Research on the Transmission Performance of a High-Temperature Magnetorheological Fluid and Shape Memory Alloy Composite

Abstract

To address the fact that the performance of magnetorheological fluid decreases with increasing temperature, a high-temperature magnetorheological fluid and shape memory alloy spring friction composite transmission method is proposed, and its transmission performance is shown to essentially maintain stability under high temperatures. We introduce a composite transmission method, performed a magnetic field finite element analysis, and present the equation of torque transmission of the composite. The results show that the amount of torque transferred by the magnetorheological fluid reached its maximum value at magnetic saturation, but decreased with increasing temperature, down to 33.41%, whereas the frictional torque generated by the shape memory alloy spring increased with increasing temperature. When the temperature reached 100 °C, the frictional torque effectively compensated for the decrease in the magnetorheological fluid’s performance.

1. Introduction

Magnetorheological fluid (MRF) is a solid–liquid two-phase material formed by micron-sized magnetic particles uniformly distributed in a base fluid by the action of additives. Under the action of the magnetic field, the magnetic particles in the MRF are arranged into chains in the magnetic field direction. The liquid fluidity becomes poor, and the MRF shows solid-like mechanical properties. When the magnetic field disappears, the magnetic particles quickly return to a disordered state, and the MRF returns to a viscous liquid. MRF can complete the reversible transformation from Newtonian fluid to Bingham fluid in milliseconds under the action of the magnetic field, and the properties of the material, such as damping and shear yield stress, change continuously with the change in the magnetic field. Based on these characteristics, MRF can be used in such engineering fields as transmission and vibration damping [1,2]. Shape memory alloys (SMAs) are a class of smart materials with superelasticity and a shape memory effect whose internal metallurgical structure interchanges between low-temperature martensite and high-temperature austenite when the temperature changes. The shape memory effect of SMAs can be used for the purpose of adaptive control and stroke output [3].

The application of MRF and SMAs in transmission has received a lot of attention from a large number of researchers. Huang et al. [4] derived the torque equation for MRF brakes, providing a theoretical basis for the design of MRF actuators. Rizzo [5] proposed a multi-gap cylindrical magnetorheological (MR) brake to improve the braking torque by increasing the working area of the magnetorheological fluid. Dai et al. [6] investigated a disc-type MR clutch with a composite cylinder, and increased the torque compared with the disc-type MR clutch by 1.45 times. Pilon et al. [7] designed an MR clutch with a helical fluid flow channel, which improved the durability of the clutch by 42%. Hegger et al. [8] designed a tapered-gap MR drive and enhanced the performance of the MRF by axial squeezing, which combined shear and frictional forces for torque transmission.

To control the total volume of the MR transmission device while improving the transmission performance, Sun et al. [9] proposed a cycloidal corrugated MR transmission device with composite shear and squeeze modes, which enhanced the anti-damage capability of the magnetic chain. The torque equation was established based on the Metal Plastic Deformation Theory, which greatly improved the torque transmission capacity of the device at the same volume. Yu et al. [10] proposed an MR damper based on the helical flow mode, and the torque equation was derived based on the consideration of the fluid flow pressure. An experimental analysis was also conducted on the torque performance at different rotational speeds, and the experimental results showed that the new MR damper increased the torque density of the transmission while limiting the volume.

SMAs are usually machined into spring shapes for transmission applications. Zhou et al. [11] established incremental constitutive equations between stress, strain, and temperature and proposed a method for finite element simulation of complex SMA structures. Malukhin et al. [12] proposed an SMA actuator with the nonlinear dynamic properties of human muscles and established an analytical expression for the damping coefficient of the actuator. Hwang et al. [13] introduced a design concept for an SMA rotary actuator in order to transform the linear motion of SMA springs into high-torque rotary motion and explained its thermodynamic behavior based on experiments.

