鄧國紅，李 飛*，楊鄂川，歐 健，張 勇

1.重慶理工大學(xué)車輛工程學(xué)院，重慶 400054 2.重慶理工大學(xué)機(jī)械工程學(xué)院，重慶 400054

1 Introduction

With the rapid development of automobile industry and the rapid increasing of car ownership，car collision accidents happened under various traffic conditions also show a trend of raising，among them，the low-speed collision is one of the highest frequency of traffic accidents.But automobile parts of energy absorption play an important role during a low-speed collision，so，improving the energy absorption characteristics of components in low-speed collision can effectively improve the passive safety performance of car.Compared with energy absorption mechanism and responsiveness of traditional bumpers，magnetorheological damper can better play its cushioning effect with fast response time，continuous adjustable damping force.However，at present the design of magnetorheological dampers mainly bases on the quasi static models［1-3］，which are only applicable to low speed environment，when magnetorheological dampers are applied to high speed impact occasions(such as car crash)，it is necessary to adopt a new design theory and method.So，on the basis of Bingham-plastic nonlinear flow modified model(BPM)［4］，regarding the maximum damping force and the damping dynamic range as the optimization goals，the NSGA-II algorithm of modeFRONTIER was used to design the structural parameters of the magnetorheological damper for multiobjective optimization，and the optimization results were analyzed.

2 The working p rincip le of m agnetorheological dam per

Magnetorheological dampers make mainly use of magnetorheological fluid flowing properties to control the adjustable damping force.Under the action of magnetic field，MRF can transform quickly and reversibly Newtonian fluid with good liquidity into Bingham plastic solid with high yield strength，low liquidity.When the current size of excitation coil changes，applied magnetic field changes gradually，magnetorheological fluid viscosity also changes gradually.Magnetorheological fluid’s shear yield stress changes，as magnetic induction intensity of applied magnetic field changes，so as to adjust the damping force of damper continuously and reversible.According to the different forms of stress and flow of MRF applied to dampers magnetorheological damper mainly consists of valve type，shear，extrusion and shear valve type four basic patterns.According to the background requirements of magnetorheological damper used in the front energy absorption structure in car，based on the single rod cylinder monocular lever shear valve type magnetorheological damper as the research object，the piston structure diagram is shown in Fig.1.

Fig．1 The piston structure diagram

Where，nr means cylinder diameter，d1 means piston rod diameter，d2 means piston diameter，la means piston effective length，lc means coil width，dc means the clearance of the coil and damping channel，wc means the depth of coil，dd means damping clearance.

3 Bingham-p lastic non linear flow m odified model(BPM)

The mechanical model of the current research of magnetorheological damper have glue Bingham plastic model，Bouc-Wen model，the phenomenon model of Spencer，Herschel-Bulkley model.Now，the most common used constitutive model is Bingham plasticity model［5］，referred to as BP model，BP model regards the yielded viscosity of MRF as a fixed value，so the relationship between shear stress and shear strain

When speed is low ，the flow of MRF can be approximate to laminar flow，the design of the damper adopts BP model can meet the engineering requirements，but as more and more wide engineering application of magnetorheological damper，its running speed has been improved，gradually develops to the impact，collision，drop and other fields of the high speed，at this moment，magnetorheological fluid flow of activation regional of the magnetorheological damper transforms layer flow into turbulent flow［6］，BP model applied to low speed condition becomes no longer accurate under high speed，it will produce great error when design，so，in this paper，adopting Bingham-plastic nonlinear flow modified model(BPM)［7-9］based on Bingham plastic model.According to the fluid mechanics knowledge，the maximum damping force can be obtained when magnetorheological fluid flows through the damping channel.rate is the type(1)，where，τ0means the shear yield stress;H means the magnetic field strength;˙γmeans the shear strain rate;ηmeans the liquid viscosity of yielded.

