x_i^{\prime }\right){\displaystyle \frac{{\mu }_s{T}_i}{R_{f\left(s\right)}cos\left(\alpha \right)}}\hfill \end{array}\right.$$

(3)

where $D_M$, $W_n$, b, and $\delta$ represent the center diameter, wave number, width, and thickness of the split spring, respectively. $T_i$ represents torque developed by the friction pair. The spline friction coefficient is ${\mu }_s$. The pitch circle radius and the pressure angle of the spline are $R_{f\left(s\right)}$ and $\alpha$, respectively.

2.2. Asperity Contact and Lubrication Model

2.2.1. Asperity Contact Model

The underlying assumption is that the surface roughness of the friction component is described by a Gaussian distribution with a mean of zero and mutually independent peak values in the contact model. In addition, the asperity on the friction interface is rarely subjected to plastic deformation under the action of the pressured hydraulic oil. Therefore, the effective asperity area $A_c$ and the corresponding pressure $p_c$ are presented in Equation (4) [25].

$$\left\{\begin{array}{c}{A}_c\left(H\right)=\gamma A_n{\left(\pi \beta R\sigma \right)}^2{F}_2\left(H\right)\hfill \\ p_c\left(H\right)=K^{\prime }{E}^{\prime }{F}_{5/2}\left(H\right)\hfill \\ K^{\prime }={\displaystyle \frac{8\sqrt{2}}{15}}\pi {\left(\beta R\sigma \right)}^2\sqrt{{\displaystyle \frac{\sigma }{R}}}\hfill \\ {\displaystyle \frac{1}{E^{\prime }}}={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1-v_1^2}{E_1}}+{\displaystyle \frac{1-v_2^2}{E_2}}\right)\hfill \\ F_n\left(H\right)={\int }_H^{\infty }{(x-H)}^n{\displaystyle \frac{1}{\sqrt{2\pi }}}exp\left(-{\displaystyle \frac{x^2}{2}}\right)dx\hfill \end{array}\right.$$

(4)

where $\beta$ and R represent the density and radius of the asperity, respectively. $\sigma$ and $\gamma$ represent the roughness and deformation coefficient, respectively. $H=h/\sigma$ represents the film thickness ratio. E represents Young’s modulus of the friction element. v denotes Poisson’s ratio of the friction element.

The nominal contact area comprises the asperity area and the lubrication area between the friction pair. The contact ratio C, as an essential index, is introduced to effectively determine contact states of the friction interfaces, which is detailed in Equation (5).

$$C={\displaystyle \frac{A_c}{A_n}}=\gamma {\pi }^2{\displaystyle \frac{{\left(\beta R\sigma \right)}^2}{2}}\left\{\left(1+H^2\right)\left[1-\mathrm{erf}\left({\displaystyle \frac{H}{\sqrt{2}}}\right)\right]-{\displaystyle \frac{\sqrt{2}}{\sqrt{\pi }}}Hexp\left(-{\displaystyle \frac{H^2}{2}}\right)\right\}$$

(5)

where erf $(·)$ is the error function. $A_n$ denotes the friction pair area.

The asperity bearing capacity $F_c$ on the friction interface is obtained by Equation (6).

$$F_c=A_{red}C{\mathit{\int }}_0^{2\pi }d\theta {\mathit{\int }}_{R_i}^{R_o}{p}_{c}rdr$$

(6)

where $A_{red}$ is the asperity contact area coefficient.

2.2.2. Lubrication Model

When the multi-disc clutch engages or disengages, there are three assumptions: (1) The friction component is not deformed and is always parallel. (2) The pressure distribution of the oil film is uniform. (3) The lubrication oil fills the space within the friction pair. Incorporating flow factors and the model, a modified Reynolds equation for the lubrication oil between the friction pair is derived in Equation (7) [26].

$$\left\{\begin{array}{c}{\displaystyle \frac{1}{r}}{\displaystyle \frac{d}{dr}}\left(r{\varphi }_r(h^3+12\psi d_m){\displaystyle \frac{dp}{dr}}\right)=12\eta {\displaystyle \frac{d{\overline{h}}_t}{dt}}\hfill \\ {\varphi }_r=1-0.9e^{-0.56h/\sigma }\hfill \\ {\overline{h}}_t={\displaystyle \frac{h}{2}}\left[1+\mathrm{erf}\left({\displaystyle \frac{h}{\sqrt{2}\sigma }}\right)\right]+{\displaystyle \frac{\sigma }{\sqrt{2\pi }}}{e}^{\left(-{\displaystyle \frac{h^2}{2{\sigma }^2}}\right)}\hfill \\ {\displaystyle \frac{d{\overline{h}}_t}{dt}}=\left\{{\displaystyle \frac{1}{2}}\left[1+\mathrm{erf}\left({\displaystyle \frac{h}{\sqrt{2}\sigma }}\right)\right]\right\}{\displaystyle \frac{dh}{dt}}\hfill \end{array}\right.$$

