Calculation and Verification of the Real-Time Working Characteristics of a Viscous Coupling

During operation, the shear friction between the silicone oil and the plates of a viscous coupling (VC) will generate heat and increase the temperature of the silicone oil, inflate the volume, increase the internal pressure, and eventually deliver more torque, in what is called hump operation mode. Temperature is the root cause of the change of the operation characteristics in VCs. In this paper, the heat-transfer model for a VC is established based on the thermodynamical theory. The capacity of the heat transmission of each part of the VC are calculated to obtain the temperature of silicone oil. The real-time shear torque of the VC is finally obtained. Then, the theoretical analysis on hump phenomenon was done. The internal pressure was obtained by analyzing the characteristics of ideal gas, and the maximum torque during the hump phenomenon was calculated. The simulation of the key parameters and the entire working process for the VC were carried out based on the proposed calculation model. A prototype of a viscous limited-slip differential (VLSD) was developed to test the output torque characteristics. The test results were quite consistent with the simulation results, and the accuracy of the calculation model was verified.


Introduction
A VC is a torque-transmission device that uses the shearing force of high-viscosity fluid. VCs continue to be an important part of modern automobile drive systems and provide a cost-effective and reliable traction-control system. Several million units are built into vehicles each year [1]. VCs can improve the torque-distribution characteristic of the conventional differential to increase the dynamic performance and trafficability of the vehicle [2]. A limited-slip differential (LSD) equipped with a VC will improve the traction capability of vehicle on a wet road surface, and will also increase the degree of understeering while the vehicle is being driven, especially on the split-road, and the steering characteristics will be improved significantly [3]. However, the driving torque will change the distribution between the front and rear axles and affect the vehicle's steering performance [4,5]. The effect on automobile performance caused by a VLSD depends on its own working characteristics. The application of VLSDs in vehicles requires an accurate calculation method for the delivering torque characteristics, that is, the accurate calculation of the working characteristics of the VC [6].
The two working modes of the VC are oil-film-shearing operation and hump operation [7], both of which are closely related to temperature. When a VC is operating during the oil-film-shearing operation mode, the heat generated by shearing friction will bring about an increase in the internal pressure and temperature, as well as a decline in viscosity and shear torque. Reference [7] showed the calculation method of shear torque in a certain temperature; however, the temperature was unable to be predicted. If a VC operates at a large speed difference for a long period of time, the generated heat will continue to increase the internal pressure and temperature, and eventually form a quasi-rigid connection of the driving and driven plates, the delivering torque will increase sharply, and the VC moves into hump operation mode [1]. The internal heat of the VC is transferred from the inside to the outside of shell through all parts of the VC; only by accurately calculating the heat transfer of each part can the real-time working characteristics of the coupling be obtained. In reference [8], according to the gradient of velocity and torque, the temperature change was calculated using the finite element method inside the VC, but it was too complex for real-time calculation of the working characteristics. In references [7,9,10], the flow phenomenon of silicon oil and effect of main factors on the hump phenomenon were analyzed while a VC was in operation. There is no calculating method for real-time temperature so far. Therefore, only by performing an accurate analysis of the absorbed heat and the temperature of the silicone oil, the real-time characteristics of the VC can be predicted accurately. This is one of the most important aspects of VCs, and the problem to be resolved in this paper.

The Calculation of Shear Torque
The shear torque transmitted by a VC is [6]: where v s -the kinematic viscosity of the silicone oil; ρ s -the density of the silicone oil; λ-the clearance between the plates; n-the plate number; r 1 -the inner radius of the outer plates; r 2 -the outer radius of the inner plates; and ∆ω-the speed difference between the input and output shafts. According to the equation above, the shear torque is affected by many factors: v s and ρ s represent physical properties of the silicone oil, and both are closely related to the temperature; λ, n, r 1 , and r 2 are structural parameters of the VC; and ∆ω determines the shear rate of the oil. Considering that the speed difference of the VLSD and the shear rate are both small, we suppose that the viscosity of silicone oil does not change with shear rate [6]. Consequently, the shear torque is mainly affected by temperature.
The approximate expression for the relationship between oil viscosity and temperature is [11][12][13]: where T e -the absolute temperature of the silicone oil at this moment; ν 25 -the absolute viscosity of the silicone oil at 25 • C; A-constant, the value of which is related to the viscosity of the silicone oil ( Figure 1). It can be seen in Equation (2) that the viscosity of the silicone oil decreases with an increase in the oil's temperature. According to Equation (1), when the viscosity of the silicone oil decreases, the shear torque decreases.

