CN112325713A - Analysis method for pneumatic nonlinear angular motion characteristics of double spinning bombs - Google Patents

Analysis method for pneumatic nonlinear angular motion characteristics of double spinning bombs Download PDF

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CN112325713A
CN112325713A CN202011173044.6A CN202011173044A CN112325713A CN 112325713 A CN112325713 A CN 112325713A CN 202011173044 A CN202011173044 A CN 202011173044A CN 112325713 A CN112325713 A CN 112325713A
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equation
double
spinning projectile
nonlinear
projectile
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CN112325713B (en
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沈元川
于剑桥
贾振岳
陈曦
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Beijing Institute of Technology BIT
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F42—AMMUNITION; BLASTING
    • F42B—EXPLOSIVE CHARGES, e.g. FOR BLASTING, FIREWORKS, AMMUNITION
    • F42B35/00—Testing or checking of ammunition

Abstract

The invention discloses a method for analyzing the characteristic of pneumatic nonlinear angular motion of a double spinning projectile, which comprises the following steps of: the method comprises the following steps: establishing the speed omega of the double spinning projectile relative to the attack angle alpha, the sideslip angle beta and the pitch anglezYaw rate ωyThe non-linear angular motion equation of (1); step two: determining a nonlinear complex attack angle equation of the spinning projectile according to the nonlinear angular motion equation established in the step one; step three: converting the stability analysis of the nonlinear complex attack angle equation solution into the analysis of the corresponding homogeneous equation solution; step four: determining an amplitude plane equation of the double spinning projectile under the action of the cubic static moment; step five: determining the necessary condition that the double spinning projectile generates stable conical motion under the action of cubic static moment; starting from pneumatic nonlinearity, the necessary condition that the double spinning projectile generates stable conical swing under the action of the cubic static moment is obtained, so that the reason that the generated attack angle is not attenuated and even the flight instability phenomenon is caused under the action of the nonlinear static moment is pointed out.

Description

Analysis method for pneumatic nonlinear angular motion characteristics of double spinning bombs
Technical Field
The invention relates to the technical field of analysis of angular motion characteristics of double spinning bullets, in particular to a method for analyzing the angular motion characteristics of pneumatic nonlinearity of the double spinning bullets.
Background
The rapid development of conventional projectile guidance technology is fueled by the tremendous demand in modern warfare for high-precision, low-cost, beyond-the-horizon, small collateral damage guided munitions. Aiming at the characteristic of high rotating speed of the rotation stabilization projectile, the duck rudder component is adopted to replace an original ammunition fuse to form a 'double-rotation' structure, and then the intelligent and flexible transformation of the rotation stabilization projectile is realized.
However, the double-spin projectile dynamics has the characteristics of strong coupling and strong nonlinearity, and meanwhile, the structural change and the introduction of a control system make a plurality of nonlinear motion phenomena of the projectile difficult to analyze and explain by using the theory of a linear system. The nonlinear characteristics severely restrict the application of the double-spin ammunition in the development of weaponry. Therefore, the method has important theoretical research value and engineering application significance for the research of the nonlinear dynamics analysis of the spinning projectile.
Disclosure of Invention
In view of the above, the invention provides a method for analyzing the pneumatic nonlinear lower angular motion characteristic of a double spinning projectile, and a necessary condition that the double spinning projectile generates stable conical oscillation under the action of cubic static moment is obtained by adopting an angular motion quasi-linear analysis method based on an averaging method.
The technical scheme of the invention is as follows: a method for analyzing the characteristic of the pneumatic nonlinear angular motion of a spinning projectile comprises the following steps:
the method comprises the following steps: establishing the speed omega of the double spinning projectile relative to the attack angle alpha, the sideslip angle beta and the pitch anglezYaw rate ωyThe non-linear angular motion equation of (1);
step two: determining a nonlinear complex attack angle equation of the spinning projectile according to the nonlinear angular motion equation established in the step one;
step three: converting the stability analysis of the nonlinear complex attack angle equation solution into the analysis of the corresponding homogeneous equation solution;
step four: determining an amplitude plane equation of the double spinning projectile under the action of the cubic static moment;
step five: determining the necessary condition of stable conical motion generated by the double spinning projectile under the action of the cubic static moment.