(7)

where h denotes the oil film thickness. $d_m$ represents the friction lining thickness. $\psi$ is the friction material permeability. $\eta$ denotes the dynamic viscosity. ${\varphi }_r$ represents the pressure flow factor.

Considering the structural and fluid field characteristics of the aviation clutch, the boundary condition of Equation (7) is determined as $p_{v(r=R_i)}=p_{v(r=R_o)}=0$. The average pressure of the lubrication oil between the friction pair is formulated in Equation (8) [27].

$$p_v\left(r\right)={\displaystyle \frac{3\eta }{2{\varphi }_r(h^3+12\psi d)}}\left[r^2-R_o^2+ln{\displaystyle \frac{r}{R_o}}{\displaystyle \frac{R_o^2-R_i^2}{2\left(ln{R}_i-ln{R}_o\right)}}\right]\left[1+\mathrm{erf}\left({\displaystyle \frac{h}{\sqrt{2}\sigma }}\right)\right]{\displaystyle \frac{dh}{dt}}$$

(8)

where $R_i$ corresponds to the inner radius, while $R_o$ corresponds to the outer radius.

The corresponding oil film bearing capacity $F_v$ is derived by integrating Equation (8) across the lubricated area $A_n$.

$$F_v=(1-A_{red}C){\mathit{\int }}_0^{2\pi }d\theta {\mathit{\int }}_{R_i}^{R_o}{p}_{v}rdr$$

(9)

2.3. Torque Balance Model

In the clutch engagement phase, as the hydraulic oil inflows into the piston chamber, the torque transmission mechanism gradually transforms from the viscous torque $T_v$ to the friction torque $T_c$. However, the opposite occurs in the clutch disengagement phase. The viscous torque $T_v$ and friction torque $T_c$ are described as follows.

$$\left\{\begin{array}{c}{T}_v=(1-A_{red}C){\int }_0^{2\pi }d\theta {\int }_{R_i}^{R_o}\eta ({\phi }_{fs}+{\phi }_f)r^3{\displaystyle \frac{{\omega }_{rel}}{h}}dr\hfill \\ T_c=A_{red}C{f}_c{\int }_0^{2\pi }d\theta {\int }_{R_i}^{R_o}{r}^2{p}_{c}dr\hfill \end{array}\right.$$

(10)

where ${\phi }_f$ and ${\phi }_{fs}$ are the flow factors. $f_c$ represents the friction coefficient, which can be calculated as follows [28].

$$\begin{array}{cc}\hfill f_c& =0.035+23{\mathrm{e}}^{\left(-2.6v/\left((ln{T}_{oil}-3.2)\left({\left(28.3P_{fn}\right)}^{0.4}-0.87\right)\right)-5.16\right)}\hfill \\ \hfill & +0.08\left({\mathrm{e}}^{-0.005T_{oil}}-1\right)\left({\mathrm{e}}^{-0.2v}-1\right)+{\displaystyle \frac{0.01ln(4v+1)}{{\mathrm{e}}^{0.005T_{oil}}}}-0.005ln\left(28.3P_{fn}\right)\hfill \end{array}$$

(11)

where the lubricant oil temperature is expressed by $T_{oil}$. $P_{fn}$ is the effective pressure exerted on the friction pair.

There are a friction torque and a viscous torque in the torque transmitted by the wet clutch. Consequently, the torque dynamic equation for an aviation wet clutch system based on Newton’s second law is established as follows.

$$I_f{\displaystyle \frac{d{\omega }_2}{dt}}=\sum _{i=1}^n{T}_{ci}+\sum _{i=1}^n{T}_{vi}-T_r$$

(12)

where $I_f$ is the inertia of the driven shaft. $T_r$ represents the load torque.

