CN103558009B - The subsection-linear method of supercavitating vehicle dynamical property analysis - Google Patents
The subsection-linear method of supercavitating vehicle dynamical property analysis Download PDFInfo
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Abstract
A subsection-linear method for supercavitating vehicle dynamical property analysis, it comprises step 1: set up supercavitating vehicle kinetic model; Step 2: sectional linear fitting is carried out to skidding forces function non-linear in supercavitating vehicle kinetic model, obtains linear skidding forces function; Step 3: setting supercavity kinetic parameters; Step 4: adopt piecewise linearity skidding forces F
pthe supercavity kinetic model of function obtains unique equilibrium point of supercavitating vehicle, and at equilibrium point place by system linearization process, obtain the Jacobi matrix of system, and equilibrium point place secular equation, the eigenwert of acquisition system, the equilibrium point of decision-making system is that unstable saddle is burnt.Method of the present invention adopts the piece-wise linearization of skidding forces function to simplify the kinetic model of supercavitating vehicle, the balance point position of model and stability condition is made to have succinct analytical expression, the dynamics of convenient analysis supercavitating vehicle.
Description
Technical field
The present invention relates to a kind of subsection-linear method of supercavitating vehicle dynamical property analysis, specifically, particularly relate to a kind of subsection-linear method of dynamical property analysis of supercavitating vehicle under water.
Background technology
Current, underwater navigation body technique worldwide gets the attention.Wherein, what get most of the attention is supercavitating vehicle technology under water.
Specifically, so-called " supercavitating vehicle " refers to, when sail body under water high speed operation time, due to Bernoulli effect, make the vaporizing liquid around sail body, thus produce the supercavity covering sail body most surfaces, by supercavity, sail body and water are produced and isolate, therefore, it is possible to reduce the resistance of sail body in water, substantially increase movement velocity and the distance to go of sail body.
Supercavitating vehicle under water high speed operation time, complicated non-linear skidding forces can be produced when the afterbody of sail body contacts with cavity wall, the appearance of non-linear skidding forces not only can increase the frictional resistance of sail body, also can cause vibration and impact to sail body, and then produce the non-linear phenomena of this complexity of chaos.Therefore, need to analyze this phenomenon, and process can be optimized to it by analysis result, thus obtain ideal navigation result.
In fact, for a Kind of Nonlinear Dynamical System, when systematic parameter changes within the specific limits, just there will be the physical phenomenon such as chaos and fork.Chaos and the nonlinear physics phenomenon of fork as a kind of complexity, obtain researcher in every field such as science, mathematics and engineer applied the decades in past to pay close attention to greatly, disclose about the Dynamic Modeling of concrete physical system, nonlinear physics phenomenon, the many aspects such as stability and Bifurcation achieve a large amount of achievements in research.
At present, both at home and abroad about the Research on Nonlinear Dynamics of supercavitating vehicle, the non-linear phenomena mainly caused for supercavitating vehicle open loop parameter and the research of sail body FEEDBACK CONTROL.But, for the Dynamical Characteristics of supercavitating vehicle closed-loop control, in the field of businessly still belong to blank.
Closed-loop control characteristic due to supercavitating vehicle is the important evidence of carrying out supercavitating vehicle Design of Feedback Controller, therefore needs to carry out Correct Analysis to its dynamics.
Summary of the invention
In view of the above problems, the object of the present invention is to provide a kind of subsection-linear method of the supercavitating vehicle dynamical property analysis can analyzed the dynamics of supercavitating vehicle closed-loop control, the dynamics of convenient analysis supercavitating vehicle.
