Numerical and experimental studies on a novel magneto-rheological fluid brake based on fluid–solid coupling

Abstract

The previous work of the authors indicated that the fluid–solid coupling effect of the magneto-rheological fluid and the brake disc is a necessary focus during braking process. In this study, a novel design of magneto-rheological fluid brake was proposed and studied theoretically and numerically, aiming to solve the prominent problem of heat dissipation, especially in the case of single emergency braking. First, based on the modified Bingham model, a parameter defined as the apparent equivalent viscosity was utilized to represent the relationship of magnetic field, flow field, and temperature field. The braking torque and the formula for calculating the impact factor of fluid–solid coupling employed for characterizing the associations among the thermal field and the stress field were established based on fluid–solid coupling. With a detailed explanation of simulation method, the distribution disciplinarian’s numerical simulation of each field was analyzed using COMSOL software. To validate the accuracy of the established model on the designed magneto-rheological fluid brake, the prototype was also manufactured, and results achieved experimentally which were measured on inertia test system of brake, for braking torque, motion parameters, and surface temperature in braking process, were compared with simulations. Simulation results manifested that the designed magneto-rheological fluid brake’s magnetic circuit structure is feasible based on magnetic induction intensity distribution. Finally, it has been shown that the simulations appear to be basically consistent with the experimental results, and the heat dissipation of the designed magneto-rheological fluid brake is partially improved. These results might contribute to the structure design, optimization, and improvement of magneto-rheological fluid products, extending the previous work on fluid–solid coupling analyses.

Introduction

Magneto-rheological fluid (MRF) is one type of smart material with solid–liquid two-phase, and its morphology and properties are constrained and regulated by external magnetic field. With the continuous, reversible, rapid, and easy to control rheological properties, it makes MRF have a broad range of applications including the fields of aerospace, automotive industry, hydraulic transmission, sealing, precision machining, biotechnology, and medical instrument. In addition, based on the above-mentioned characteristics, MRF has been applied to products such as dampers, clutches, brakes, polishing devices, hydraulic valves, flexible fixtures, and other sandwich plates.^1–3 As a type of new control brake that has emerged about 30 years ago,^4,5 the MRF brake was assessed in use of various fields or devices, such as precision instrument processing, aerospace, control engineering, and mechanical and automotive engineering,^6–8 due to their characteristics of simple structure, low plastic viscosity, low energy consumption, fast response time (the order of magnitude is usually millisecond), insensitive to interference with external impurities, and controllable braking torque, compared with conventional hydraulic brakes.^9,10 However, since the braking performance is limited by installation space, design volume and weight, and heat dissipation, as well as the stability of MRF properties, the industrial application of MRF brakes is still not mature, ¹¹ and the working performance cannot meet the high or demanding requirements of brake technology, especially in the automotive brakes. So, more new structural designs and advanced analysis methods are needed to get a good design of MRF brake with high performance under limited conditions.

In recent years, numerous researches on the MRF brake focus primarily on the structural design and optimization,^12–14 magnetic circuit design and analysis,^15–17 braking performance test,^18,19 multi-physics coupling analysis involving fluid–solid coupling such as thermal (or temperature) field, flow field, and stress field.^9,20–23 To be specific, Nguyen et al. ¹² proposed a new configuration of MRF brake with coils on the brake’s side housing, instead of placing the coils on the cylindrical housing, and performed the optimization for the proposed MRF brake with the consideration of the largest braking torque and the least mass of the brake. Poznic et al. ¹³ presented a new approach on combined materials application in the MRF brake, to achieve magnetic flux density path routing, which can additionally improve MRF brake’s overall braking performance. Shamieh and Sedaghati ¹⁴ introduced a novel MRF brake design with no zero-field viscous torque and studied multi-objective programming (MOP) to obtain the optimal brake parameters. For magnetic circuit design and analysis, with the use of magnetically conductive and non-conductive rings, Senkal and Gurocak ¹⁵ created a serpentine flux path for weaving the magnetic flux through the MRF, and their design can yield 2.7 times more torque with one-third smaller size, compared with a commercial MRF brake. Furthermore, the finite element analysis (FEA) can be used to analyze the electromagnetic action in a MRF braking system, for example, the work of Sarkar and Hirani. ¹⁶ Later in 2017, Meng et al. ¹⁷ developed a combined optimization design method based on the FEA and multi-objective genetic algorithm (MOGA), and the optimal magnetic circuit structure parameters were extracted through Pareto optimization. Thus, it can clearly be seen that such a MRF brake is desirable to be obtained in engineering application, which has maximum braking torque while minimizing braking time and weight of the MRF brake under space constraints.

In terms of braking performance test, the experimental apparatus which is suitable for the designed prototype was usually developed for performance evaluation, such as Shiao et al. ¹⁸ and Attia et al. ¹⁹ These experimental studies are still not sufficient to indicate its braking performance compared to industrial testing. Taking the frictional brake with friction lining used in vehicle as an example, to test the quality of the designed brake, it is necessary to conduct the road test. ²⁴ During the design stage, the brakes are not able to perform the road test and can only be tested in the lab, similar to the design of MRF brakes. However, the braking process of the brake must be required to be as consistent as possible with the process of the brake on the road test vehicle (i.e. the experimental setup’s rotational kinetic energy should be as consistent with the road test vehicle’s translational kinetic energy as much as possible). This means that the corresponding structural parts, such as flywheel components in Wang et al., ⁹ should be configured to achieve the equivalent moment of inertia. So, further works should be aimed at establishing more suitable MRF brake test-rigs with better performance evaluation that can give a comprehensive understanding of the designed MRF brake.

More recently, with multidisciplinarity in the continuous development of mechanical engineering, multi-physics coupling analysis has received significant attention from researchers, especially fluid–solid coupling mechanics involving the interactions between multi-phases media. Because the rheological characteristics involve the interaction between the fluid and solid media and the coupling effect between the fluid and solid interface, MRF brake is a typical fluid and solid coupling model, with a high degree of non-linearity. Patil et al. ²⁰ conducted the thermal analysis of one MRF brake envisaged for an e-bicycle based on FEA model and estimated the temperature rise of MRF on account of braking maneuver. Topcu et al. ²¹ developed an algorithm of particle swarm optimization (PSO) modified for rotary MRF brakes for optimizing multi-physics engineering, which exhibits a performance better than the traditional PSO. Le-Duc et al. ²² presented a multi-objective optimization procedure that combines NSGA-II (Elitist Non-dominated Sorting Genetic Algorithm) and an approach for optimizing robust multi-objective integrated with FEA, to evaluate different design objectives’ interactive relations in the MRF brake, and then established a thermal analysis model for MRF brake’s off-state and on-state conditions, to achieve the Pareto set and relative solutions. Song et al. ²³ investigated the tribological and thermal properties of a disc-type MRF brake under the condition of different working gaps and performed several heating and wear tests. The results showed that the smaller the working clearance, the higher the temperature, and this thermal characteristic leads to the decrease of braking torque. To the best of our knowledge, in all of the previous works, to establish a suitable model it is vital to simulate, design, and optimize the MRF braking system correctly, especially to manifest the complex coupling characteristics of an entire system.

