Jing-hi Hui ,Min Go ,Ming Li ,Ming-rui Li ,Hui-hui Zou ,Gng Zhou ,

a Northwest Institute of Nuclear Technology,Xi’an,710025,China

b Shijiazhuang Campus,Army Engineering University,Shijiazhuang,050003,China

Keywords: Actuator Trajectory correction fuze Impact loadings Optimized Latin hypercube design Kriging model Optimization algorithm

ABSTRACT This paper presents an actuator used for the trajectory correction fuze,which is subject to high impact loadings during launch.A simulation method is carried out to obtain the peak-peak stress value of each component,from which the ball bearings are possible failures according to the results.Subsequently,three schemes against impact loadings,full-element deep groove ball bearing and integrated raceway,needle roller thrust bearing assembly,and gaskets are utilized for redesigning the actuator to effectively reduce the bearings’stress.However,multi-objectives optimization still needs to be conducted for the gaskets to decrease the stress value further to the yield stress.Four gasket’s structure parameters and three bearings’peak-peak stress are served as the four optimization variables and three objectives,respectively.Optimized Latin hypercube design is used for generating sample points,and Kriging model selected according to estimation result can establish the relationship between the variables and objectives,representing the simulation which is time-consuming.Accordingly,two optimization algorithms work out the Pareto solutions,from which the best solutions are selected,and veri fied by the simulation to determine the gaskets optimized structure parameters.It can be concluded that the simulation and optimization method based on these components is effective and ef ficient.

1.Introduction

In the 1960s,the US Army first introduced an ammunition to improve the accuracy of the attack guidance system,which consists of sophisticated electronic components.As a research target of this paper,the trajectory correction fuze proposed by the United States in 2006 is one such guidance system.In this fuse,an actuator within rotating micro-generator supplies electrical equipment.In addition,it controls the relative rotation between the projectile and the canard,changing the aerodynamics of the projectile during flight to correct its trajectory,thus improving the accuracy of the projectile.

During launch process,the impact loadings imposed on the fuze is over 15000 G’s,causing the actuator and other equipment to fail.Therefore,most researchers have focused on reducing stress of equipment that is affected by impact loadings.Thus,to find means to reduce the impact loadings applied to the actuator is very important.The impact resistance of the electronic components in the electromagnetic launch projectile was analyzed by Cui et al.,and the mathematical model of the two-phase pulse alternator had been also established[1].The optimization algorithm was also used to optimize the structure design[2].The electronic devices subjected to impact loadings could be out of operation.In order to reduce the device’s stress produced by impact loads,the buffer structure,like foam,stiffeners and dampers were adopted[3-7].

Major researchers adopt the buffer structure as the traditional and effective method to reduce the impact loadings on the fuze’s devices.However,it is always neglected that the dimension and material properties of the buffer affect the impact loads imposed on the device,even the stress.As everyone knows,the launch is a highly nonlinear process,FEM(finite element modeling)and simulation can describe the dynamic response,such as velocity,displacement,acceleration and stress.It always takes much more time to conduct the simulation,causing the optimization with simulation inef ficiency.The DOE(design of experiment)method and approximation modeling can be used to obtain the sample points to establish the relationship between the structure parameters and dynamic response[8].Furthermore,on the basis of approximation model,the optimal design can be achieved by optimization algorithm such as genetic algorithm[9].Response surface methodology and Kriging model are the common use in the optimization process[10,11].Buyuk et al.conducted an approximate optimization methodology to optimize a projectile-whipple shield system which is subjective high impact loadings under the hypervelocity conditions[12].Anderson et al.used the response surface model to optimize the composite[13].Lee et al.also adopted the response surface methodology to study the effect of mixture compositions on the accelerator,verifying the results with the test and simulation[14].Lin et al.employed the software,ADAMS and ISIGHT,to carry out the simulation and optimize with multi-objective genetic algorithm[15].Chiandussi and Avalle optimize a tapered tubular steel component which can absorb the energy to reduce the device’s impact loads[16,17].Khalkhali et al.use the MLF-type neural network,also belong to one of the approximation models,to optimize the projectile tip’s shape[18].The honeycomb structure was optimized with multi-objective optimization method by Zarei and Kroger[19,20].