In order to achieve the above object, the subsection-linear method of supercavitating vehicle dynamical property analysis of the present invention is such:
A subsection-linear method for supercavitating vehicle dynamical property analysis, it comprises the following steps:
Step 1: set up supercavitating vehicle kinetic model;
Suppose that motive power keeps balance in navigation process, sail body general speed V remains unchanged, and the kinetic model of supercavitating vehicle is as follows:
formula (1)
formula (2)
Wherein, in formula (1), described w is transverse velocity, described θ is the angle of pitch, described q is rate of pitch, described L is sail body length, described m is density ratio (ρ
m/ ρ), described F
planingfor skidding forces, described F
gravityfor the gravity of sail body centroid position, described z are upright position, described δ
ewith described δ
cbe respectively control inputs; In formula (2), described V is sail body general speed, described h ' is supercavitating vehicle tail end submergence, described R ' is cavity radius R
cbe supercavitating vehicle submergence angle with the normalized value of the difference of sail body radius R, described α;
Step 2: sectional linear fitting is carried out to skidding forces function non-linear in supercavitating vehicle kinetic model, obtains linear skidding forces function;
Described w
t0for being positioned at the positive w value of transition point;
Step 3: setting supercavity kinetic parameters;
Step 4: adopt piecewise linearity skidding forces F
pthe supercavity kinetic model of function obtains unique equilibrium point of supercavitating vehicle, and at equilibrium point place by the process of supercavitating vehicle system linearization, obtain the Jacobi matrix of described supercavitating vehicle system, and equilibrium point place secular equation, obtain the eigenwert of described supercavitating vehicle system, judge that the equilibrium point of described supercavitating vehicle system is burnt as unstable saddle, i.e. index 2 equilibrium point.
Preferably, in the formula (1) of described step 1, g is acceleration of gravity, and n is empennage efficiency; Each matrix of coefficients M
0, A
0, B
0with gravity F
gravitycan be expressed as respectively
Wherein, described R is sail body radius, described R
nfor cavitation device radius, described σ are cavitation number; Defined parameters C is
In formula, C
x0for lift coefficient.
Preferably, in the formula (2) of described step 1, the expression formula of described R ' is:
Described h ' is expressed as
Here,
f(w)＝2w+(w+w
t0)tanh[-k(w+w
t0)]
+(w-w
t0)tanh[k(w-w
t0)]
In formula, w
t0=(R
c-R) V/L is the positive w value being positioned at transition point, k be one for selecting to control the constant of approximate error, generally have k=300.
Described alpha expression formula is:
In formula, the cavity radius R at distance cavitation device distance L place
cand expansion and contraction
have
In formula,
Preferably, in described step 2,
Based on nonlinear function
Make c
1=2 (RV-R
cv+LR '
c)/(RR '), b
1=-c
1w
t0,
and b
2=F
planing(5w
t0)-5w
t0c
2, obtain piecewise linearity skidding forces F
p.
Preferably, the supercavity kinetic parameters in described step 3, the feedback control gain of supercavitating vehicle is respectively kz=15, k θ=30 and kq=0.3.
Preferably, in described step 4, the secular equation at described equilibrium point place is:
det(1λ-J
S)＝0
λ
1，2＝-21.11±j31.95，λ
3＝272.68±j345.1。
Beneficial effect of the present invention is: adopt the former non-linear skidding forces function of piecewise linearity skidding forces Function Fitting for supercavitating vehicle kinetic model under water.The piece-wise linearization of skidding forces function simplifies the kinetic model of supercavitating vehicle, makes the balance point position of model and stability condition have succinct analytical expression, the dynamics of convenient analysis supercavitating vehicle.
Accompanying drawing explanation
Fig. 1 a illustrates structure and the physical dimension of supercavitating vehicle;
Fig. 1 b illustrates the acting force of supercavitating vehicle;
Fig. 1 c illustrates the submergence of supercavitating vehicle;
Fig. 2 illustrates piecewise linearity skidding forces F
pwith non-linear skidding forces F
planingbetween relation curve;
Fig. 3 a illustrates and adopts non-linear skidding forces F
planingthe bifurcation graphs about state variable w that the kinetic model of function changes with cavitation number σ;
Fig. 3 b illustrates and adopts piecewise linearity skidding forces F
pthe bifurcation graphs about state variable w that the kinetic model of function changes with cavitation number σ;
Fig. 4 a-4d illustrates running orbit and the projection of chaotic attractor in each plane of sail body, and wherein Fig. 4 a represents q-θ-w plane, and Fig. 4 b represents z-θ-q plane, and Fig. 4 c represents w-θ plane, and Fig. 4 d represents θ-q plane;
Fig. 5 illustrates the Poincaré map corresponding to Fig. 4.