Among them:

Where，F(xiàn)maxmeans maximum damping force produced by piston structure;Foffmeans the viscous force of piston compression stroke without magnetic field;Fsmeans the damping force of the shear mode without magnetic field;Fτmeans the coulomb damping force with magnetic field;D means the damping dynamic range;Poffmeans pressure loss caused by the viscous damping;Apmeans the piston cross-sectional area;η means MRF viscosity;V1means MRF average flow velocity in work gap;V2means MRF average flow velocity in coil and work cylinder clearance;τymeans MRF shear yield stress with magnetic field;Pηmeans the pressure loss of MRF flows through the piston outside diameter and the inner wall of the work cylinder;Pmlmeans the expansion and compression pressure loss of MRF flows through the coil and the inner wall of the work cylinder;Pcoilmeans the pressure loss of MRF flows through the coil and the inner wall of the work cylinder;Peemeans the expansion and compression pressure loss of MRF flows through the piston outside diameter and working cylinder inner diameter;ρ means the density of magnetorheological fluid;f1and f2mean respectively Darcy friction coefficient related to the Reynolds number of the piston outside diameter and working cylinder inner diameter，the coil and working cylinder inner diameter;Kentryand Kexitmean respectively inlet pressure loss coefficient and the outlet pressure loss coefficient，respectively 0.5 and 1;Kscand Ksemean respectively inlet pressure loss coefficient and outlet pressure loss coefficient，Admeans the sectional area of the piston outside diameter and working cylinder inner diameter;Acmeans the sectional area of the coil and working cylinder inner diameter.

4 M u lti-ob jective op tim ization of magnetorheological dam per

4．1 Damper design requirements

In order to design reasonable magnetorheological damper，it is necessary to estimate the range of control damper damping force and stroke，it is assumed that the vehicle collides with fixed rigid wall at a certain speed，the collision process does not occur secondary collision，all the kinetic energy is dissipated by two magnetorheological dampers，ignoring the friction between the tires and the ground，car is 1 421 kg，the initial velocity of the collision is 3 m/s(10.8 km/h)，collision time is approximate56 ms，the required average damping force of magnetorheological damper is 38 kN calculated by the impulse theorem in head-on collision，other dimensions are estimated according to the limiting conditions of damper replaces crash box:working cylinder outside diameter is 60 mm，inner diameter is 54 mm，piston length is 50 mm，the damper’s biggest length is320 mm.The proposed design of this paper is that the magnetorheological damper’s biggest damping force is greater than 38.5 kN，the damping dynamic range is greater than 1.2 when the maximum current is 2 A at 3 m/s.

Considering the requirements of the magnetorheological damper applied to frontal crash.Setting the maximum speed of the piston as 3 m/s.The adopted magnetorheological fluid is SG-MRF2035 developed by a company，the fitting relationship of the shear yield stressτyand the magnetic flux density B is that:

The materials of piston and cylinder are industrial pure iron，the piston rod is 45#steel，the wire is enamelled copper withφ0.8 mm，the relative permeability is 1.

4．2 Optimization design variables，constraints and objective functions

When the performance parameters of magnetorheological fluid are confirmed，the size of the magnetorheological damper damping channel and the size of the excitation magnetic field will have a decisive effect on the magnetorheological damper performance.The size of excitation magnetic field is related to coil loading current and the size of piston，loading current can be controlled through constant current power，but the piston sizes need to be optimized.So，setting la，lc，dc，wc，dd as optimization variables.Optimization variables and scope is shown Table 1.

Table 1 Optimization variables and scope

The initial value for the initial variables are:dd＝1 mm，dc＝2 mm，wc＝5 mm，la＝30 mm，lc＝20 mm.

Other parameters values are wr＝60 mm，nr＝54 mm，d1＝24 mm.

The objective functions:to make Fmaxand D the largest.

4．3 Establishing optimization model

This optimization model is established using multiobjective optimization analysis software modeFRONTIER，first simulating the magnetic flux density produced by piston and coil by using magnetic field analysis module of ANSYS and optimization variables［10］，inputting the magnetic flux density into the Matlab，and then Matlab brings the optimization variables and the magnetic flux density into the mechanical model(BPM)for magnetorheological damper to calculate the maximum damping force and the damping dynamic range.According to the need to select 30 groups optimization variables，setting optimization variables according to the scope of table 1，setting optimization initial conditions combined with the constraints.Choosing NSGA-II algorithm for the multi-objective optimization，the maximum generation is 200，crossover rate is 0.9，mutation rate is 0.1，multi-objective optimization model is set up as shown in Fig.2.