The above-mentioned studies mainly focus on the application of MRF or SMAs alone, and less on the combined application of the two materials. Furthermore, since the shear stress of MRF decreases with increasing temperature, especially at high temperatures, its performance decreases significantly [14]. In addition, the magnetic saturation of MRF also limits the torque, which means it cannot be increased continuously [15]. The torque of the transmission device designed based on the shear mode is low, which restricts the application and development of MRF in the transmission field [16]. This paper proposes a high-temperature MRF and SMA spring friction composite transmission method for the above-mentioned problems. The SMA spring was added to the MR drive with the aim of increasing the structural flexibility of the system. The transmission performance of the composite transmission method is guaranteed to have a certain level of stability under high temperatures. In the design process of the transmission device, the working area of the MRF was increased with a number of circular arcs to improve the transmission torque of the device. As the temperature rises, the performance of the MRF begins to decline. At this time, the SMA spring, under the effect of heat, squeezes the hydraulic fluid to push the input disc and output housing contact friction to generate additional torque. The additional frictional torque is used to compensate for the performance loss of the MRF, so that the transmission performance of the device can remain basically stable even at high temperatures.

2. Design of the MR Transmission Device

Figure 1 shows a schematic diagram of the high-temperature MRF and SMA spring friction compound transmission device. The MR transmission device consists of an input shaft, an output shaft, a coil, an input disc, an output housing, an SMA spring, a driving piston, a friction ring, and other components. The coil is connected to the electric slip ring through the wire hole on the output housing. The input disc is circumferentially fixed to the input shaft by a slide key, and the friction ring is fixedly connected to the input disc by a screw. The surface of both the input disc and the output housing is a semi-circular arc, and the arc gap between the two parts is filled with MRF. The shape of the MRF working gap is shown in Figure 1b.

The total torque of the compound transmission consists of two parts: MR torque and SMA compensation torque. The operating principle of the transmission is as follows.

First, in the initial state, the input shaft starts to rotate under the action of the input torque from the power source. At this time, no current is passed into the coil, and the torque transmitted by the zero-field viscosity of the magnetorheological fluid alone is not sufficient to drive the output housing to rotate.

Second, after the excitation coil is energized, the magnetic flux generated by the coil passes perpendicularly through the gap of the magnetorheological fluid, and the magnetic particles in the MRF are arranged into a chain-like structure in the direction of the magnetic flux. The shear yield stress of the magnetic chain transmits torque to drive the output housing to rotate synchronously. The current continues to increase until the MRF is magnetically saturated, the magnetic flux cannot be further increased, and the shear torque reaches its maximum value.

Finally, when the system needs to stop the power input, the coil is disconnected, and the shear yield stress of the MRF decreases rapidly. Then, the input disc and the output housing are disconnected, and the output shaft stops rotating.

3. Analysis of Torque

3.1. Finite Element Model Simplification

In order to improve the maximum torque of the composite transmission device, the MRF working area was designed to be a disc-type gap consisting of multiple circular arc surfaces, and a 2D axisymmetric steady-state magnetic field finite element analysis was performed on the high-temperature MRF and SMA spring friction composite transmission device, the simplified model and specific dimensions of which are shown in Figure 2 and

Table 1. Dimensions of the transmission device (mm).

R1	R2	R3	R4	R5	L1	L2
87	77	65	35	25	15	20

, respectively.

The materials of the input disc and the drive housing are Steel-1010 and Steel-1030, respectively, and the material of the input shaft is Steel-1020. The surrounding air was set as the boundary, and the magnetorheological fluid model used was MRF-5. The parameters and the magnetic curve of MRF-5 are shown in

Table 2. Magnetorheological fluid (MRF-5) performance parameters.

Volume Fraction/%	Additives Ratio/%	Base Fluid	Zero Field Viscosity/Pa∙s
45	1.4	silicone oil	2.9

and Figure 3, respectively [14] (pp. 287–290).

According to the data in Figure 3, the relationship between magnetic induction density and magnetic shear yield stress was fitted to the relational equation, which can be expressed as

$${\tau }_y(T,B)={0.9}^{\frac{T-20}{20}}\left(0.09B^3-2.51B^2+23.17B-2.06\right)$$

(1)

where B is the magnetic induction density in the working gap, T is the steady-state temperature of the MRF, and τ_y (T, B) is the magnetic shear yield stress of the MRF.

The number of turns of the excitation coil was set to 100 and the current was 0~3.5 A. The Ansoft Maxwell software was used, and the solution was obtained using axisymmetric steady-state magnetic field analysis. The magnetic properties of the various materials used in the simulation are shown in Figure 4 [17] (pp. 362–365). Considering the magnetic saturation of the material, in order for all materials to achieve the actual maximum performance, we set the maximum magnetic flux density of the steel to 1.8 Tesla.