Embodiment
Incorporated by reference to consulting Fig. 1 a, 1b, 1c.As shown in Figure 1a, in figure, the sail body of length L is formed by two sections: the cylindrical section of rear end length 2/3L and radius R, and the conical section of front end length 1/3L for its structure of supercavitating vehicle and physical dimension.For this sail body model, center of gravity CG is 17/28L from the distance of head.Sail body has a cavitation device in head gear, is equipped with empennage in rear end.Cavitation device can see a radius R as
ndisk.
Act on advocating on sail body and will have lift F on cavitation device
cavitator, lift F on empennage
fins, afterbody and cavity wall the skidding forces F that produces of interphase interaction
planing, and the gravity F of sail body centroid position
gravity.
Supercavitating vehicle Dynamic Modeling adopts one of four states variable to describe the dynamics of supercavitating vehicle, is respectively upright position z, transverse velocity w, pitching angle theta and rate of pitch q.Definition transverse velocity w be positioned at cavitation device place and with sail body axes normal, line speed V and sail body axis being parallel before definition.
Supercavitating vehicle is provided with feedback controller, and its control inputs is respectively δ
eand δ
c, generally select δ
e=0, δ
c=k
zz-k
θθ-k
qq, k
z, k
θand k
qbe respectively the feedback gain of control variable z, θ and q.
The subsection-linear method of supercavitating vehicle dynamical property analysis of the present invention is as follows:
(1) supercavitating vehicle kinetic model is set up
Suppose that motive power keeps balance in navigation process, sail body general speed V remains unchanged, and the kinetic model of supercavitating vehicle is as follows:
formula (1)
Wherein, m is density ratio (ρ
m/ ρ), g is acceleration of gravity, and n is empennage efficiency; Each matrix of coefficients M
0, A
0, B
0with gravity F
gravitycan be expressed as respectively
Defined parameters C is
In formula, C
x0for lift coefficient.
formula (2)
In formula (2), the expression formula of R ' is:
Supercavitating vehicle tail end submergence parameter h ' represents, can be expressed as
Here,
f(w)＝2w+(w+w
t0)tanh[-k(w+w
t0)]
+(w-w
t0)tanh[k(w-w
t0)]
In formula, w
t0=(R
c-R) V/L is the positive w value being positioned at transition point, k be one for selecting to control the constant of approximate error, generally have k=300.
α is supercavitating vehicle submergence angle, and expression formula is:
In formula, the cavity radius R at distance cavitation device distance L place
cand expansion and contraction
have
In formula,
(2) sectional linear fitting is carried out to skidding forces function non-linear in supercavitating vehicle kinetic model, obtain linear skidding forces function.The system parameter values of supercavitating vehicle is listed in shown in table 1.Based on the system parameter values of table 1, adopt the non-linear skidding forces F that formula (1) is expressed
planing, then non-linear skidding forces F
planingand relation curve as shown in Figure 2 between transverse velocity w.
