Interval uncertain optimization for damping fluctuation of a segmented electromagnetic buffer under intensive impact load

Zi-xuan Li,Guo-lai Yang,Feng-jie Xu,Li-qun Wang

School of Mechanical Engineering,Nanjing University of Science and Technology,Nanjing,PR China

Keywords:EMB Intensive impact load Segmented Sensitivity analysis Interval uncertain Optimization

ABSTRACT Aiming at the problems of demagnetization effect of electromagnetic buffer(EMB)caused by high velocity under intensive impact load and the difficulty and error of machining composite thin-walled long tube,a segmented EMB is proposed.The inner tube and air-gap are divided into initial segments and the traversing segments.Through theoretical analysis,impact test and simulation,it can be found that the RRF curve has two peaks.Firstly,in order to reduce the resultant resistance force(RRF)peaks,the sensitivity analysis based on optimal Latin hypercube design(OLHD)and polynomial regression was performed.The results show that the smallest contribution ratio to the dynamic response is the seventh and ninth segments of the inner tube,which are less than 1%.Then,fully considering the uncertain factors,important parameters are selected for uncertain optimization after sensitivity analysis.The interval order and interval probability degree methods are used to establish interval uncertain optimization model of the RRF considering robustness.The model was solved using an interval nested optimization method based on radial basis function(RBF)neural network.Finally,the Pareto front is obtained and numerical simulation is performed to verify the optimal value.It indicates that the two kinds of RRF peak is obviously reduced,and the optimization object and strategy are effective.

1.Introduction

In the dynamic system,the motional eddy currents can be induced due to relative motion of the field source and conductor.The generated eddy currents circulate in such a way that they induce new field with opposite polarity of the original field causing a repulsive force proportional to the relative velocity.The electromagnetic buffers(EMBs)produce damping forces without contact,additional stiffness and friction[1-4]different from almost any other passive damping mechanisms.Therefore,applications of eddy currents for damping purposes have been investigated for more than three decades,such as braking systems[5]vibration control[4,6],magnetic levitation[7],machining technology[8]and other fields.

The permanent magnet EMB has the advantage of no need to provide external electric energy.Improving the energy density and damping characteristics of permanent magnet EMBs can effectively maintain the effectiveness and reliability of the buffering process.A permanent magnet with high maximum magnetic energy product[9]called NdFeB was introduced and patented independently by Sumitomo Special Metals and General Motors(later Magnequench)in 1983[10].Applying NdFeB to the EMB effectively increases the damping coefficient.

At the same time,many researchers have improved the resistance by changing the structure of the EMB and the arrangement of the permanent magnets.Jang et al.[11,12]obtained that the Halbach magnetization of linear EMB can produce the highest braking force compared to horizontal magnetization and vertical magnetization when the permanent magnets are limited to the same volume.But the fact is that the radially magnetized PMs are rare in the market[13].Sodano et al.[14]proposed a passive EMB with controllable position of permanent magnet.The damping force is significantly increased by maximizing the relative velocity of the permanent magnets and the conductor.Zuo et al.[15]divide the permanent magnet into several blocks,and a conductor plate is placed between each two permanent magnets with the same magnetization direction in each column.The advantage of the EMB is that the braking force density is large,and the disadvantage is poor heat dissipation and difficult processing.To reduce the structural vibrations of a mechanical system used in aerospace,Pan et al.[16]proposed a new sketch of EMB with high damping,which is significantly superior to the one-plate EMB with the same structure and dimensions.Perez-Diaz et al.[17]presented a passive EMB with enhanced performance for use in high temperatures by means of impedance matching,which is designed,manufactured and experimentally demonstrated in a temperature range from 25°C to 200°C.