3.2. The Effect of the Gap Shape on the Magnetic Circuit

To investigate the effect of the gap shape on the magnetic circuit, a transmission drive with the same radial planar disc working gap size was used for comparison. r_a is the arc radius, α is the arc circle angle, N is the number of turns of the excitation coil, and I is the coil current. The magnetic field distribution in both gaps when both the arc radius r_a and the working gap thickness h are 1 mm and the number of ampere-turns NI is 300 A is shown in Figure 5.

The upper and lower surfaces are the contact surfaces between the output housing, the input disc, and the MRF, respectively. We use the magnetic flux density in Region 2 as an example of the effect of the gap shape on the magnetic field. From Region 2 in Figure 5a,b, it can be found that the magnetic flux density distribution of the semi-circular gap is more uneven compared with the planar gap. The magnetic resistance at the transition of the two arcs is low. It can be observed from the magnetic boundary conditions that the direction of the magnetic flux density will be deflected after it passes through different magnetic materials [18] (pp. 252–253). The relationship between the deflections θ₁ and θ₂ of the magnetic flux density on both sides and the relative magnetic permeability of the material is

$$\frac{\tan {\theta }_1}{\tan {\theta }_2}=\frac{{\mu }_{r1}}{{\mu }_{r2}}$$

(2)

where μ_r1 and μ_r2 are the relative magnetic permeabilities of the materials on each side, respectively. The magnetic field boundary conditions are illustrated in Figure 6.

As can be seen in Figure 5, the magnetic saturation region appears only near the cusp of the model. However, there is no magnetic saturation in either the Steel-1010 or the Steel-1030 in the entire working gap region. When the magnetic field from the Steel-1010 passes through the MRF, μ_r1 and μ_r2 are 620 and 5 (NI = 300 A), respectively. Bringing the relative magnetic permeability into Equation (2) yields θ₂ as follows:

$${\theta }_2=\arctan (\frac{5\tan {\theta }_1}{620})$$

(3)

From Equation (2), the magnetic field will deflect at a lower inclination angle θ₂ when it crosses the boundary at any angle (except 90°), and the same applies to the deflection that occurs when the magnetic field penetrates the MRF gap. According to the Flux Continuity Theorem [18] (pp. 255–257), the magnetic induction line will pass through the path of least magnetic resistance. That is, the magnetic induction within the working gap is in the same direction as the radius of the lower surface arc.

Define h_z as a coordinate from 0 to h = 1 mm along the normal axis, which indicates the distance between the arc plane where the MRF is located and the input disc. The relationship between the magnetic flux density at different thicknesses h within the MRF gap and the circular center angle α of the arc is shown in Figure 7.

As can be seen from Figure 7a, the magnetic flux density at the transition of the arc increases as h_z increases when the current is the same, which is the opposite trend to that at the center of the arc. To clarify the magnetic induction density corresponding to different angles, the surface at an MRF thickness of h/2 was taken as the neutral surface. The magnetic induction density at the neutral surface was evaluated and the results are shown in Figure 7b.

When comparing the magnetic flux densities in Figure 7b, one can observe that the maximum value in the semi-circular gap is 0.542 Tesla, whereas the maximum value in the planar gap is 0.5 Tesla. Compared with the planar gap, the maximum magnetic flux density in the semi-circular gap increased by 8.4%. However, over all angles, the average B for the semi-circular gap is smaller than the same value for the planar gap. When the current rises, the MRF in the semi-circular gap is more likely to become magnetically saturated in local areas. If the degree of unevenness in the magnetic field distribution is too high in the entire gap, it will be detrimental to torque control.

The equivalent length l_g of the unilateral semi-circular gap increases compared with the planar gap as follows:

$$l_{\mathrm{g}}=n_{\mathrm{m}}{r}_{\mathrm{a}}(\pi -2)$$

(4)

where n_m is the total number of arcs. Combined with the equivalent length l_g, at r_a = 1 mm, the equivalent working length of a single semi-circular gap is (π − 2) mm longer than that of a planar gap with the same radial distance (about a 57.1% increase). Although the average B of the semi-circular gaps is smaller, the equivalent working length increases more. Altogether, the semi-circular gap can provide more torque.

The effect of the arc radius r_a on the magnetic field is explored below, and the magnetic flux density at the neutral surface is shown in Figure 8.