The parameter of table 1 supercavitating vehicle
Adopt a piecewise linearity skidding forces F
pfunction carrys out matching formula complex nonlinear skidding forces F
planingfunction:
Based on nonlinear function
Make c
1=2 (RV-R
cv+LR '
c)/(RR '), b
1=-c
1w
t0,
and b
2=F
planing(5w
t0)-5w
t0c
2, then piecewise linearity skidding forces F
pfunction can be expressed as
(3) set supercavity kinetic parameters, refer to Fig. 3 a, 3b.When the feedback control gain of supercavitating vehicle is respectively kz=15, k θ=30 and kq=0.3, namely when δ e=0, δ c=15z-30 θ-0.3q, adopt piecewise linearity skidding forces F
pthe bifurcation graphs that the kinetic model (i.e. system 4) of function changes with cavitation number σ with adopt non-linear skidding forces F
planingthe bifurcation graphs that the kinetic model (i.e. system 1) of function changes with cavitation number σ is analyzed, and the variation tendency of two kinds of bifurcation graphs is basically identical, illustrates thus and adopts piecewise linearity skidding forces F
pthe former non-linear skidding forces F of Function Fitting
planingthe rationality of function.Incidentally, piecewise linearity skidding forces F is adopted
pthe former non-linear skidding forces F of Function Fitting
planingfunction, just forms system 7.
(4) piecewise linearity skidding forces F is adopted
pthe supercavity kinetic model of function obtains unique equilibrium point of supercavitating vehicle, and at equilibrium point place by system linearization process, obtain the Jacobi matrix of system, and equilibrium point place secular equation, the eigenwert of acquisition system, the equilibrium point of decision-making system is that unstable saddle is burnt, i.e. index 2 equilibrium point.Utilize jacobi method to calculate its Lyapunov exponent, and draw the running orbit of sail body and corresponding Poincare is penetrated.From the Lyapunov exponent of system, phase rail figure and Poincaré map, system is a four dimensional chaos system, can generate chaotic attractor.
Specifically, when supercavity model selection table 1 parameter value, fixing σ=0.0313 and δ
e=0, and adopt piecewise linearity skidding forces F
pduring function, the kinetic model that can obtain supercavitating vehicle closed-loop control is
Piecewise linearity skidding forces F
pfunction can be reduced to
(1) stability of balance point
Make system 4
both
When σ=0.0313, get δ
c=k
zz-k
θθ-k
qq, δ
e=0, system 4 only has an equilibrium point in 1.38 < w < 1.82
S[1.3986,0,0.0190,(0.0190k
θ+0.0525)/k
z]
System 7 in equilibrium point S place's linearization, obtaining Jacobi matrix is
For canonical parameter k
z=15, k
θ=30 and k
q=0.3, obtain the equilibrium point of system:
S＝[1.3986,0,0.0190,0.0415]
Equilibrium point place is carried out linearization to system 4, obtains Jacobi matrix
The secular equation at equilibrium point place is
det(1λ-J
S)＝0
λ
1，2＝-21.11±j31.95，λ
3＝272.68±j345.1
Therefore, equilibrium point S is that unstable saddle is burnt, i.e. index 2 equilibrium point, meets the necessary condition forming chaotic attractor.
(2) chaotic attractor
Work as δ
c=15z – 30 θ – 0.3q, as shown in Figure 4, corresponding Poincaré map as shown in Figure 5 for the running orbit of sail body and the projection of chaotic attractor in each plane.Utilizing Jacobi method to calculate its Lyapunov index is L
1=13.7054, L
2=– 3.6816, L
3=– 29.0609 and L
4=– 46.3403.From the phase rail figure of system 7, Poincaré map and Lyapunov index, system 7 is a four dimensional chaos system, can generate chaotic attractor.
The subsection-linear method of the supercavitating vehicle dynamical property analysis of above the application of the invention, to the kinetic description of supercavitating vehicle, have employed the skidding forces function of a sectional linear fitting, construct the kinetic model of supercavitating vehicle closed-loop control, thus to obtain with feedback control gain be the four dimensional chaos system of variable element.Take feedback control gain as variable element, utilize conventional dynamic analysis instrument, have studied the dynamics of supercavitating vehicle closed-loop control.