Preferably, in the first step:
selecting x ═ alpha beta omegaz ωy]TAccording to Newton's second law and momentum moment theorem, the equation of the nonlinear angular motion of the double spinning projectile is obtained as follows:
Figure RE-GDA0002838041360000021
in the formula (I), the compound is shown in the specification,
Figure RE-GDA0002838041360000022
Figure RE-GDA0002838041360000023
wherein ρ, S, L, g,
Figure RE-GDA00028380413600000210
m is air density, characteristic area of the double spinning projectile, characteristic length of the double spinning projectile, gravity acceleration of the double spinning projectile, pitch angle of the double spinning projectile, mass of the double spinning projectile, Jy＝Jz，JyAnd JzAre equatorial moments of inertia, JxIs the polar moment of inertia;
Figure RE-GDA0002838041360000024
and
Figure RE-GDA0002838041360000029
respectively representing the lift force, the static moment, the damping moment, the Magnus moment and the control moment generated by the attack angle.
Preferably, in the second step:
consider mΔNon-linearly induced conical motion, selecting
Figure RE-GDA00028380413600000211
The complex variables delta beta + i alpha and delta are defined by a complex analysis method in Murphy stability theoryy+iδzEliminating ω from formula (1)z、ωyAnd the derivative thereof, the nonlinear complex attack angle equation of the spinning projectile is obtained as follows:
Figure RE-GDA0002838041360000026
wherein H is cΔ-mΩDamping characteristics characterizing angular movement, M ═ MΔ+mΩcΔDetermines the frequency of angular movement, T ═ mω/P+cΔMiddle mωWhich is a magnus moment, affects the stability of the angular movement.
Preferably, in the third step:
m is expressed as M ═ M0+M2Δ2Δ ═ Δ |, the stability of the solution of equation (2) from the ordinary differential equation is determined by its corresponding solution of the homogeneous equation:
Figure RE-GDA0002838041360000027
preferably, in step four:
let the two-circle motion form of the solution of the formula (3) be:
Figure RE-GDA0002838041360000028
defining a damping factor lambdai：
Figure RE-GDA0002838041360000031
Substituting the formula (4) and the formula (5) into the formula (3) to obtain:
Figure RE-GDA0002838041360000032
on both sides of formula (6), is divided by
Figure RE-GDA0002838041360000033
And order
Figure RE-GDA0002838041360000034
Is amplified at
Figure RE-GDA0002838041360000035
Averaged over one period of (a):
Figure RE-GDA0002838041360000036
omitting its derivative
Figure RE-GDA0002838041360000037
While omitting a small amount lambda1(λ1+ H) separating the real and imaginary parts of equation (7) to obtain:
Figure RE-GDA0002838041360000038
Figure RE-GDA0002838041360000039
resolving the frequency from the formula (8)
Figure RE-GDA00028380413600000310
And
Figure RE-GDA00028380413600000311
the substitution into formula (9) gives:
Figure RE-GDA00028380413600000312
obtaining the following by the same method:
Figure RE-GDA00028380413600000313
the stability problem of the fourth-order system is reduced to the stability problem of the second-order system, and the equations (10) and (11) are combined to construct
Figure RE-GDA00028380413600000314
Figure RE-GDA00028380413600000315
Amplitude plane equation for coordinate plane:
Figure RE-GDA00028380413600000316
preferably, in the step five:
when in use
Figure RE-GDA00028380413600000317
When it is used, order
Figure RE-GDA00028380413600000318
Then, the equations (10) and (11) can be simplified to obtain:
Figure RE-GDA00028380413600000319
Figure RE-GDA0002838041360000041
here, λ1And λ2The expression is meaningful to satisfy:
Figure RE-GDA0002838041360000042
at K2＜＜K1Under the condition (2), the necessary condition for the double-spin projectile to form a conical motion is that
Figure RE-GDA00028380413600000413
With only one singular point on the axis
Figure RE-GDA0002838041360000043
The singularity should be a zero damping factor curve lambda10 and
Figure RE-GDA0002838041360000044
intersection of axes, let λ1Obtaining when the yield is 0:
Figure RE-GDA0002838041360000045
for normal flying positive spin double spin ammunition, there is P>0，H>0, then singular point
Figure RE-GDA0002838041360000046
The following constraints should be satisfied:
Figure RE-GDA0002838041360000047
to make coordinate changes
Figure RE-GDA0002838041360000048
Moving the origin of the coordinate system to an odd point, performing coordinate transformation by the equation (12), and substituting the coordinate transformation into the equations (13) and (14) to obtain a new amplitude plane equation:
Figure RE-GDA0002838041360000049
(18) equation Jacobian matrix at equilibrium point (0,0)
Figure RE-GDA00028380413600000410
Wherein b is1＝0，c1＝0，
Figure RE-GDA00028380413600000411
Figure RE-GDA00028380413600000412
The criterion of Lyapunov stability is known, when a1<0，d1<0, equilibrium point (0,0) is stable; the requirements for obtaining stable conic motion of double spin elasticity by combining the formulas (15) and (17) are as follows:
Figure RE-GDA0002838041360000051
all in one
Figure RE-GDA0002838041360000052
When deriving the
Figure RE-GDA0002838041360000053
The requirements for stable conical motion of the double spin projectile are as follows:
Figure RE-GDA0002838041360000054
wherein the content of the first and second substances,
Figure RE-GDA0002838041360000055
Figure RE-GDA0002838041360000056
has the advantages that:
(1) the invention starts from pneumatic nonlinearity, obtains the necessary condition that the bispin bomb generates stable conical swing under the action of cubic static moment, thereby indicating the reasons that the generated attack angle is not attenuated and even the flight instability phenomenon is caused under the action of nonlinear static moment, and having guiding significance for the structure and pneumatic design of the bispin bomb.