In our previous work, we have investigated a double-coil side-mounted MRF brake^5,17,25 and conducted a fluid–solid coupling analysis using COMSOL Multiphysics software. ⁹ It has been found that the MRF brake’s fluid–solid coupling effect is very significant in the course of braking, that is, there is a strong nonlinear relationship between temperature field and stress field, and also reflects a prominent problem of heat dissipation. In this study, we focus on improving the braking performance of the MRF brake under the constraints of space size and solving the problem of heat dissipation in the course of braking. So, a novel design of MRF brake with multi-grooves in the brake disc is studied theoretically and numerically, using fluid–solid coupling approach, and then, the formula for calculating the impact factor of fluid–solid coupling (δ) is proposed, aiming to express comprehensively the coupling relationships of the entire system more realistically, by analyzing the influenced factors of each physical field simultaneously. Finally, the preliminary experiments are performed on the brake’s inertia test system, and the results achieved experimentally are compared with the simulations.

The layout of this work is as follows: in section “Mathematical model,” the mathematical model is established for a novel MRF brake based on fluid–solid coupling, consisting of constitutive equation of rheological properties, braking torque, and the formula for calculating the impact factor of fluid–solid coupling (δ); in section “Numerical simulation,” the modeling methodology for numerical simulation and the solution for temperature and stress fields are presented in detail, and the mathematical expression of the impact factor of fluid–solid coupling (δ) has been calculated and obtained; subsequently, in section “Experiments,” the prototype of MRF brake proposed in this article is manufactured and experimental results are measured on the inertia test system of the brake, in terms of braking torque, motion parameters, and surface temperature during braking, and some comparison analyses are performed; finally, some conclusions are drawn, which are useful for the design and application of the MRF brake.

Mathematical model

Here, a novel design of the MRF brake that may be used in light belt conveying equipment has been proposed and studied, aiming at a prominent problem of heat dissipation in braking process, especially in the case of single emergency braking,^26,27 through the analysis of the influencing factors and the relationship between the constitutive equations of the physical field. The mathematical model and the method of fluid–solid coupling simulation were established and explained as below.

Structural design of MRF brake

The configuration of the designed MRF braking system is shown in Figure 1. This MRF brake primarily consists of a drive shaft, double brake discs, three magnet exciting coils, two end covers, a shell, sealing rings, MRF, and non-magnetic bobbins. One end of the drive shaft is fixed to a flange plate and connected to the experimental apparatus, and the other end is connected to the bearing pedestal. Multi-grooves are set up for the brake disc to increase heat dissipation and improve braking torque. Note that the limitation in number of the grooves must meet the strength requirements. The mutual coupling of each magnetic circuit is designed to increase magnetic induction intensity in effective working gap under the same space conditions. Using the rheological properties of MRF, when electric current is supplied through the magnetic exciting coils, a magnetic field is generated at the effective working gap, and then the MRF changes from Newtonian fluid to solid-like state, which plays a role as the friction lining as described in Wang et al. ²⁷ or mentioned in any other friction brake.

Figure 1.

Structure schematic of the designed MRF braking system.

Constitutive equation of rheological properties

From the recognition of MRF reported in literature, and considering the designed MRF braking system’s structure and fluid properties, whereas with our designed MRF brake example, the assumptions below are proposed:

1. The MRF cannot be compressible, and it is evenly distributed in working gaps, while the sedimentation effect of suspended particles is not taken into consideration.

2. The frictional resistance moment generated by the seal is ignored.

3. The magnetic field generated by the magnetic exciting coil passes through the MRF uniformly, and its viscosity is isotropic.

4. For MRF brakes, the MRF usually operates in a shear mode, and its rheological properties can be described by the Bingham viscoplastic model or the Herschel–Bulkley model. Though both models have their own advantages, the computational complexity of the Bingham viscoplastic model is relatively small. So, in this work, it is adopted for the fluid–solid coupling analysis and will be modified in order to solve the non-differentiable problem of constitutive equation in the unyielded region of MRF.

As is well known, the constitutive relationship of MRF characterizes the change of shear stress under different shear strain rate and magnetic flux density. The following constitutive equation of Bingham viscoplastic model of MRF is

$$\tau ={\tau }_y\mathrm{sgn}(\stackrel{·}{\gamma })+\eta (T)\stackrel{·}{\gamma }$$

(1)

where τ_y is the yield stress of MRF (Pa), η is the viscosity of MRF (Pa s), T is the instantaneous temperature of the brake disc surface (K), $\stackrel{·}{\gamma }$ is the shear strain rate of MRF (1/s), and τ is the shear stress of MRF (Pa).

The yield stress of MRF, τ_y, can be written as a exponential function of magnetic field intensity, H, which is also confirmed to be nonlinear with the magnetic induction intensity (B) and to follow H–B curve, while the particles in the MRF are not fully saturated

$${\tau }_y={\alpha }_1{H}^{\alpha }$$

(2)

where α₁ and α are the proportional coefficient and the exponent, respectively.

Using the Barus equation, ²⁸ the viscosity of MRF with temperature can be expressed in the following equation

$$\eta ={\eta }_0{e}^{-\beta (T-T_0)}$$

(3)

where β is the temperature viscosity coefficient, η₀ is the viscosity when temperature is T₀ (Pa s), and T₀ is the ambient temperature (K).

In an ideal state, when ignoring any pressure-driven flow, the shear strain rate can be approximately expressed as

$$\stackrel{·}{\gamma }=\frac{\omega ·r}{d}$$

(4)

where d is the thickness of working gap (mm), r is the working radius of MRF (mm), and ω is the angular velocity of brake disc (rad/s).

For the non-differentiable problem of yield stress in the unyielded region of MRF, which makes the calculation complicated and non-convergent, a small constant and a hyperbolic function have been introduced to modify the constitutive equation of Bingham viscoplastic model, ⁵ that is to say, the unyielded region is replaced by the minimally yielded region, thereby the rheological behavior of MRF can be interpreted by the modified Bingham model. In this article, MRF-140CG (Lord Corporation, USA)—a mixture with micron-sized magnetizable particles suspended in a carrier fluid—was adopted as the working medium for the designed MRF braking system. The basic properties of MRF-140CG had been provided in our previous work. ⁹ Then, the close approximation expression of Bingham viscoplastic model, that is, the modified Bingham model provided in Meng et al., ⁵ can be defined as below

$$\tau =(\frac{{\tau }_y\tanh \left(\xi \stackrel{·}{\gamma }\right)}{\sqrt{{\zeta }^2+{\stackrel{·}{\gamma }}^2}}+\eta )\stackrel{·}{\gamma }$$

(5)

where ζ is a small constant, which approaches to zero and in this case is set to ζ = 0.0001, and ξ is a constant such that the curve is close to the case, where ξ is equal to infinity.