The Temperature Calculation of the VC at Working Course
The working temperature of the VC is affected by the working environment temperature and the heat generated during working process. Because the former is relatively stable, the latter is the main factor. Part of the heat generated by the VC is transferred to the outside air through the plates, housing shell, and shaft, enabling the heat exchange between the inside and outside of the shell.
The structure of the VC is complicated, and it needs to be simplified to develop a model. In the simplified structure model of the VC, the main parts, such as the shaft, shell, end cover, and blade are considered, while the small parts, such as the sealing parts of coupling, are ignored. In order to simplify the calculation, the influence of surface morphology of each part, especially blades, is ignored. The schematic diagram of the simplified structure of the VC is shown in Figure 2.
The heat absorbed by the VC is transferred between the seven parts of the VC and the outside air. Figure 3 shows the heat-transfer model of the VC. It can be seen in the figure that the quantity of heat Q entering the VC is absorbed by silicone oil, then transfers to each part of the VC and to the outside air. The heat transfer between the seven parts of the VC and the outside air is xy Q ( xy indicates heat transferred from x to y ; the numbers 0-7 in the subscript correspond to outside air, silicone oil, case, left cap, right cap, axle, inner plate, and outer plate, respectively). The arrows in the figure indicate the direction of the heat transfer between each part of the model. The following assumptions are made for this model: (1) Ignoring the thermal conductivity between the shell and the left and right caps, that is, 23 24 53 54 , , , Q Q Q Q are all zero.
(2) Ignoring the thermal conductivity between the seals and the silicone oil, and with the sealed parts. The heat is exchanged between the plates and the silicone oil. The plates are metal parts and good conductors of heat generally. The thickness of the plates is less than 1 mm generally, and therefore the heat conduction from the plates' surface to the inside can be ignored. While the VC works, the temperature rises slowly, considering

The Temperature Calculation of the VC at Working Course
The working temperature of the VC is affected by the working environment temperature and the heat generated during working process. Because the former is relatively stable, the latter is the main factor. Part of the heat generated by the VC is transferred to the outside air through the plates, housing shell, and shaft, enabling the heat exchange between the inside and outside of the shell.
The structure of the VC is complicated, and it needs to be simplified to develop a model. In the simplified structure model of the VC, the main parts, such as the shaft, shell, end cover, and blade are considered, while the small parts, such as the sealing parts of coupling, are ignored. In order to simplify the calculation, the influence of surface morphology of each part, especially blades, is ignored. The schematic diagram of the simplified structure of the VC is shown in Figure 2.   The heat absorbed by the VC is transferred between the seven parts of the VC and the outside air. Figure 3 shows the heat-transfer model of the VC. It can be seen in the figure that the quantity of heat Q entering the VC is absorbed by silicone oil, then transfers to each part of the VC and to the outside air. The heat transfer between the seven parts of the VC and the outside air is Q xy (xy indicates heat transferred from x to y; the numbers 0-7 in the subscript correspond to outside air, silicone oil, case, left cap, right cap, axle, inner plate, and outer plate, respectively). The arrows in the figure indicate the direction of the heat transfer between each part of the model. The following assumptions are made for this model: (1) Ignoring the thermal conductivity between the shell and the left and right caps, that is, Q 23 , Q 24 , Q 53 , Q 54 are all zero. (2) Ignoring the thermal conductivity between the seals and the silicone oil, and with the sealed parts. (3) The inner and outer plates are pieces of metal immersed in silicone oil. There is no heat exchanging between the plates, the shell, and the axle, so Q 65 , Q 75 are zero. The heat is exchanged between the plates and the silicone oil. The plates are metal parts and good conductors of heat generally. The thickness of the plates is less than 1 mm generally, and therefore the heat conduction from the plates' surface to the inside can be ignored. While the VC works, the temperature rises slowly, considering the heat exchange between the plates and the silicone oil being completed instantaneously at the moment of temperature difference. That is, the temperatures of the plates and the silicone oil are always equal. (4) Ignoring the radiation heat transfer from the shell and the cap to the outside; namely, the heat transmission from the cap and the cap to the outside is only the convective heat transfer. (5) The heat transmission of all parts of the VC is a steady heat transfer process.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 17 the heat exchange between the plates and the silicone oil being completed instantaneously at the moment of temperature difference. That is, the temperatures of the plates and the silicone oil are always equal. (4) Ignoring the radiation heat transfer from the shell and the cap to the outside; namely, the heat transmission from the cap and the cap to the outside is only the convective heat transfer. (5) The heat transmission of all parts of the VC is a steady heat transfer process.     Regarding α 1 as the coefficient of the average convectional heat transfer, we see:

The Convective Heat Transfer Coefficient for the Parts of VC
where In the above, γ s -the coefficient of average convectional heat transfer; β s -the volume expansion coefficient of the silicone oil; C ps -the specific heat coefficient of the silicone oil at constant pressure; t s -the temperature of the silicone oil; ω 2 -the angular velocity of the shaft; N u -Nusselt number; R e -Reynolds number; G r -Grashof number; P r -Prandtl number; D a -the outside diameter of the axle; and t a -the temperature of the axle.

The Coefficient of Average Convectional Heat Transfer between the Silicone Oil and the Shell
Regarding α 2 as the coefficient of the average convectional heat transfer, we see: In the above, D hi -the inner diameter of the shell; t hi -the temperature of the shell; and ω 1 -the angular velocity of the shell. Others are the same as in previous equations.

The Coefficient of Average Convectional Heat Transfer between the Outside Air and the Shell
Regarding α 3 as the coefficient of the average convectional heat transfer, we see: where In the above, D ho -the outer diameter of the shell; t ho -the outer surface temperature of the shell; t A -the temperature of the air; β A -the thermal expansion coefficient of the air; ν A -the kinematic viscosity of the air; C pA -the specific heat coefficient of the air at constant pressure; ρ A -the density of the air; and γ A -the thermal conductivity coefficient of the air.

The Coefficient of Average Convectional Heat Transfer between the Outside Air and the Axle
Regarding α 4 as the coefficient of the average convectional heat transfer, we see: In the above, D ao -the outside diameter of the axle; and t ao -the outer surface temperature of the axle.

The Coefficient of Average Convectional Heat Transfer between the Silicone Oil and the Cap
Regarding α 5 as the coefficient of the average convectional heat transfer, we see: where In the above, D c -the diameter of the cap; t ci -the temperature of the inside face of the cap (which contacts the silicone oil).

The Coefficient of Average Convectional Heat Transfer between the Cap and the Air
Regarding α 6 as the coefficient of the average convectional heat transfer, we see: In the above, D c -the diameter of the cap; t co -the temperature of the outside face of the cap (which contacts the air).

The Heat Transfer Calculation of VC Parts
The VC transmits torque by shearing the silicone oil with the plates. The rotating speed is different from that of the input and the output shafts. Therefore, there is energy loss during the VC's operation. Supposing the torque transmitted when the VC works is T v , the input speed is ω 1 , and the output speed is ω 2 , the speed difference is: The loss of power is: Supposing that the lost power is all converted to heat, the total quantity of heat within t (the length of working time) is: The heat transfer of each part of the VC can be determined by using the heat-transfer coefficient above.
The heat transfer of silicon oil by shell includes three links: (a) A convectional heat transfer between the silicone oil and the shell's inner wall; (b) A convectional heat conduction between the shell's outer wall and inner wall; and (c) A convectional heat transfer between the outside air and the shell's outer wall.
It is same for the heat flow through a series of links under a steady-state condition. The heat flow of the above three are: where r ho , r hi -the inside and outside diameter of the shell; and F ho , F hi -the inside and outside surface area of shell. Equation (36) may then be converted to the forms of temperature and pressure, namely: The heat transfer of the silicone oil through the left and right end covers includes three links in series: For the left end cover, the heat flow of the above three are: where F cli , F clo -the inside and outside surface area of the left end cover; t cli , t clo -the inside and outside surface temperature of the left end cover; and H cl -the thickness of the left end cover. Equation (43) may then be converted to the forms of temperature and pressure, namely: The above equations are also applicable to the right end cover. The heat transfer of the silicone oil through the shaft includes four links in series: (a) The convective heat transfer between the silicone oil and the axle inside the shell; (b) The convectional heat conduction between the axle and the surface; (c) The convectional heat conduction between the axle and one outside shell; and (d) The convectional heat transfer between the outside air and the outer wall of the outside shell.
When the convectional heat conduction between the axle and the surface is ignored, and then the temperature of each point that is perpendicular to the cross-sectional area of the axle is equal. So, the heat flow of each link is: where F ai , F ao -the inside and outside surface areas of the shell; and F d -the cross-sectional area of the axle. They then can be converted to the forms of temperature and pressure, namely:

The Temperature Calculation of VC Parts
Except for the above-mentioned capacity of heat, the remaining energy is: Supposing that the heat is overall absorbed by the silicone oil and the plates, their temperature is always the same. The ratio of the absorbed heat between them is R a , which can be determined when they raise the same temperature: In the above, C ps , m s are the specific heat coefficient of fixed pressure and the mass of the silicone oil; and C pp , m p are the specific heat coefficient of fixed pressure and the mass of the plates.
The heat that the silicone oil absorbed is: The temperature increase of the silicone oil is: The temperature of the silicone oil in time I is: If we substitute t si into the heat-transfer model, the transferred heat of all parts and the temperature can be calculated. Thus, the parameters such as the instantaneous density, the volume, the instantaneous viscosity of the silicone oil, and the filling rate of the VC are determined. If we substitute these instantaneous values of parameters into the torque calculation in Equation (1), the instantaneous torque of VC can be calculated. Therefore, the real-time torque-transmission characteristic of the viscous LSD can be calculated.
The variation of the internal temperature of the VC during operation is shown in Figure 4. The torque characteristic of the VC is shown in Figure 5. All the characteristics were derived from a simulation of the above model. As shown in the figures, the temperature of the silicone oil rose rapidly at the beginning, and the torque declined rapidly as well. Then, the temperature of VC parts rose, the heat of the VC dissipated fast, and the temperature variation of the silicone oil became increasingly smaller. The torque transmitted by the VC tended to be stable. well. Then, the temperature of VC parts rose, the heat of the VC dissipated fast, and the temperature variation of the silicone oil became increasingly smaller. The torque transmitted by the VC tended to be stable.

Theoretical Analysis and Calculation of the Hump Mode of the VC
When the VC operates at a large speed difference for a long time, the dissipation of the generated heat is insufficient. The excess heat will increase the temperature, inflate the volume, and increase the pressure, and the delivered torque will increase sharply. This is the hump phenomenon. There are several explanations for formation mechanism of the hump phenomenon in reference [7]. No final conclusion has been reached on it so far, but the theory in reference [14] is well accepted.
When the hump phenomenon occurs, the internal pressure is very high and difficult to measure. This necessitates the theoretical evaluation. Based on the generation mechanism of the hump phenomenon, the following assumptions are made to evaluate the internal pressure: (a) The solubility of silicone oil to air is very low. The proportion of air is considered to be quite small (ignorable) inside the silicone oil; (b) When the hump phenomenon occurs, the gas inside the VC is considered to be an ideal gas, and thus follows the physical law of ideal gases; (c) During the hump phenomenon, the pressure of gas inside the VC is the same as that of the other parts.
Based on the above assumptions, the combined gas law is applied to calculate the internal air pressure, which will be used as the pressure inside the shell of the VC.
From the combined gas law:

Theoretical Analysis and Calculation of the Hump Mode of the VC
When the VC operates at a large speed difference for a long time, the dissipation of the generated heat is insufficient. The excess heat will increase the temperature, inflate the volume, and increase the pressure, and the delivered torque will increase sharply. This is the hump phenomenon. There are several explanations for formation mechanism of the hump phenomenon in reference [7]. No final conclusion has been reached on it so far, but the theory in reference [14] is well accepted.
When the hump phenomenon occurs, the internal pressure is very high and difficult to measure. This necessitates the theoretical evaluation. Based on the generation mechanism of the hump phenomenon, the following assumptions are made to evaluate the internal pressure: (a) The solubility of silicone oil to air is very low. The proportion of air is considered to be quite small (ignorable) inside the silicone oil; (b) When the hump phenomenon occurs, the gas inside the VC is considered to be an ideal gas, and thus follows the physical law of ideal gases; (c) During the hump phenomenon, the pressure of gas inside the VC is the same as that of the other parts.
Based on the above assumptions, the combined gas law is applied to calculate the internal air pressure, which will be used as the pressure inside the shell of the VC.
From the combined gas law: The initial fill-up rate of the VC is C L0 , the volume of the shell is V, the initial air volume is V 1 , the initial pressure inside the shell is P 1 , the initial temperature is T 1 ; at time t, the volume of the air is V 2 , the internal pressure of the shell is P 2 , and internal temperature is T 2 .
Equation (60) yields: Therefore: Substitute the above two equations into Equation (61), yielding Therefore: As long as the temperature T 2 at time t is known, the internal pressure P 2 can be obtained.
According to the Coulomb friction law, the maximum torque transmitted during the hump mode is: In the above, f -the coefficient of friction between plates; and K-the coefficient of contact distribution between the plates.
The hump phenomenon appears when the volume of silicone oil is equal to the volume of the internal shell, so the moment of occurrence can be obtained through the calculation of the instantaneous filling rate of the silicone oil.
The instantaneous filling rate is: In the above, C L -the instantaneous filling rate of the oil; V s -the instantaneous volume of the oil.
The filling rate of the oil is almost 100% when the hump phenomenon occurs, namely, C L = 1. So, the trigger temperature is: The variation of internal pressure caused by the rising temperature during different speed differences is shown in Figure 6, it can be seen that for a certain speed difference, the increase of the pressure was nonlinear. The bigger the speed difference was, the faster it increased. The variation of internal pressure caused by the rising temperature during different speed differences is shown in Figure 6, it can be seen that for a certain speed difference, the increase of the pressure was nonlinear. The bigger the speed difference was, the faster it increased. A simulation of the whole working process (from the beginning to the hump) of the VC was carried out based on the proposed model; the simulation curve is shown in Figure  7. The torque curve from 0 s to 30 s in Figure 7 is similar to that in Figure 5. It reflects that, due to the existence of the speed difference, the temperature of the silicone oil was increased and the viscosity was reduced, resulting in a decline in the shear torque. Then, the shear torque began to rise after 30 s, because the continuous accumulation of heat greatly increased the temperature, which inflated the volume and increased the internal pressure, so the shear torque increased gradually, and then at an increasing rate. Finally, the hump phenomenon occurred at 76 s, when the peak torque approached 700 Nm. A simulation of the whole working process (from the beginning to the hump) of the VC was carried out based on the proposed model; the simulation curve is shown in Figure 7. The torque curve from 0 s to 30 s in Figure 7 is similar to that in Figure 5. It reflects that, due to the existence of the speed difference, the temperature of the silicone oil was increased and the viscosity was reduced, resulting in a decline in the shear torque. Then, the shear torque began to rise after 30 s, because the continuous accumulation of heat greatly increased the temperature, which inflated the volume and increased the internal pressure, so the shear torque increased gradually, and then at an increasing rate. Finally, the hump phenomenon occurred at 76 s, when the peak torque approached 700 Nm.

