CN105160108B - A kind of definite method of sounding rocket aerodynamic configuration design object function - Google Patents

A kind of definite method of sounding rocket aerodynamic configuration design object function Download PDF

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CN105160108B
CN105160108B CN201510574018.7A CN201510574018A CN105160108B CN 105160108 B CN105160108 B CN 105160108B CN 201510574018 A CN201510574018 A CN 201510574018A CN 105160108 B CN105160108 B CN 105160108B
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msub
msubsup
trajectory
rocket
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江振宇
彭科
张士峰
胡凡
向敏
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National University of Defense Technology
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Abstract

To overcome the shortcomings of that the existing object function of sounding rocket Design of Aerodynamic Configuration optimization process determines method in optimization efficiency and optimum results performance, the present invention provides a kind of definite method of sounding rocket aerodynamic configuration design object function.The present invention is with the minimum starting point of the loss of momentum caused by aerodynamic drag along trajectory, it is proposed that the sounding rocket aerodynamic configuration design object function comprised the steps of determines method：(1) rocket ascent stage initial trajectory data are provided；(2) initial trajectory data prediction, determines characteristic point of trajectory；(3) each characteristic point weighting coefficient is calculated；(4) it is used as sounding rocket aerodynamic configuration design object function so that characteristic point of trajectory weighted resistance coefficient is minimum.Efficiency invention significantly improves sounding rocket aerodynamic configuration design, the performance for improving optimum results, suitable for all kinds of sounding rockets, the configuration design of each design phase, and the object function that can be applied to other flight vehicle aerodynamic profile optimizations determines problem.

Description

A kind of definite method of sounding rocket aerodynamic configuration design object function
Technical field
The invention belongs to shape design for aerocraft to optimize field, and in particular to a kind of mesh of sounding rocket aerodynamic configuration design Scalar functions determine method.
Background technology
Sounding rocket is to carry out space exploration and the indispensable effective tool of scientific experimentation, is pushed away from U.S.'s jet in 1945 Since launching first piece of sounding rocket Wac Corporal into laboratory (JPL), the mankind have launched thousands of pieces of sounding rockets, in gas As detection, Space environment detection, microgravity experiment, Bioexperiment and space new technology verification etc. field achieve extensive application into Fruit.With the continuous progress of fast development with the space science research of near space vehicle, sounding rocket use pattern will Constantly expand, play and be more and more widely used in space exploration and scientific experimentation field.Rationally design optimization sounding rocket gas Dynamic shape is most important to improving overall performance, its core objective is to meet stable constraint, the condition of Aerodynamic Heating constraint Under reduce to greatest extent air drag, improve apex of the trajectory height, Design of Aerodynamic Configuration optimization problem object function it is reasonable really Surely it is the basic assurance for reaching this target, directly determines design optimization result performance, and optimization efficiency is had a significant impact. Sounding rocket Design of Aerodynamic Configuration optimization object function determines that method is mainly divided to two classes both at home and abroad at present：First, imitated with reference to trajectory Very, using the most high ballistic characteristic data of apex of the trajectory height as object function, second, choosing some characteristic point of trajectory weighted resistances Coefficient minimum is used as object function, and the profile optimization result that the former obtains can improve apex of the trajectory height to greatest extent, improve fire Arrow performance, but single object function calculating process need to be sequentially completed calculation of aerodynamic characteristics and be integrated with trajectory, take longer；The latter's mesh Scalar functions obtain and pertain only to calculation of aerodynamic characteristics, more convenient, and there are larger subjectivity and randomness for weighting coefficient selection, it is impossible to Apex of the trajectory height is improved to greatest extent.
The content of the invention
For overcome the existing object function of sounding rocket Design of Aerodynamic Configuration optimization process determine method optimization efficiency with it is excellent Change the deficiency in result performance, the present invention provides a kind of definite method of sounding rocket aerodynamic configuration design object function.This hair It is bright with the minimum starting point of the loss of momentum caused by aerodynamic drag along trajectory ascent stage, based on it is stringent derive determine characteristic point of trajectory with Weighting coefficient, sounding rocket aerodynamic configuration design object function is used as so that characteristic point of trajectory weighted resistance coefficient is minimum.