Along with the increase in the arc radius, the inhomogeneity of the magnetic flux distribution increases as the magnetic flux density B decreases at the transition between the two arcs. In addition, the magnetic flux density within the working gap of a single arc is symmetrically distributed along the geometric axis. In accordance with the data presented in Figure 8, it can be found that the flux distribution within the working gap is more uniform when r_a = 0.5 mm. As the chain formation process of magnetic particles is influenced by the magnetic flux density’s magnitude, the MRF in the gap will not be sufficient when the inhomogeneity of the magnetic field distribution increases [19]. The shear yield stress at the circular gap is low, which leads to a decrease in the transmission device’s ability to transfer torque. Therefore, r_a = 0.5 mm was chosen as the optimal size.

3.3. Effect of the Current Value on the Magnetic Field

In Section 3.2, we explored the effect of gap shape on the magnetic circuit. In the following section, we explore the effect of current on the magnetic field. The coil current was continuously changed from 0 A to 3.5 A. The magnetic flux distribution clouds for the planar and semi-circular drives when the current was 3.0 A are shown in Figure 9.

Figure 9 shows that the magnetic flux density of the MRF working gap has an increasing trend in the radial direction. Disregarding the effect of heat generation, when the radial working gap lengths are all 30 mm, the equivalent distances R for the semi-circular gap and the planar gap are 47.12 mm and 30 mm, respectively. The magnetic flux density caused by different currents is shown in Figure 10.

The equivalent distance R is defined by equating the length of the curved disc-type MR drive to the length corresponding to the straight disc-type MR drive. As the coil of the disc-type drive is located on the outside of the working gap, the magnetic field generated by the coil decays in the direction away from the coil. From Figure 10, we can observe that the magnetic flux density increases as the current increases, but it changes more slowly with the radius of the working gap, and the data show an overall wavy growth trend. The magnetic flux density at the transition of the arc is lower, the maximum value appears at the circular angle α = 45°, and the maximum fluctuation rate is 28%. The data on the magnetic flux density B in the working gap are shown in

Table 3. The range of variation in the magnetic flux density for the two working gaps.

Current I/(A)	Magnetic Flux Density B/(Tesla)
	Multi-Arc		Planar
	Min	Max	Min	Max
1.0	0.209	0.456	0.340	0.382
2.0	0.251	0.721	0.465	0.553
3.0	0.297	0.863	0.501	0.606
3.5	0.313	0.920	0.513	0.628

.

As can be noticed from Table 3, when the current exceeds 3 A, the magnetic flux density of the MRF increases slowly as the current increases. The MRF approached magnetic saturation after a current of 3.0 A, so the maximum current was set to 3.0 A. By comparison, it can be observed that the magnetic flux density within the semi-circular gap has a wider range of variation when the working radius is the same, and the tunability of the torque is better than that of the planar disc-type drive.

3.4. Torque Equation for Curved Disc-Type MR Drives

To establish the torque equation for curved disc-type MR drives, the MR torque within a single semi-circular gap was analyzed in cylindrical coordinates (ρ, θ, z). The equivalent model is shown in Figure 11.

Taking an infinitesimal arc ds in the semi-circular gap, the shear yield stress dF generated by the infinitesimal arc is

$$\mathrm{d}F=2\pi y\tau \mathrm{d}s=2\pi y\tau r_{\mathrm{a}}\mathrm{d}\theta$$

(5)

where y is the distance between the infinitesimal arc and the center of rotation, O is the center of the semi-circular arc, L_n is the distance between the center of the n-th arc and the center of rotation, θ is the angle of the corresponding infinitesimal arc, and r_a is the radius of the semi-circular arc.

Then, the distance between the infinitesimal arc and the center of rotation y is

$$y=L_n-r_{\mathrm{a}}\cos \theta$$

(6)

The Bingham model is used to describe the rheological properties of the MRF, and its constitutive relationship is

$$\{\begin{array}{cc}\tau ={\tau }_y(T,B)\mathrm{sgn}(\dot{\gamma })+\eta \dot{\gamma }& \left|\tau \right|\ge {\tau }_y\\ \dot{\gamma }=0& \left|\tau \right|<{\tau }_y\end{array}$$

(7)

where η is the zero-field viscosity and $\dot{\gamma }$ is the shear strain rate of the MRF.