2.4. Thermal Field Numerical Model and Solution Method

2.4.1. Heat Conduction Equation

When the clutch engages with an excessive rotation speed difference, the considerable thermal load generated by friction is primarily absorbed by the friction element, thereby increasing the temperature of the component in the clutch. Under the assumption that the material properties remain constant with temperature and with the guidance of heat transfer theory, the transient heat transfer model for the aviation wet clutch is established as follows [29].

$$\rho c{\displaystyle \frac{\partial T}{\partial t}}={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial }{\partial r}}\left(\zeta r{\displaystyle \frac{\partial T}{\partial r}}\right)+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{\partial }{\partial \theta }}\left(\zeta {\displaystyle \frac{\partial T}{\partial \theta }}\right)+{\displaystyle \frac{\partial }{\partial z}}\left(\zeta {\displaystyle \frac{\partial T}{\partial z}}\right),R_i\le r\le R_o,0\le z\le H_{s\left(f\right)d},t>0$$

(13)

where $\zeta$, $\rho$, and c are the thermal conductivity, density, and specific heat of the friction component, respectively.

The axisymmetric temperature distribution of the friction element is a result of the axisymmetric structure characteristics and boundary conditions [20]. Therefore, a two-dimensional thermal model for the aviation wet multi-disc clutch is proposed in this paper to calculate the transient temperature distribution in both the thickness and radial directions. In this heat transfer model, there are boundary conditions for heat flux input and for convective heat transfer between friction pairs, which involve grooves and asperity contact regions between the friction pair for convective heat transfer. Additionally, there are boundary conditions for convective heat transfer at the inner and outer radii of the friction element. Given a friction disc as a case in point, the meshing and boundary conditions of the friction disc at different regions are exhibited in Figure 3.

2.4.2. Heat Flux Density

As the clutch engages, the action of friction torque and the excessive rotation speed difference between the drive disc and the driven disc result in a heat flux, which is primarily absorbed by the friction component. As a result, the heat flux density fed into the friction disc and steel disc is determined as follows in Equation (14) [10,11].

$$\left\{\begin{array}{c}{q}_{st}=k_{qst}·f_c·P_{fn}·{\omega }_{rel}·r\hfill \\ q_{fc}=k_{qfc}·f_c·P_{fn}·{\omega }_{rel}·r\hfill \\ k_{qst}={\displaystyle \frac{\sqrt{{\zeta }_{st}{\rho }_{st}{c}_{st}}}{\sqrt{{\zeta }_{st}{\rho }_{st}{c}_{st}}+\sqrt{{\zeta }_{fc}{\rho }_{fc}{c}_{fc}}}}\hfill \\ k_{qfc}={\displaystyle \frac{\sqrt{{\zeta }_{fc}{\rho }_{fc}{c}_{fc}}}{\sqrt{{\zeta }_{st}{\rho }_{st}{c}_{st}}+\sqrt{{\zeta }_{fc}{\rho }_{fc}{c}_{fc}}}}\hfill \\ P_{fn}={\displaystyle \frac{F_c+F_v}{\pi \left(R_o^2-R_i^2\right)}}\hfill \end{array}\right.$$

(14)

where $q_{st}$, $q_{fc}$, ${\rho }_{st}$, ${\rho }_{fc}$, $c_{st}$, $c_{fc}$, ${\zeta }_{st}$, and ${\zeta }_{fc}$ are the input heat flow density, density, specific heat, and thermal conductivity of the steel disc and friction disc, respectively. $P_{fn}$ is the applied pressure. $f_c$ is the friction coefficient. ${\omega }_{rel}$ represents the rotation speed difference.

2.4.3. Convective Heat Transfer

In the aviation clutch engagement and disengagement phase, the primary paths for frictional heat dissipation involve convection heat transfer, which occurs between the inner and outer cylindrical surfaces and the lubrication oil, between the lubrication oil and the groove surface, and between the lubrication oil and the friction interface.

The convective heat transfer between the inner and outer cylindrical surfaces is accomplished by sweeping the lubricant oil across the hollow cylindrical surface, and the convective heat transfer coefficient is obtained from Equation (15).

$$\left\{\begin{array}{c}{h}_{in}=0.193{\displaystyle \frac{{\zeta }_{oil}}{2R_i}}{\left({\displaystyle \frac{2v_i{R}_i{\rho }_{oil}}{\eta }}\right)}^{0.618}{P}_r^{{\displaystyle \frac{1}{3}}}\hfill \\ h_{out}=0.193{\displaystyle \frac{{\zeta }_{oil}}{2R_o}}{\left({\displaystyle \frac{2v_o{R}_o{\rho }_{oil}}{\eta }}\right)}^{0.618}{P}_r^{{\displaystyle \frac{1}{3}}}\hfill \\ P_r={\displaystyle \frac{\eta c_{oil}}{{\zeta }_{oil}}}\hfill \end{array}\right.$$

(15)

where ${\zeta }_{oil}$ represents the thermal conductivity of the lubrication oil. $v_i$ and $v_o$ are the velocity differences between the inner and outer cylindrical surfaces of the friction element, respectively. $c_{oil}$ is the specific heat of the lubrication oil. $P_r$ is the Prandtl number.