In order to solve the above technical problems, the technical solution adopted by the present invention is：
A kind of definite method of sounding rocket aerodynamic configuration design object function, specifically includes following steps：
The first step：Rocket ascent stage initial trajectory data are provided, rocket ascent stage initial trajectory data include time, height Degree, Mach number, the angle of attack, dynamic pressure parameter；
Second step：Rocket ascent stage initial trajectory data prediction, rejects the less Ballistic Number of ballistic flight later stage dynamic pressure According to determining characteristic point of trajectory；
The trajectory dynamic pressure q of sounding rocket is in the rocket ascent stage [0, ts] later stage trajectory dynamic pressure numerical value is relatively small, such as Fig. 1 institutes Show, the less inflight phase rocket loss of momentum of dynamic pressure numerical value is also smaller, in definite sounding rocket aerodynamic configuration design object function During can not consider.If moment tcMeet following formula：
In formula, C is to be approximately equal to 1 but the constant less than 1, generally can be taken as the constant between 0.99~0.999.
The present invention only considers [0, t in definite sounding rocket aerodynamic configuration design object functionc] period ballistic data, will The period is divided into n parts, must be segmented moment t0,t1,t2…,tn-2,tn-1,tn, moment (ti-1+tiThe corresponding trajectory point in)/2 is Characteristic point of trajectory, whereiniFor 1,2,3 ..., n.
3rd step：[0, t obtained by step 2c] trajectory time period, dynamic pressure parameter calculate the weighting of each characteristic point of trajectory Coefficient wi；
Each characteristic point of trajectory weighting coefficient calculates according to the following formula：
In formula,
Step 4：Take CxiFor the rocket resistance coefficient of each characteristic point of trajectory in step 3, obtaining object function is
In the present invention, the object function expression formula provided is：
C in formulaxiFor characteristic point of trajectory rocket resistance coefficient.For multistep rocket, different characteristic point of trajectory resistance coefficient Cxi Corresponding rocket configuration may be different, but resistance coefficient area of reference SrefIt should be noted that take same numerical value.
To be best understood from the present invention, the derivation of object function, that is, formula (3) is described as follows：
The basic goal of sounding rocket shape drag reduction design is to reduce momentum caused by ascent stage aerodynamic drag is integrated along trajectory Loss, the loss of momentum are represented by：
In formula, D is rocket aerodynamic drag；CxFor rocket resistance coefficient；Q is the trajectory dynamic pressure of sounding rocket；SrefFor resistance Coefficient area of reference.
According to the analysis of step 2, [tc,ts] relatively small, the integrated value in formula (4) of period trajectory dynamic pressure valuePhase It is rightAlso it is smaller, it can be neglected, obtain in processing procedure：
In formula,With PlossApproximately equal, according to second step, by [0, tc] period is divided into n parts, Moment t must be segmented0,t1,t2…,tn-2,tn-1,tn, moment (ti-1+tiThe corresponding trajectory point in)/2 is characteristic point of trajectory, formula (5) It can turn to：
C in formulaxiFor (ti-1+tiThe moment rocket resistance coefficient of)/2, as Δ t=ti-ti-1Level off to zero when, above formula about equal sign Take equal sign.
By formula (6) the right and left with divided by：
In formula：
The dynamic pressure course difference of each stage trajectory scheme of sounding rocket scheme is relatively small, i.e., for rocket initial trajectory With the trajectory after the completion of design, the w being calculated according to formula (8)iValue difference is different smaller, can be according to formula (7), with reference to initial trajectory number It is as follows according to the object function for calculating rocket configuration design optimization：
The beneficial effects of the invention are as follows：
(1) object function directly represent the loss of momentum caused by aerodynamic drag is integrated along the trajectory ascent stage, can be to greatest extent Apex of the trajectory height is improved, improves rocket performance；
(2) object function calculates and pertains only to pneumatic subject, easy to operate, beneficial to raising optimization efficiency；
(3) it is suitable for all kinds of sounding rockets, the configuration design of each design phase；
(4) object function that can be applied to other flight vehicle aerodynamic profile optimizations determines problem.