Working-Characteristics Test of the VC
In order to test the working characteristics of the VC, a VLSD, which sets the VC and the differential in one body, was designed and manufactured. The assembly layout and structure are shown in Figure 8. The working-characteristics test of VLSD was carried out. The actual conditions of the test field are shown in Figure 9. The input speed of the axle was 1500 r/min, and the speed difference of two outputs remained 400 r/min. The limited-slip torque of the VLSD in the test is shown in Figure 10.

Working-Characteristics Test of the VC
In order to test the working characteristics of the VC, a VLSD, which sets the VC and the differential in one body, was designed and manufactured. The assembly layout and structure are shown in Figure 8. The working-characteristics test of VLSD was carried out. The actual conditions of the test field are shown in Figure 9. The input speed of the axle was 1500 r/min, and the speed difference of two outputs remained 400 r/min. The limited-slip torque of the VLSD in the test is shown in Figure 10.  As shown in Figure 10, the limited-slip torque decreased slightly during the first 30 s of the test. Then, the increasing trend grew fast, and reached 200 Nm at 60 s. The limitedslip torque reached the top at around 76 s, when the peak value was about 680 Nm.
When comparing the two curves in Figure 10, apart from some differences between 40 s and 70 s, the trends of the torque curves are consistent with each other, and the time of the hump phenomenon's occurrence and the peak value of torque are basically the same on the two curves. So, the accuracy of the calculation model established in this paper is verified. The main reason for the differences is that the plates in the VC are movable during operation, which was ignored in the calculation model. This deficiency deserves further research.  As shown in Figure 10, the limited-slip torque decreased slightly during the first 30 s of the test. Then, the increasing trend grew fast, and reached 200 Nm at 60 s. The limitedslip torque reached the top at around 76 s, when the peak value was about 680 Nm.
When comparing the two curves in Figure 10, apart from some differences between 40 s and 70 s, the trends of the torque curves are consistent with each other, and the time of the hump phenomenon's occurrence and the peak value of torque are basically the same on the two curves. So, the accuracy of the calculation model established in this paper is verified. The main reason for the differences is that the plates in the VC are movable during operation, which was ignored in the calculation model. This deficiency deserves further research. As shown in Figure 10, the limited-slip torque decreased slightly during the first 30 s of the test. Then, the increasing trend grew fast, and reached 200 Nm at 60 s. The limited-slip torque reached the top at around 76 s, when the peak value was about 680 Nm.
When comparing the two curves in Figure 10, apart from some differences between 40 s and 70 s, the trends of the torque curves are consistent with each other, and the time of the hump phenomenon's occurrence and the peak value of torque are basically the same on the two curves. So, the accuracy of the calculation model established in this paper is verified. The main reason for the differences is that the plates in the VC are movable during operation, which was ignored in the calculation model. This deficiency deserves further research.

Conclusions
The rising temperature of the silicone oil is the basis of the change in operating characteristics of the VC. The heat-transfer model of the VC was proposed based on the thermodynamic theory. The heat-transfer coefficient and the capacity of heat transmission of each heat-transfer part were calculated. The calculation method for the heat the silicone oil absorbed and the change in the silicone oil's temperature were obtained. This is the theoretical foundation for real-time analysis of the operating characteristics for the VC. A method to calculate the working characteristic of the VC in real-time that considered the temperature and pressure change was originally proposed. The simulation was carried out based on the calculation model. The prototype of a limited-slip differential with a VC was made, and the working-characteristics test was carried out, especially to determine the working characteristics of the hump phenomenon. The test result and the simulation result of the model were in good alignment, which verifies the accuracy of the model. However, the limitation of the VC's simplified model and the ignorance of movable plates introduced some errors in the simulation, which needs to be improved in the future. A roadway test of vehicles equipped with a VLSD could be another interesting study in the future.