Shock and Vibration

Volume 2015 (2015), Article ID 382541, 8 pages

http://dx.doi.org/10.1155/2015/382541

## Frequency Dependent Spencer Modeling of Magnetorheological Damper Using Hybrid Optimization Approach

Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, Canada H3G 1M8

Received 23 December 2014; Revised 5 March 2015; Accepted 5 March 2015

Academic Editor: Weihua Li

Copyright © 2015 Ali Fellah Jahromi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Magnetorheological dampers have been widely used in civil and automotive industries. The nonlinear behavior of MR fluid makes MR damper modeling a challenging problem. In this paper, a frequency dependent MR damper model is proposed based on Spencer MR damper model. The parameters of the model are identified using an experimental data based hybrid optimization approach which is a combination of Genetic Algorithm and Sequential Quadratic Programming approach. The frequency in the proposed model is calculated using measured relative velocity and relative displacement between MR damper ends. Therefore, the MR damper model will be function of frequency. The mathematical model is validated using the experimental results which confirm the improvement in the accuracy of the model and consistency in the variation damping with the frequency.

#### 1. Introduction

The semiactive control system provides both features of passive and active devices in terms of reliability and adaptability. Using semiactive system, the rate of energy dissipation becomes controllable, while, in active control devices, the energy can be added to the system to control the dynamic response. Magnetorheological (MR) damper is a semiactive control device which is commonly used in vehicle industries and structural applications. The MR damper contains MR fluid instead of regular oil. The MR fluid is a smart material which contains micron sized magnetic polarized metal particles which provide variable viscous damping with changing magnetic field [1]. The application of MR fluid is dependent on three different operational modes: flow, squeeze-flow, and shear [2]. For instance, MR dampers and servovalves are designed based on flow mode of MR fluid [3]. The squeeze-flow mode of MR fluid is utilized in the application of impact control dampers for large forces [3]. The shear mode can be used for brakes, clutches, and damping layer of sandwich structures [3]. In order to describe the dynamic behavior of the MR damper, different mathematical models have been proposed in both discrete and continuous time domain.

Modeling of MR damper by using black box nonlinear models is carried out in discrete time domain [4] where MR damper hysteresis, function of displacement, and velocity are modeled by Neural Network (NN). In this model, three parameters are indicated based on the collected experimental input and output data. However, the nonmodel based parameter identification is only valid for the system operation range during which the experimental data are collected and used for training the NN model. Therefore the accuracy of model cannot be guaranteed for extrapolating the range of operation.

The Bingham viscoplastic MR damper model [5] is built in continuous time domain and it describes the dynamic behavior of MR damper based on measuring the shear stress and the shear strain rate. The Bingham model consists of a Coulomb friction element parallel with viscous damping. Using Bingham model, the storage energy in the MR damper cannot be modeled. Moreover, the difference between the simulated force and the real force increases when the velocity is near zero.

The modified Bingham MR damper model proposed by Gamota and Filisko [6] is a viscoelastic-plastic model. The so-called Gamota and Filisko model is a Bingham model which is in series with a parallel set of spring and viscous damper. The Gamota and Filisko MR damper model improves the accuracy of the model in describing the hysteresis loop and storage energy of the MR damper. However, the simulation of this modified Bingham model needs step size in the order of 10^{−6} which is the main drawback of this model [1].

The Bouc-Wen model [7] is a continuous, viscoelastic-plastic model which can describe a wide range of hysteresis behavior [1]. The hysteresis behavior of Bouc-Wen model is described by an evolutionary variable with three coefficients of velocity which results in smoothness of transition from the preyield to the postyield region. The roll-off effect cannot be simulated using Bouc-Wen model in the region of the small magnitude of the velocity where the velocity and acceleration have opposite directions [1].

In the Spencer MR damper model [1], a spring and a viscous damping element are added to the Bouc-Wen model to simulate the roll-off effect at small velocities. Therefore, the other pair of damping and stiffness elements can be adjusted for small velocities or high frequency region. The Spencer model is capable of simulating the roll-off effect in all velocity and acceleration regions. In the Spencer model, the assigned damping coefficients only depend on the changing current. However, the MR damper viscosity depends on the frequency of excitation [8] and temperature of MR fluid [9]. And in the literature, MR damper models cannot describe such frequency dependent behavior.

The present study deals with the variation of MR fluid viscosity with the frequency of the excitation in Spencer MR damper model. The viscosity of the MR damper is modeled using two viscous damping elements for large and small velocities. The MR model is identified by minimizing the error between the experimental data and simulated data of the proposed model. The hybrid optimization approach is used for the identification which is a combination of Genetic Algorithms (GA) and Sequential Quadratic Program (SQP). The excitation frequency in real application can be calculated by measuring the velocity and the displacement of the MR damper. Therefore, the viscosity of MR fluid in Spencer model is described by exponential and Gaussian equations which are the functions of velocity and displacement for small and large velocity regions, respectively.

The rest of the paper is organized as follows. The modeling of MR damper is presented in Section 2. The experimental set-up and procedure are explained in Section 3. The optimization approach and characterization of MR damper model are discussed in Section 4. A comparison between the proposed model and experimental data is presented in Section 5. Finally the conclusions and future work are presented in Section 6.

#### 2. Spencer Magnetorheological Damper Model

Due to the nonlinearity in dynamic behavior of MR damper, the accuracy and validity of the Spencer MR damper model over wide range of frequencies are not consistent. The schematic of the Spencer MR damper model is shown in Figure 1. The governing equations of Spencer MR damper model are presented in the following [1]: where and are the MR damper accumulator stiffness and viscous damping coefficients for small velocities, respectively. is the initial displacement associated with spring . Further, and are accumulator stiffness and viscous damping coefficients for large velocities, respectively.