Brief description of the drawings
M- dynamic pressure curve when Fig. 1 is certain sounding rocket ascent stage；
Fig. 2 determines method flow diagram for sounding rocket aerodynamic configuration design object function proposed by the present invention；
Fig. 3 is the ballistic curve after the completion of pretreatment；
Fig. 4 is characteristic point of trajectory state parameter；
Fig. 5 is each characteristic point of trajectory respective weights coefficient；
Fig. 6 is the method for the present invention and existing methods optimization efficiency and optimum results performance comparison.
Embodiment
The sounding rocket aerodynamic configuration design object function that this hair proposes determines that method flow diagram is as shown in Figure 2.The present invention Below in conjunction with the accompanying drawings, the embodiment of the present invention is further described.
Step 1：Provide rocket ascent stage initial trajectory data
The rocket ascent stage initial trajectory data for including the parameters such as time, height, Mach number, the angle of attack, dynamic pressure are provided, with certain Exemplified by type single-stage is without control meteorological rocket, it is as shown in Figure 1 that its ascent stage dynamic pressure changes over time curve.
Step 2：Initial trajectory data prediction, determines characteristic point of trajectory
If moment tcMeet following formula：
In formula, C is to be approximately equal to 1 but the constant less than 1, generally can be taken as the constant between 0.99~0.999.Moment tcAfterwards Trajectory dynamic pressure numerical value is relatively small, and the corresponding rocket loss of momentum is also smaller, and the present invention is in definite rocket configuration design optimization target [0, t is only considered during functionc] period ballistic data.
According to initial trajectory data, C=0.99 is taken, t is calculated to obtain using formula (11)c=32.28s.In definite rocket configuration Only consider [0, t during design optimization object functionc] period ballistic data (as shown in figure 3, the trajectory whole process angle of attack be 0 °), during by this Section is divided into 10 parts, must be segmented moment t0,t1,t2…,t8,t9,t10, take moment (ti-1+tiThe corresponding trajectory point in)/2 is trajectory Characteristic point, it is as shown in Figure 4 to obtain 10 characteristic point of trajectory height, Mach number, angle of attack parameters.
Step 3：Each characteristic point weighting coefficient calculates
Calculate according to the following formula each characteristic point of trajectory weight coefficient is as shown in Figure 5.
Obtain object function：
C in formulaxiFor characteristic point of trajectory rocket resistance coefficient.
On the premise of design variable, constraints are consistent, sounding rocket profile optimization object function proposed by the present invention Determine the correspondings apex of the trajectory height of shape scheme that method and following two existing object functions determine that method obtains and optimize effect Rate is as shown in Figure 6.
Existing method one：Characteristic point of trajectory shown in Fig. 4 is taken, object function takes each characteristic point resistance coefficient average value.
Existing method two：With reference to Ballistic Simulation of Underwater, using the most high ballistic characteristic data of apex of the trajectory height as object function.
It will be appreciated from fig. 6 that the object function calculating process of the method for the present invention is with " having method one " equally, pertaining only to pneumatic spy Property calculate, it with optimization total time-consuming is only that " the 1/3 of existing method two ", optimization efficiency is higher that single object function, which calculates time-consuming,；This The apex of the trajectory that inventive method obtains highly is 91.981km, with " having method two " unanimously, than " the existing gained of method one " knot Fruit is high by 4.11%.
The present invention is determined with the minimum starting point of the loss of momentum caused by aerodynamic drag along trajectory ascent stage based on stringent derive Characteristic point of trajectory and weighting coefficient, sounding rocket aerodynamic configuration design target is used as so that characteristic point of trajectory weighted resistance coefficient is minimum Function, has optimization efficiency height, the remarkable advantage of optimum results function admirable, suitable for all kinds of sounding rockets, each design rank The configuration design of section, and the object function that can be applied to other flight vehicle aerodynamic profile optimizations determines problem.
In conclusion although the present invention is disclosed above with preferred embodiment, so it is not limited to the present invention, any Those of ordinary skill in the art, without departing from the spirit and scope of the present invention, when can make it is various change and retouch, therefore this hair Bright protection domain is subject to the scope defined depending on claims.