Fig.1.Simulation flowchart.

This paper carries out a nonlinear structure optimization design against the high impact loadings imposed on actuator used for trajectory correction fuze.For an initial designed actuator,to find out which component is likely to failure under impact loadings,a simulation method,is proposed and conducted to obtain the stress at maximum stress element of each component.Subsequently,three schemes against impact loadings are proposed,from which the third scheme should be optimized with multi-objectives optimization technique presented in this paper,including design of experiment(DOE),approximation modeling and optimization algorithms,together utilized for nonlinear structure optimization,which has not been used in the existing literature.

The remainder of this paper can be divided into four sections.Section 2 presents a simulation method,and stress of the initial designed actuator’s each component is acquired to be determined which component is apt to be failure under such impact loadings.Section 3 proposes three schemes against impact loadings imposed on the actuator mechanism which is redesigned based on the previous simulation results.Section 4 conducts a multi-objectives nonlinear structure optimization for the redesigned actuator on the basis of the proposed schemes.In section 5,we conclude the paper.

2.Modeling and simulation

As research continued,experimental test is too expensive and time-consuming to conduct and there are some uncontrollable factors during test leading to failure of demonstrating theoretical and experiential analysis.Therefore,the researchers in this field gradually adopted numerical models such as finite-element models.In this section,implicit-explicit dynamic simulation with fuze-projectile-barrel coupling will be carried out,whose process is exhibited in Fig.1.

2.1.Solid model

Fig.2.Projectile and trajectory correction fuze.

Fig.2 shows the correction fuze screwed on the projectile with the thread.The actuator mechanism depicted in Fig.3 is comprised of fixed collars,bearings,casing,shaft and bottom bolt.Those two bearings are the deep groove ball bearing(ball bearing)mounted inside the actuator.The actuator is capable of relative spin between the canards and projectile.When the projectile moves out of the muzzle,the air imposes on the canards and makes it spin reverse respect to the casing,thereby the machine in the actuator can power for the electrical devices in the fuze.Hence,the actuator mechanism plays a signi ficant role for the fuze’s work in projectile’s fight.

Fig.3.Actuator mechanism.

Fig.4.Initial state of launch.

Fig.5.Zoomed in version of the barrel’s ri fling and chamber throat.

During projectile’s launch process,as is well known that each component of the fuze is subjected high impact loadings,which give rise to the deformation and failure of the fuze’s components.To find out which component of the actuator has the potential of failure,stress of those components must be obtained with valid method.However,launch is a highly nonlinear process involving material,geometry and boundary conditions,achieving the dynamic response for impact loadings in the launch process with analytical method is practically feasible.Generally,the model of projectile and devices is always simpli fied to more effectively conduct simulation,and the application of some boundary condition improve the simulation technology to make the model close to reality[21,22].A simulation method,implicit-explicit dynamic with fuze-projectile-barrel coupling.Fig.4 exhibits the initial state of launch.Generally,smooth and ri fled gun bore are assembled with the howitzer,from which the later kind(as Fig.5 shows)is considered and modeled in this paper to establish the real launch process as much as possible.Moreover,it is necessary to simplify the model to improve the computer ef ficiency.The fixed canards are not needed to modeled,however,the mass of electronic devices,such as generator,inside the fuze,including canards cannot be neglected.Thereby,a payload substitutes the mass,attaching to the shaft is modeled(as Fig.6 shows).

Fig.6.Simpli fied fuze model.

2.2.Finite element model

Consisted of shell,charge and band,which must be meshed by re fined element,the projectile is contacted with the barrel through eroding between band and ri fling.Fig.7 depicts the projectile modeled by finite element.

Fig.7.Projectile and band’s FEM.