The EMB parameters are also coordinated through optimization.Canova et al.[18,19]investigated the three-dimensional analytical models of the permanent magnets radial EMB and verified the effectiveness of finite elements.The multi-objective optimization based on genetic algorithm is proposed to improve braking torque with less quality and cost.Combined with the Rosenbrock’s method and the experimental design method,Takahashi et al.[20]proposed an optimization method for the permanent magnet radial EMB,which improved the braking torque and considerably reduced the CPU time.Aberoomand et al.[21]examined the best set of components for the double-sided PM axial eddy-current couplers through genetic algorithm with respect to some significant dynamic criteria such as system bandwidth.In order to achieve the desired torque-speed characteristic and the minimized weight,ASL et al.[22]implemented multi-objective optimization for pole number and geometric dimensions of EMB.It is found that the mass and volume of the initial prototype can be halved by implementing this optimization algorithm.

Unfortunately,there are many parasitic effects in these applications,such as demagnetization effect,skin effect,eddy currents edge effect,magnetic saturation,and they are becoming the limiting factor for their performance.At the same time,there are many uncertain factors in the machining,assembly and impact process.

Therefore,this paper proposes a segmented permanent magnet cylindrical EMB.The machining difficulty and error of the composite thin-walled long tube is reduced by segmenting the inner tube and air-gap.The proposed EMB has high-energy density to accommodate the buffering of intensive impact load and can give different damping force at different positions,a situation rarely studied in the past.The sensitivity analysis of the EMB system is studied based on optimal Latin hypercube design(OLHD)and polynomial regression.In order to reduce the resultant resistance force(RRF)peaks,the interval order and interval probability degree methods are carried out to establish optimization model considering robustness solved by interval nested method based on RBF neural network.

2.Nonlinear irreversible demagnetization primarysecondary eddy currents coupling dynamics model for EMB system

2.1.Configuration of the EMB

As denoted Fig.1,the inner tube and air-gap of the segmented EMB proposed in this paper are divided into 10 segments.The initial segmentsb1ande1are the length of the size of the primary.The traversing segmentsb2～b10ande2～e10are divided into one segment every 100 mm.

The studied EMB mainly consists of two parts:(1)the primary part,which consists of a moving rod combined with a sequence of ring-shaped,axially magnetized permanent magnets with magnetic poles of the same polarity facing each other separated by pure iron poles,and(2)the secondary part,which consists of outer tube and segmented inner tube.

Fig.1.Partial schema of the proposed segmented EMB and the equivalent magnetic circuit.

2.2.Load force

Damping of the segmented EMB driven by impact loads which is calculated by the following method.According to the pressure distribution in the container,the Lagrange quadratic interpolation is used to obtain the average pressure in the container at any time.Further,the resultant forceFptis obtained.In order to obtain the accurate dynamics model for the EMB system and buffering mechanism,five impact loads with different peaks are calculated and applied to the primary of the EMB,as shown in Fig.2.

In addition to the impact load,the recuperator force is applied to act on the buffering process and return the EMB to the initial position after the buffering is completed,which can be calculated as:

Fig.2.Different impact loads.

As denoted Fig.3,the recuperator force provided by the elastic medium is only controlled by the recoil travel after the elastic medium and structural parameters are determined.The calculated impact loads are functions of time.In order to introduce the impact load and the recuperator force data into the dynamics model simultaneously,the method of piecewise linear with linear extrapolation onxis used.

Fig.3.Recuperator force.

Electromagnetic damping occurs when the primary and secondary of the EMB move relative to each other under the impact loads.This process is accompanied by a variety of parasitic effects,such as demagnetization effect,skin effect,eddy currents edge effect,magnetic saturation.

2.3.Demagnetization effect

The demagnetization effect is obvious at high velocity caused by impact load.Demagnetization effect is divided into two parts:eddy current demagnetization and permanent magnet selfdemagnetization.