Figure 2 presents the shear stress versus the shear strain rate for the different value, ξ. The yield stress of MRF under the action of the magnetic field is 40 kPa, and the exciting current of magnetic exciting coils is 5 A, which is consistent with the simulation setting in subsection “Modeling and parameters setting”. As explained in section “Experiments,” the actual shear strain rate at the edge of the brake disc, $\stackrel{·}{\gamma }$ (r = 150 mm), is calculated by equation (4) to be about 13,000 s⁻¹; then, the shear stress can be up to 43.64 kPa, as shown by the green dot in Figure 2. When ξ = 6.4, the modified Bingham approximation model can be a good substitute for the Bingham viscoplastic model. So, based on the Bingham viscoplastic model, as well as Meng et al., ⁵ the apparent equivalent viscosity can be written as

$$\eta (\stackrel{·}{\gamma },H(B))=\frac{{\tau }_y\tanh \left(\xi \stackrel{·}{\gamma }\right)}{\sqrt{{\zeta }^2+{\stackrel{·}{\gamma }}^2}}+\eta$$

(6)

Figure 2.

Shear stress versus shear strain rate for modified Bingham model, with ζ = 0.0001.

As can be seen in equation (6), the apparent equivalent viscosity of MRF is a function of magnetic field intensity and shear strain rate, which not only clearly represents the relationship of magnetic field (H), flow field (N–S equation), and temperature field (T), but introduces a bridge between related physical parameters too.

Braking performance

Unlike the braking torque of a double-coil side-mounted MRF braking system reported in literature,^5,9,17 the total braking torque M in this work consists of four parts. The first item is the shear moment M₁ between the MRF and the end face of the brake disc, the second item is the shear moment M₂ between the MRF and the cylinder surface of the brake disc, the third item M₃ is the shear moment of cylinder surfaces of all grooves at one brake disc, and the last item M₄ is the shear moment of end surfaces of grooves for one brake disc at the middle working gap. Figure 3 shows a schematic of structure parameters for the designed MRF braking system. Here, only the final formulation has been presented; details for derivation method can be found in Wang et al. ⁹

Figure 3.

Structure parameters of the designed MRF brake with multi-grooves.

The total braking torque of the designed MRF braking system, M, can be calculated as

$$M=2M_1+M_2+2M_3+2M_4$$

(7)

with

$$\begin{array}{c}\multicolumn{1}{c}{M_1=\frac{2\pi }{3}{\tau }_r(r_2^3-r_1^3)+\frac{2\eta \pi \omega r_2^4}{(n+3)d}{\left(\frac{r_2\omega }{d}\right)}^{n-1}[1-{\left(\frac{r_1}{r_2}\right)}^{n+3}]}\\ \multicolumn{1}{c}{M_2=2\pi r_2^2(L-L_1)[\tau \prime +\eta {\left(\frac{\omega ·r_2}{d}\right)}^n]}\\ \multicolumn{1}{c}{M_3=\sum _{i=3}^{5}2\pi {(r_i-\frac{h}{2})}^{2}h[{\tau }_r+\tau ″+2\eta {\left(\frac{\omega }{d}(r_i-\frac{h}{2})\right)}^n]}\\ \multicolumn{1}{c}{+2\pi {(r_i+\frac{h}{2})}^{2}h[{\tau }_r+\tau ″+2\eta {\left(\frac{\omega }{d}(r_i+\frac{h}{2})\right)}^n]}\\ \multicolumn{1}{c}{M_4=\sum _{i=3}^5\frac{2\pi }{3}\tau ″[{(r_i+\frac{h}{2})}^3-{(r_i-\frac{h}{2})}^3]}\\ \multicolumn{1}{c}{+\sum _{i=3}^5\frac{2\eta \pi \omega }{(n+3)d}{(r_i+\frac{h}{2})}^4{\left(\frac{\omega }{d}(r_i+\frac{h}{2})\right)}^{n-1}[1-{\left(\frac{r_i-\frac{h}{2}}{r_i+\frac{h}{2}}\right)}^{n+3}]}\end{array}$$

where h is the width and depth of one groove (mm); r₃, r₄, and r₅ are the radii of the groove in turn (mm); ${\tau }_r$ , $\tau \prime$ , and $\tau ″$ are average shear stresses in the working gap where the MRF is located; and n denotes the flow coefficient of the MRF.

The relationship between the braking torque and the angular velocity of the brake disc can be calculated as

$$M=-J_e\frac{d\omega }{\mathrm{dt}},\omega (0)={\omega }_0$$

(8)

where J_e is the equivalent moment of inertia (kg m²), which can be calculated by the law of conservation of energy, and ω₀ is the initial angular velocity of the MRF brake (rad/s).

By means of separation variable method, the angular velocity of the brake disc can be derived as

$$\omega =D_1{e}^{-D_{2}t}+D_3$$

(9)

where D₁, D₂, and D₃ are constants which can be determined by equation (8), based on the structure parameters of the designed MRF brake and the associated motion parameters of the applied object.

For any brake, its braking process is to reduce the angular velocity under the action of the braking torque until the angular velocity satisfies the terminate condition. In the case of single emergency braking, the end condition is that the angular velocity reaches zero. Thus, from equation (9), the braking time, t_s, can be calculated by

$$t_s=\frac{1}{D_2}\ln (-\frac{D_1}{D_3})$$

(10)

It should be noted that equations (9) and (10) are only to describe a theoretical analysis of the two motion parameters during braking process, that is, the angular velocity of the brake disc, ω, and the braking time, t_s. They may have a certain role in numerical solution, but for FEA technology, these two parameters are dynamic solution parameters. Therefore, there is no need to give a derivation of specific related intermediate quantities, such as D₁, D₂, and D₃, and of course, interested readers can refer to Meng et al. ⁵

Fluid–solid coupling

Research shows that multi-physical fields’ coupling and interaction are of great significance for the practical engineering application of MRF braking system. Considering the working principle of the designed MRF braking system, the heat dissipation capability and the braking performance are two key technical indicators, and the analysis of thermal field and stress field involves the fluid–solid coupling relationship of the entire system. In what follows, a more complete understanding of the fluid–solid coupling analysis will be explained in detail.

According to the symmetry of the structure of the designed MRF brake, a 2D model can be employed for analyzing the magnetic field. Therefore, since the field quantity is independent of time, the Maxwell’s equation can be expressed as

$$\{\begin{array}{c}\nabla ·\mathit{B}=0\hfill \\ \multicolumn{1}{c}{\nabla \times \mathit{H}=\mathit{J}}\end{array}$$

(11)

with

$$\mathit{B}={\mu }_0(1+\chi )\mathit{H}$$

(11)

where B is the magnetic induction intensity vector (T), J is the electric current density vector (A/m²), H is the magnetic field intensity vector (A/m), ∇ is the Hamilton operator, μ₀ is the permittivity of free space (H/m), and χ is the magnetic susceptibility.