Three-dimensional geometry model should be accurately created to obtain the accurate,reliable results.While,the ri fling is a kind of spiral model which is too complex and time-consuming to generate element and simulate.Therefore,the modeling commands,such as spiral line,sweep and array can be applied to construct the barrel,then imported to the meshing software to conduct the pre-process.The completed barrel FEM is shown in Fig.8,and the partial enlarged view of ri fling and chamber throat is exhibited as Fig.9 shows.

Fig.8.FEM of the barrel.

Fig.10 shows the global view of actuator model.As mentioned above,to reduce the stress of the bearings,accurate stress value must be achieved,thus re fining the bearings’mesh like band.

2.3.Simulation

The properties of material assigned to each model’s component are listed as Table 1 shows.Actually,the ball bearings are installed onto the casing with the manner of interference fit,whose the contact type should be defined with tied surface to surface in preprocess.Furthermore,tha band is eroded with the barrel’s ri fling,which called engraving in the first stage during launch process.Fig.11 shows the initial state of launch,which the band clings to the chamber throat.

Table 1 Material properties.

In addition,like ri fling modeling,gravity effect cannot be neglected should be imposed on the model during both the implicit and explicit simulation.For both the two stages,the gravity imposed on the model with z(7.723 m/s2)and y(6.033 m/s2)coordinate direction(as Fig.12 shows)for the reason that the barrel is elevated to 52°QE(quadrant elevation)in reality.Secondly,regarded as the thrust loads,the base pressure produced from burning propellant is imposed on the projectile’s bottom.Fig.13 depicts the time history of base pressure.

As Fig.14 shows,the implicit simulation results indicate that the barrel’s deformation at the muzzle is 2.032 mm.As the input condition,this value is imported to the pre-process of explicit simulation,accordingly,the stress of actuator’s each component can be calculated with explicit solver.Then the peak-peak value of stress time history at the maximum stress element can be obtained.

With the simulation method,each component’s stress curve at the maximum stress element are acquired and exhibited as Fig.15 shows in the following.

It can be obtained the peak-peak value of each component’s stress from the curves,from which the casing,shaft,bottom bolt and fixed collars’peak-peak stress are not over the material’s yield stress.Therefore,those actuator’s components cannot be deformed under the impact loadings during launch.However,from Fig.15(c),the two ball bearings peak-peak stress are 661.9 MPa and 588.9 MPa,respectively,both exceed the yield stress of bearing’s material yield stress.Overall,the results indicate that such impact loadings cause the two ball bearings’plastic deformation or cracks,leading to the actuator got stuck,and cannot rotate smoothly during the launch process,while the other components are not in fluenced by the high impact loadings.Accordingly,the bearings in the actuator should be protected from the impact loadings with the actuator’s structure redesign and optimization.Next section,the actuator mechanism is redesigned with the structure against impact loadings.

Fig.9.Patial enlarged view of FEM of the ri fling and chamber throat.

Fig.10.FEM of actuator.

Fig.11.Initial state of band clings to the chamber throat.

Fig.12.Model’s coordinate.

Fig.13.Base pressure.

Fig.14.Barrel’s deformation due to gravity at 52°QE.

3.Schemes against impact loadings

To prevent the bearings in the actuator from plastic deformation or cracks,the initial actuator structure should be redesigned with the structure against impact loadings.The schemes adopted must improve the capacity of impact loading resistant for the bearings to ensure the actuator rotating smoothly.In this way,three schemes are applied for the redesigned structure and described in this section.

3.1.Full-element deep groove ball bearing and integrated raceway

3.1.1.Full-element deep groove ball bearing

In the ball bearing,each roller contacts the outer and inner raceway with point contact manner.On the one hand,obviously,the quantity of the roller affects the loading resistant of bearing.More rollers can reduce each roller’s stress,thus avoiding the plastic deformation or crack,ensuring the actuator rotating in high speed.On the other hand,the two ball bearings in initial actuator are contact to the casing and shaft with interference fit,which is tend to lead to the slide between the bearing and casing or shaft under high axial impact loadings.It may also cause the vibration shock,giving rise to the crash between the bearing’s roller and raceway.To settle this problem,an integrated raceway can be used for actuator mechanism to preventing the bearing’s slide between the casing and shaft.

A kind of full-element deep groove ball bearing which has maximum quantity of roller can be utilized for the actuator,whose the structure and mounting state are shown in Fig.16.The mounting slots in each bearing’s ring are stagger in operation status while aligned in the mounting status.In the actuator mechanism,the left ball bearing is changed to this bearing’s type.

Fig.15.Stress curve at the maximum stress element of actuator’s each component.

Fig.16.Full-element-deep groove ball bearing.

3.1.2.Integrated raceway

The integrated raceway manner can be seen in Fig.17.The bearing’s outer and inner rings are tied to the casing and shaft with laser welding,respectively,to be an integrated raceway of the bearing.The inner ring and shaft constitute the bearing’inner raceway,while the outer ring and casing compose the outer raceway.

Fig.17.Integrated raceway.

3.2.Needle roller thrust bearing assembly

Compared with the ball bearing,needle roller thrust bearing has more likely to resist to the high impact loadings in axial direction.Such the advantage can be utilized to decrease the loadings which is imposed on the ball bearings,which decrease the ball bearing’s stress as much as possible.The constitution of the thrust bearing and its install location are exhibited in Fig.18.

3.3.Buffer structure with the gasket

As the buffer structure used in the actuator,the gaskets can reduce the loadings imposed on the two ball bearings during projectile launching process.Two gaskets are needed to add inside the actuator mechanism.As Fig.19 shows,one gasket is put between the right ball bearing’s outer ring and bottom bolt,the other gasket is located at the space between the ball bearing’s inner ring and thrust bearing.

To sum up,the actuator mechanism can be redesigned with the three schemes against impact loadings resistant,as Fig.20 shows,which is constituted with the shaft(including the left ball bearing’s inner raceway),rollers,casing(including left ball bearing’s outer raceway),right ball bearing,thrust bearing,gaskets and bottom bolt.

3.4.Simulation for the redesigned structure

The implicit-explicit dynamic simulation with fuze-projectilebarrel coupling is also carried out for the redesigned actuator mechanism to obtain the bearings’stress at the maximum stress element,whose the stress curves are depicted in Fig.21.It is obviously seen from the curves that the redesign with the structure against impact loadings effectively decrease the stress.The peakpeak stress values of the two ball bearings at the maximum stress element are up to 520 MPa and 513.2 MPa,respectively.

Fig.18.Thrust bearing and the install location.

Fig.19.Gaskets in the actuator.

However,the thrust bearing is also subject to the impact loadings,whose the stress and plastic deformation should also be taken into consideration.Similarly,the thrust bearing’s stress curve at the maximum stress element can be acquired and shown in Fig.22.The peak-peak value is up to 567.5 MPa,which is over the bearing’s material’s yield stress,indicating that the thrust bearing appears the plastic deformation.Furthermore,the deformation of thrust bearing is also likely to in fluence the high-speed rotation of actuator.Accordingly,the actuator’s redesigned structure of the loading resistant need to be optimized.

Fig.22.The stress of thrust bearing at the maximum stress element.

4.Multi-objectives nonlinear structure optimization

To a certain extent,three schemes reduce the bearing’s stress which subject to impact loadings successfully,however,it cannot be reduced all the three bearings’stress to the value which is lower to the bearing’s material’s yield stress.For the first schemes,the bearing which has full rollers cannot be changed for its the geometry parameters are belong to the standard parts.In addition,it is unpredictable that the bearing’s geometry which has been optimized installed on this actuator mechanism can has the same capacity of impact loading resistant or not when it is installed on the other type of mechanism.The second scheme,the thrust bearing is also regarded as a standard part,whose the parameters cannot be changed.Therefore,the two gaskets,as the third scheme,can be selected by the buffer material and geometry parameters.Optimization can be conducted for the two gaskets’structure parameters.The three bearings’peak-peak values of stress at the maximum stress element are served as the optimization objectives,forming a multi-objectives optimization issue.Furthermore,it is the nonlinear relationship existed in the bearings’stress and gaskets’structure parameters,thus direct optimization through mathematical model based on the physics cannot be achieved.In this section,nonlinear structure optimization for the gaskets in the actuator mechanism will be carried out,including the DOE method,approximation and optimization algorism.In addition,the simulation is also used for the sample points and veri fication of optimization results.

