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Edited by: Miao Yu, Chongqing University, China

Reviewed by: Xufeng Dong, Dalian University of Technology, China; Yu Tian, Tsinghua University, China

This article was submitted to Smart Materials, a section of the journal Frontiers in Materials

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The present study aims at the development of a magnetorheological elastomer (MRE) based semi-active seat suspension isolator and its adaptive control using a neural network (NN) control scheme. Isotropic MRE samples with 25% volume fraction of iron particles have been fabricated and then characterized under shear mode using a rotary magneto-rheometer to obtain MRE's viscoelastic properties (shear storage and loss moduli) under different levels of applied magnetic flux density. Results reveal a significant change in the storage and loss moduli with respect to the varied magnetic field. The viscoelastic properties of the MRE are then utilized to design an MRE-based seat suspension isolator in order to attenuate the transmitted vibration to the driver. For this purpose, the modeling of the seat incorporated with the MRE-based isolator is derived and subsequently, a novel NN control scheme is proposed for the semi-active control of the MRE-based isolator. The convergence and stability of the proposed control strategy have been mathematically verified using the Lyapunov method. Finally, the performance of the proposed control strategy is compared with those obtained using passive and widely used sky-hook controllers under different types of excitation including harmonic motion, road bump, and random profile. It is shown that the proposed NN controller considerably mitigates the vibration of the driver seat and outperforms the passive and skyhook controllers over the frequency range of interest.

Long-term exposure to the low frequency and large amplitude vibrations from car seats can lead to severe adverse health effects on the drivers (Wilder et al.,

Semi-active isolators featuring smart magnetorheological (MR) materials can effectively utilize the adjustable viscoelastic properties to develop the required control forces. This unique adaptability feature combined with their inherent fail-safe design and low power requirements makes MR-based isolators attractive adaptive devices which can attenuate transmitted vibrations in a wide range of applications. Magnetorheological elastomers (MREs) are solid analogs of MR fluids which can provide both variable stiffness and damping under varying applied magnetic fields. This unique feature can be effectively utilized for the development of novel and practical semi-active isolators. MREs consist of micron-sized ferromagnetic particles dispersed into an elastomeric medium. Thus, they do not encounter the limitations often posed by MR fluids, such as sedimentation and leakage (Li et al.,

While substantial efforts have been made on the development, characterization, modeling, and design of MRE-based devices (Kallio et al.,

The present study firstly addresses the development of an MRE-based seat suspension isolator design considering the constraints on the magnetic field density, stroke, and static deformation of the MRE. The MRE samples with 25% volume fraction of carbonyl iron particles are fabricated and then characterized using a rotary magneto-rheometer. Secondly, a novel adaptive NN control scheme is developed to mitigate the transmitted vibration of the developed MRE-based seat suspension isolator. The convergence and stability of the proposed control system are verified using the Lyapunov stability theory. Finally, the superior performance of the proposed controller over the passive control and ON-OFF control is demonstrated.

The MRE samples with 25% volume fraction of magnetic particles were fabricated in the laboratory at room temperature using silicon rubber,

In the present study, an advanced MRE testing system equipped with a rotary rheometer (

The hysteresis loop, applied force vs. displacement, results of the MRE sample under various levels of applied magnetic flux density are presented in

The force with respect to displacement under different levels of applied magnetic flux density and fixed frequency of 1 Hz.

Storage and loss moduli in the presence of varied magnetic flux density with the fixed frequency of 2 Hz and shear strain amplitude of 15% and its corresponding curve fitting results.

Using the experimental data shown in

where _{1} = −234.3, _{1} = 396.7, _{1} = 10.94, and _{1} = 63.04 are identified using the least square method. The approximated function of the loss modulus vs. magnetic flux density is also derived by using a cubic polynomial function, which can be described as:

where _{2} = −103.2, _{2} = 151.1, _{3} = 7.79, and _{4} = 13.27 are identified using the least square method. The results obtained using the models presented in Equations (1) and (2) are compared with those measured experimentally in

The schematic diagram of the MRE-based seat suspension isolator operating in shear mode is presented in

