Experimental analysis and mathematical modelling for novel magnetorheological damper design

Abstract

Considering the particular demands for a large damping force in special vehicles, a new design for a parallel disc slotted magnetorheological damper is presented. In this damping generator, the direction of liquid flow between disc gaps is completely perpendicular to the magnetic flux direction, and the gaps can be cascaded to enhance the damping effect, thereby introducing a larger damping force based on the magnetorheological effect. In this paper, the damping characteristics of the new damper design are experimentally analysed first. Based on the experimental results, a dynamic model for the damper is constructed. The mathematical model is obtained by combining the Bingham model and the Herschel-Bulkley model, and accurately and clearly describes the characteristics of magnetorheological fluid. For parameter identification, two methods are utilized: a trial-and-error method and a recursive least squares method. Analysis shows that both methods have drawbacks. Lastly, a new method is proposed for combining the trial-and-error method with recursive least squares method. By comparing the theoretical curves obtained from the experimental curves and combined methodology, the results show that the combined methodology exhibits the highest accuracy as well as efficiency. The mathematical model established in this paper can accurately describe the characteristics of damping force. This model lays down the theoretical foundations for application and research on new parallel disk slotted magnetorheological dampers.

Keywords

Magnetorheological damper damping characteristics mathematical model parameters identification

1. Introduction

A magnetorheological (MR) damper is a damper which can adjust its damping parameters for variable working conditions. The MR damper’s core component is the damping generator [1]. At present, most MR dampers will adopt a cylinder piston structure design, where the MR fluid will flow through a cylindrical annular gap in order to establish a pressure difference that forms the damping force. Because of difficulties in heat dissipation as well as the low utilization rate of the magnetic field in such a design, the structure presents difficulties in meeting the needs of special vehicles [2–4]. In view of this, an alternative parallel disk slotted MR damper is presented in this study. In this novel design, the flow direction of the MR fluid in the damping generator is completely perpendicular to the direction of the magnetic flux [5,6]. The MR effect can be utilized in this design to generate a damping force to its maximum extent.

Test results show that the dynamic characteristics of the new MR damper design displays strong nonlinearity and hysteresis. However, the stability and the effectiveness of its structure control processes mean there are higher demands for the MR damper’s mechanical model [7–11]. Therefore, constructing an accurate mechanical model for MR dampers has become an area of intense research. Many have put forward different models and methodologies in this respect. Examples include the least squares algorithm, the stochastic approximation algorithm, and the prediction error algorithm [12–15].

In this study, a trial-and-error method as well as a recursive least squares methodology are used to identify the parameters of the model system. The trial-and-error method is based on the physical model, and obtains parameter values by combining numerical calculations with experimental measurements. The shortcoming of this methodology is that precision will not be high enough. In order to improve the accuracy of parameter identification, the recursive least squares method is introduced for this study.

The recursive least squares method is an improved version of the least squares method. By using only the latest single data entry as well as the previous step’s fitting result, the method can achieve the effect of using all historical data fitting. As such, the method can be more efficient and up-to-date. Although classical parameter fitting methods such as the recursive least squares method can achieve high accuracy, the values of some of the parameters obtained will not be practical. This is also a problem that has been presently difficult to solve in parameter identification studies.

2. Magnetorheological damper design

The structural principles of the MR damper prototype are presented in Fig. 1. The MR damper has high efficiency, high sensitivity, good heat dissipation, and a simple structure. The MR fluid in the damping cylinder produces a unidirectional and forced flow with no dead angles that does not easily precipitate.

Fig. 1.

Structural principles of the magnetorheological damper prototype.

Fig. 2.

The basic structure of the damping generator.

The basic structure of the design’s damping generator is presented in Fig. 2. The generator is designed according to the flow principles between the parallel disc gaps. The liquid flow between disc gaps is completely perpendicular to the device’s magnetic field, and the gap can be cascaded to enhance the damping effect.

3. Damping characteristic experiment

The study’s experimental schematic diagram is presented in Fig. 3. The inner diameter of the damping cylinder is 80 mm. The outer diameter of the piston rod is 40 mm. The stroke of the piston is 192 mm. The gap of the disc is 3 mm, while the diameter of the inlet hole is 8 mm and the diameter of the outlet hole is 36 mm. The damping characteristics of the MR damper prototype were tested using an MTS systems experimental machine with an applied sinusoidal excitation. The excitation frequencies used were 0.1 Hz, 0.2 Hz, and 0.5 Hz. The amplitudes used were 10 mm, 20 mm, and 30 mm. The input currents used were 0 A, 1 A, 1.5 A, 2 A.