Claims (1)

  1. A kind of 1. definite method of sounding rocket aerodynamic configuration design object function, it is characterised in that comprise the following steps：
    The first step：Rocket ascent stage initial trajectory data are provided, rocket ascent stage initial trajectory data include time, height, horse Conspicuous number, the angle of attack, dynamic pressure parameter；
    Second step：Rocket ascent stage initial trajectory data prediction, rejects the less ballistic data of ballistic flight later stage dynamic pressure, really Determine characteristic point of trajectory；
    The trajectory dynamic pressure q of sounding rocket is in the rocket ascent stage [0, ts] later stage trajectory dynamic pressure numerical value is relatively small, dynamic pressure numerical value is smaller The inflight phase rocket loss of momentum it is also smaller, during definite sounding rocket aerodynamic configuration design object function can not examine Consider；If moment tcMeet following formula：
    <mrow> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>c</mi> </msub> </msubsup> <mi>q</mi> <mi>d</mi> <mi>t</mi> <mo>/</mo> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>s</mi> </msub> </msubsup> <mi>q</mi> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mi>C</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
    In formula, C takes the constant between 0.99~0.999；
    Moment tcTrajectory dynamic pressure numerical value afterwards is relatively small, and the corresponding rocket loss of momentum is also smaller, therefore in definite sounding rocket gas Only consider [0, t during dynamic profile optimization object functionc] period ballistic data；The period is divided into part, moment t must be segmented0,t1, t2…,tn-2,tn-1,tn, moment (ti-1+tiThe corresponding trajectory [0, t in)/2c] point as trajectory spy n signs point, wherein i is 1,2, 3…,n；
    3rd step：[0, t obtained by step 2c] trajectory time period, dynamic pressure parameter calculate each characteristic point of trajectory weighting coefficient wi；
    Each characteristic point of trajectory weighting coefficient calculates according to the following formula：
    <mrow> <msub> <mi>w</mi> <mi>i</mi> </msub> <mo>=</mo> <msubsup> <mo>&amp;Integral;</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>t</mi> <mi>i</mi> </msub> </msubsup> <mi>q</mi> <mi>d</mi> <mi>t</mi> <mo>/</mo> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>c</mi> </msub> </msubsup> <mi>q</mi> <mi>d</mi> <mi>t</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>
    In formula,
    Step 4：Take CxiFor the rocket resistance coefficient of each characteristic point of trajectory in step 3, obtaining object function is The derivation of wherein object function is as follows：
    The basic goal of sounding rocket shape drag reduction design is to reduce the loss of momentum caused by aerodynamic drag is integrated along trajectory, the momentum Loss is represented by：
    <mrow> <msub> <mi>P</mi> <mrow> <mi>l</mi> <mi>o</mi> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>s</mi> </msub> </msubsup> <mi>D</mi> <mo>&amp;CenterDot;</mo> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>s</mi> </msub> </msubsup> <msub> <mi>C</mi> <mi>x</mi> </msub> <msub> <mi>qS</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msub> <mi>S</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>s</mi> </msub> </msubsup> <msub> <mi>C</mi> <mi>x</mi> </msub> <mi>q</mi> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msub> <mi>S</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mrow> <mo>(</mo> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>c</mi> </msub> </msubsup> <mrow> <msub> <mi>C</mi> <mi>x</mi> </msub> <mi>q</mi> <mi>d</mi> <mi>t</mi> </mrow> <mo>+</mo> <msubsup> <mo>&amp;Integral;</mo> <msub> <mi>t</mi> <mi>c</mi> </msub> <msub> <mi>t</mi> <mi>s</mi> </msub> </msubsup> <mrow> <msub> <mi>C</mi> <mi>x</mi> </msub> <mi>q</mi> <mi>d</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>
    In formula：D is rocket aerodynamic drag；CxFor rocket resistance coefficient；Q is the trajectory dynamic pressure of sounding rocket；SrefFor resistance coefficient Area of reference；