Detailed Description
The present invention will be described in detail below with reference to examples.
The embodiment provides a method for analyzing pneumatic nonlinear lower angular motion characteristics of a double-spinning projectile, and the method is based on an angular motion quasi-linear analysis method of an averaging method, so that the necessary condition that the double-spinning projectile generates stable conical swing under the action of cubic static moment is obtained.
The analysis method mainly comprises the following steps:
the method comprises the following steps: establishing the speed omega of the double spinning projectile relative to the attack angle alpha, the sideslip angle beta and the pitch anglezYaw rate ωyThe non-linear angular motion equation of (1);
to characterize angular motion, assume linear velocity V of the dual spinning projectile and rotational velocity ω of the rear bodyxSelecting x ═ alpha beta omega as slow variablez ωy]TIs the state quantity of the nonlinear angular motion of the double spinning projectile, and alpha and beta are relative to V and omega in the stable flying process of the double spinning projectilexThe fast variable is a small variable (when the attack angle and the sideslip angle are small, they can be approximated in the derivation process, that is, sin (α) ═ α, sin (β) ═ β, cos (α) ═ 1, cos (β) ═ 1), and the magnus force and the control surface control force are smaller than the aerodynamic force generated by the attack angle by two or more orders (i.e., the influence of the magnus force and the control surface control force on the angular motion of the double-spinning projectile can be ignored), so that the equation of the double-spinning projectile nonlinear angular motion can be obtained according to the second newton's law and the theorem of the moment of momentum as follows:
Figure RE-GDA0002838041360000061
in the formula (I), the compound is shown in the specification,
Figure RE-GDA0002838041360000062
Figure RE-GDA0002838041360000063
wherein ρ, S, L, g,
Figure RE-GDA00028380413600000610
m is air density, characteristic area of the double spinning projectile, characteristic length of the double spinning projectile, gravity acceleration of the double spinning projectile, pitch angle of the double spinning projectile and mass of the double spinning projectile respectively, JyAnd JzAre equatorial moments of inertia, J is due to the axial symmetry of the double spin projectiley＝Jz，JxIs the polar moment of inertia;
Figure RE-GDA0002838041360000064
and
Figure RE-GDA0002838041360000065
respectively representing a lift force, a static moment, a damping moment, a Magnus moment and a control moment generated by an attack angle;
step two: determining a nonlinear complex attack angle equation of the double spinning projectile;
for convenience of theoretical analysis, for the equation of the nonlinear angular motion of the double-spinning projectile described by the formula (21), the present embodiment mainly performs analysis on the phenomenon of the nonlinear motion of the double-spinning projectile from the pneumatic nonlinearity, and for the pneumatic nonlinearity, because the moment applied to the double-spinning projectile plays a dominant role in the angular motion, the cubic static moment applied to the double-spinning projectile is considered in a critical manner;
introducing horizontal ballistic assumptions, i.e. selection
Figure RE-GDA0002838041360000069
Defining complex variables by using a complex analysis method in Murphy stability theoryThe quantities Δ β + i α and δy+iδzWherein, deltazFor pitching rudder deflection angle, deltayFor yaw rudder angle, the ω is eliminated from the equation (21)z、ωyAnd the derivative thereof, the nonlinear complex attack angle equation of the spinning projectile is obtained as follows:
Figure RE-GDA0002838041360000066
in the formula (I), the compound is shown in the specification,
Figure RE-GDA0002838041360000067
and
Figure RE-GDA0002838041360000068
specifying the first and second derivatives of a complex-defining variable Δ, respectively, H ═ cΔ-mΩH characterizes the damping characteristic of angular motion, M ═ MΔ+mΩcΔM determines the frequency of the angular movement, T ═ Mω/P+cΔT is related to the Magnus moment (m)ωMagnus moments) that affect the stability of angular motion;