Fig. 3.

Experimental schematic diagram.

4. Experiment results

Fig. 4.

Experimental characteristics (0.5 Hz, 30 mm, different state of electricity).

The “dynamic ratio” is defined as the ratio of the damping force under the maximum magnetic field and the damping force in the absence of magnetic field to characterize the dynamic characteristics of the controllable damping effect. Through the calculation of the experimental data, the dynamic ratio reaches 35.

As can be seen in Fig. 4, the experimental results show that there is no empty travel during motion, and that the indicator curve is full. The tensile damping force is larger than the compression damping force. The maximum damping force can range up to 15 kN.

5. Mathematical modelling

5.1. Theoretical model

By combining the Bingham model with the Herschel-Bulkley model [16], the force-velocity model for the study design’s parallel disk slotted damper is established as follows: $\begin{eqnarray}\displaystyle F_{f} & = & \displaystyle (c+{\zeta}({\dot{x}})){\dot{x}}+F_{{\tau}}sign({\dot{x}})\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle F_{{\tau}} & = & \displaystyle {\beta}\frac{{\tau}_{0}}{h}A(r_{2}-r_{1})\end{eqnarray}$ (2)

F _τ: Controllable damping force; ${\zeta}({\dot{x}})$ : Non-linear viscous damping coefficient; c: Linear viscous damping coefficient of the MR damper; k: Viscosity coefficient; R: Damping cylinder radius; r: Outside diameter of the piston rods; r ₁: Inside diameter of the inlet hole; r ₂: Outside diameter of the outlet hole; h: Disk clearance thickness; 𝛽: Characteristic index.

Fig. 5.

Comparison of theoretical and experimental characteristics (with a frequency of 0.1 Hz, an amplitude of 30 mm, and a current of 3 A).

Fig. 6.

Comparison of theoretical and experimental characteristics (with a frequency of 0.5 Hz, an amplitude of 30 mm, and a current of 3 A).

5.2. Parameter identification

When the structural parameters of the study’s MR damper design were determined, its damping characteristics were also found. According to the principles of the study’s design, the theoretical model can be formulated as follows: $\begin{eqnarray}\displaystyle F_{f}=\left\{\begin{array}{@{}cc@{}}c_{1}{\dot{x}}+{\xi}{\dot{x}}^{m}+F_{{\tau}1}+f_{1} & {\dot{x}}\geq 0\\ c_{2}{\dot{x}}-f_{2}-F_{{\tau}2} & {\dot{x}}<0\end{array}\right. & & \displaystyle\end{eqnarray}$ (3)

c ₁: Tensile damping coefficient; c ₂: Compression damping coefficient; 𝜉: Pseudo-damping coefficient; m: Velocity exponent; f ₁: Friction force during tension; F _τ1: Controllable damping force during tension; f ₂: Friction force under compression; F _τ2: Controllable damping force under compression.

Table 1

Empirical parameters for various current with a frequency of 0.1 Hz and amplitude of 30 mm

Parameters	c1	c2	𝜉	F _τ1	F _τ2	f ₁	f ₂
current (1 A)	35.0e-3	5.0e-3	0.2e-4	2.65	1.12	0.1	0.2
current (3 A)	100.0e-3	15.0e-3	0.2e-4	7.5	2.75	0.1	0.2

Table 2

Empirical parameters for a frequency of 0.5 Hz, amplitude of 30 mm, and current of 3 A

Parameters	c1	c2	𝜉	F _τ1	F _τ2	f ₁	f ₂
Values	50.0e-3	7.5e-3	0.2e-4	7.3	2.65	0.1	0.2

The model’s parameters are identified using a trial-and-error methodology and the recursive least squares method. Tables 1 and 2 present the parameter values obtained using the trial-and-error method.

–

The tensile and compression damping coefficients c₁ and c₂

The parameters increase in a non-linear fashion as excitation current increases. Because of the increase in current, the magnetic field increases and the core flow area increases as well. The viscous flow region becomes compressed as a result, and the damping coefficients increase in turn. The tensile and compression damping coefficients decrease with an increase in the excitation frequency.

–

The pseudo-damping coefficient 𝜉 and velocity index m

They reflect the characteristics of non-linear viscous damping in the MR damper. There are a set of constant parameters for when power is not engaged, and another set of parameters for when the power is switched on. These parameters are unaffected by the excitation current, excitation frequency, and excitation amplitude.

