YU Hao(郁浩)，WANG Xiao-tian(王曉天)，LI Xin(李欣)，LI Ji-min(李紀(jì)敏)

(Baicheng Ordnance Test Center of China，Baicheng 137001，Jilin，China)

Introduction

After launch，if the cargo projectile flies to desired distance and height，its fuse is stimulated，and the expelling charge is ignited.Under the gas pressure，the sub-projectiles and other parts are ejected from the cargo projectile.The cargo projectile’s ejection height［1］can be defined as the vertical height of the projectile at this time.It is a very important performance index for the cargo projectile.Some factors can influence it，such as the product performance fluctuation，test error and conditions.For the same production conditions，the variations of ejection height led by the performances，such as fuse starting error，muzzle speed error and quality change，can be considered as random errors.The ejection height is measured by a high-speed photography.Its precision can be up to centimeters.Thus，the error led by the test instrument is very small.This paper adopts multivariate statistical analysis method to excavate relevant information on ejection height in different test conditions，such as environment conditions and launch angle，and finds out the influence factors on the ejection height.

1 Statistical Analysis on Influence Factors for Ejection Height

Under n test conditions，m ejection height data are measured in each condition，and they can be expressed as

Then，the observation vector under the test condition i is X(i)=(xi1，xi2，…xim)，i=1，2，…n.Analysis of influence factors on the ejection height is to consider the relation among different rows of the matrix actually.The test conditions close each other are regarded as a group and all conditions in the group have the same influence on the ejection height.In this way，the analysis for influence factors can be transformed as a cluster analysis problem in the multivariate statistics.The cluster analysis is named as the group analysis also.It is a multivariate statistical method used to study the sorting problem for samples or indexes.The group is a set of similar elements popularly［2］.As the congener elements have great similarity，in other words，the dis-tances between the congener elements should be short，and therefore the distance statistical value is selected as the sorting basis.

1.1 Distances Between Samples Under Different Conditions

For the observation vector in Eq.(1)，n samples can be regarded as n points in m dimension space.dijstands for the distance between X(i)and X(j).According to Ref.［2］，the requests can be expressed as

(1)dij≥0，for i，j;as dij=0?X(i)=X(j);

(2)dij=dji，for i，j;

(3)dij≤dik+dkj，for i，j，k(k stands for one point in n points in m-dimension space.)(the triangle infinitive).

For the quantificational variable，the distances used commonly are Minkowski distance，Canberra distance，Mahalanobis distance and skew space distance.Because the variance of the ejection height under different test conditions is small，Minkowski distance can be adopted as follows

where r is a given positive real number，taken as 1，2 or∞.If r=1，we can get the absolute value distance;if r=2，we can get Euclidean distance;if r=∞，we can get Chebyshev distance.

1.2 Hierarchical Cluster Method

After distances between different samples are obtained，the samples can be sorted according to the distance.The cluster methods are numerous，but the common method is the hierarchical cluster method［3］.Its primary idea can be described as follows.Firstly，suppose that each sample is a group，so the distances between samples are the distances between groups.Then，combine the two closest groups as a new group;and calculate the distances between the new group and other groups，and combine the two closest groups as a new group again.So the rest may be deduced analogically，as far as all of samples become a group.Finally，the number of group and the last group can be determined according to the demand and the given critical distance.The hierarchical cluster rule depends on the distance between samples and the distance between groups.Many definitions describe the distance between groups.Commonly used definitions［4］are following.In the following text，Dpqstands for the distance between the group Gpand the group Gq.

1)Minimum distance

2)Maximum distance

3)Center distance

According to the property of the hierarchical clustering method，the minimum distance has great concentration in space;the maximum distance has expansibility in space，and the number of samples including in a group is not considered;the center distance has great representativeness，but it does not use data of each sample sufficiently;the class average distance which uses the square distance between two samples included in two different groups as the distance is neither concentrated nor expansionary，and it is monotonic，so it is applied extensively and has a good cluster result.The definition of the group is a very difficult and fuzzy problem，and now it is not solved satisfactorily.According to the trait of the problem，we think that the hierarchical graph method is suitable.It can use the experience of analysis people and it is a good method.In the Ref.［2］，the sort rule based on the hierarchical graph is given.It is put forward by Bemirmen in 1972.First，the center distance between each group must be very large.Second，in groups，the number of elements in each group is not numerous.Third，the number of group should accord with the aim of practicality.Fourth，if different cluster methods are applied，the same group should be found in each cluster graph.

