Lin-fang Qian,Guang-song Chen

Nanjing University of Science&Technology,200 Xiaolingwei,Nanjing,Jiangsu 210094,China

The uncertainty propagation analysis of the projectile-barrel coupling problem

Lin-fang Qian*,Guang-song Chen

Nanjing University of Science&Technology,200 Xiaolingwei,Nanjing,Jiangsu 210094,China

A R T I C L E I N F O

Article history:

28 June 2017

Accepted 30 June 2017

Available online 8 July 2017

Uncertainty propagation

Projectile-barrel coupling

Bootstrap

Maximum entropy

Interval

Based on the model which couples the projectile and gun barrel during an interior ballistic cycle,the uncertainty propagation analysis of the model is presented caused by the uncertainty of the input parameters.The Bootstrap method is employed to calculate the statistical moments(i.e.the mean,variance, skewness coef ficient and kurtosis coefficient)of the parameters of the projectile.Meanwhile,the maximum entropy method is used to estimate the probability density function(PDF)and the cumulative density function(CDF),the interval of the parameters of the projectile are also given.Moreover,the results obtained are compared to the results calculated by Monte Carlo(MC)method to verify the effectiveness of the presented method.Finally,the rule and the uncertainty propagation model of the projectile-barrel coupling system are given with the variation of the uncertainties of the input parameters.

?2017 Published by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http:// creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

The projectile,which is acted by high-temperature and highpressure propellant gas during an interior ballistic cycle,moves along the barrel at high-speed velocity.Due to the uncertainty of the clearance between the projectile and the barrel,the contact stiffness and the velocity of the engraving process,the projectile moves with uncertainty,resulting in the disturbance of the state parameters of the projectile at muzzle,which is the corresponding initial conditions of the exterior ballistics,directly related to the accuracy of the gun fire.Therefore,it is of great importance for improving the accuracy of gun fire to conduct uncertainty analysis [1,2]of the projectile-barrel coupling system based on the established coupling dynamic model[3],aiming at the prediction about the disturbance of the state parameters of the projectile at muzzle.

Due to the existence of contact-impact events[4]and highspeed friction[5]phenomena,the calculation results always present strong nonlinearities,and even sometimes mutations,which is dif ficult to establish the high-precision appropriate models[6]to conduct uncertainty propagation analysis.While the interval method[7,8]owns good applicability and practicality for the uncertainty propagation of projectile-barrel coupling.It only needs a small amount of information to obtain the upper and lower bounds of the variables,and does not need to establish the appropriate model.

Based on the established model which couples the projectile and gun barrel during an interior ballistic cycle,the statistical moments of the parameters of the projectile are calculated using the Bootstrap method[9].

Meanwhile,the maximum entropy method[10]is employed to estimate the PDF and the CDF,and the intervals of the projectile parameters are also given through the first-order Taylor expansion. Finally,the applicability of the presented method on projectilebarrel coupling is demonstrated.

2.Uncertainty propagation analysis of the projectile-barrel coupling model

Based on the multi-body dynamic theory,the projectile-barrel coupling dynamic model is established by utilizing the relative coordinate method,which considers the relative motion parameters of the part in gun launching dynamic system as the generalized coordinates.And the Lagrange multiplier method is used to introduce the constraint equations.Finally,the virtual power principle isemployed to obtain the gun launching dynamic equations.In the model,the flexible model of barrel is adopted,and a rigid- flexible coupling dynamic model of barrel based on the Timoshenko beam theory is established and embedded into the multi-body dynamic model.The coupling effect between the projectile and barrel is mainly re flected by the contact-impact events occurred between the front and rear bourrelet,the rotating band of the projectile and the inner bore of the barrel.

The input and output parameters of the projectile-barrel coupling dynamic model are expressed in the following form

wherexi(i=1,...,n1)represents theith input parameter,n1is the number of input parameters,yj(y=1,...,n2)represents thejth output parameter,andn2denotes the number of output parameters.Due to the independence of the output parameters,this paper only makes some analyses about the distribution interval of single output parameter.

The corresponding output parameters can be calculated by the projectile-barrel coupling model for input parameters,as shown in Fig.1.However,the uncertainty of the input parameters,such as the clearance between the projectile and barrel and the engraving velocity etc.,may lead to disturbances of the output parameters of projectile-barrel coupling model,i.e.the state parameters of the projectile at muzzle.

Assuming the interval of the input parameterxiis[sLi,sRi],and the width of the interval is de fined asThe probability density function of the parameterxiis de fined asfi(x).If the independence of the input parameters,the joint probability density function of x can be written as follows

msamples of input parameters can be obtained by using Latin Hypercube sampling method in the interval of input parameters, expressed as

where xk(k=1,2,...,m)∈ denotes thekth sample of the input parameters of these samples.

By substituting themsamples of the input parameters into the projectile-barrel coupling dynamics model,msamples of single output parameter can be obtained as follows

whereYk(k=1,2,...,m)denotes thekth output parameter corresponding to thekth input parameter of X.

Withoutanyassumptions,theBootstrapmethod isan nonparametric method,which only depends on the given information.A new sample(X,Y)1,which consists ofmsamples of the input and output parameters,can be obtained by using the Bootstrap method,while each sample of the input and output parameters is individually sampled from the original samples(X,Y).

Repeating this process byBtimes andBnew samples(X,Y)pcan be obtained,wherep=1,2,...,B.The meanμ,the varianceσ2,the coef ficient of skewnessγand the coef ficient of kurtosisκof each sample can be estimated respectively as

Fig.1.The diagram about projectile-barrel coupling process.