step three: converting the stability analysis of the nonlinear complex attack angle equation solution into the analysis of the corresponding homogeneous equation solution;
considering that the solution of the homogeneous equation is equivalent to the stability characterized by the solution of the nonlinear complex attack angle equation, and the solution of the homogeneous equation is easier to solve, meanwhile, the embodiment considers the influence of the cubic static moment on the angular motion, and M can be expressed as M ═ M0+M2Δ2Wherein M is0Is a linear term of moment, M2For a non-linear term of moment, Δ ═ Δ |, as can be derived from the knowledge of ordinary differential equations, the stability of the solution of equation (22) can be determined by its corresponding solution of the following homogeneous equation:
Figure RE-GDA0002838041360000071
step four: determining an amplitude plane equation of the double spinning projectile under the action of the cubic static moment;
since the amplitude reflects the magnitude of the angular motion, let the form of the two-circle motion of the solution of equation (23) be:
Figure RE-GDA0002838041360000072
wherein, K1And K2Is a die with two-circle motion,
Figure RE-GDA0002838041360000073
and
Figure RE-GDA0002838041360000074
the phase of the two-circle motion;
defining a damping factor lambdai：
Figure RE-GDA0002838041360000075
Figure RE-GDA0002838041360000076
Is KiA first derivative with respect to time;
substituting the formula (24) or (25) into the formula (23) to obtain:
Figure RE-GDA0002838041360000077
wherein the content of the first and second substances,
Figure RE-GDA0002838041360000078
and
Figure RE-GDA0002838041360000079
are each lambdai，λi，
Figure RE-GDA00028380413600000710
And
Figure RE-GDA00028380413600000711
a first derivative with respect to time;
Figure RE-GDA00028380413600000712
and
Figure RE-GDA00028380413600000713
is composed of
Figure RE-GDA00028380413600000714
And
Figure RE-GDA00028380413600000715
a second derivative with respect to time;
on both sides of formula (26), divided by
Figure RE-GDA00028380413600000716
And order
Figure RE-GDA00028380413600000717
Is amplified at
Figure RE-GDA00028380413600000718
Averaged over one period of (a):
Figure RE-GDA00028380413600000719
due to lambda1Slowly changing, omitting its derivatives
Figure RE-GDA00028380413600000720
While omitting a small amount lambda1(λ1+ H) separating the real and imaginary parts of equation (27) to obtain:
Figure RE-GDA00028380413600000721
Figure RE-GDA00028380413600000722
resolving the frequency from equation (28)
Figure RE-GDA00028380413600000723
And
Figure RE-GDA00028380413600000724
the substitution into the formula (29) gives:
Figure RE-GDA00028380413600000725
the same can get:
Figure RE-GDA0002838041360000081
the stability problem of the fourth-order system is reduced to that of the second-order system, and combined with the equations (30) and (31), the system can be constructed to
Figure RE-GDA0002838041360000082
Amplitude plane equation for coordinate plane:
Figure RE-GDA0002838041360000083
step five: determining the necessary condition that the double spinning projectile generates stable conical motion under the action of cubic static moment;
for the motion stability of the double spinning projectile, only the mode K of the two-circle motion is required1、K2If the double-spin projectile does not diverge, the motion is stable, and when the double-spin projectile makes two-circle motion, the mode K of two partial motions of the complex attack angle delta1、K2With K1＜＜K2Or K2＜＜K1Without setting K2＜＜K1(ii) a On the other hand, in the prior art, two angular movement frequencies
Figure RE-GDA0002838041360000084
And
Figure RE-GDA0002838041360000085
generally, the difference is large, and the present embodiment will be right
Figure RE-GDA0002838041360000086
And
Figure RE-GDA0002838041360000087
respectively discussing;
when in use
Figure RE-GDA0002838041360000088
When it is used, order
Figure RE-GDA0002838041360000089
Then, the equations (30) and (31) can be simplified to obtain:
Figure RE-GDA00028380413600000810
Figure RE-GDA00028380413600000811
here, λ1And λ2The expression is meaningful to satisfy:
Figure RE-GDA00028380413600000812
at K2＜＜K1Under the condition (2), the necessary condition for the double-spin projectile to form a conical motion is that
Figure RE-GDA00028380413600000813
With only one singular point on the axis
Figure RE-GDA00028380413600000814
KeFor the final convergence of the conic motion of the spinning projectile, the singularity should be the zero damping factor curve λ10 and
Figure RE-GDA00028380413600000815
intersection of axes, let λ10 can result in:
Figure RE-GDA00028380413600000816
for normal flying positive spin double spin ammunition, there is P>0，H>0, then singular point
Figure RE-GDA00028380413600000817
The following constraints should be satisfied:
Figure RE-GDA00028380413600000818
for ease of discussion, coordinate changes are made
Figure RE-GDA0002838041360000091
Moving the origin of the coordinate system to an odd point, performing coordinate transformation by the equation (12), and substituting the coordinate transformation into the equations (13) and (14) to obtain a new amplitude plane equation:
Figure RE-GDA0002838041360000092
(38) equation Jacobian matrix at equilibrium point (0,0)
Figure RE-GDA0002838041360000093
Wherein b is1＝0，c1＝0，
Figure RE-GDA0002838041360000094
Figure RE-GDA0002838041360000095
The criterion of Lyapunov stability is known, when a1<0，d1<0, equilibrium point (0,0) is stable; the requirements for obtaining stable conic motion of double spin can be obtained by combining the formulas (35) and (37):
Figure RE-GDA0002838041360000096
all in one
Figure RE-GDA0002838041360000097
When deriving the
Figure RE-GDA0002838041360000098
The requirements for stable conical motion of the double spin projectile are as follows:
Figure RE-GDA0002838041360000099
wherein the content of the first and second substances,
Figure RE-GDA00028380413600000910
Figure RE-GDA0002838041360000101
in summary, the equations (39) and (40) constitute the requirement for stabilizing the conical motion of the double spinning projectile under the action of the cubic static moment.
In summary, the above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (6)

1. A method for analyzing the pneumatic nonlinear angular motion characteristics of a spinning projectile is characterized by comprising the following steps:
the method comprises the following steps: establishing the speed omega of the double spinning projectile relative to the attack angle alpha, the sideslip angle beta and the pitch anglezYaw rate ωyThe non-linear angular motion equation of (1);
step two: determining a nonlinear complex attack angle equation of the spinning projectile according to the nonlinear angular motion equation established in the step one;
step three: converting the stability analysis of the nonlinear complex attack angle equation solution into the analysis of the corresponding homogeneous equation solution;
step four: determining an amplitude plane equation of the double spinning projectile under the action of the cubic static moment;
step five: determining the necessary condition of stable conical motion generated by the double spinning projectile under the action of the cubic static moment.
2. The method for analyzing the characteristic of the pneumatic nonlinear lower angular motion of the spinning projectile as set forth in claim 1, wherein in the first step:
selecting x ═ alpha beta omegaz ωy]TAccording to Newton's second law and momentum moment theorem, the equation of the nonlinear angular motion of the double spinning projectile is obtained as follows:
Figure FDA0002747894630000011
in the formula (I), the compound is shown in the specification,
Figure FDA0002747894630000012
Figure FDA0002747894630000013
wherein ρ, S, L, g, θ and m are respectively air density, characteristic area of the double-spinning projectile, characteristic length of the double-spinning projectile, gravitational acceleration of the double-spinning projectile, pitch angle of the double-spinning projectile, mass of the double-spinning projectile, and Jy＝Jz，JyAnd JzAre equatorial moments of inertia, JxIs the polar moment of inertia;
Figure FDA0002747894630000014
and
Figure FDA0002747894630000015
respectively representing the lift force, the static moment, the damping moment, the Magnus moment and the control moment generated by the attack angle.
3. The method for analyzing the characteristic of the pneumatic nonlinear lower angular motion of the spinning projectile as set forth in claim 2, wherein in the second step:
consider mΔSelecting theta as 0 degree, and defining complex variables delta as beta + i alpha and delta as delta by using a complex analysis method in Murphy stability theoryy+iδzEliminating ω from formula (1)z、ωyAnd the derivative thereof, the nonlinear complex attack angle equation of the spinning projectile is obtained as follows:
Figure FDA0002747894630000021
wherein H is cΔ-mΩDamping characteristics characterizing angular movement, M ═ MΔ+mΩcΔDetermines the frequency of angular movement, T ═ mω/P+cΔMiddle mωWhich is a magnus moment, affects the stability of the angular movement.
4. A method for analyzing the characteristics of the pneumatic nonlinear lower angular motion of a spinning projectile as claimed in claim 2 or 3, wherein the third step is:
m is expressed as M ═ M0+M2Δ2Δ ═ Δ |, the stability of the solution of equation (2) from the ordinary differential equation is determined by its corresponding solution of the homogeneous equation:
Figure FDA0002747894630000022
5. the method for analyzing the characteristics of the pneumatic nonlinear lower angular motion of the spinning projectile as recited in claim 4, wherein in the fourth step:
let the two-circle motion form of the solution of the formula (3) be:
Figure FDA0002747894630000023
defining a damping factor lambdai：
Figure FDA0002747894630000024
Substituting the formula (4) and the formula (5) into the formula (3) to obtain:
Figure FDA0002747894630000025
on both sides of formula (6), is divided by
Figure FDA0002747894630000026
And order
Figure FDA0002747894630000027
Is amplified at
Figure FDA0002747894630000028
Averaged over one period of (a):
Figure FDA0002747894630000029
omitting its derivative
Figure FDA00027478946300000210
While omitting a small amount lambda1(λ1+ H) separating the real and imaginary parts of equation (7) to obtain:
Figure FDA00027478946300000211
Figure FDA00027478946300000212
resolving the frequency from the formula (8)
Figure FDA00027478946300000213
And
Figure FDA00027478946300000214
the substitution into formula (9) gives:
Figure FDA0002747894630000031
obtaining the following by the same method:
Figure FDA0002747894630000032
the stability problem of the fourth-order system is reduced to the stability problem of the second-order system, and the equations (10) and (11) are combined to construct
Figure FDA0002747894630000033
Figure FDA0002747894630000034
Amplitude plane equation for coordinate plane:
Figure FDA0002747894630000035
6. the method for analyzing the characteristic of the pneumatic nonlinear lower angular motion of the spinning projectile as set forth in claim 5, wherein in the fifth step:
when in use
Figure FDA0002747894630000036
When it is used, order
Figure FDA0002747894630000037
Then, the equations (10) and (11) can be simplified to obtain:
Figure FDA0002747894630000038
Figure FDA0002747894630000039
here, λ1And λ2The expression is meaningful to satisfy:
Figure FDA00027478946300000310
at K2＜＜K1Under the condition (2), the necessary condition for the double-spin projectile to form a conical motion is that
Figure FDA00027478946300000313
With only one singular point on the axis
Figure FDA00027478946300000316
The singularity should be a zero damping factor curve lambda10 and
Figure FDA00027478946300000314
intersection of axes, let λ1Obtaining when the yield is 0:
Figure FDA00027478946300000311
for normal flying positive spin double spin ammunition, there is P>0，H>0, then singular point
Figure FDA00027478946300000315
The following constraints should be satisfied:
Figure FDA00027478946300000312
to make coordinate changes
Figure FDA00027478946300000411
Moving the origin of the coordinate system to an odd point, performing coordinate transformation by the equation (12), and substituting the coordinate transformation into the equations (13) and (14) to obtain a new amplitude plane equation:
Figure FDA0002747894630000041
(18) equation Jacobian matrix at equilibrium point (0,0)
Figure FDA0002747894630000042
Wherein b is1＝0，c1＝0，
Figure FDA0002747894630000043
Figure FDA0002747894630000044
The criterion of Lyapunov stability is known, when a1<0，d1<0, equilibrium point (0,0) is stable; the requirements for obtaining stable conic motion of double spin elasticity by combining the formulas (15) and (17) are as follows:
Figure FDA0002747894630000045
all in one
Figure FDA0002747894630000049
When deriving the
Figure FDA00027478946300000410
The requirements for stable conical motion of the double spin projectile are as follows:
Figure FDA0002747894630000046
wherein the content of the first and second substances,
Figure FDA0002747894630000047
Figure FDA0002747894630000048
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