Bringing Equations (6) and (7) into Equation (5), we can obtain the MR torque generated by the MRF infinitesimal arc as

$$\mathrm{d}M=2\pi r_{\mathrm{a}}{\left(L_n-r_{\mathrm{a}}\cos \theta \right)}^2[{\tau }_y(T,B)+\eta \dot{\gamma }]\mathrm{d}\theta =2\pi r_{\mathrm{a}}{\left(L_n-r_{\mathrm{a}}\cos \theta \right)}^2\left[{\tau }_y(T,B)+\eta \frac{\mathrm{d}{v}_{\theta }}{\mathrm{d}z}\right]\mathrm{d}\theta$$

(8)

where v_θ is the shear linear velocity of the MRF infinitesimal arc in the direction of rotation, i.e., v_θ = ωy.

Bringing in Equation (8) yields

$$\begin{array}{c}\mathrm{d}M=2\pi r_a{(L_n-r_a\cos \theta )}^2[{\tau }_y(T,B)+\eta y\frac{\mathrm{d}\omega }{\mathrm{d}z}]\mathrm{d}\theta \\ \hspace{1em}\hspace{1em} =2\pi r_a{(L_n-r_a\cos \theta )}^2[{\tau }_y(T,B)+\eta y\frac{({\omega }_1-{\omega }_2)}{h}]\mathrm{d}\theta \end{array}$$

(9)

where ω₁ and ω₂ are the input and output shaft angular velocities, respectively. Bringing Equation (6) into Equation (9) yields

$$\mathrm{d}M=2\pi r_{\mathrm{a}}{\left(L_n-r_{\mathrm{a}}\cos \theta \right)}^2\left[{\tau }_y(T,B)+\eta \left({\omega }_1-{\omega }_2\right)\frac{\left(L_n-r_{\mathrm{a}}\cos \theta \right)}{h}\right]\mathrm{d}\theta$$

(10)

By integrating Equation (10) over the entire arc interval, the total MR torque transmitted by double-disc curved MR drives can be obtained as

$$M_{\mathrm{m}1}={\displaystyle \sum _{n=1}^{n_{\mathrm{m}}}2}{\pi }^2{r}_{\mathrm{a}}{\tau }_y(T,B)\left(2L_n^2+r_{\mathrm{a}}^2\right)+\frac{2\pi r_{\mathrm{a}}\eta L_n}{h}\left({\omega }_1-{\omega }_2\right)\left(2\pi L_n^2+3\pi r_{\mathrm{a}}^2\right)$$

(11)

where n is the current number of arcs and n_m is the total number of arcs.

Similarly, the total MR torque of the double-disc planar MR drive is [6]

$$M_{\mathrm{m}2}=2{\displaystyle {\int }_{R_4}^{R_3}2}\pi r^2\tau dr=\frac{4\pi }{3}\left(R_3^3-R_4^3\right){\tau }_y(T,B)+\frac{\pi }{h}\left(R_3^4-R_4^4\right)\left({\omega }_1-{\omega }_2\right)\eta$$

(12)

3.5. Analysis of the MR Torque Transmission Performance of the Transmission Device

When the inner and outer diameters of the MRF working gap are 35 mm and 65 mm, respectively, the thickness of the working gap h is 1 mm, the viscosity η is 2.9 Pa∙s, the speeds of the input and output shafts are 30 rad/s and 29 rad/s, respectively, n_m = 30, and r_a = 0.5 mm, the equivalent working length of the unilateral disc increases by l_g = 17.124 mm.

In order to highlight the beneficial effects of the curved disc-type MR drive, the points with the same magnetic flux density within the working gap were selected for equivalence. The MR torque of the curved disc-type drive was compared with the experimental torque values of the existing planar disc-type drive [20], and the results are shown in Figure 12.

From the data shown in Figure 12, it can be observed that the numerical and experimental torques of the planar disc-type drive are 43.626 N∙m and 44.944 N∙m, respectively, when the current is 1.5 A. The difference between the finite element numerical and experimental values is 3.02%, and the accuracy of the numerical calculation is within an acceptable range. At this time, the torque of the curved disc-type drive is 74.805 N∙m, which is 66.44% higher than that of the planar disc-type drive. When the current is 3 A, the torque of the curved and planar disc-type drives is 82.627 N∙m and 46.430 N∙m, respectively, and the torque is increased by 77.96%. The curved disc-type MR drive significantly improved the transmission performance compared with the traditional planar disc-type MR drive, and the torque improvement effect became more obvious as the current increased.