Fig.20.Constitution of the actuator mechanism’s structure against impact loadings.

Fig.21.The stress of modi fied and initial actuator’s ball bearings at the maximum stress element.

4.1.Selection for optimization variables of the gasket

According to energy principle,the gasket can reduce the energy transmitted from shaft to casing,being absorbed by bearings.In this way,it can protect bearings subjected to impact loadings from failure.LetQwbe the value of total energy,QbandQbe the value of energy absorbed by bearing and the gasket’s deformation energy,respectively.The relationship of those three variables can be defined as

whereQcan be also defined in the following:

whereAandhare the gasket’s sectional area and axial thickness.The deformation of gasket involves two stages,elasticity and plasticity.For the first stage,letεbe the gasket’s strain andσbe the stress which is further defined as:

VariableErefers to the elastic modulus of the gasket.Thus,energyQbcan be further modi fied as

On the basis of Eq.(4),the larger theQ=Ahvalue,the smaller theQbwould be.So the bearing’s stress can be decreased.

Due to the fuze’s size,the gasket’s diameter cannot be changed.Thereby,its axial thickness can be varied to research the effect to bearing’s stress.Furthermore,Eq.(4)indicates that the elastic modulus also affects the impact loadings imposed on the bearing.In a word,the axial thickness and elastic modulus are regarded as the design parameters to optimize the gasket performance against impact loadings.Eqs.(3)and(4)only describe the gasket’s elastic deformation,however,the plastic deformation of gasket which is like to happen also should not be neglected.In accordance with material’s strain-stress curve,the yield stress of the gasket material determines the maximal strain of elastic range.Accordingly,it should not be ignored that the effect of gasket material’s yield stress on the energy absorbing.Thereby the yield stress should be also reserved as one of the optimization variables of the gasket.To sum up,two gaskets’axial thickness,elastic modulus and yield stress of gasket material are selected as the optimization variables.

Constrained by the fuze’s size condition,the value range of gasket’s thickness is from 1 mmto 5 mm.Furthermore,according to the buffer material in common use,value range of its elastic modulus and yield stress are from 1 GPa to 100 GPa,and from 1 MPa to 100 MPa,respectively.

4.2.DOE method for gasket’s parameters

Full-factorial,one of the DOE methods,is adopted for experiment which has two or less variables.For the experiment which has three or more variables,the Orthogonal experiment and Latin hypercube design can be employed.For the multi-objectives optimization issue in this section,four variables,including axial thickness of the inner gasket and outer gasket(LiandLo),elastic modulusE,and yield stressFs,are to be optimized,and regarded as the four factors for DOE method.Optimized Latin hypercube design(OLDH)modi fied by Latin hypercube design is used,whose the number of levels for each factor equal to number of points with combinations optimized to evenly spread points within n-dimensional space defined by n factors.The OLHD allows many more points and more combinations to be studied for each factor.Experiment points are spread evenly,allowing higher order effects to be captured[23-26].Engineer has total freedom in selecting the number of designs to run as long as it is greater than the number of factors.The experiment points composed of four factors is summarized as Table 2 shows,including the corresponding results,three bearings’stress are also included through simulation.Sl,SrandStdonate the peak-peak stress value at maximum stress element of the left,right ball bearing and thrust bearing,which are served as the optimization objectives,respectively.

Table 2 Experiment points of DOE for gasket’s optimization variables.