The time-varying eddy current magnetic field interacts with the original magnetic field,resulting in the distortion of magnetic lines and weakening of the original field.It is assumed that the magnetic field strength(H)of the iron poles and the outer tube is zero.The magnetic induction of the eddy current field can be obtained by the Ampere circuital theorem:

The magnetic Reynolds number is introduced to characterize the eddy current demagnetization,which can be calculated by:B0is assumed to be a fixed value in the conductor.Therefore,the net flux density is defined as

km(T)is introduced to characterize the degree of eddy current demagnetization.In the design of EMB,different structural parameters will affect the magnitude of the eddy current magnetic field and at the same time change the working point of the permanent magnet.The relationship between the quiescent operation point of the permanent magnet and the load line can be expressed as:

The increase of the rate permeance of the external magnetic path will cause the load line and the quiescent operation point to be close to they-axis,ie,the self-demagnetization is attenuated.It can be found through magnetic path analysis that the ferromagnetic material has negligible influence on the reluctance of the external magnetic path compared with the air-gap and the conductor tube.And the inner tube and the air-gap are equivalent in terms of the reluctance of the outer magnetic path and the resulting quiescent operation point.Therefore,the parameterkm(T)can be assumed askw(T)andqare considering the movement of the operation point.Also the demagnetization effect is not uniformly present in permanent magnets,where reversible demagnetization may occur in a large area accompanied by local irreversible demagnetization.

Therefore,a demagnetization model considering the true constitutive relation with nonlinear relative permeability is selected to fully consider the influence of the demagnetization effect under the impact loads,as shown in Fig.4.

Fig.4.Demagnetization curve of NdFeB and Magnetization relationship of iron.

2.4.Skin effect

The skin effect is the tendency of an alternating current induced by moving magnetic field or alternating magnetic field in a conductor tube to be distributed such that the eddy current density on the surface of the conductor is much stronger than the eddy current inside the conductor.Due to the permanent magnets with magnetic poles of the same polarity facing each other and the magnetic path structure,the skin effect in the EMB with linear motion still occurs.The eddy currents in the secondary are induced by the primary field.The fundamental frequency of the induced currents in the secondary is limited by the velocity of the EMB,which is obtained by computing

The penetration depth is defined as the depth below the conductor surface at which the current density decreases to 1/eof the current density at the surface.The penetration depth of the secondary conductor is calculated as follows:

The skin effect causes the equivalent cross-section of the conductor to decrease with high current frequency,which greatly increases the effective resistance.The current density in a conductor decreases exponentially with depthdfrom the surface,represented by

2.5.Eddy currents edge effect

The eddy currents given by Eq.(9)is valid for only an infinite conducting plate,indicating that zero eddy current density at the outer tube edge where the eddy currents edge effect occurs is not taken into account,which is written as

If the edge effect is ignored,the predicted force is overestimated[23].Since the inner tube is surrounded by the outer tube,the edge effect only appears in the outer tube.To accurately represent the distribution of eddy currents in the outer tube the image method is employed,as demonstrated in Fig.5.Therefore,the imaginary eddy current density is written as

Fig.5.The infinite and the imaginary eddy current.

The net eddy current density at the outer tube can be expressed as

Therefore,the motional eddy current damping force is modified by computing

2.6.Magnetic saturation and hysteresis

There are two main reasons for the magnetic saturation in the iron pole and the outer tube.First,the accumulation of the primary field generated by the permanent magnets cause magnetic saturation at the edge of the iron pole.Most importantly,the original field will be distorted by the interaction of the eddy current field under intensive acceleration and high velocity,resulting in local magnetic saturation of the iron pole and the outer tube.The hysteresis effect can be described as:

Since the hysteresis loop of the soft magnetic material is narrow,the hysteresis effect processed into polyline in the engineering calculation can be neglected,as shown by the dashed line in Fig.4.

2.7.The coupling of primary and secondary eddy currents

The variable secondary field excites the primary vortex-like eddy current with the direction opposite to the eddy current of the secondary.Moreover,the primary eddy current and the secondary eddy current interact and couple each other.This paper sets the eddy current effect on the overall model of the EMB.The triangle element is used to perform fine mesh on the eddy current concentrated part,and the vector magnetic potential function is constructed based on the linear interpolation method in the element.