There is a convective heat transfer between the lubrication oil film and the friction contact interface. Hence, the convective heat transfer coefficient for the lubricant oil film in the local contact region is formulated as

$$\left\{\begin{array}{c}{h}_l\left(r\right)={\displaystyle \frac{{\zeta }_{oil}Nu}{r}}\hfill \\ Nu=0.664R_e^{{\displaystyle \frac{1}{2}}}{P}_r^{{\displaystyle \frac{1}{3}}}\hfill \\ R_e={\displaystyle \frac{{\rho }_{oil}\overline{v}{L}_r}{\eta }}\hfill \\ \overline{v}={\left({\left({\displaystyle \frac{Q}{2\pi rh}}\right)}^2+{\left({\omega }_{rel}·r\right)}^2\right)}^{{\displaystyle \frac{1}{2}}}\hfill \end{array}\right.$$

(16)

where $R_e$ and Q are the Reynold’s number and the lubrication oil flow, respectively.

In addition, the convective heat transfer coefficient for the lubricant oil in the groove is calculated using Equation (17).

$$h_g=0.064{\displaystyle \frac{2{\zeta }_{oil}}{R_o-R_i}}{R}_e^{{\displaystyle \frac{1}{2}}}{P}_r^{{\displaystyle \frac{1}{3}}}$$

(17)

The convective heat transfer coefficient within the friction pair involves the convective heat transfer for the lubricant oil film in the local contact zone and for the lubricant oil in the groove, which is expressed as

$$h_b=h_g+{\displaystyle \frac{A_{red}}{1-A_{red}}}{h}_l$$

(18)

2.4.4. Boundary and Initial Conditions

The contribution of thermal radiation to the temperature field is considered insignificant, and only the roles of heat conduction and heat convection in the temperature field of friction elements are considered. In addition, the structural characteristics and boundary conditions of the clutch friction element are symmetrical. Therefore, only half of the clutch disc needs to be simulated to reduce calculation costs. It is assumed that the lubricant oil is fully circulated and the initial temperature of the component in the wet clutch is $T_{\infty }$. In addition, the regime at the convection boundary is laminar. The inlet temperature of lubrication oil is constant $T_{\infty }$. The outlet pressure is atmospheric. Therefore, the proposed thermal model satisfies the following boundary conditions and initial conditions [10].

The boundary conditions for convective heat transfer that occurred at the inner and outer cylindrical surfaces are

$${\left.\zeta {\displaystyle \frac{\partial T(r,z,t)}{\partial r}}\right|}_{r=R_i,0\le z\le H_d}=+h_i\left(T(r,z,t)-T_{\infty }\right)$$

(19)

$${\left.\zeta {\displaystyle \frac{\partial T(r,z,t)}{\partial r}}\right|}_{r=R_o,0\le z\le H_d}=-h_o\left(T(r,z,t)-T_{\infty }\right)$$

(20)

The boundary conditions for convective heat transfer between the friction pair are

$${\left.\zeta {\displaystyle \frac{\partial T(r,z,t)}{\partial z}}\right|}_{R_i\le r\le R_o,z=0}=-\left(q_{fc}-h_b\left(T(r,z,t)-T_{\infty }\right)\right)$$

(21)

$${\left.\zeta {\displaystyle \frac{\partial T(r,z,t)}{\partial z}}\right|}_{R_i\le r\le R_o,z=H_d}=q_{fc}-h_b\left(T(r,z,t)-T_{\infty }\right)$$

(22)

The initial condition to be satisfied by the temperature field of the friction component is

$${\left.T(r,z,0)\right|}_{R_i\le r\le R_o,0\le z\le H_d}=T_{\infty }$$

(23)

2.4.5. Solution Method for Thermal Model

It is assumed that the temperature distribution of the friction element is axisymmetric [20]. The transient heat transfer equation for the aviation wet clutch is established in Equation (24) based on heat transfer theory.

$${\displaystyle \frac{\partial T}{\partial t}}={\displaystyle \frac{\zeta }{\rho c}}\left({\displaystyle \frac{{\partial }^{2}T}{\partial r^2}}+{\displaystyle \frac{{\partial }^{2}T}{\partial z^2}}\right)$$

(24)

As presented in Figure 3, the transient temperature field solution region (r, z, t) of the friction element is divided into a grid with step sizes $dr$, $dz$, and $dt$. Subsequently, after discretizing Equation (24), one gets

$${\displaystyle \frac{T_{i,j}^{N+1}-T_{i,j}^N}{dt}}={\displaystyle \frac{\zeta }{\rho c}}\left({\displaystyle \frac{T_{i+1,j}^N-2T_{i,j}^N+T_{i-1,j}^N}{d{r}^2}}+{\displaystyle \frac{T_{i,j+1}^N-2T_{i,j}^N+T_{i,j-1}^N}{d{z}^2}}\right)$$

(25)

The transient heat transfer equation in explicit difference form is deduced as

$$T_{i,j}^{N+1}=T_{i,j}^N\left(1-2{\displaystyle \frac{\zeta }{\rho c}}dt\left({\displaystyle \frac{1}{{\left(dr\right)}^2}}+{\displaystyle \frac{1}{{\left(dz\right)}^2}}\right)\right)+{\displaystyle \frac{\zeta }{\rho c}}dt\left({\displaystyle \frac{T_{i+1,j}^N+T_{i-1,j}^N}{{\left(dr\right)}^2}}+{\displaystyle \frac{T_{i,j+1}^N+T_{i,j-1}^N}{{\left(dz\right)}^2}}\right)$$

(26)

3. Numerical Simulation and Comparative Analysis

3.1. Model Parameters and Simulation Solution

MATLAB/Simulink 2022b is employed to calculate the axial–rotational-coupled dynamical mathematical model and the thermal-fluid-dynamic model for aviation wet friction clutches. The calculation flow chart is presented in Appendix A. Firstly, the proposed coupled model is solved by initially employing the initial state or the results calculated from the prior time step to derive the asperity bearing force and the oil film bearing force on the component. Secondly, the balance equations for the axial force and torque of the component are solved to obtain the corresponding displacements, velocities, and accelerations. Next, the obtained displacements and velocities of the friction component are employed to derive the friction pair gaps. To solve the proposed thermal-fluid-dynamic model, the initial thermal field of the friction element is first established. Then, the heat flux density and convective heat transfer coefficients are derived using the kinetic results in the coupled model. The boundary conditions and heat transfer equations are then employed to calculate the transient temperatures at the mesh nodes. Finally, the aforementioned simulation steps are cycled until the termination requirements are satisfied. The parameters required for solving the proposed rotational–axial-coupled kinetic model and thermal-fluid-dynamic model are enumerated in

Table 1. Parameters required for models.

Parameter	Value	Parameter	Value	Parameter	Value
$R_o$	0.086 m	$I_f$	0.2 kg·m2	$c_{st}$	487 J·(kg·°C)−1
$R_i$	0.072 m	$H_{fd}$	0.003 m	$c_{fc}$	536 J·(kg·°C)−1
n	10	$H_{sd}$	0.002 m	$c_{oil}$	2231 J·(kg·°C)−1
$k_r$	1.52 × 106 N·m−1	$\beta$	7 × 107 m−2	${\zeta }_{st}$	46 W·(m·°C)−1
$m_s$	0.2 kg	R	8 × 10−4 m	${\zeta }_{fc}$	9.3 W·(m·°C)−1
$m_f$	0.3 kg	$\sigma$	8.41 × 10−6 m	${\zeta }_{oil}$	0.3 W·(m·°C)−1
$h_o$	5 × 10−4 m	$E^{\prime }$	4.7 × 109 Pa	${\rho }_{st}$	7800 kg·m−3
$T_r$	300 N·m	$A_{red}$	0.78	${\rho }_{fc}$	5500 kg·m−3
$\psi$	4 × 10−13 m2	$d_m$	0.001 m	${\rho }_{oil}$	875 kg·m−3

.