The result data from Table 2 will be used for the approximation modeling,while the four variables and three objectives should be non-dimensioned beforehand.Note thatx1,x2,x3andx4,y1,y2andy3(x1,x2refer to the outer and inner gasket’s dimensionless axial thickness.x3,x4refer to elastic modulus and yield stress,respectively.y1,y2andy3refer to the left and right ball bearings’and thrust bearing’s dimensionless stress of peak-peak value,respectively)are defined in the following formulas:

4.3.Approximation modeling

The approximation model will be created based on the result data from DOE method to substitute the simulation process and predict the simulation results.Among approximation models,including response surface model(RSM),RBF neural network and Kriging model are in common use.In this section,those three approximation models will be created and one of the models which can better predict the simulation results will be selected according to estimated coef ficientR2defined in the following:

whereyianddonate the simulation value and predicted value,respectively.refers to the average value of the simulation results of sample points.In general,the value ofR2must be over 0.9.

RSM,RBF neural network and Kriging model are created by fitting the simulation results with experiment points.To visualize the fitting performance of each model,the scatter diagrams produced by the test points listed in Table 3 and corresponding simulation result show in Figs.23-25.The distribution of predicted points by approximation model related to the simulation result points can be apparently seen from those scatter diagrams.

The values ofR2for the three approximation models are calculated and the results are listed in Table 4.

It can be seen from Table 4 thatR2values of three objectives yielded from RSMare not all over 0.9.Meanwhile,R2value ofy2by RBF does not also exceed 0.9.Those two models,whose the fitting precision are also made out from Figs.23 and 24,cannot be used to fit the simulation results.However,the Kriging model,whose the objectives’R2value all lager than 0.9,is selected to use for the optimization with algorithm in the next section.

Table 3 Test points.

Table 4 Estimated coef ficient approximation model.

4.4.Multi-objectives optimization with algorithm

Because of three optimization objectives,there is no explicit optimization design for the gasket with the best value of all the objectives.Pareto solutions can be used for multi-objectives optimization which has many outstanding designs making good tradeoff among those objectives exist.In this section,on the basis of the approximation created in the previous section,the Pareto solutions for gasket’s optimization will be searched with two kinds of algorithm,MOPSO and NSGA-II.In addition,it should be noted that the final solutions achieved by the algorithm may be not the optimum solutions and selection of an appropriate index to evaluate the quality of the solutions is not a simple task.Accordingly,we will conduct a comparison of the two optimum solutions from the two kinds of algorithm.

The optimization mathematical model based on the performance indexes can be presented by:

4.4.1.Optimization with MOPSO

Fig.23.Scatter diagram generated by RSM.

Fig.24.Scatter diagram generated by RBF neutral network.

Multi-objectives particle swarm optimization(MOPSO)mimics the social behavior of animal groups such as flocks of birds or fish shoals.The process of finding an optimal design point is likened to the food-foraging activity of these organisms.Particle swarm optimization is a population-based search procedure where individuals(called particles)continuously change position(called state)within the search area.In other words,these particles‘fly’around in the design space looking for the best position.The best position encountered by a particle and its neighbors along with the current velocity and inertia are used to decide the next position of the particle.

Fig.25.Scatter diagram generated by Kriging model.

Given dimension of search space whose value isW,the number of particlesn,the position of particleican be written byxi=（xi1，xi2，…，xiW）,and the best position of particleifrom the beginning to now is expressed bypbesti=（pi1，pi2，…，piW）.The best position of the whole population from the beginning to now can be presented bygbesti=（gi1，gi2，…，giW）.The velocity of particleiis written by vi=（vi1，vi2，…，viW）,therefore the particle’s velocity and position in each dimension can be expressed in the following:each objective parameter is treated separately.Standard genetic operation of mutation and crossover are performed on the designs.Selection process is based on two main mechanisms, “non-dominated sorting” and “crowding distance sorting” .By the end of the optimization run a Pareto set is constructed where each design has the “best” combination of objective values and improving one objective is impossible without sacri ficing one or more of the other objectives.