2.8.The EMB system

In addition to the electromagnetic damping force,the static part of the electromagnetic buffer is also subjected to a variety of forces to form an EMB system,which is defined as the RRF,given by

The differential equation of the EMB system is established to dynamically analyze the EMB as follows:

By considering the above factors,a nonlinear irreversible demagnetization primary-secondary eddy current coupling timestep finite element model(FEM)is established for subsequent optimization.

3.Impact tests

A prototype EMB system has been manufactured and tested under the five different intensive impact load to verify the accuracy of the established FEM through the data acquisition device,as given in Fig.6.

Fig.6.Experimental set-up for the fabricated EMB.

Fig.7.Comparison of experimental(a)distance-time,(b)and(c)velocity-time with that calculated by simulation.

Table 1Comparison of numerical simulation and impact test.

Table 2Determination coefficients of the linear polynomial regression models.

The thickness of the inner tube and the air-gap used in the impact tests and FEM are set to 1 mm and 0.5 mm,respectively.The velocity and displacement curves under the five different impact loads obtained by the nonlinear irreversible demagnetization primary-secondary eddy currents coupling dynamics model are in good agreement with the impact tests as demonstrated in Fig.7.The tested maximum displacement,maximum velocity and its time node is consistent with the FEM and the errors are extremely small,as denoted in Table 1.

When the impact 1,2,and 3 are applied to the EMB,the time of damping force peak is basically the same as that of velocity peak(see Table 2).The maximum damping force starts to appear earlier than the maximum velocity under the impact load 4.When the impact load 5 is applied to the EMB,the experimental and FEM results show that the time of damping force peak is significantly earlier than that of velocity peak under the impact load 5,which indicates that EMB has exceeded the critical velocity due to the excessive demagnetization effect of high velocity.In order to ensure the accuracy of the test results,the acceleration data is also given.More importantly,the acceleration curves shows very good credibility of the five different impact and buffering processes.

There are two reasons for the fluctuation of the buffering displacement,velocity and the RRF.First,the vibration of the sensor and signal line causes the test data to fluctuate slightly.More importantly,the machining and assembly error of inner tube and the air-gap exist and are inevitable.Generally,the dynamic responses of FEM are in agreement with those collected by the impact test under the impact 1,2,3,4 and 5,as reflected in Figs.7 and 8 and Fig.9.Therefore,the Established FEM is adapted to the low,medium and high velocities.

4.Sensitivity analysis of the EMB system responses

4.1.The EMB system responses and uncertain parameter under intensive impact load

Electromagnetic damping and the corresponding dynamic responses of the EMB system generated under intensive impact load are different from the quasi-static response.And we are mainly concerned with the responses of maximum damping force and displacement.The previous chapter showed that the RRF curve is saddle shape with concave center and high ends,that is,there are two RRF peakFe_maxandFf_max.Fe_maxis generated with the arrival of the critical velocity.Ff_maxis mainly caused by the increasing recuperator force with the displacement.

The segmented EMB reduces the difficulty and error of machining composite thin-walled long tube.But,there are still dimensional errors in the manufacturing process and uncertain factors under intensive impact load.Therefore,the studied uncertain parameters includeb1-b10,e1-e10,c1,l1.Keeping the radial length of the primary constant,the size of the air-gap is achieved by changing the inner diameter of the inner tube,while the thickness of the inner tube is changed by the outer diameter.The outer tube changes uniformly based on the outer diameter of inner tube.Each of parameters is independent.

Fig.8.Comparison of experimental RRF 1,2,3,4,5-time characteristics with that calculated by simulation.

Fig.9.Comparison of experimental acceleration 1,2,3,4,5-time characteristics with that calculated by simulation.

Fig.10.The sensitivity analysis process for the EMB system buffering based on OLHD and polynomial regression analysis.