3.2. Transient Thermal Behavior and Comparative Analysis

The control oil pressure, as depicted in Figure 4a, begins with a linear rise from 0 to a maximum of 1.7 MPa within the time range of 0.2 s to 0.35 s. Subsequently, the pressure level of 1.7 MPa is maintained until 2 s. Then, a linear decrease from 1.7 MPa to 0 occurs between 2 s and 2.3 s, and ultimately, a steady state of zero is always maintained beyond 2.3 s. The input rotation speed of the clutch drive shaft is 2200 rpm, and the initial conditions for friction components and the lubrication oil are 30 °C. In addition, the lubrication oil flow is 1 L/min. In the solution for the proposed numerical model, there are 41 mesh nodes along the axial direction and 201 mesh nodes along the radial direction in the friction element mesh. Meanwhile, a comparative analysis of the transient thermal behavior of the aviation wet clutch is conducted using the commercial software COMSOL Multiphysics 6.2. The simulation parameters, initial and boundary conditions, are identical to those of the proposed numerical model. The simulation results obtained from the numerical method and the COMSOL method are presented in Figure 4, Figure 5 and Figure 6.

In Figure 4, as the applied pressure rises, the friction pair gap sharply decreases; the viscous torque initially experiences an increase and subsequently fades to 0. In contrast, the friction torque rises and then maintains a steady state. The driven shaft speed increases to 2200 rpm and subsequently remains constant at 2200 rpm. The third friction disc is selected to explore the transient thermal behaviors in the following analysis. In the clutch engagement phase, there is a large rotation speed difference between the drive disc and the driven disc, which generates a significant amount of frictional heat under the action of the friction torque. The frictional heat is mostly absorbed by the friction element and the lubricating oil, resulting in a substantial temperature increase, up to 101.6 °C at 0.59 s. However, in the wet clutch disengagement, with a decrease in the applied pressure, the friction torque incrementally reaches 0, the friction pair gap enlarges, and the viscous torque initially rises before attenuating. Meanwhile, the output rotation speed incrementally decreases from 2200 rpm back to 0. The increased friction pair gap allows the lubrication oil to enter the friction pair more effectively. The frictional heat generated at the friction interface is quickly absorbed and carried away, sharply decreasing the temperature of the friction element to the level of the lubrication oil at the inlet. Additionally, the results of Figure 4c,d indicate that the temperature variation trends and peak temperatures at different radial and axial nodes, obtained from the numerical method and the COMSOL method, are consistent. Furthermore, the proposed numerical model can be employed to analyze the transient thermal behavior of the aviation clutch reliably.

Figure 5 illustrates the transient thermal behaviors of the third friction disc in the whole clutch operation cycle. In the clutch engagement phase, with a rise in control oil pressure, the heat flux flowing into the friction interface increases, resulting in the friction interface temperature rising, and then the frictional heat is transferred to the inside of the friction element. The temperature of friction elements is approximately symmetrical about the center plane ($z=H_{fd}/2$). In addition, an expansion in the radial results in an increase in the heat flux density and a corresponding rise in temperature, reaching a maximum of 101.6 °C at the outer ring. In the clutch disengagement phase, a decrease in control oil pressure facilitates the widening of the friction pair gap and an increase in speed difference, which favors the entry of more lubricant oil into the friction interface. The frictional heat is conducted from the inside to the surface of the friction element and is then absorbed and carried away by the lubricant oil, resulting in a lower temperature at the component surface compared to that on the inside. Eventually, the friction disc temperature falls back to the level of the lubricant oil at the inlet.

Figure 6 presents the transient thermal behavior of the third friction disc obtained via the COMSOL method at different times. In the clutch engagement, the temperature steadily rose to a peak of 93.9 °C at 0.6 s, with the high-temperature zone gradually expanding. In the clutch disengagement, the temperature steadily decreased to 35.5 °C at 4 s, approaching the initial temperature. Furthermore, the temperature field distribution along the radial direction exhibited near-linear characteristics. At the inner and outer diameters, temperatures were lower than the interior due to cooling effects from the lubricant oil. The temperature variation trends and distribution patterns presented in Figure 6 are nearly identical to those obtained using the numerical method in Figure 5. The result further demonstrates that the numerical model proposed in this paper can be employed to analyze the transient thermal behavior of the aviation clutch friction discs.

4. Results and Parameter Analysis

4.1. Effect of Rotation Speed

To explore the effect of the input rotation speed on transient thermal behaviors, only this parameter is modified in the simulation. The corresponding parameter is assigned the values of 2000 rpm, 2200 rpm, and 2400 rpm. The control oil pressure is 1.7 MPa. The lubrication flow is 5 L/min. The resulting temperature dynamic response behaviors are depicted in Figure 7.