3.6. Influence of Temperature on MR Torque

The rheological properties of MRF, such as shear stress and viscosity, decrease significantly under high-temperature conditions, and the performance decays significantly. To describe the relationship between MRF temperature and torque, an expression for the relationship between temperature and MRF shear yield stress was established.

Figure 13 shows the relationship between the MR torque of the curved disc-type MR drive and the temperature according to Equation (1) when the current was 3.0 A and the temperature increased from the ambient temperature of 20 °C to 100 °C. As can be seen from Figure 13, the MR shear torque of the drive follows an approximately linear and decreasing trend as the temperature increases, and the maximum torque decreases from 82.627 N∙m to 55.021 N∙m (a decrease of 33.41%).

4. Analysis of SMA-Compensated Torque Performance

4.1. Driving Force of the SMA Spring

To compensate for the degradation in the performance of the MRF under high-temperature conditions and to maintain the stability of the transmission performance, SMA springs were used to output additional friction torque. Figure 14 shows a schematic diagram of the SMA spring. The structural parameters of the SMA (Ni51Ti49 (at. %)) spring are a spring middle diameter D₀ = 12.5 mm, a spring wire diameter d₀ = 1.95 mm, an effective number of turns n_SMA = 6, and an inclination angle Φ = 6°. The material parameters of the spring are shown in

Table 4. Material parameters of the shape memory alloy spring (Ni51Ti49 (at. %)) [21].

AS/°C	AP/°C	AF/°C	GA/GPa
39.85	60.85	80.85	200
MS/°C	MP/°C	MF/°C	GM/GPa
31.85	26.85	22.85	70

.

M_S, M_F, A_S, and A_F are the starting and finishing transformation temperatures of martensite and austenite, respectively, as shown in Table 4. G_M and G_A are the shear moduli of martensite and austenite, respectively.

The SMA spring is restrained during the phase transformation, and the spring exerts a force on the restrained body, which is the restoring force of the SMA spring F_r. When the temperature of the SMA spring is between the finishing temperature of the martensitic phase transformation M_F and the finishing temperature of the austenitic phase transformation A_F, the relationship between the temperature and the restoring force is [22]

$$F_r=\frac{G(T)d_0^4}{8D_0^3{n}_{\mathrm{SMA}}}{\delta }_L$$

(13)

where δ_L is the axial expansion of the spring when the temperature is lower than A_s.

G(T) is the shear modulus of the SMA, which is temperature dependent and takes the value of

$$G(T)=G_{\mathrm{M}}+\frac{G_{\mathrm{A}}-G_{\mathrm{M}}}{2}\left[1+\sin \psi \left(T-T_m\right)\right]$$

(14)

where T_m = (A_S + A_F)/2 and ψ = π/(A_F − A_S).

The restoring force of the SMA spring in the interval from 20 °C to 100 °C can be obtained from Equation (13) and was compared with the temperature–force data on the SMA measured in the previous experiment [21] as shown in Figure 15. It can be seen from the experimental values that the restoring force F_r increases rapidly from 7.93 N to 63.66 N at temperatures from 40 °C to 80 °C. When the temperature is greater than 80 °C, the high-temperature austenite phase transformation inside the SMA ends and the restoring force is generated by the high-temperature expansion and deformation of the material. The value remains basically constant, and the maximum restoring force is 67.28 N.

4.2. Friction Torque of the Drives

The composite transmission device pushes the piston to squeeze the hydraulic fluid via the spring restoring force, so that the friction ring of the input disc squeezes the output housing to produce additional friction torque. A schematic diagram of the hydraulic transmission friction is shown in Figure 16.

According to Pascal’s principle, the fluid pressure thrust can be expressed as

$$F_2+F_3=n_2{\eta }_m{F}_S\left(\frac{D_1^2-D_2^2}{D_3^2}\right)$$

(15)

where D₁ and D₂ are the inner diameter and the outer diameter of the input disc ring piston, respectively, D₃ is the diameter of the driving piston, n₂ is the number of SMA springs, and F_s is the thrust force on the piston, that is, F_s = F_r. The hydraulic thrust forces F₂ and F₃ are equal, and η_m is the hydraulic transmission efficiency.