4.2.The sensitivity analysis based on OLHD and polynomial regression analysis

Fig.10 signifies the calculation process of the sensitivity analysis method for dynamic responses of the EMB system based on DOE sampling methods and polynomial regression analysis,which is illustrated as follows:

Step 1:The parameter vector X=(X1X2…Xk…Xn)is generated according to the number of design variables,and the upper and lower limits of the design variables areXiminandXimax,respectively.In order to satisfy the projection spatial uniformity of parameter design matrix,the OLHD method with improved uniformity by means of an additional criterion that changes the order of combination of the random combinations of LHD,such as the integrated mean squared error criterion[24]and the entropy criterion[25].

Step 2:The dynamic calculation model considering nonlinear magnetization,irreversible demagnetization of permanent magnets and coupling of primary-secondary eddy current is parameterized.Then,according to the parameter design matrixDm×n,the csv format file is linked to the parameterized model.

Step 3:Update meshing and reconstruct the new irreversible demagnetization dynamics model based on parameter design matrixDm×nuntil the simulation results of all samples are obtained.

Step 4:The linear polynomial regression model is used for sensitivity analysis of EMB system responses,which is mathematically expressed as follows:

β0，β1，β2，…，βk，σ2can be estimated by ordinary least squares.The principle is to minimize the sum of the squares of the residuals:

The coefficients of the linear polynomial regression model can be obtained by solving the partial derivative of Eq.(18)and setting it to 0.

Step 5:The contribution ratio of the dynamic response is used to visually realize the quantification of the influence degree of each parameter.Therefore,the coefficients of the linear polynomial regression model are converted into percentage contributions using Eq.(19).

Step 6:The sensitivity results of the dynamic responsesFe_max,Ff_maxandλmaxare taken as absolute values respectively.Then,the total contribution ratio of each parameter can be calculated according to Eq.(20).

Changing the parameterl1can get the contribution ratio of 25 iron poles.Therefore,in the statistical analysis,the iron pole coefficient is written asβ=βi×25.Then the percentage contribution is calculated to get the contribution ratio of one iron pole.Through the sensitivity calculation process,the sensitivity analysis results of each parameter toFe_max,Ff_max,λmaxand the total contribution ratio are presented,as demonstrated in Fig.11.It can be seen thatFe_maxis obviously affected by the entire outer tube,the first segment of the air-gap and inner tube.However,the contribution ratio of the last and fourth segments of the air-gap,and the sixth and ninth segments of the inner tube toFe_maxwas less than 1%.The sensitivity results of the dynamic responseFe_max,andFf_max,are opposite.The first segment of the inner tube becomes a factor with minimal influence because of the weak relationship between the dynamic response of the later buffering and the original initial segment.Similarly,the first segment of the inner tube and air-gap is highly sensitive toλmax,in general,which contribute the most to the three dynamic responses,reaching 13.14% and 19.01%.The minimum contribution ratio accounting for less than 1% is the seventh and ninth segments of the inner tube.Therefore,these two parameters can be ignored in the subsequent optimization.Then,the optimized parameters are given as follows:b1～b6,b8,b10,e1～e10,c1,l1.Its nominal values and tolerance intervals are demonstrated in Table 3.

Fig.11.Sensitivity analysis results of the electromagnetic buffering system responses under intensive impact load.

Table 3Nominal values and tolerance intervals.

5.Uncertain optimization considering robustness for EMB system responses under intensive impact load

5.1.Evaluation criterion

Firstly,in order to evaluate the buffering process reasonably,the evaluation criteria are defined.

(a)Peak of RRF

In the previous analysis of the EMB buffering system,the RRF making the moving EMB return to static state is a force opposite to the intensive impact load.Therefore,we can control other dynamic responses by changing the RRF.The smaller the fluctuation of the RRF,the more stable the buffering process.At the same time,the RRF curve is saddle shape with concave center and high ends.Therefore,there are two kinds of RRF peaks.It is necessary to decrease the two peaks together under the condition that the buffering displacement changes little to weaken the fluctuation of RRF curve.The peak of RRFFr_maxcan be expressed mathematically as:

Fig.12.Schema of fullness.