The schematic diagram of the driver seat equipped with the proposed MRE-based seat suspension isolator and its equivalent mechanical system are shown in _{MRE} and _{MRE}, respectively. An extra linear spring _{b} is added in parallel to the MREs to reduce the static deformation of the MRE layers due to the weight of the seat and driver. The dynamics modeling of the MRE-based seat suspension system can be described as:

Let us define:

where _{min} and _{min} are the equivalent stiffness and damping of the MRE-based isolator in absence of applied magnetic flux density, respectively. △_{MRE}, and damping, _{MRE}, can be described as Li et al. (

where

Substituting _{MRE} and _{MRE} from Equations (6) and (7) into Equation (3) yields:

where _{MRE} is the generated actuation force induced by MREs in the presence of the applied field and is described as:

Using Equations (1) and (2), and knowing that _{MRE} can be obtained as:

It can be realized from Equation (14) that the proposed MRE-based seat suspension isolator has complex non-linear dynamics, and the actuation force of the MRE can be adjusted by the ratio of

Magnetostatic analysis is also performed to further examine the capability of the proposed MRE-based isolator to supply the required magnetic flux density. For this purpose, the magnetostatic finite element (FE) model of the isolator is developed using an open-source finite element software (Meeker, ^{2} and 1.6 cm. The simulation results for the distribution of the magnetic flux density under 3.0 A current input to the coils are presented in

FE magnetic analysis of the MRE-based isolator;

_{d}) with respect to the excitation frequencies in the presence of various controllers.

Magnetic flux density at the center of the MRE layers under different levels of applied current to the coils.

B = 2 mT | B = 95 mT | B = 275 mT | B = 430 mT | B = 566 mT | B = 700 mT | B = 806 mT |

In this section, the NN control law is developed to isolate the transmitted vibration to the seat under varying base excitations. The convergence and stability of the proposed control scheme are subsequently verified using the Lyapunov theory.

NN controller is a powerful technique used to address complex non-linear systems under uncertainties (Ge et al.,

where

where _{k} are the width of the radial basic functions. The estimated weights

where

The control objective is to attenuate the transmitted vibration to the seat frame and, subsequently, to the driver. Let us choose the NN input as:

where γ and β are positive constants. The proposed control law is designed as:

where τ and ϑ are positive constant. It is worth noting that the proposed control scheme is model-free. As the accurate dynamic of the MRE-based isolator is generally unavailable in the real application, the model-free control strategy would be beneficial for practical implementation.

The unknown function to be approximated is defined as:

Substituting Equations (20) and (19) along with Equation (17) into Equation (12), we can obtain:

Then substituting Equation (15) into Equation (21), we may write:

In the following, Lyapunov theory has been utilized to prove the convergence and stability of the proposed control scheme. The first Lyapunov function candidate may be selected as:

where φ is a positive constant. Taking the derivative of the first Lyapunov function yields:

The updated law is proposed as:

where σ is a positive constant. Substituting the proposed updated law along with Equation (17) into Equation (24), yields:

Since

Choosing the second Lyapunov function as:

and taking its derivative:

and then substitute Equation (22) into Equation (29) yields:

Finally, combining the above two Lyapunov functions, we can write:

Taking derivative of the above Lyapunov function, and considering inequality in Equations (27) and (30), it can be shown that:

The above equation can be simplified as:

where ρ and ϖ are defined as:

Choosing ^{ρt}, we can obtain:

Integrating Equation (36), yields:

Using Equations (23), (28), and (37), we have the followings identities:

Hence, in the closed-loop system, _{s} remain in the compact set _{s}, respectively. Considering above, the following theorem can be stated:

Using the proposed control scheme and considering that the current of 3.0 A can reach to the maximum applied magnetic flux density (_{max} = 806 _{min} = 2

where _{est} is obtained from Equation (14). The control parameters τ and ϑ are identified as 10 and 2, respectively, and the control parameters γ and β in Equation (18) are chosen as 4 and 1, respectively. The centers of the nodes are evenly distributed in [−0.5, 0.5] and the width of the centers _{k} is fixed at 2. The initial weights are chosen as zero. The parameters φ and σ in the updated law are chosen as 2,000 and 0.1, respectively. Two hundred nodes are used for the NN approximation. Note that the provided control parameters comply with the control law design and stability analysis described in section Development of the Adaptive Controller Using Neural Network (NN), and a trial and error method is adopted to select the control parameters to achieve satisfactory control performance.