Fig．2 The optimization model

5 Op tim ization resu lts

Seeing optimized Pareto frontier is very clear(as shown in Fig.3)，it’s impossible for two optimization objectives D and Fmaxto achieve the optimal at the same time，the relationship of them is inversely proportional，so only to choose eligible optimized solutions in many frontier solutions.

Through the correlation coefficient matrix analysis of five optimization variables and two optimization targets(as shown in Fig.4)，seeing that the two main factors of influencing the maximum damping force are the clearance of damping channel dd and the piston effective length la，the correlation coefficient are respectively-0.909 and 0.128;The two main factors of influencing the damping dynamic range are the clearance of damping channel dd and the clearance of coil and damping channel dc，the correlation coefficient are respectively 0.630 and-0.433.

Fig．3 Pareto solutions for the chart

Fig．4 The correlation coefficient matrix

When analyzing the relationship of multiple independent variables and dependent variables，it’s more intuitive to observe the effect of each variable on the dependent variables through the response surface analysis.Fig.5 is the response surface analysis of the clearance of damping channel dd and the piston effective length la about the maximum damping force.Fig.6 is the response surface analysis of the clearance of damping channel dd and the clearance of coil and damping channel dc about the damping dynamic range.

Fig．5 The influence of dd，la to the F max

Fig．6 The influence of dd，dc to D

As can be seen from the Fig.5:the change of the maximum damping force along the clearance of damping channel dd direction is faster than the piston effective length la direction，i.e，the influence of the clearance of damping channel on the maximum damping force is greater than the piston effective length，the damping clearance is more smaller and the maximum damping force is more bigger，but the piston effective length makes the maximum damping force decrease within a certain scope.

From Fig.6 can be obtained:the change of the damping dynamic range along the clearance of damping channel dd direction is faster than the clearance of coil and damping channel dc direction，i.e，the influence of the clearance of damping channel on the damping dynamic range is greater than the clearance of coil and damping channel，the clearance of damping channel is more bigger and the damping dynamic range is more bigger，but the clearance of coil and damping channel is more bigger，the damping dynamic range is more smaller.

Considering the application background of automotive magnetorheological damper，table 2 lists the five groups typical Pareto solutions which can satisfy the maximum damping force and the damping dynamic range，making the effective length of piston la as big as possible when it’s possible for the two optimization goals to obtain the maximum value，at the same time，according to the design requirements of vehicle under low-speed collision(it is high speed for magnetorheological damper)，choosing the fourth group solution as the final optimization solution.

Simulating the magnetic induction intensity produced by piston coil for before optimization and optimized structure parameters of magnetorheological damper in magnetic field analysis module of ANSYS，as shown in Figs.7，8，the simulation results showed that the optimized magnetic field distribution is more concentrated and reasonable.

Table 2 Typical Pareto solutions

Fig．7 Magnetic induction intensity before optimization

Fig．8 The optimized magnetic induction intensity

6 Conclusions

The method of using the non dominated sorting genetic algorithm(NSGA II)to optimize the structure parameters of the magnetorheological damper for multiobjective is feasible.Optimization results showed that it’s impossible for two optimization goals to achieve the optimal at the same time，only to choose the optimized solutions that meet the conditions from all the Pareto solutions，the magnetic field distribution of optimized magnetorheological damper is more concentrated and reasonable.The NSGA-II algorithm can well solve the magnetorheological damper structure parameters optimization problem.

Applying the optimized magnetorheological damper to the automobile front energy-absorbing structure to study its energy absorption characteristics will be the work that next step need to be done.

Acknow ledgem ent

This paper is supported by Graduate Student Innovation fund of Chongqing University of Technology(YCX2014201).

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