The input disc is extruded and rubbed by the output housing under the action of fluid pressure, and the stress on the unilateral friction ring is

$$\sigma =\frac{F_2}{\pi \left(r_2^2-r_1^2\right)}$$

(16)

where r₁ and r₂ are the inner and outer diameters of the input disc friction ring, respectively.

The bilateral friction torque M_f generated by the hydraulic thrust is

$$M_{\mathrm{f}}=2{\displaystyle {\int }_{r_1}^{r_2}\mu }r\sigma \mathrm{d}S=\frac{4\mu F_2\left(r_2^2+r_1{r}_2+r_1^2\right)}{3\left(r_1+r_2\right)}$$

(17)

where μ is the friction coefficient.

We set the friction coefficient μ = 0.29, the number of springs n₂ = 6, and the hydraulic transmission efficiency η_m = 0.85. The piston diameters D₁, D₂, D₃ were 70 mm, 50 mm, and 20 mm, respectively, and the inner diameter r₁ and outer diameter r₂ of the friction ring were 66 mm and 75 mm, respectively. The SMA spring compensation torque was derived from Equation (17).

4.3. Total Torque of the Composite Transmission Drive

By adding the MR torque and SMA compensation torque at different temperatures, the total torque of the composite transmission drive can be obtained as

$$M=M_{\mathrm{m}}+M_{\mathrm{f}}$$

(18)

Figure 17 shows the relationship between the MR torque, SMA compensation torque, and total torque of the composite transmission device with temperature when the current I = 3 A. From Figure 17, it can be seen that the torque transferred by the MRF decreased as the temperature increased, but the trend of the SMA compensation torque was the opposite. The compensation torque provided by the SMA spring was low at temperatures from 20 to 45 °C. When the temperature was in the range of 45 to 80 °C, the compensation torque increased more rapidly from 3.216 N∙m to 25.727 N∙m, with a maximum compensation torque of 27.69 N∙m.

The total torque of the high-temperature MRF and SMA spring friction composite transmission device was finally maintained at 82.627 N∙m, which is basically the same as that of the MR torque at 20 °C.

To facilitate the measurement of the torque performance of the transmission device, the torque fluctuation rate β of the composite transmission device was defined as

$$\beta =\frac{M-82.627}{M}\times 100\%$$

(19)

The torque fluctuation curve in Figure 17 shows that the total torque of the transmission device changes smoothly, the fluctuation in the torque is the largest when the temperature is 77 °C, and the total torque is 88.711 N∙m, which is 8.16% higher than the MR torque at 20 °C. When the temperature is 53 °C, the fluctuation in the torque is the smallest, the total torque is 82.141 N∙m, and the torque error is only 0.15%.

In summary, the compensating torque of the shape memory alloy at high temperatures is exactly equal to the torque lost as a result of the MRF’s degradation. That is, the torque of the composite drive can be basically maintained at a stable level.

5. Conclusions

In this article, we proposed a high-temperature MRF and SMA spring friction composite transmission device that can effectively improve the MR torque performance by means of several semi-circular arcs that increase the MRF working gap.

The structure and working principle of the composite transmission device were described. Based on the Flux Continuity Theorem of magnetic fields, the magnetic flux density of the MRF working gap was calculated. The torques generated by the MRF and SMA springs at different temperatures and the composite torque were calculated. The results show that:

(1) The shear yield stress of the MRF decreased approximately linearly with the increase in the temperature, and the maximum decrease in torque was 33.41%. The restoring force of the SMA spring started to increase significantly when the temperature was 40 °C and reached the maximum when the temperature increased to 100 °C;

(2) The SMA spring effectively compensated for the decline in the high-temperature MRF’s performance. The torque of the composite transmission device was relatively stable, with a maximum torque fluctuation rate of 8.16% and a minimum torque error of 0.15%;

(3) When the temperature increases, the thermal effect of the SMA spring can be used to compensate for the decrease in the performance of the MRF, which solves the problems with the previous MR transmission devices that have a narrow torque adaptation range and a transmission performance that is greatly affected by temperature. This study also provides a theoretical reference for high-temperature MR transmission devices;

(4) The performance of the curved disc-type MR drive was significantly improved compared with the planar disc-type MR drive, with a maximum torque increase of 77.96%.