Claims (6)
1. a subsection-linear method for supercavitating vehicle dynamical property analysis, it comprises the following steps:
Step 1: set up supercavitating vehicle kinetic model; The kinetic model of supercavitating vehicle is as follows:
formula (1)
formula (2)
Wherein, in formula (1), described w is transverse velocity, described θ is the angle of pitch, described q is rate of pitch, described L is sail body length, described m is density ratio (ρ
m/ ρ), described F
planingfor skidding forces, described F
gravityfor the gravity of sail body centroid position, described z are upright position, described δ
ewith described δ
cbe respectively control inputs; In formula (2), described V is sail body general speed, described h ' is supercavitating vehicle tail end submergence, described R ' is cavity radius R
cbe supercavitating vehicle submergence angle with the normalized value of the difference of sail body radius R, described α;
Step 2: sectional linear fitting is carried out to skidding forces function non-linear in supercavitating vehicle kinetic model, obtains linear skidding forces function;
Described w
t0for being positioned at the positive w value of transition point;
Step 3: setting supercavity kinetic parameters;
Step 4: adopt piecewise linearity skidding forces F
pthe supercavity kinetic model of function obtains unique equilibrium point of supercavitating vehicle, and at equilibrium point place by the process of supercavitating vehicle system linearization, obtain the Jacobi matrix of described supercavitating vehicle system, and equilibrium point place secular equation, obtain
The eigenwert of described supercavitating vehicle system, judges that the equilibrium point of described supercavitating vehicle system is burnt as unstable saddle, i.e. index 2 equilibrium point.
2. the subsection-linear method of supercavitating vehicle dynamical property analysis according to claim 1, is characterized in that: in the formula (1) of described step 1, g is acceleration of gravity, and n is empennage efficiency; Each matrix of coefficients M
0, A
0, B
0with gravity F
gravitycan be expressed as respectively
Wherein, described R is sail body radius, described R
nfor cavitation device radius, described σ are cavitation number; Defined parameters C is
In formula, C
x0for lift coefficient.
3. the subsection-linear method of supercavitating vehicle dynamical property analysis according to claim 1, is characterized in that: in the formula (2) of described step 1, the expression formula of described R ' is:
Described h ' is expressed as
Here,
f(w)＝2w+(w+w
t0)tanh[-k(w+w
t0)]
+(w-w
t0)tanh[k(w-w
t0)]
In formula, w
t0=(R
c-R) V/L is the positive w value being positioned at transition point, k be one for selecting to control the constant of approximate error,
Described alpha expression formula is:
In formula, the cavity radius R at distance cavitation device distance L place
cand expansion and contraction
have
In formula,
4. the subsection-linear method of supercavitating vehicle dynamical property analysis according to claim 1, is characterized in that: in described step 2,
Based on nonlinear function
Make c
1=2 (RV-R
cv+LR '
c)/(RR '), b
1=-c
1w
t0,
and b
2=F
planing(5w
t0)-5w
t0c
2, obtain piecewise linearity skidding forces F
p.
5. the subsection-linear method of supercavitating vehicle dynamical property analysis according to claim 1, it is characterized in that: the supercavity kinetic parameters in described step 3, the feedback control gain of supercavitating vehicle is respectively kz=15, k θ=30 and kq=0.3.
6. the subsection-linear method of supercavitating vehicle dynamical property analysis according to claim 1, is characterized in that: in described step 4, and the secular equation at described equilibrium point place is:
det(1λ-J
S)＝0
λ
1，2＝-21.11±j31.95，λ
3＝272.68±j345.1。
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| CN110411709B (en) * | 2019-08-27 | 2020-12-15 | 中国船舶科学研究中心（中国船舶重工集团公司第七0二研究所） | Dynamic sliding force measurement test device for tail of supercavitation navigation body |
| CN112149362A (en) * | 2020-07-08 | 2020-12-29 | 北京电子工程总体研究所 | Method for judging motion stability of supercavitation navigation body |
| CN114323552B (en) * | 2021-11-18 | 2022-10-21 | 厦门大学 | Method for judging stability of water entering and exiting from cross-medium navigation body |
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