(b)Fullness

TheFr～xcurve with good fullness can make the buffer process more stable.At present,the fullness of RRF generally depends on the experience of designers,and there is no quantitative evaluation method.

As illustrated in Fig.12,Scis the area enclosed byFr～xandxaxis under different parameter conditions.Sris the rectangular area formed by the straight line of the peak of RRFFr_maxandλmax.Therefore,the quantitative evaluation of fullnessαis tentatively defined as:

The fullness value is[0,1].Largeαmeans the fullness ofFr～xcurve is better.It is found that the fullness can be controlled by theFe_maxandFf_max.By reducing the two peaks,excellent fullness can be obtained.

5.2.An interval uncertain optimization model of RRF considering robustness

Although the error of the inner tube and the air-gap is small,it will cause huge fluctuations of the RRF.Therefore,it is necessary to consider the uncertain parameters in the buffering process.They can be described as intervals since their range of fluctuation is easily determined although their probability distribution is difficult to give.TheFe_max(X)andFf_max(X)are taken as optimization objectives to get better fullness and buffering performance,which is described as a function of the interval uncertain vector.The upper and lower bounds arerespectively.The uncertain objectives can be transformed into three deterministic objectives by using the interval order method[26]:reflect the average design performance under uncertainty in engineering.is the sum of the interval radii of the two optimization objectives.It can reduce the sensitivity of the objective function to the uncertainty,so as to ensure the robustness of the optimization.

The buffering displacement of the EMB cannot exceed the maximum allowable value.It can be expressed as:denotes the constraint function range of the maximum displacement.Its upper and lower bounds arerespectively.

The deterministic inequality constraints are obtained by the deterministic transformation of the model considering all the interval possibilities[27]:

η1is determined by the designer in[0,1].If the constraint is strict.Its value should be increased.With designed large displacement,the impact collision between the mechanisms at the end of the buffering will be caused.Therefore,it is necessary to take a more strict constraintη1=0.9.

Through the treatment of objective functions and constraints,the interval optimization model of RRF considering robustness based on interval order and interval possibility can be described as follows:

5.3.Interval nested optimization method based on RBF neural network

The simulation of the nonlinear irreversible demagnetization primary-secondary eddy currents coupling time-step FEM is timeconsuming,cost-effective and requires superior hardware.After adding the optimization module,especially the nested interval optimization module,it is difficult to obtain the optimization result in a reasonable time.Using RBF neural network to establish the approximate model of the mapping relationship between the optimization parameters and the output response can ensure accuracy while saving calculation costs.Therefore,the following optimization process can be given in Fig.13.

Fig.13.Interval nested optimization method.

Table 4R2 of the neural network models.

Step 1:It is necessary to test the reliability to ensure the validity of the approximate model of RBF neural network.The reliability test can be carried out by calculating the determination coefficientR2which represents the fitting degree of dependent variable and independent variable.

TheR2is between[0,1].LargeR2represents the higher fitting accuracy.The OLHD method is used to obtain the test samples in the design space.As illustrated in Table 4,theR2value of the output parameter calculated by Eq.(27)is greater than 0.9,therefore,the approximate model is acceptable.

Step 2:After establishing the approximate model,a nested interval optimization method is used to solve the optimization model of Eq.(24).NSGA-II is selected as the outer optimizer of this optimization method for multi-objective optimization.In the inner operator,genetic algorithm is used to calculate the interval of uncertain objective function and constraint which are obtained by calling inner operator twice.

Step 3:When the external optimizer reaches the maximum number of iterations,the optimal solution set is obtained;otherwise,update the individuals of the population.