In the 2D polar coordinate system, a vector of magnetic potential, A , is defined to form an independent differential equation, so as to simplify the finite element method (FEM) calculation. For the 2D axis-symmetric model, A only has a component in the φ direction, and then the magnetic induction intensity vector B can be written as

$$\mathit{B}=\nabla \times \mathit{A}=\left(\begin{array}{c}-\frac{\partial {\mathit{A}}_{\varphi }}{\partial z}\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{\frac{{\mathit{A}}_{\varphi }}{r}+\frac{\partial {\mathit{A}}_{\varphi }}{\partial r}}\end{array}\right)$$

(12)

Without consideration of the phenomenon of magnetic flux leakage, the boundary condition of parallel flux is set onto the outer layer of the model. After setting the relevant parameters of magnetic exciting coils, such as turns, conductivity of coil, cross-section area of coil wire, and current, the magnetic induction intensity vector, B , can be obtained from the above formulations. Besides, the velocity field also can be solved, by adopting the method of sequential coupling, ²⁹ and then the distribution of magnetic field is transformed into that of magnetic body force of magnetic particles in MRF, which resists the flow of fluid.

The fluid flow equation of MRF can be determined by Navier–Stokes equation

$$\rho \frac{\partial \mathit{u}}{\partial t}+\rho \mathit{u}·\nabla \mathit{u}=-\nabla \rho +\nabla ·\eta (\stackrel{·}{\gamma },H(B))(\nabla \mathit{u}+{(\nabla \mathit{u})}^T)+{\mathit{F}}_m$$

(13)

where u denotes the velocity vector of MRF (m/s), ρ is the density of MRF (kg/m³), and F _m is the magnetic body force (N).

Considering the non-uniformity of the magnetic field, the magnetic body-fore acting on the magnetic particles in MRF is

$${\mathit{F}}_m=\frac{{\mu }_0\chi \nabla {\mathit{H}}^2}{2}$$

(14)

with the formulation of magnetic susceptibility, χ, provided in Gorodkin et al. ³⁰

$$\chi =(0.668D_p+0.961)\phi$$

(14)

where D_p is the diameter of magnetic particles (m) and $\phi$ is the volume fraction of magnetic particles in MRF.

Thus, from equations (11)–(14), the magnetic body force, F _m , acting on fluid elements can be rewritten as

$${\mathit{F}}_m=\frac{\chi }{{\mu }_0{(1+\chi )}^2}\{\begin{array}{c}\frac{\partial {\mathit{A}}_{\varphi }}{\partial z}·\frac{\partial }{\partial r}\left(\frac{\partial {\mathit{A}}_{\varphi }}{\partial z}\right)+(\frac{{\mathit{A}}_{\varphi }}{r}+\frac{\partial {\mathit{A}}_{\varphi }}{\partial r})(\frac{1}{r}·\frac{\partial {\mathit{A}}_{\varphi }}{\partial r}-\frac{{\mathit{A}}_{\varphi }}{r^2}+\frac{{\partial }^2{\mathit{A}}_{\varphi }}{\partial r^2})\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{\frac{\partial {\mathit{A}}_{\varphi }}{\partial z}·\frac{{\partial }^2{\mathit{A}}_{\varphi }}{\partial z^2}+(\frac{{\mathit{A}}_{\varphi }}{r}+\frac{\partial {\mathit{A}}_{\varphi }}{\partial r})(\frac{1}{r}·\frac{\partial {\mathit{A}}_{\varphi }}{\partial z}+\frac{\partial }{\partial z}\left(\frac{\partial {\mathit{A}}_{\varphi }}{\partial r}\right))}\end{array}$$

(15)

Details for the source of equations (14) and (15) have been provided in Meng et al. ⁵ After solving the distribution pattern of magnetic field of the MRF braking system, the flow field of MRF can be performed by setting relevant parameters in the solution domain, such as density, apparent equivalent viscosity, magnetic body force, and the boundary conditions of flow field.

A large amount of heat produced by the shearing friction will increase the temperature of the working gap filled with MRF during the braking process. This means that the rise in temperature not only changes the viscosity of MRF, that is, viscosity-temperature characteristics as shown in the Barus equation, ²⁸ but also makes the brake disc generate the corresponding thermal stress and then forms a mutual coupling relationship between the temperature field and the stress field. From the aspect of energy flow, the brake disc’s kinetic energy will be transformed into thermal energy during this period, resulting in intense heating of the contact surface, so as to cause the interactions between associated physical fields and changes in parameters, such as the heat conduction, the thermal expansion, and the heat dissipation.^9,27,31 As a result, the variation in temperature during braking can be governed by conjugate heat transfer equation:^32,33

$$\rho C_p\frac{\partial T}{\partial t}+\nabla ·(-\lambda ·\nabla T)=Q_e+Q_v+W_p-\rho C_p\mathit{u}·\nabla T$$

(16)

with

$$Q_e=I^{2}R;Q_v=\tau :\frac{1}{2}(\nabla \mathbf{u}+{(\nabla \mathbf{u})}^T);W_p=-\frac{T}{\rho }\frac{\partial \rho }{\partial T}(\frac{\partial \rho }{\partial t}+(\mathbf{u}·\nabla )\rho )$$

(16)

where C_p is the specific heat capacity (J/(kg K)), λ is the fluid thermal conductivity (W/(m K)), Q_e is the produced heat of magnetic exciting coils (W/m³), Q_v is the heat generated by fluid viscosity (W/m³), W_p is the work done by pressure changes (W/m³), I is the exciting current (A), and R is the resistance of magnetic exciting coils (Ω).

Setting the brake disc as the research object, the produced heat per unit of the contact area in time t can be calculated as

$$q=\tau ·r·({\omega }_0+\overline{\alpha }t)$$

(17)

where q is the produced heat (J) and $\overline{\alpha }$ is the angular acceleration (rad/s²), $\overline{\alpha }=-\stackrel{·}{\omega }$ .

Obviously, the heat generation coexists with the heat dissipation. For the designed MRF brake, the heat dissipation can be described primarily as both convection and radiation. So, the expression of heat dissipation from the contact surface of one brake disc to the surrounding environment, that is, MRF, can be written as

$$Q=[-h(T-T_0)-\epsilon k(T^4-T_0^4)]A\prime$$

(18)

with

$$h_c=\frac{0.037\lambda }{2r_2}{\left(\frac{2\rho ·r_{2}u}{\mu }\right)}^{0.8}{\left(\frac{C_p\mu }{\lambda }\right)}^{0.33}$$

where Q is the dissipated heat (J); h_c is the film coefficient of convective heat transfer (W/(m² K)); ε is the coefficient of radiation heat transfer; k is the Stefan–Boltzmann constant (W/(m² K ⁴ )), k = 5.67 × 10⁻⁸; $A\prime$ is the area of contact surface of one brake in friction (m²); and μ is the dynamic viscosity of MRF selected (Pa s).