Initial population is given the number of 100 randomly and the maximum generation is set to 100.The crossing probability is 0.8.With algorithm searching completed,10000 result points which are on behalf of 10000 solutions are obtained,however,among those points,1677 points are regarded as the Pareto solutions depicted in scatter diagram shown in Fig.27.Similarly,the best solution from NSGA-II is summarized in Table 6.

4.5.Multi-objectives optimization with algorithm

The optimization results produced by the two algorithms and the corresponding simulation results are listed in Table 7.

The optimization result by MOPSO is consistent with that by NSGA-II.Furthermore,the corresponding simulation’s peak-peak

wherec1andc2are the learning factors.r1andr2are the random numbers.The initial position and velocity of the particle swarm are yielded randomly.

As for the optimization of gasket’s structure parameters,given the maximum interaction is 100,and the initial number of particle is 100.With algorithm searching completed,10000 result points which are on behalf of 10,000 solutions are obtained,however,among those points,1398 points are regarded as the Pareto solutions depicted in scatter diagram shown in Fig.26.

Fig.26.Scatter diagram of Pareto solutions by MOPSO.

The best solution selected from Pareto solutions depicted by the scatter diagram according to the optimization target principle as Eq.(7)shows,thus selecting the solution,which the two bearings’peak-peak value of stress are minimum.The best solution is summarized in Table 5.

4.4.2.Optimization with NSGA-II

In the Non-dominated Sorting Genetic Algorithm(NSGA-II),stress values are closed to those of algorithms,and not exceeding the material’s yield stress.For the integrated variation of stress curves by simulation from Fig.28 after optimization,except for right ball bearing,it can be clearly seen that the left ball bearing and thrust bearing’s stress value optimized by MOPSO are lower than those optimized by NSGA-II.Accordingly,the gaskets’optimization parameters by MOPSO are selected to be the best scheme of gasket structure against impact loadings,whose the parameters listed in Table 5.

Table 5 Optimization result with MOPSO.

Fig.27.Scatter diagram of Pareto solutions by NSGA-II.

Table 6 Optimization result with NSGA-II.

Table 7 Optimization result with NSGA-II.

Fig.28.Stress curves of bearings by simulation after optimization.

5.Conclusion

To precisely describe the projectile’s launch process and acquire the stress response of the actuator’s components for the high impact loadings,a simulation method,implicit-explicit dynamic simulation with fuze-projectile-barrel coupling is presented in this paper.Peak-peak values extracted from stress curves of each component’s maximum stress element are obtained.Consequently,ball bearings in the actuator are vulnerable to failure.Subsequently,three schemes for redesigned actuator are proposed.Full-element deep groove ball bearing and integrated raceway,the needle roller thrust bearing assembly,and the buffer structure with gaskets are utilized for effectively improve the actuator mechanism’s capacity of impact loadings.However,peak-peak stress values of three bearings in the redesigned actuator exceed the yield stress,which indicated that the redesigned actuator should be also optimized with the proposed schemes.

For the optimization conducted for the third scheme,the gaskets have four parameters regarded as the four optimization variables to be further optimized.Two ball bearings and one thrust bearing’s peak-peak stress values are served as the three objectives to be optimized.Optimized Latin hypercube design and Kriging model are used for establishing relation between the optimization variables and objectives,representing the time-consuming simulation.The MOPSO and NSGA-II can find many outstanding designs making good tradeoff among those objectives.Finally,the best solution is selected from the Pareto fronts according to the mathematical model.Finally,optimization results are veri fied by simulation and the best solution by MOPSO is determined as the ultimate gasket scheme.

The necessary components,such as DOE method,approximation model,MOPSO and NSGA-II,are not new and have been utilized in some studies.However,they are merged together in this paper to optimize the rotating actuator mechanism.It can be concluded that the simulation and optimization method based on these methods is effective and ef ficient.

Declaration of conflicting interest

The authors declare that there is no con flict of interest.

Declaration of competing interest

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

Acknowledgements

The authors would like to acknowledge National Defense Pre-Research Foundation of China(Grant No.41419030102)to provide fund for conducting experiments.