Combining the maximum entropy method and the Lagrange multiplier method,the PDF of output parameters can be expressed as follows

whereaq(q=0,1,2,3,4)is the unknown coef ficient and can be calculated through the following formulations

The CDF of the output parameters can be obtained through integrating the PDF of the output parameters

Table 1 Intervals of the input parameters.

The maximum value and the minimum one of the output parameters in the original samples areYmaxandYmin,respectively. The first order Taylor expansion of the CDF functionC(y)of the output parameters is performed atYmax.According to the relationship between CDF and PDF,the values of PDF atYmaxis equaled to that of the slope of the CDF atC(Ymax).It is the upper bound of the interval of the single output parameter just when the value of CDF equals to 1.The upper bound of the interval of the output parameters can be obtained as follows

Similarly,the value of PDFatYmincan be obtained fromthe slope of the CDF at pointC(Ymin).The lower bound of the interval of the output parameters can be obtained when the value of CDF equals to 0

where[ymin,ymax]denotes the interval of instantaneous state parameters of the projectile at muzzle obtained by the proposed method.

3.Simulation example

The input parameters of the existing projectile-barrel coupling dynamic model and the intervals are given as shown in Table 1.

The output parameters of the existing projectile-barrel coupling dynamic model are expressed by v0,φa1,φa2,φb1,φb2,˙φb1,˙φb2,where v0is the instantaneous velocity of the projectile at the muzzle,φa1is the pitching angle of v0,φa2is the direction angle of v0,φb1is the pitching angle of the projectile axis,φb2is the direction angle of the projectile axis,˙φb1is the pitching angle velocity of the projectile axis,˙φb2is the direction angle velocity of the projectile axis.

Sampling 100 inputs in the interval of mass center offset of the projectile,100 outputs were calculated by the projectile-barrel coupling dynamic model.Through standardized processing,the results obtained by this method compared with Monte Carlo(MC) method are shown as following:

Fig.2.The effect of the mass center offset of projectile.

As shown in Figs.2-13,the intervals of the 7 state parameters of the projectile at the muzzle calculated by the proposed method include the intervals estimated by MC method sharply,which demonstrates the effectiveness of this method for the uncertainty propagation analysis of the projectile-barrel coupling problem. Furthermore,it is further illustrated that the proposed approach is ef ficiency by using only 100 samples.

Fig.3.The effect of the mass of projectile.

Fig.4.The effect of the mass of muzzle brake of the driving band.

Fig.5.The effect of the equivalent stiffness.

Fig.6.The effect of the contact stiffness of steel.

Fig.7.The effect of the clearance between projectile and barrel.

Fig.8.The effect of the clearance between barrel and cradle.

4.Conclusions

Fig.9.The effect of the axial velocity of projectile.

Fig.10.The effect of the vertical velocity of projectile.

Fig.11.The effect of the transverse velocity of projectile.

Based on the established model which couples the projectile and barrel during an interior ballistic cycle,the statistical moments of the parameters of the projectile are calculated using the Bootstrap method.Meanwhile,the maximum entropy method is employed to estimate the PDF and the CDF,and the intervals of the projectile parameters are also given through the first-order Taylor expansion.Fromthe simulation results,it can be concluded that the range of the projectile parameter obtained is reliable and effective and the presented approach can be thus applied to the uncertainty propagation analysis of the projectile-barrel coupling problem.

Fig.12.The effect of the vertical angular velocity of projectile.

Fig.13.The effect of the transverse angular velocity of projectile.

[1]Le M,Tre OP,Najm HN,et al.Multi-resolution analysis of wiener-type uncertainty propagation schemes.J Comput Phys 2004;197(2):502-31.

[2]Kanoulas D,Tsagarakis NG,Vona M.Uncertainty analysis for curved surface contact patches.In:Ieee-Ras,International conference on humanoid robots. IEEE;2017.

[3]Chen GS.Study on the projectile-gun couple system dynamic and the key parameters.Nanjing University of Science&Technology;2016.

[4]Ma J,Qian LF,Chen GS,et al.Dynamic analysis of mechanical systems with planar revolute joints with clearance.Mech Mach theory 2015;94:148-64.

[5]Wei ZG,Batra RC.Modelling and simulation of high speed sliding.Int J Impact Eng 2010;37:1197-206.

[6]Brazier FMT,Jonker CM,Treur J.Compositional design and reuse of a generic agent model.Springer Netherlands;2002.p.491-538.

[7]Lhommeau M,Jaulin L,Hardouin L.Capture basin approximation using interval analysis.Int J Adapt Control Signal Process 2015;25(3):264-72.

[8]Zhang W,Liu J,Cho C,et al.A hybrid parameter identi fication method based on Bayesian approach and interval analysis for uncertain structures.Mech Syst Signal Process 2015;s 60-61:853-65.

[9]Efron B.Bootstrap methods:another look at the jackknife.Breakthroughs in statistics.Springer New York;1992.p.1-26.

[10]Xia X,Zhu W,Sun L,et al.Reliability evaluation for zero-failure data of rolling bearings based on Bootstrap maximum entropy method.Bearing 2016;53(7): 32-6(in Chinese).

13 June 2017

*Corresponding author.

E-mail address:lfqian@vip.163.com(L.-f.Qian).

Peer review under responsibility of China Ordnance Society

http://dx.doi.org/10.1016/j.dt.2017.06.005

2214-9147/?2017 Published by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

in revised form