As mentioned in Wang et al., ⁹ for most substances, the thermal expansion usually happens with the temperature changes. As can be seen from the structural design of the proposed MRF braking system in this work, the surface of the brake disc is almost fully contacted with MRF, in addition to the cross-section of drive shaft. Thus, during braking, the temperature rise caused by a large amount of frictional heat will lead to a prominent thermal expansion of the brake disc. To solve the thermal stress, the coefficient of thermal expansion, a, is a key important parameter. Since the coefficient of thermal expansion, a, does not remain a constant in the large temperature range, ³⁴ the thermal stress calculated from the inherent properties of the material is only an approximation. Subsequently, based on the relationship between the stress and the strain, the approximate thermal stress of the brake disc can be written as follows

$$\{\begin{array}{c}\Delta {\epsilon }_1=a(T-T_0)\hfill \\ \multicolumn{1}{c}{{\sigma }_1=E·\Delta {\epsilon }_1}\end{array}$$

(19)

where σ₁ is the thermal stress (MPa), Δε₁ is the thermal strain, a is the coefficient of thermal expansion (1/K), and E is the Young’s modulus of elasticity (GPa).

In this article, the impact factor of fluid–solid coupling (δ), that is, a weight coefficient, is used to characterize the mutual relations between the thermal field and the stress field, by analyzing the contribution of each stresses, namely, the thermal stress and the frictional stress. Detailed definitions are as follows: the total stress caused by fluid–solid coupling is σ_T, and the stress induced by the shear stress of MRF is σ₂. Figure 4 shows the procedures to achieve the stress parameters. Thus, the impact factor of fluid–solid coupling (δ) is proposed and can be calculated by the formula

$${\sigma }_T=\delta {\sigma }_1+(1-\delta ){\sigma }_2$$

(20)

Figure 4.

Flowchart to characterize the mutual relations of fluid–solid coupling analysis.

As illustrated in Figure 4, MRF properties are the central node to conduct the coupling analyses. The red continuous box represents the variable parameters of the constitutive equation employed to describe the rheological properties of MRF. Then, the Bingham viscoplastic model is modified by introducing a small constant and a hyperbolic function to solve the non-differentiable problem of yield stress in the unyielded region of MRF. Meanwhile, the apparent equivalent viscosity of MRF not only clearly represents the relationship of magnetic field (H), flow field (N–S equation), and temperature field (T), but introduces a bridge between related physical parameters too. Before the braking occurs, the brake disc together with drive shaft is first accelerated to the initial angular velocity (ω₀). When switching on the current of the magnetic exciting coils, the MRF in working gap by magnetic field lines through happens the rheological properties, resulting in similar solid properties and thereby generating the shear stress (i.e. the braking occurs). Here, Maxwell’s equations have been used to solve the magnetic induction intensity (B), and with Navier–Stokes equation, the distribution of magnetic field is transformed into that of magnetic body force (F_m) of magnetic particles in MRF. The red center box stands for the stress field during the braking. Under the action of braking torque (M) caused by the shear stress at the contact surfaces, the angular velocity of brake disc (ω) is decelerated, and the changes of angular velocity have an effect on the shear strain rate of MRF $(\stackrel{·}{\gamma })$ . With the extension of the braking time, considerable frictional heat produced by the shearing friction increases the temperature (T). The red dashed box represents the temperature field during the braking. The rise of temperature causes the brake disc to generate corresponding thermal stress (σ₁) and leads to changes in viscosity of MRF (η), which in turn affects the shear stress (τ). So, the conjugate heat transfer equation has been adopted to characterize the time-variation law of temperature. By solving equations (16)–(19) simultaneously, the transient temperature and thermal stress can be obtained. Finally, the impact factor of fluid–solid coupling (δ) depending on the braking time can be solved numerically using simulation results, due to the expressions’ complexity and non-linearity.

Numerical simulation

In this section, we specifically focused on finding the coupling relationship of stress field to temperature field, and the numerical simulation of fluid–solid coupling was performed using COMSOL Multiphysics software. The multi-physics modeling methodology was described in detail, which was based on Maxwell’s equations, Naiver–Stokes equation, and conjugate heat transfer equation. Then, the quality of magnetic circuit design was indicated. Moreover, the solution for temperature and stress fields was presented, and the mathematical expression for the impact factor of fluid–solid coupling (δ) depending on the braking time was obtained by a numerical analysis.

Modeling and parameters setting

The materials of each part in the designed MRF braking system are as follows. The brake disc is made of soft magnetic iron (DT4E type, and its material properties are reported in Wang et al ⁹ ). The drive shaft and the bearing cap are made of stainless steel (UNS S30400) and medium-carbon steel (AISI 1045), respectively. The material of the magnetic exciting coil and the isolation ring is pure copper (UNS C12700) and simple brass (UNS C26000), respectively.

Figure 5 shows the meshing of 2D axis-symmetric model. Before building the finite element model of the designed MRF brake, the model should be reasonably simplified in order to reduce the amount of computation.^25,35 Hence, some unrelated parts can be neglected, such as seals and rolling bearings. Then, the 2D axis-symmetric model of all parts of the braking system is built using COMSOL software. Finally, we set the established model as the assembly and set the pair type as the identity pair and then set the element type of mesh as the triangular elements split into 19,496 units totally (i.e. the elements size is extra fine).

Figure 5.

Finite element mesh of 2D model.

In what follows, the methodology of numerical simulation based on fluid–solid coupling will be explained in detail. The associated physics module needs to be selected and added to the COMSOL software, and the aim is to establish the mutual relations between each physical field. Note that the capitalized phases below represent the items in the operation menu of COMSOL software, except for terminology. Besides, details for the values of above-mentioned parameters, such as specific heat capacity, thermal conductivity, coefficient of radiation heat transfer, and basic material properties, have been provided in Wang et al. ⁹ First, under the Global Definitions node found in the model wizard, enter values in the Parameters table to define parameters used throughout the model making it possible to parameterize, for instance, the geometry size and related operating parameters of the designed MRF brake, as well as the basic material properties of all parts.

AC/DC module setting

The interface of Magnetic Fields has the Ampère’s Law, Multi-Turn Coil Domain, Magnetic Insulation, and External Current Density for modeling magnetic fields, which was found beneath the AC/DC branch in the model wizard. It solves for the magnetic vector potential. The material properties of each part of the designed MRF brake was set in the Ampère’s Law, except for the magnetic exciting coils, and set the boundaries in Magnetic Insulation. For the Multi-Turn Coil Domain feature, we set the ambient temperature T₀, the turns of the magnetic exciting coil as 20, the conductivity of coil as 6e7 (S/m), the cross-sectional area of coil wire as 1e–6 (m²), current as the type of excitation, and the current of coil as 5 (A). Subsequently, in the Domain Selection of External Current Density, the isolation ring is selected, and each vector of external current density is 0.