As depicted in Figure 7, with an elevation in input rotation speed, the maximum temperature at the mesh node (198, 40) increases from 85.36 °C to 94 °C, and the maximum temperature at the mesh node (100, 40) increases from 79.50 °C to 87.35 °C. In the clutch engagement, as the input rotation speed increases, the slip stage is extended, and the excessive frictional heat cannot be immediately dissipated, causing a sharp rise in the temperature of the friction pair. However, in the clutch disengagement, as the input rotation speed increases, the flow velocity of the lubrication oil also rises, the frequency of heat transfer between the friction element and the lubrication oil increases, and the heat dissipation rate increases, which ultimately leads to a rapid decrease in temperature.

The maximum radial and axial temperature differences with different input rotation speeds are recorded in

Table 2. Maximum radial and axial temperature differences with different rotation speeds.

Rotation Speed	Maximum Radial Temperature Difference	Maximum Axial Temperature Difference
2000 rpm	11.0017 °C (0.5682 s)	44.3499 °C (0.4695 s)
2200 rpm	11.9621 °C (0.6047 s)	45.1875 °C (0.4881 s)
2400 rpm	12.9010 °C (0.6438 s)	45.5721 °C (0.5087 s)

and Figure 8. As shown in Figure 8a, Figure 8b, and Table 2, an elevation in rotation speed causes the maximum radial and axial temperature differences to increase, reaching 12.9010 °C at 0.6438 s and 45.5721 °C at 0.5087 s, respectively. Compared to radial temperature differences, the rotational speed has a negligible effect on axial temperature differences. In addition, the peak values of the radial and axial temperature differences of the friction disc become larger as the rotational speed increases, which deteriorates the uniformity of the temperature field. The combined results of Figure 8c and Figure 8d illustrate the temperature dynamic response behaviors for an input shaft speed of 2400 rpm in the radial direction (node (:, 40)) and axial direction (node (198, :)), respectively. The axial temperature of the friction disc increases in a parabolic manner, and the radial temperature of the friction disc varies linearly.

4.2. Effect of Control Oil Pressure

To explore the effect of control oil pressure on transient thermal behaviors, only this parameter is modified in the simulation. The corresponding parameter is assigned maximum values of 1.5 MPa, 1.7 MPa, and 1.9 MPa. The rotation speed is 2200 rpm. The lubrication flow is 5 L/min. The resulting temperature response curves are illustrated in Figure 9. The relationship between the heat flux density and the applied pressure is directly proportional. As the maximum control oil pressure increases, it can accelerate the clutch engagement process. Simultaneously, the contribution of the heat flux input to the friction interface is significantly enhanced. Hence, the maximum temperature and temperature rise rate of the friction disc increase substantially. The maximum temperature at node (198, 40) increases from 81.95 °C to 97.4 °C. The maximum temperature at node (100, 40) increases from 76.17 °C to 90.33 °C.

The maximum radial and axial temperature differences with different control oil pressures are reported in Figure 10 and

Table 3. Maximum radial and axial temperature differences with different control oil pressures.

Control Oil Pressure	Maximum Radial Temperature Difference	Maximum Axial Temperature Difference
1.5 MPa	10.5284 °C (0.6698 s)	36.1188 °C (0.5042 s)
1.7 MPa	11.9621 °C (0.6047 s)	45.1875 °C (0.4881 s)
1.9 MPa	13.2700 °C (0.5596 s)	54.0000 °C (0.4730 s)

. Figure 10a, Figure 10b, and Table 3 show that an increase in control oil pressure promotes an increase in the maximum radial and axial temperature differences, reaching 13.27 °C at 0.5596 s and 54 °C at 0.473 s, respectively. Compared to radial temperature differences, the control oil pressure has a significant effect on axial temperature differences. Figure 10c and Figure 10d illustrate the thermal dynamic response behaviors for a control oil pressure of 1.9 MPa in the radial direction (node (:, 40)) and axial direction (node (198, :)), respectively. The axial temperature in the friction disc increases parabolically, and the radial temperature in the friction disc varies linearly. Moreover, an increase in oil pressure simultaneously exacerbates both radial and axial temperature differentials while accelerating transient thermal processes. The axial thermal imbalance proves more severe than the radial imbalance, making it more prone to inducing thermal stress issues.