Step 4:Import the preferred parameters of the optimal solution set into the established dynamics model of EMB system to verify the accuracy of the solution set and solve the accurate solution.

5.4.Optimization results

Fig.14.Pareto front.

Table 5Detailed information on the Pareto front.

Table 6Nominal values and tolerance intervals of the optimized parameters.

The interval uncertainty optimization program is written and solved by Matlab.A set of non-dominated solutions organized in a Pareto-optimal front are obtained,as depicted in Fig.14.The solutions in Pareto-optimal front satisfy the constraints.Details of these solutions are shown in Table 5.It can be seen that there is an obvious contradiction between the optimization objectivesThe larger interval midpoint always has a smaller interval radius.The optimal design and error scheme of variable combination are obtained.The nominal value and interval of each design variable are demonstrated in Table 6.These parameters can provide reference of mechanical tolerance for the design of air-gap and inner tube.Optimized variables with smaller interval radius are biased.Therefore,on the premise of considering the radius of the resistance peak,this means that the robustness of the optimization result is considered,the point circled in Fig.14 is selected from the Pareto-optimal front.

The preferred value variables and their upper and lower bounds are imported into the established FEM to verify the effectiveness of the optimization and make further comparative analysis.Through numerical calculation,the results of RRF,damping force,velocity,and displacement are denoted in Fig.15.

Fig.15.Comparison of(a)RRF-displacement,(b)damping force-displacement,(c)velocity-time,and(d)displacement-time before and after optimization.

Fig.15(a)depicts the change of RRF before and after optimization when the optimized parameters are nominal values and tolerance intervals.It can be seen that the RRF curve fluctuates seriously before optimization.The two kinds of RRF peakFe_maxandFf_maxare significantly reduced and the fullness of the RRF curve is significantly improved from 86.71%to 91.98%.The demagnetization effect was successfully controlled.The dynamic response of the endpoint of the interval and the nominal values fluctuate slightly,indicating that the selection of the optimal interval is reasonable.As denoted in Fig.15(b),the fluctuation of damping curve is also effectively suppressed.

The comparison of displacement and velocity curve is demonstrated in Fig.15(c)and Fig.15(d).It can be seen that the maximum displacement after optimization is increased by 13.72 mm,which is far within the allowable range of EMB.The displacement caused by the upper bound values is slightly increased compared with the nominal values.After optimization,the velocity decline process is faster first and then slower than the initial value.It shows that the optimization variables,objectives and methods selected in this paper are feasible and the optimization effect is obvious.

6.Conclusion

In this paper,in order to solve the problems of demagnetization effect on RRF and the error of machining composite thin-walled long pipes,a segmented EMB is first proposed.The segmented EMB is divided into one initial segment and nine traversing segments.Each segment has a different inner and outer diameter of the inner tube.The proposed EMB can be applied to intensive impact loads and can give different damping force at different positions.

However,when the parameters of each segments are not designed properly,there will be two kinds of RRF due to demagnetization effect and restoring force peak.Therefore,the sensitivity analysis of segmented parameters based on OLHD and polynomial regression was performed to classify the important and secondary factors that affect the damping force.The results indicate that the contribution ratio of the seventh and ninth segments of the inner tube to the dynamic response of the EMB system is less than 1%.Therefore,these two parameters can be ignored in optimization.From the absolute value analysis,the contribution ratio of the initial segments of inner tube and air-gap is the largest,reaching 13.14%and 19.01%.

The interval uncertain optimization model of the RRF based on interval order and interval probability degree methods is carried out to reduce the RRF fluctuations,which can fully consider the influence of the uncertain factors and robustness.The two kinds of RRF peakFe_maxandFf_maxare significantly reduced and the fullness of the RRF curve is significantly improved from 86.71% to 91.98%.The Pareto-optimal front obtained by the interval nested optimization method based on RBF neural network can effectively meet the demand of gentle resistance curve.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

The authors would like to acknowledge the National Natural Science Foundation of China(grant number 301070603).