Fluid Flow module setting

The Laminar Flow interface, found under the Single-Phase Flow branch in the Fluid Flow module, is added to model and simulate the flow behavior of the incompressible and stable fluid (i.e. MRF). The fluid properties of MRF such as density and dynamic viscosity can be defined in the feature of Fluid Properties. For the steady state, use the default density and the default dynamic viscosity from material, but for the braking state, select user defined to define a different value or expression. The Wall feature including a set of boundary conditions is adopted to describe the fluid flow condition at a wall. The inner surface of the top shell and the left/right cover is selected as the default boundary condition, that is, No slip, for a stationary solid wall. Then, define the end surface and the circumferential surface of the brake disc as the sliding wall, and enter the φ-component of the velocity of the moving wall V_w = ω·r.

Heat Transfer module setting

The Heat Transfer interface is added, which has the equations, boundary conditions, and sources for modeling conductive and convective heat transfer and solving for the temperature. Use the Heat Flux node to add heat flux across boundaries, which include the surface of the brake disc, drive shaft, top shell, and left/right end cover; then, set surface-to-ambient radiation to boundaries under the Surface-to-Ambient Radiation node. Thus, the heat dissipating capacity can be calculated. Under the Heat Source node, enter the distributed heat source in the form of general source, which is used to describe the produced heat of magnetic exciting coils; the surface of brake disc is used to calculate the heat generated by fluid viscosity, and set it to the Boundary Heat Source. Furthermore, choose the Translational Motion section to enter the velocity component of the brake disc during braking process, that is, v_x = 0 and v_y = ω·r.

Structural Mechanics module setting

The Solid Mechanics interface, found under the Structural Mechanics branch, has been used for stress analysis. For this module, we only focus on the response characteristics of the brake disc. Define the brake disc’s thermal expansion coefficient from the Thermal Expansion section under the Linear Elastic Material feature. The braking torque equals to the shear stress generated by the MRF, and set it to the Boundary Load.

Multiphysics Modeling and Solver setting

Within COMSOL software, it can solve the established equations, taken from various areas of physics (i.e. magnetic field, flow field, temperature field, and stress field), as one fluid–solid coupling system. The Laminar Flow and the Heat Transfer are coupled in the multi-physics interfaces to realize the bidirectional coupling. As a final procedure, the Stationary Solver is selected for the coupling of the Magnetic Fields, the Laminar Flow, and the Heat Transfer; the Transient Solver is suitable for thermo-mechanical coupling analysis. Then, based on the coupling feature, we choose BICGSTAB or PARDISO as the solver configuration for solving the impact factor of fluid–solid coupling (δ). Step 1 is to calculate the thermal stress $({\sigma }_1)$ only caused by temperature rise, and Step 2 is to calculate the stress $({\sigma }_2)$ induced by the shear stress of MRF. Subsequently, Step 3 can get the total stress $({\sigma }_T)$ based on fluid–solid coupling.

Analysis for magnetic circuit

Figure 6 shows the distribution of magnetic induction intensity. It can be seen from Figure 6 that the magnetic induction intensity appears as an approximately symmetrical distribution with respect to axial center plan of the middle gap. Though there is a flux leakage phenomenon near the left/right end cover, it has no influence on the magnetic induction intensity at the working gaps. Compared with other energization modes of the magnetic exciting coil (i.e. current direction), the distribution of magnetic induction intensity in this work is more reasonable, thereby to meet the design requirements on the magnetic circuit structure of designed MRF brake. Subsequently, we carry out the further analysis about the distribution of magnetic induction intensity along paths at the radial and axial working gaps, that is, AB (or DC), BC, and EF, as shown in Figure 7.

Figure 6.

Nephogram of magnetic induction intensity.

Figure 7.

Distribution of magnetic induction intensity along paths.

We know that the variation range of magnetic induction intensity at working gaps is 0.491–1.48 T, and the maximum value is no more than 2 T. The location for the maximum value is at the top gap, near the left/right magnetic exciting coil, that is, the point B or C; the minimum value is at the middle gap, at a radius slightly larger than the radius of the drive shaft, that is, the point E. There is a mutation in magnetic induction intensity near the multi-grooves and also occurring at both ends of the top working gap. For the distance BC, it is symmetric with respect to the EF plane, and then for distance AB (DC) and EF, except for the mutation domain, the curves show an increasing trend.

Distribution of temperature field

At the onset of braking starts, the brake disc along with the drive shaft should be accelerated to the initial angular velocity and stabilized for a period of time. Therefore, we define this procedure as the steady running state without external magnetic field. The ambient temperature is set to 19.7°C, which is consistent with the actual ambient temperature of the experiment. In Figure 8, the steady temperature distribution of MRF brake without external magnetic field is shown. During the steady running state, the temperature distribution is relatively uniform excluding the drive shaft, with the maximum temperature difference no more than 5°C. The highest temperature in working gaps is 66.5°C, which is located at the top gap, near the EF plane. From here, we see that the temperature rise in the MRF brake will occur even at a zero magnetic field, due to the shear model of MRF.

Figure 8.

Steady temperature distribution of the MRF brake without external magnetic field.

After getting the steady running state of MRF brake, it will be adopted to analyze the transient temperature distribution of MRF brake during braking process. Figure 9 shows the transient temperature distribution at the end of braking. Similarly, the further analysis about the transient temperature distribution along paths is shown in Figure 10.

Figure 9.

Transient temperature distribution of the MRF brake during the braking process.

Figure 10.

Variation curves of transient temperature along paths: (a) radial on the left plane, (b) radial on the right plane, (c) radial on the middle gap, and (d) axial on the top gap.

As the braking process is of short duration, the transient temperature rises rapidly. The transient temperature nephogram is identical also for the case when the external magnetic field is not exerted. Of course, the effect of heat dissipation through the brake disc and the other parts can also be seen. As shown in Figure 10, to study on the time-variation law of temperature distribution in both radial and axial, seven points in radial working gap (40, 60, 75, 90, 105, 120, and 150.5 mm) and in axial working gap (79.5, 89.5, 99.5, 110, 120.5, 130.5, and 140.5 mm) are taken, respectively. For the distance AB and DC, the behavior of temperature rise at different points appears to be almost same with a tiny difference. Except for the points at the location of multi-grooves (i.e. 60, 90, and 120 mm), it shows a trend of increasing first and then decreasing. This is because the heat generation by the viscous friction at the beginning of braking leads to the temperature increase rapidly, and then the temperature begins to decline with the decrease of the shear strain of MRF. At this time, the frictional heat is smaller than the heat conduction, heat convection, and heat radiation. Certainly, the settings of multi-grooves do improve the heat dissipation during the braking process. At various points in distance EF, the trend of temperature rise and drop occurs almost consistently, but except for the points at both ends (i.e. point E and F), other points show that there can be little differences between results from the seemingly similar curve. The average relative error is less than 2%, that is, the produced heat and the heat dissipation at the middle gap are relatively uniform. Moreover, for the points at the top gap, the temperature variation of the symmetry point is the same with a tiny difference. The closer to the point F, the greater the temperature; but for the point F, the value is only larger than that of both ends. Essentially, it also reveals that the frictional braking at the middle gap is approximately uniform. In addition, as can be seen from the above analysis, this indirectly shows that the magnetic circuit design of the brake disc is feasible in combination with the distributing disciplinarian of magnetic induction intensity in Figure 7.

Distribution of stress field

For this study, the thermal stress and the total stress of the brake disc surface at the end of braking, that is, t = 2.5 s, are shown in Figures 11 and 13, respectively. Meanwhile, by extracting the data from the simulation results, a clear analysis of the stress distribution on the contact surface between the brake disc and the MRF was obtained. The circumference of the brake disc surface with a radius of 120 mm is selected as the circumferential direction, and the vertical diameter of the brake disc surface from bottom point to top point is defined as the radial direction. Figures 12 and 14 show the thermal stress distribution and the total stress distribution of the brake disc surface along paths, respectively.

Figure 11.

Thermal stress distribution of brake disc: (a) end-cover side and (b) middle side.

Figure 12.

Variation curves of thermal stress along paths: (a) circumferential and (b) radial.

Figure 13.

Total stress distribution of brake disc: (a) end-cover side and (b) middle side.

Figure 14.

Variation curves of total stress along paths: (a) circumferential and (b) radial.

It can be seen from Figure 11 that there is only a tiny difference for the thermal stress distribution, even though the magnetic induction intensity is different at each location, and the maximum absolute error is 0.503 MPa, that is, the frictional braking is comparatively uniform between MRF and the brake disc. The result for the thermal stress along the circumferential direction is shown in Figure 12(a). As a result of MRF subjected to the centrifugal force, the thermal stress distribution in circumferential direction presents the approximate calabash-type distribution, except for the top showing a notching. When the circumferential angle is in the range of 83°–103°, the thermal stress at the middle side is almost the same as that of end-cover side. However, at other angles, the thermal stress at end-cover side is slightly greater than that of middle side. It should be noted that there is sudden drop at 90° position, because of the effects of gravity and centrifugal force, but the maximum difference is less than 1%. Figure 12(b) shows the thermal stress along the radial direction. It can be seen that the thermal stress appears to decrease with the increase in the radial length until it tends to be gradually steady and slow, and similarly, the thermal stress at end-cover side is slightly greater than that of middle side, but the thermal stress in the lower part has obvious fluctuations.

From Figure 13, we know that the total stress distribution and the thermal stress distribution of the brake disc surface present consistency, showing a similar distribution trend, which indicates the coupling characteristics between the temperature field and the stress field. It can be concluded that the fluid–solid coupling effect of the MRF and the brake disc is very significant during braking process as stated in Wang et al. ⁹ Moreover, it is almost the same with a tiny difference for the total stress distribution as the end-cover side or the middle side, and the maximum absolute error is 1.67 MPa. Figure 14 shows the results for the total stress along paths. It is interesting that the various trends of the total stress along paths have nearly the same pattern as the thermal stress (Figure 12). The above simulation analysis has been performed for more than 50 times, and the same pattern can be observed for all of the cases. As a consequence, this also illustrates the necessity of the proposed impact factor of fluid–solid coupling, δ, to characterize comprehensively the coupling relationships of the whole system.

Impact factor of fluid–solid coupling

Figure 15 depicts the variation of impact factor of fluid–solid coupling δ depending on the brake time t, which can be calculated by the formula (20) and can be considered as the extent of fluid–solid coupling affecting the frictional braking. It can be seen that the impact factor of fluid–solid coupling δ decreases with t, and the variation also manifests that the thermal stress is in the opposite direction to the total stress because of negative value. And the trend of the curve shows the nonlinear relation of the temperature field and the stress field. At the end of the braking, that is, t = 2.5 s, the impact factor of fluid–solid coupling is δ = –096369 ≠ –1, showing that the thermal stress still has a loss with the continuous dissipation of heat during the braking process compared with our previous conclusion about the double-coil side-mounted MRF braking system. However, the relative error is reduced to 3.631%, less than that of the double-coil side-mounted MRF braking system, which indicates that the heat dissipation of the designed MRF brake has been improved partially. Obviously, the relative error is much less than 10%–15%, representing the heat radiation and the heat conduction as a percentage of the overall amount of heat dissipation during braking process using friction lining material.^36,37 In other words, it reflects the superiority of the MRF brake from the side. Similar to the method of least-squares curve fitting used in Wang et al.,^9,27 a mathematical expression on the impact factor with time can be solved as follows

$$\delta =-0.01144t^3-1.40161\times {10}^{-4}{t}^2-0.30995t+0.00296$$

(21)

where the correlation coefficient (adj. R²) is 0.98458, indicating that the fitting curve is closely related. Besides, the residual sum of squares is 0.2853 × 10⁻⁴, showing that the residual variance is small, and the fitting curve is very effective.

Figure 15.

Variation of impact factor of fluid–solid coupling δ depending on braking time t.

In addition, according to the fitting curve and its mathematical expression, the impact factor of fluid–solid coupling (δ) shows a monotonous decreasing trend with the change of braking time (t), and there is no inflection point in the function δ of t. So, combined with the result provided in Wang et al., ⁹ the performance of one MRF brake is indeed closely related to fluid–solid coupling due to the working pattern of MRF.

Experiments

To validate the accuracy of the established model on the designed MRF brake, the dynamic braking performance tests are carried out on the inertial test system of the brake. Specially, all the experimental data are the cases of single braking process (i.e. single emergency braking), and the aim is mainly to concentrate the temperature rise, the braking torque, and the motion parameters such as the angular velocity and the braking time. In addition, to ensure the accuracy of results achieved experimentally and reduce the influence of systematic error, all braking events are independent of each other.

Inertia test system of brake

The experimental setup, shown in Figure 16(a), has been built for the performance evaluation of the designed MRF brake. The test platform mainly consists of industry control computer, testing base, driving motor, flywheel components, sliding system, and several sensors. The required data for experimental analysis including temperature, braking torque, and angular velocity are collected by sensors (T-type thermocouple, non-contact infrared thermometer, and HBM T40FM torque transducer). The DC regulated power supply DP310 (MESTEK Corporation, Shenzhen Mestek Tools Co., LTD, China) was chosen for controlling the coil current. This power supply can output a current of 0–10 A. Both ends of the torque transducer are connected to clutch and shaft coupling, respectively. As the designed MRF brake is still in the preliminary design stage, which cannot match the road test, the clutch is used to disconnect the flywheel components in order to reduce the effect of the moment of inertia of the rotational parts when braking occurs. Figure 16(b) shows the prototype of the designed MRF brake.

Figure 16.

Experimental apparatus: (a) inertia test system of the brake and (b) prototype of the designed MRF brake.

In this study, the single emergency braking process is as follows. First, accelerate the brake disc of the MRF brake to the initial angular velocity (ω₀) and keep it running steadily for a period of time. Then, at onset of a braking event, turn off the driving motor and disconnect the clutch, while the magnetic exciting coil is energized to generate a magnetic field. After that, the MRF undergoes the rheological effect under the action of magnetic field, which presents solid-like state. At the moment, the brake disc is decelerated by the shear stress generated by MRF in the shear mode. Finally, after the angular velocity meets the end condition, the single braking will be completed.

Figure 17 shows the layout position of thermocouples used to measure the surface temperature of the brake disc during the braking process. The diameter of hole mounting the T-type thermocouple is 2 mm and the depth is 1.8 mm away from the inner surface of shell and cover. So, the measuring value can actually be regarded as the average value within the diameter 2 mm. The position parameters of the arranged thermocouples are expressed as follows. For the left/right end cover, there are eight measuring points arranged at the radius 135 mm in the circumferential direction with the gap of 45°, and five measuring points are arranged in the radial direction from 90 to 149 mm with the gap of 15 mm (note that one point is set at the radius 149 mm, since the diameter of the brake disc is 300 mm). For the top shell, six symmetrical measuring points are organized in the axial direction with the gap of 6.5 mm. In addition, as the thickness of working gap filled with MRF is only 1 mm, the temperature of inner surface can be employed for approximating the brake disc’s surface temperature.

Figure 17.

Schematic of the layout position of thermocouples on the end cover and shell.

Results and comparison

In what follows, the braking torque, the angular velocity derived from the rotational speed, and the surface temperature during one braking event has been analyzed in detail, as shown in Figures 18 –20, respectively. The following experimental data are given: the voltage of the experimental apparatus is 406.8 V, the rated speed of driving motor is 1000 r/min (actual output speed: 834 r/min), the ambient temperature is 19.7°C, the transmission efficiency is 85%, and the diameter of the brake disc is 300 mm.

Figure 18.

Comparison between the simulation and the experimental results for braking torque.

Figure 19.

Comparison between the simulation and the experimental results for angular velocity.

Figure 20.

Comparison between the simulation and the experimental results for the surface temperature at the end of braking: (a) radial, (b) circumferential, and (c) axial.

Figure 18 shows the braking torque of the designed MRF brake during braking. Simulation results reveal that although there is a significant fluctuation in the braking torque, it can be approximated as a constant value, that is, 440.14 Nm. The braking torque obtained from the experiment shows that the braking torque rises rapidly and reaches the maximum value 432.28 Nm at the beginning of braking. This reflects that the MRF has the advantage of fast response time when the magnetic exciting coil is energized. After that, as the braking time increases, the braking torque decreases along with the phenomenon of fluctuations, suggesting that the temperature rise caused by the shear friction makes the viscosity of MRF vary with time, which affects the braking torque. When the braking is nearing the end, the braking torque drops sharply. This is because the rheological effect has a certain response time when the external magnetic field is removed, that is, the braking process has a delay phenomenon based on the feedback control of angular velocity. In addition, in the non-braking state, a fluctuating braking torque can also be detected, which is caused by the static energy of the inertia test system of the brake, that is, it mainly represents the equivalent moment of inertial of the rotating components converted to the drive shaft and the dynamic viscosity of the fluid.

Compared with the experimental results, the simulation results do not show a similar variation curve. This is because the simulation process cannot describe the on–off physical process of exciting current of magnetic exciting coils, that is, even with the development of computers allowing accurate simulation to perform, still there are many actual physical processes that cannot be objectively described, and also there are many assumptions such as neglecting the effect of frictional resistance moment generated by the seal, and the sedimentary stability of selected MRF, which are also the sources of error. So, we can indirectly judge the quality of the simulation from the maximum value of the braking torque achieved experimentally. The relative error between the simulation results and the experimental results for the braking torque was 1.8%. Certainly, more advanced simulation methods are probably the core study in multi-physics coupling. The average braking torque in effective braking time is 344.167 Nm based on the mean value theorems for definite integrals, which is improved compared to previous double-coil side-mounted MRF brake provided in Wu et al. ²⁵ and Adamowicz. ³⁷ What’s more, this is not the maximum braking torque of the designed MRF brake.

Figure 19 depicts the angular velocity of the brake disc during the braking process. It can be seen that although the actual angular velocity during braking is larger than the simulation results, the derivative of the curve is almost the same except for both the ends. So, the reliability of the simulation method is demonstrated indirectly. The experiment of braking time has a delay of about 0.22 s compared with the simulation results.

Figure 20 displays the comparison between the simulations and the experimental results for the surface temperature in different directions. As shown in Figure 20, the actual surface temperature at the end of braking is slightly larger than the simulation results except for the point at the location of r = 105 mm in the radial direction (Figure 20(a)), and the maximum absolute error is 8.01°C at the location of top point in the circumferential direction (Figure 20(b)), and there existed no remarkable difference between the results achieved experimentally and the simulation results. In addition, the average relative error was no more than 5%, indicating that the simulation results showed high consistency with the results achieved experimentally.

Conclusion

In this study, a new design of MRF brake has been proposed and studied, aiming at a prominent problem of heat dissipation during braking process, especially in the case of single emergency braking, and the mathematical model and the simulation method of fluid–solid coupling were established and explained in detail. Then, based on fluid–solid coupling, the impact factor of fluid–solid coupling (δ) was used for characterizing the associations between the thermal field and the stress field. Finally, to validate the accuracy of the established model on the designed MRF brake, the prototype of MRF brake has also been manufactured and experimental results were measured on inertia test system of the brake, in terms of braking torque, motion parameters, and surface temperature in braking, and were compared with simulations.

Simulation results indicated that the designed MRF brake’s magnetic circuit structure is feasible based on magnetic induction intensity distribution. Through the analysis of the temperature field and the stress field, it also revealed that the fluid–solid coupling effect of the MRF and the brake disc is very significant during the braking process. In comparing with the experimental results, it can be concluded that the simulations appear to be basically consistent with the experimental results. Moreover, the experiment of braking time has a delay of about 0.22 s compared with the simulation results. From the analysis about the impact factor of fluid–solid coupling (δ), the heat dissipation of the designed MRF brake has been partially improved. It should be noted that this work is only a preliminary experimental study on the single emergency braking case. Further researches may include a better understanding of the inertia test system of the brake described herein. It will be more interesting to develop a comprehensive study with complex test conditions for the MRF brake, such as the normal braking or frequency interval braking cases.