–

The tensile friction force f₁ and the compression friction force f₂

f ₁ and f ₂ reflect the coulomb friction caused by the relative motion of the MR damper itself. The variables f ₁ and f ₂ are constant and not affected by the excitation current, excitation frequency, and excitation amplitude. These two parameters can be determined from testing.

–

The tensile controllable damping force F_τ1 and the compression controllable damping force F_τ2

These two parameters reflect the controllability of the MR damper. The two parameters are found to increase with the increase in electric current. However, the compression controllable damping force was unstable during testing, as it is affected significantly by the performance of the MR damper design’s bottom valve during compression.

Figures 5 and 6 present a comparison between the curve of the theoretical model that was obtained by the trial-and-error method and experimental curve. It can be seen that the theoretical model obtained by trial-and-error method basically reflects the damping characteristics of the MR damper. Specifically, the trial-and-error method combines experimental measurements with numerical calculations to obtain parameter values. The results are reliable, but lack precision.

In order to improve the accuracy of the model, the recursive least squares method is introduced into this study. In the process of parameter identification via the recursive least squares method, the range of the parameters c ₁ and c ₂ as well as F _τ1 and F _τ2 is limited according to the parameter values obtained by the trial-and-error method and when considering the practical significance of the parameters.

Fig. 7.

Comparisons of theoretical and experimental characteristics (0.1 Hz, 30 mm, current of 1 A).

Fig. 8.

Comparisons of theoretical and experimental characteristics (0.1 Hz, 30 mm, current of 3 A).

Fig. 9.

Comparisons of theoretical and experimental characteristics (0.5 Hz, 30 mm, current of 3 A).

Tables 3 and 4 show the parameters that were obtained by the recursive least squares method. When compared with the parameter values that were obtained by the trial-and-error method, the uncontrollable damping coefficients c ₁ and c ₂ were found to have smaller values. The controllable damping force variables F _τ1 and F _τ2, on the other hand, were found to have greater values. Theoretically, the mathematical model obtained using the recursive least squares model is superior. When using the recursive least squares model, computing time is about half the time it takes when using the least squares method to calculate historical data comprehensively. When the amount of data is large, the recursive least squares will significantly shorten computation time and thereby improve efficiency.

Table 3

Identification parameters for various currents (with a frequency of 0.1 Hz and an amplitude of 30 mm)

Parameters	c1	c2	𝜉	F _τ1	F _τ2	f ₁	f ₂
current (1 A)	19.7e-3	2.0e-3	0.00092126	2.57	0.93	0.07	0.22
current (3 A)	34.4e-3	2.09e-3	0.005808	7.49	2.55	0.1	0.2

Table 4

Identification parameters for a frequency of 0.5 Hz, an amplitude of 30 mm, and a current of 3 A

Parameters	c1	c2	𝜉	F _τ1	F _τ2	f ₁	f ₂
Values	32.4e-3	6.24e-3	0.00071353	6.85	2.47	0.08	0.22

As presented in Figs 7, 8, and 9, theoretical curves are obtained by the recursive least squares method and compared with the experimental curves under three different working conditions. It can be seen that the fitting rate for the theoretical curve and the experimental curve, as obtained by the recursive least squares method, is very high. Moreover, the data indicate that the fitting rate is over 99.8%. These findings show that the recursive least squares method can improve fitting accuracy and that the mathematical model can accurately reflect damping in MR dampers.

6. Conclusion

In this paper, a novel parallel disk slotted MR damper design was presented. The damper design presents advantages in high efficiency, high sensitivity, and good heat dissipation. The design’s tensile damping force is larger when compared with its compression damping force, the maximum damping force can reach up to 15 kN, and the dynamic ratio reaches 35, meeting design requirements. A dynamic model for the damper was compiled. In this, the relationship between damping force and speed was established first by combining the Bingham model with the Herschel-Bulkley model. Thereafter, parameter identification was carried out to determine the parameter values for the dynamic model.

In parameter identification, a method of trial-and-error was combined with the recursive least squares method. This methodology improved the precision of the mathematical model. Moreover, the methodology solved a problem regarding obtained parameters not fitting the reality of the model, thereby realizing greater levels of accuracy and efficiency. By comparing experimental curves with theoretical model curves, it was shown that the theoretical model in this study accurately reflects the damping characteristics of the prototype parallel disk slotted MR damper. This study lays the theoretical foundations for application and research on new parallel disk slotted magnetorheological dampers.

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