    According to step 2, [tc,ts] relatively small, the integrated value in formula (3) of period trajectory dynamic pressure valueRelativelyAlso it is smaller, it can be neglected, obtain in processing procedure：
    <mrow> <msub> <mi>P</mi> <mrow> <mi>l</mi> <mi>o</mi> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>s</mi> </msub> </msubsup> <msub> <mi>C</mi> <mi>x</mi> </msub> <mi>q</mi> <mi>d</mi> <mi>t</mi> <mo>&amp;ap;</mo> <msub> <mi>S</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>c</mi> </msub> </msubsup> <msub> <mi>C</mi> <mi>x</mi> </msub> <mi>q</mi> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mrow> <mi>l</mi> <mi>o</mi> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>
    In formula,With PlossApproximately equal, according to second step, by [0, tc] period is divided into n parts, it must be segmented Moment t0,t1,t2…,tn-2,tn-1,tn, moment (ti-1+tiThe corresponding trajectory point in)/2 is characteristic point of trajectory, and formula (4) can turn to：
    <mrow> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mrow> <mi>l</mi> <mi>o</mi> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>c</mi> </msub> </msubsup> <msub> <mi>C</mi> <mi>x</mi> </msub> <mi>q</mi> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msub> <mi>S</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msubsup> <mo>&amp;Integral;</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>t</mi> <mi>i</mi> </msub> </msubsup> <msub> <mi>C</mi> <mi>x</mi> </msub> <mi>q</mi> <mi>d</mi> <mi>t</mi> <mo>&amp;ap;</mo> <msub> <mi>S</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>C</mi> <mrow> <mi>x</mi> <mi>i</mi> </mrow> </msub> <msubsup> <mo>&amp;Integral;</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>t</mi> <mi>i</mi> </msub> </msubsup> <mi>q</mi> <mi>d</mi> <mi>t</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
    C in formulaxiFor (ti-1+tiThe moment rocket resistance coefficient of)/2, as Δ t=ti-ti-1Level off to zero when, above formula about equal sign takes Number；
    By formula (5) the right and left with divided by：
    <mrow> <mfrac> <msub> <mover> <mi>P</mi> <mo>~</mo> </mover> <mrow> <mi>l</mi> <mi>o</mi> <mi>s</mi> <mi>s</mi> </mrow> </msub> <mrow> <msub> <mi>S</mi> <mrow> <mi>r</mi> <mi>e</mi> <mi>f</mi> </mrow> </msub> <mrow> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>c</mi> </msub> </msubsup> <mrow> <mi>q</mi> <mi>d</mi> <mi>t</mi> </mrow> </mrow> </mrow> </mfrac> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>C</mi> <mrow> <mi>x</mi> <mi>i</mi> </mrow> </msub> <mfrac> <mrow> <msubsup> <mo>&amp;Integral;</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>t</mi> <mi>i</mi> </msub> </msubsup> <mi>q</mi> <mi>d</mi> <mi>t</mi> </mrow> <mrow> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>c</mi> </msub> </msubsup> <mi>q</mi> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>w</mi> <mi>i</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>x</mi> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    In formula：
    <mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>w</mi> <mi>i</mi> </msub> <mo>=</mo> <mfrac> <mrow> <msubsup> <mo>&amp;Integral;</mo> <msub> <mi>t</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>t</mi> <mi>i</mi> </msub> </msubsup> <mi>q</mi> <mi>d</mi> <mi>t</mi> </mrow> <mrow> <msubsup> <mo>&amp;Integral;</mo> <mn>0</mn> <msub> <mi>t</mi> <mi>c</mi> </msub> </msubsup> <mi>q</mi> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>w</mi> <mi>i</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>
    The dynamic pressure course difference of each stage trajectory scheme of sounding rocket scheme is relatively small, i.e., for rocket initial trajectory with setting Trajectory after the completion of meter, the w being calculated according to formula (7)iValue difference is different smaller, can be according to formula (6), with reference to initial trajectory data meter The object function of rocket configuration design optimization is calculated, it is as follows：
    <mrow> <mi>min</mi> <mi> </mi> <mi>f</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>w</mi> <mi>i</mi> </msub> <msub> <mi>C</mi> <mrow> <mi>x</mi> <mi>i</mi> </mrow> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> <mo>.</mo> </mrow>
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