4.3. Effect of Lubrication Oil Flow

To investigate the effect of lubrication oil flow on transient thermal behaviors, only this parameter is modified in the simulation. The corresponding parameter is assigned values of 1 L/min, 5 L/min, and 10 L/min. The rotation speed is 2200 rpm. The control oil pressure is 1.7 MPa. The resulting temperature response curves are illustrated in Figure 11.

As the lubrication oil flow rises, the maximum temperature at node (198, 40) decreases from 97.87 °C to 84.84 °C, and the maximum temperature at node (100, 40) decreases from 91.45 °C to 78.17 °C. An increase in the lubrication oil flow facilitates the convective heat transfer between the friction disc surface and the lubricant oil. The frictional heat is efficiently absorbed by the low-temperature lubricant oil, causing a considerable depression in the temperature of friction discs.

The maximum radial and axial temperature differences with different lubrication oil flows are presented in Figure 12a,b and

Table 4. Maximum radial and axial temperature differences with different lubrication oil flows.

Lubrication Oil Flow	Maximum Radial Temperature Difference	Maximum Axial Temperature Difference
1 L/min	12.2472 °C (0.6074 s)	50.3731 °C (0.5016 s)
5 L/min	11.9621 °C (0.6047 s)	45.1875 °C (0.4881 s)
10 L/min	11.6101 °C (0.6015 s)	41.9843 °C (0.4794 s)

. The lubricant oil flow exhibits an inverse correlation with both the maximum radial and maximum axial temperature differences. Specifically, the maximum radial temperature difference reaches 12.2472 °C at 0.6074 s, while the maximum axial temperature difference reaches 50.3731 °C at 0.5016 s. Compared to axial temperature differences, the lubricant oil flow has a negligible effect on radial temperature differences. Figure 12c and Figure 12d illustrate the temperature dynamic response behaviors for a lubrication oil flow of 5 L/min in the radial direction (node (:, 40)) and axial direction (node (198, :)), respectively. The axial temperature of the friction disc increases parabolically, while the radial temperature of the friction disc varies linearly. Furthermore, an increase in the lubricant oil flow significantly enhances convective heat transfer, which further helps to reduce the temperature gradient, improve temperature uniformity, and ultimately depress the temperature of the friction element.

5. Conclusions

A comprehensive numerical model, coupled with a dynamic model that considers the spline friction and split spring and a thermal model, was proposed in this paper. The effects of operating parameters on the transient thermal behavior of the aviation wet clutch were analyzed. The results are synthesized, and the corresponding conclusions are briefly summarized as follows:

(1)

Throughout the entire cycle of the aviation wet clutch, from slip phase to synchronization engagement and disengagement, the temperature field of the friction element successively increases and then decreases, eventually reaching the temperature of the inlet lubricant oil. In addition, the axial temperature in the friction element increases parabolically, and the radial temperature in the friction element varies linearly.

(2)

As the rotation speed rises from 2000 rpm to 2400 rpm, there is a corresponding 10.1% increase in maximum temperature of the friction disc, raising it from 85.36 °C to 94 °C (the error is approximately ±0.01 °C). The elevation in control oil pressure from 1.5 MPa to 1.9 MPa contributes to a 19.4% increase in the maximum temperature of the friction disc (from 81.6 °C to 97.4 °C, with an error of approximately ±0.1 °C). Moreover, the lubrication oil flow increased from 1 L/min to 10 L/min, and the contribution of that to the maximum temperature of the friction disc is diminished significantly, causing a reduction of 14.5% from 91.45 °C to 78.17 °C (the error is approximately ±0.01 °C).

(3)

In improving temperature uniformity, the input rotation speed has an insignificant influence on temperature uniformity. Increasing the lubricant oil flow significantly helps to narrow the temperature gradient (maximum temperature difference decreased from 50.3731 °C to 41.9843 °C) and improves the temperature uniformity. In contrast, an elevation in control oil pressure deteriorates the temperature uniformity.

(4)

To effectively control temperature escalation and improve temperature field uniformity of the friction element in a wet multi-disc clutch system for helicopters, the lubricating oil flow should be increased appropriately, the shifting interval should be reasonably extended, and the control oil pressure should be moderately minimized. These adjustments should be performed while the clutch can achieve the operational condition requirements.

This study provides a reliable theoretical basis for guiding the rational selection of operating parameters for aviation wet clutches, ensuring their safe and efficient operation. In the future, research should be conducted on the thermodynamic characteristics of clutches after repeated engagement under extreme operating conditions, especially in the aviation industry.