2 Analyses for Influence Factors on Ejection Height

2.1 Graph of Ejection Height

In order to analyze the influences of the environment conditions and launch angle on the ejection height，the test is carried out in 13 conditions.Each test condition includes normal temperature，high temperature，low temperature or service state and launching angle of 50°，60°，70°or 75°.In order to understand the test data intuitively，we can use the graph to describe relations of groups.One or two-dimension graph can be used to express one or two-dimension data easily.We can draw three-dimension graph also，but it is not easy.A series of methods to express data more than 3 dimensions appeared in current several decades.But there is not an acknowledged method.A harmonic curve diagram is adopted in this paper.It is a good multi-dimension data graph expression method in math and put forward by Andcews in 1972.It considers one point in a multi-dimension space as a curve in a twodimension plane.For multi-dimension data X=(x1，x2，…，xm)'，the corresponding curve can be expressed as

When t changes in an interval(－π，π)，its track is a curve.For different conditions，a set of harmonic curve diagrams can be obtained.These diagrams are helpful for cluster analysis.If the cluster statistical value is selected as the distance，the curves in the same group are screwed into a bundle，and the curves in different groups are screwed into different bundles.The graph is very intuitive.Fig.1 is the harmonic curve graph of the ejection height.It can be seen from the figure that the harmonic curves in different conditions have been not screwed into a bundle，in other words，the ejection height is influenced by environment factors.

Fig.1 Harmonic curve graph of ejection height of a cargo projectile

2.2 Cluster Analysis of Ejection Height

Firstly，we need to obtain the distance matrix Dhybetween ejection heights under different conditions.Then，we can obtain the distance Dhlbetween different groups by using hierarchical clustering method.Finally，the cluster graph can be drawn，as shown in Fig.2.

Fig.2 Cluster graph of ejection height

In Fig.2，N stands for“normal temperature”;H stands for“high temperature”;L stands for“l(fā)ow temperature”;D stands for“duty”.

It can be known from the cluster graph that the launch angle has a little influence on high temperature，low temperature and service performance.They can be sorted as a group.The ejection heights under normal temperature and different launch angles are distributed into other environment conditions and it does not accord with the rupture rule.

For a certain cargo projectile，the influence factor on the ejection height is local test condition.The meteorological factors，such as wind speed and direction，are important for the normal temperature different from high temperature，low temperature and service state.The meteorological data acquired in the test is shown in Fig.3 －6.It can be found that the wind speed and direction change largely in normal temperature test，while they are stable in other test conditions.It is shown that the great change of normal temperature data is led by the wind speed and direction.Therefore，the normal temperature data in cluster analysis is distributed in other environment conditions.That is，in a certain meteorological condition，the environment condition is the main factor to influence the ejection height.In the same environment condition，the launch angle has little influence on the ejection height.

Fig.3 Meteorological trend in normal temperature test

Fig.4 Meteorological trend in high temperature test

Fig.5 Meteorological trend in low temperature test

Fig.6 Meteorological trend in service temperature test

3 Conclusions

The influence factors on the cargo projectile’s ejection height are analyzed statistically according to test results in this paper.It can provide the basis for optimal design in different stages of the next product development.The factors which have no influence on the performance of the product can be given up.In this way，the test cost and period can be reduced.On the other hand，we can use other information obtained in the test to distinguish factors which have influence on the product performance effectively.The results may be helpful for the application of the product.

［1］Editorial committee of“Dictionary of Weapon Industry Science and Technology”.Dictionary of weapon industry science and technology—projectile［M］.Beijing:National Defence Industry Press，1991:3－4.(in Chinese)

［2］GAO Hui-xuan.Applied multivariate statistical analysis［M］.Beijing:Peking University Press，2007:221 －232.(in Chinese)

［3］CHEN Yong-liang，LI Xue-bin.Kernel-based hierarchical cluster analysis［J］.Journal of Jilin University:Earch Science Edition，2010，40(5):1211 －1216.(in Chinese)

［4］ZENG Feng，ZHANG Xiao-ning.Application of cluster analysis to preventive maintenance scheme design of pavement［J］.Journal of Harbin Institute of Technology，2009，16(4):581－586.(in Chinese)