In order to evaluate the effectiveness of the proposed control scheme, its performance is compared with those of the two most widely used vibration isolation control approaches, namely passive control and sky-hook control strategies. In the case of passive control, the MRE-based isolator operates in an OFF- or ON-state in which the applied magnetic flux density is set to its minimum (0.0 T) or maximum (0.8 T) values. From Equation (13), it is clear that in passive control, the stiffness and damping of the MRE-based isolator are constant and the actuation force of a passive system is zero. In sky-hook control, the actuation force is generally described as Gu et al. (

The parameters of the MRE-based seat suspension system are provided in _{b} is added in parallel to the MRE isolator in order to limit the static deformation of the MREs, and two cuboid MREs operating in shear mode with the effective area of ^{2} and the thickness of

In the following, the effectiveness of the proposed control is demonstrated under different types of base excitations.

The performance of the proposed controller is investigated under harmonic excitation. The simulation is conducted in the frequency range of 0.5 to 5 Hz, which is the typical frequency range of a vehicle seat suspension system. Five cycles with a constant amplitude of 5 mm are considered for each excitation frequency. The Root-Mean-Square (RMS) values of the displacement and acceleration with respect to frequencies are shown in

The control input to the MRE-based isolator for harmonic excitation;

Comparison of the performance of various control schemes under harmonic excitation, bump shock, and random excitation.

Harmonic excitation | Displacement (cm) | 2.17 | 1.98 | 1.83 | 1.66 |

Acceleration (m/s^{2}) |
3.60 | 3.84 | 3.28 | 3.06 | |

Transmissibility | 6.13 | 5.63 | 5.17 | 4.67 | |

Bump shock | Displacement (mm) | 6.00 | 6.14 | 5.74 | 4.19 |

Acceleration (m/s^{2}) |
2.28 | 2.20 | 2.13 | 2.06 | |

Random excitation | Displacement (mm) | 8.7 | 9.4 | 7.6 | 6.3 |

Acceleration (m/s^{2}) |
1.45 | 1.83 | 1.22 | 1.06 |

In this section, the performance of the adaptive NN controller on the transient response of the designed MRE-based seat suspension isolator is evaluated using the bump shock excitation described as:

where _{0} = 0.01 m denotes the height of the bump shock transferred to the seat and _{b} = 20 is a constant determining the width of the bump. The displacement responses of the seat suspension system to the bump shock using OFF and ON states passive control, on-off skyhook control, and the proposed adaptive NN strategies are provided in

_{d}) for bump shock input in the presence of various controllers.

The control input to the MRE-based isolator for bump shock;

In this case, the random excitation is provided to further investigate the control performance of various control schemes. The displacement responses of the designed seat suspension system under random excitation using OFF and ON states passive control, on-off skyhook control, and the proposed adaptive NN strategies under the random excitation are provided in

_{d}) for a random excitation case in the presence of various controllers.

The control input to the MRE-based isolator for random excitation;

In this study, MRE samples with 25% volume fraction of carbonyl iron particles were fabricated and then experimentally characterized to evaluate their viscoelastic properties and their variation with respect to the applied magnetic field. An MRE-based seat suspension isolator has been developed considering the constraints on the applied magnetic flux density, stroke limit, damping effect, and static deformation. Adaptive NN control was proposed for the MRE-based seat suspension isolator to alleviate unwanted vibration. The stability and convergence of the proposed control scheme were proven using the Lyapunov method. The superior control performance of the proposed control scheme has then been verified under various base excitation profiles through the comparison with passive control and On-Off control strategies. Results suggest the superior performance of the proposed adaptive NN-based control strategy.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation, to any qualified researcher upon request.

CL, MH, RS, and GW contributed conception and design of the study. CL developed the modeling of MRE-based isolator and semi-active adaptive neural network controller. MH developed magnetic analysis, MRE fabrication, testing, and characterization. RS and GW supervised the research study. CL wrote the first draft of the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and National Natural Science Foundation of China (11832009) are gratefully acknowledged.

The Supplementary Material for this article can be found online at: