CN103558009A - Piecewise linear method for analyzing supercavitation navigation body kinetic characteristics - Google Patents

Abstract

A piecewise linear method for analyzing supercavitation navigation body kinetic characteristics includes the steps that step1, a supercavitation navigation body kinetic model is built; step2, piecewise linear fitting is performed on a non-linear sliding force function in the supercavitation navigation body kinetic model to obtain a linear sliding force function; step3, parameters of the supercavitation kinetic model are set; step4, the supercavitation kinetic model with the piecewise linear sliding force Fp function is adopted for obtaining a sole balance point of a supercavitation navigation body, linearization is performed on a system at the balance point to obtain a jacobian matrix of the system and a characteristic equation at the balance point, characteristic values of the system are obtained, and the balance point of the system is judged to be an unstable saddle focus. By the adoption of the piecewise linearization of the sliding force function, the supercavitation navigation body kinetic model is simplified, so that the balance point position and stability conditions of the model have concise analytical expressions, and the supercavitation navigation body kinetic characteristics are analyzed more conveniently.

Description

The subsection-linear method of supercavitating vehicle dynamical property analysis

Technical field

The present invention relates to a kind of subsection-linear method of supercavitating vehicle dynamical property analysis, particularly, relate in particular 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.What wherein, get most of the attention is supercavitating vehicle technology under water.

Particularly, so-called " supercavitating vehicle " refers to, when sail body navigates by water at a high speed under water, due to Bernoulli effect, make sail body vaporizing liquid around, thereby produce the supercavity that covers sail body most surfaces, by supercavity, sail body and water generates are isolated, therefore can reduce the resistance of sail body in water, greatly improve movement velocity and the distance to go of sail body.

When supercavitating vehicle navigates by water at a high speed under water, the afterbody of sail body can produce the complicated non-linear power of sliding while contacting with cavity wall, the appearance of the non-linear power of sliding not only can increase the frictional resistance of sail body, also can cause vibration and impact to sail body, and then produce this complicated non-linear phenomena of chaos.Therefore, need to analyze this phenomenon, and can to it, be optimized processing by analysis result, thereby obtain comparatively desirable 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 phenomenons such as chaos and fork.Chaos and fork are as a kind of nonlinear physics phenomenon of complexity, in the decades in past, in every field such as science, mathematics and engineering application, obtain researcher and paid close attention to greatly, about the many aspects such as Dynamic Modeling, the announcement of nonlinear physics phenomenon, stability and Bifurcation of concrete physical system, obtained a large amount of achievements in research.

At present, the domestic and international Research on Nonlinear Dynamics of relevant supercavitating vehicle, is mainly the non-linear phenomena that causes for supercavitating vehicle open loop parameter and the research of sail body FEEDBACK CONTROL.Yet, for the Dynamical Characteristics of supercavitating vehicle closed-loop control, the blank that still belongs in the field of business.

Because the closed-loop control characteristic of supercavitating vehicle is the important evidence of carrying out supercavitating vehicle Design of Feedback Controller, therefore need 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 supercavitating vehicle complex nonlinear to slide the subsection-linear method of force function, the present invention also aims to simplify supercavitating vehicle kinetic model, 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:

$\begin{array}{c}{M}_0\left[\begin{array}{c}\stackrel{.}{w}\\ \stackrel{.}{q}\end{array}\right]=A_0\left[\begin{array}{c}w\\ q\end{array}\right]+{\mathrm{<figure-callout\; id="0"\; label="B"\; filenames="HDA0000407510310000014.png"\; state="\{\{state\}\}">B</figure-callout>}}_0\left[\begin{array}{c}{\mathrm{\&delta;}}_e\\ {\mathrm{\&delta;}}_c\end{array}\right]+F_{\mathrm{gravity}}+\frac{1}{\mathrm{mL}}\left[\begin{array}{c}1\\ L\end{array}\right]F_{\mathrm{planing}}\\ \stackrel{.}{\mathrm{\&theta;}}=q\\ \stackrel{.}{z}\stackrel{}{=w-\mathrm{V\&theta;}}\end{array}$ Formula (1)

$F_{\mathrm{pianing}}=-V^2[1-{\left(\frac{R^{\&prime;}}{h+R^{\&prime;}}\right)}^2]\left(\frac{1+h^{\&prime;}}{1+2h^{\&prime;}}\right)\mathrm{\&alpha;}$ Formula (2)

Step 2: the non-linear force function that slides in supercavitating vehicle kinetic model is carried out to sectional linear fitting, obtain linearity and slide force function;

$F_p=\left\{\begin{array}{c}{c}_{2}w+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA0000407510310000014.png,HDA0000407510310000021.png"\; state="\{\{state\}\}">b</figure-callout>}}_2,w>-(b_1-b_2)/(c_1-c_2)\\ c_{1}w+{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000407510310000011.png,HDA0000407510310000014.png"\; state="\{\{state\}\}">b</figure-callout>}}_1,{\mathrm{<figure-callout\; id="0"\; label="w\; t"\; filenames="HDA0000407510310000014.png"\; state="\{\{state\}\}">w</figure-callout>}}_t<w\&le;-(b_1-b_2)/(c_1-c_2)\\ 0,-{\mathrm{<figure-callout\; id="0"\; label="w\; t"\; filenames="HDA0000407510310000014.png"\; state="\{\{state\}\}">w</figure-callout>}}_t\&le;w\&le;w_{t0}\\ c_{1}w-{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000407510310000011.png,HDA0000407510310000014.png"\; state="\{\{state\}\}">b</figure-callout>}}_1,(b_1-b_2)/(c_1-c_2)<w\&le;-w_{t0}\\ c_{2}w-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA0000407510310000014.png,HDA0000407510310000021.png"\; state="\{\{state\}\}">b</figure-callout>}}_2,w\&le;(b_1-b_2)/(c_1-c_2)\end{array}\right.$

Step 3: set supercavity kinetic parameters;

Step 4: adopt piecewise linearity to slide power F _pthe supercavity kinetic model of function obtains unique equilibrium point of supercavitating vehicle, and at equilibrium point place, system linearization is being processed, 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 points.

As preferred implementation, in the formula of described step 1 (1), m is density ratio (ρ m/ ρ), and 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

${\mathrm{<figure-callout\; id="0"\; label="M"\; filenames="HDA0000407510310000014.png"\; state="\{\{state\}\}">M</figure-callout>}}_0=\left[\begin{array}{cc}\frac{7}{9}& \frac{17L}{36}\\ \frac{17L}{36}& \frac{11}{60}{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA0000407510310000014.png,HDA0000407510310000021.png"\; state="\{\{state\}\}">R</figure-callout>}}^2+\frac{133}{405}{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA0000407510310000014.png,HDA0000407510310000021.png"\; state="\{\{state\}\}">L</figure-callout>}}^2\end{array}\right]$

$A_0=\mathrm{CV}\left[\begin{array}{cc}-\frac{(1+n)}{\mathrm{mL}}& -\frac{n}{m}\\ -\frac{n}{m}& -\frac{\mathrm{nL}}{m}\end{array}\right]+V\left[\begin{array}{cc}0& \frac{7}{9}\\ 0& \frac{17L}{36}\end{array}\right]$

$F_{\mathrm{gravity}}=\left[\begin{array}{c}\frac{7}{9}\\ \frac{17L}{36}\end{array}\right]g$

Defined parameters C is

$C=0.5C_{x0}(1+\mathrm{\&sigma;}){\left(\frac{R_n}{R}\right)}^2$

In formula, C _x0for lift coefficient.

As preferred implementation, in the formula of described step 1 (2), the expression formula of R ' is:

$R^{\&prime;}=\frac{R_c-R}{R}$

Supercavitating vehicle tail end submergence represents with parameter h ', can be expressed as

$h^{\&prime;}=\tanh \left(\mathrm{kw}\right)\frac{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA0000407510310000014.png,HDA0000407510310000021.png"\; state="\{\{state\}\}">L</figure-callout>}}{2\mathrm{RV}}f\left(w\right)$

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 that is 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:

$\mathrm{\&alpha;}=\frac{w}{V}-\tanh \left(\mathrm{kw}\right)\frac{{\stackrel{.}{R}}_c}{V}$

In formula, apart from the cavity radius R at cavitation device distance L place _cand expansion and contraction

have

$R_c=R_n\sqrt{0.82\frac{1+\mathrm{\&sigma;}}{\mathrm{\&sigma;}}}{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="HDA0000407510310000014.png,HDA0000407510310000021.png"\; state="\{\{state\}\}">K</figure-callout>}}_2$

${\stackrel{.}{R}}_c=\frac{-\frac{20}{17}{\left(0.82\frac{1+\mathrm{\&sigma;}}{\mathrm{\&sigma;}}\right)}^{0.5}V(1-\frac{4.5\mathrm{\&sigma;}}{1+\mathrm{\&sigma;}}){\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="HDA0000407510310000011.png,HDA0000407510310000014.png"\; state="\{\{state\}\}">K</figure-callout>}}_1^{23/17}}{K_2(\frac{1.92}{\mathrm{\&sigma;}}-3)}$

In formula,

${\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="HDA0000407510310000011.png,HDA0000407510310000014.png"\; state="\{\{state\}\}">K</figure-callout>}}_1=\frac{L}{R_n(\frac{1.92}{\mathrm{\&sigma;}}-3)}-1$

${\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="HDA0000407510310000014.png,HDA0000407510310000021.png"\; state="\{\{state\}\}">K</figure-callout>}}_2=\sqrt{1-(1-\frac{4.5\mathrm{\&sigma;}}{1+\mathrm{\&sigma;}}){\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="HDA0000407510310000011.png,HDA0000407510310000014.png"\; state="\{\{state\}\}">K</figure-callout>}}_1^{40/17}}.$

As preferred implementation, in described step 2,

Based on formula (2) nonlinear function $F_{\mathrm{pianing}}=-V^2[1-{\left(\frac{R^{\&prime;}}{h+R^{\&prime;}}\right)}^2]\left(\frac{1+h^{\&prime;}}{1+2h^{\&prime;}}\right)\mathrm{\&alpha;},$ Order

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 and slide power F _p.

As preferred implementation, the supercavity kinetic parameters in described step 3, the feedback control gain of supercavitating vehicle is respectively k _z=15, k _θ=30 and k _q=0.3.

As preferred implementation, 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: for supercavitating vehicle kinetic model under water, adopt piecewise linearity to slide the former non-linear force function that slides of force function matching.The kinetic model of supercavitating vehicle has been simplified in the piece-wise linearization of sliding force function, makes the equilibrium 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 and slides power F _pwith the non-linear power of sliding F _planingbetween relation curve;

Fig. 3 a illustrates and adopts the non-linear power of sliding 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 to slide power F _pthe bifurcation graphs about state variable w that the kinetic model of function changes with cavitation number σ;

The running orbit that Fig. 4 a-4d illustrates sail body is the projection of chaotic attractor in each plane, 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 corresponding Poincare mapping with 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 forms by two sections for its structure of supercavitating vehicle and physical dimension: the cylindrical section of rear end length 2/3L and radius R, and the conical section of front end length 1/3L.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, in rear end, is equipped with empennage.Cavitation device can be seen a radius R as _ndisk.

Act on advocating on sail body and will have the lift F on cavitation device _cavitator, the lift F on empennage _fins, the power that the slides F that interacts and produce between afterbody and cavity wall _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 is positioned at cavitation device place and vertical with sail body axis, and before definition, line speed V is parallel with sail body axis.

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) 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:

$\begin{array}{c}{M}_0\left[\begin{array}{c}\stackrel{.}{w}\\ \stackrel{.}{q}\end{array}\right]=A_0\left[\begin{array}{c}w\\ q\end{array}\right]+{\mathrm{<figure-callout\; id="0"\; label="B"\; filenames="HDA0000407510310000014.png"\; state="\{\{state\}\}">B</figure-callout>}}_0\left[\begin{array}{c}{\mathrm{\&delta;}}_e\\ {\mathrm{\&delta;}}_c\end{array}\right]+F_{\mathrm{gravity}}+\frac{1}{\mathrm{mL}}\left[\begin{array}{c}1\\ L\end{array}\right]F_{\mathrm{planing}}\\ \stackrel{.}{\mathrm{\&theta;}}=q\\ \stackrel{.}{z}\stackrel{}{=w-\mathrm{V\&theta;}}\end{array}$ Formula (1)

Wherein, m is density ratio (ρ m/ ρ), and 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

${\mathrm{<figure-callout\; id="0"\; label="M"\; filenames="HDA0000407510310000014.png"\; state="\{\{state\}\}">M</figure-callout>}}_0=\left[\begin{array}{cc}\frac{7}{9}& \frac{17L}{36}\\ \frac{17L}{36}& \frac{11}{60}{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA0000407510310000014.png,HDA0000407510310000021.png"\; state="\{\{state\}\}">R</figure-callout>}}^2+\frac{133}{405}{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA0000407510310000014.png,HDA0000407510310000021.png"\; state="\{\{state\}\}">L</figure-callout>}}^2\end{array}\right]$

$A_0=\mathrm{CV}\left[\begin{array}{cc}-\frac{(1+n)}{\mathrm{mL}}& -\frac{n}{m}\\ -\frac{n}{m}& -\frac{\mathrm{nL}}{m}\end{array}\right]+\mathrm{<figure-callout\; id="0"\; label="V"\; filenames="HDA0000407510310000014.png"\; state="\{\{state\}\}">V</figure-callout>}\left[\begin{array}{cc}0& \frac{7}{9}\\ 0& \frac{17L}{36}\end{array}\right]$

${\mathrm{<figure-callout\; id="0"\; label="B"\; filenames="HDA0000407510310000014.png"\; state="\{\{state\}\}">B</figure-callout>}}_0={\mathrm{CV}}^2\left[\begin{array}{cc}-\frac{n}{\mathrm{mL}}& -\frac{n}{\mathrm{mL}}\\ -\frac{\mathrm{<figure-callout\; id="0"\; label="n\; m"\; filenames="HDA0000407510310000014.png"\; state="\{\{state\}\}">n</figure-callout>}}{m}& 0\end{array}\right]$

$F_{\mathrm{gravity}}=\left[\begin{array}{c}\frac{7}{9}\\ \frac{17L}{36}\end{array}\right]g$

Defined parameters C is

$C=0.5C_{x0}(1+\mathrm{\&sigma;}){\left(\frac{R_n}{R}\right)}^2$

In formula, C _x0for lift coefficient.

$F_{\mathrm{pianing}}=-V^2[1-{\left(\frac{R^{\&prime;}}{h+R^{\&prime;}}\right)}^2]\left(\frac{1+h^{\&prime;}}{1+2h^{\&prime;}}\right)\mathrm{\&alpha;}$ In formula (2) formula (2), the expression formula of R ' is:

$R^{\&prime;}=\frac{R_c-R}{R}$

Supercavitating vehicle tail end submergence represents with parameter h ', can be expressed as

$h^{\&prime;}=\tanh \left(\mathrm{kw}\right)\frac{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA0000407510310000014.png,HDA0000407510310000021.png"\; state="\{\{state\}\}">L</figure-callout>}}{2\mathrm{RV}}f\left(w\right)$

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 that is 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:

$\mathrm{\&alpha;}=\frac{w}{V}-\tanh \left(\mathrm{kw}\right)\frac{{\stackrel{.}{R}}_c}{V}$

In formula, apart from the cavity radius R at cavitation device distance L place _cand expansion and contraction

have

$R_c=R_n\sqrt{0.82\frac{1+\mathrm{\&sigma;}}{\mathrm{\&sigma;}}}{\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="HDA0000407510310000014.png,HDA0000407510310000021.png"\; state="\{\{state\}\}">K</figure-callout>}}_2$

${\stackrel{.}{R}}_c=\frac{-\frac{20}{17}{\left(0.82\frac{1+\mathrm{\&sigma;}}{\mathrm{\&sigma;}}\right)}^{0.5}V(1-\frac{4.5\mathrm{\&sigma;}}{1+\mathrm{\&sigma;}}){\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="HDA0000407510310000011.png,HDA0000407510310000014.png"\; state="\{\{state\}\}">K</figure-callout>}}_1^{23/17}}{K_2(\frac{1.92}{\mathrm{\&sigma;}}-3)}$

In formula,

${\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="HDA0000407510310000011.png,HDA0000407510310000014.png"\; state="\{\{state\}\}">K</figure-callout>}}_1=\frac{L}{R_n(\frac{1.92}{\mathrm{\&sigma;}}-3)}-1$

${\mathrm{<figure-callout\; id="2"\; label="K"\; filenames="HDA0000407510310000014.png,HDA0000407510310000021.png"\; state="\{\{state\}\}">K</figure-callout>}}_2=\sqrt{1-(1-\frac{4.5\mathrm{\&sigma;}}{1+\mathrm{\&sigma;}}){\mathrm{<figure-callout\; id="1"\; label="K"\; filenames="HDA0000407510310000011.png,HDA0000407510310000014.png"\; state="\{\{state\}\}">K</figure-callout>}}_1^{40/17}}$

(2) the non-linear force function that slides in supercavitating vehicle kinetic model is carried out to sectional linear fitting, obtain linearity and slide force function.Adopt a piecewise linearity to slide power F _pfunction comes matching formula (2) complex nonlinear to slide power F _planingfunction:

Based on formula (2) nonlinear function $F_{\mathrm{pianing}}=-V^2[1-{\left(\frac{R^{\&prime;}}{h+R^{\&prime;}}\right)}^2]\left(\frac{1+h^{\&prime;}}{1+2h^{\&prime;}}\right)\mathrm{\&alpha;},$ 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, piecewise linearity slides power F _pfunction can be expressed as

$F_p=\left\{\begin{array}{c}{c}_{2}w+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA0000407510310000014.png,HDA0000407510310000021.png"\; state="\{\{state\}\}">b</figure-callout>}}_2,w>-(b_1-b_2)/(c_1-c_2)\\ c_{1}w+{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000407510310000011.png,HDA0000407510310000014.png"\; state="\{\{state\}\}">b</figure-callout>}}_1,{\mathrm{<figure-callout\; id="0"\; label="w\; t"\; filenames="HDA0000407510310000014.png"\; state="\{\{state\}\}">w</figure-callout>}}_t<w\&le;-(b_1-b_2)/(c_1-c_2)\\ 0,-{\mathrm{<figure-callout\; id="0"\; label="w\; t"\; filenames="HDA0000407510310000014.png"\; state="\{\{state\}\}">w</figure-callout>}}_t\&le;w\&le;w_{t0}\\ c_{1}w-{\mathrm{<figure-callout\; id="1"\; label="b"\; filenames="HDA0000407510310000011.png,HDA0000407510310000014.png"\; state="\{\{state\}\}">b</figure-callout>}}_1,(b_1-b_2)/(c_1-c_2)<w\&le;-w_{t0}\\ c_{2}w-{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="HDA0000407510310000014.png,HDA0000407510310000021.png"\; state="\{\{state\}\}">b</figure-callout>}}_2,w\&le;(b_1-b_2)/(c_1-c_2)\end{array}\right.$

The parameter of table 1 supercavitating vehicle

The system parameter values of supercavitating vehicle is listed in shown in table 1.System parameter values based on table 1, the non-linear power of the sliding F that employing formula (1) is expressed _planing, the non-linear power of sliding F _planingand between transverse velocity w, as shown in Figure 2, Fig. 2 also illustrates piecewise linearity simultaneously and slides power F relation curve in addition _pand relation curve between transverse velocity w.According to Fig. 2, can find out the non-linear power of sliding F _planingslide power F with piecewise linearity _pbetween have good degree of fitting.

(3) set supercavity kinetic parameters, refer to Fig. 3 a, 3b.When the feedback control gain of supercavitating vehicle is respectively k _z=15, k _θ=30 and k _q=0.3, i.e. δ _e=0, δ _cduring=15z-30 θ-0.3q, adopt piecewise linearity to slide power F _pthe bifurcation graphs that the kinetic model of function (being system 4) changes with cavitation number σ and the non-linear power of the sliding F of employing _planingthe bifurcation graphs that the kinetic model of function (being system 1) changes with cavitation number σ is analyzed, and the variation tendency of two kinds of bifurcation graphs is basically identical, and explanation adopts piecewise linearity to slide power F thus _pthe former non-linear power of the sliding F of Function Fitting _planingthe rationality of function.

(4) adopt piecewise linearity to slide power F _pthe supercavity kinetic model of function obtains unique equilibrium point of supercavitating vehicle, and at equilibrium point place, system linearization is being processed, 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 points.Utilize jacobi method to calculate its Lyapunov exponent, and running orbit and the corresponding Poincare of drawing sail body are penetrated.From Lyapunov exponent, phase rail figure and the Poincare mapping of system, system is a four dimensional chaos system, can generate chaotic attractor.

Specifically, as supercavity model selection table 1 parameter value, fixedly σ=0.0313 and δ _e=0, and adopt piecewise linearity to slide power F _pduring function, the kinetic model that can obtain supercavitating vehicle closed-loop control is

$\left\{\begin{array}{c}\stackrel{.}{w}=15w+78.27q-903.86{\mathrm{\&delta;}}_c+9.81-1.23F_P\\ \stackrel{.}{q}=-13w-5.72q+721.75{\mathrm{\&delta;}}_c+1.45F_P\\ \stackrel{.}{\mathrm{\&theta;}}=q\\ \stackrel{.}{z}=w-73.4\mathrm{\&theta;}\end{array}\right.$

Piecewise linearity slides power F _pfunction can be reduced to

$F_p=\left\{\begin{array}{c}-36.9w-192.3,w>1.82\\ -579w+796.2,1.38<w<1.82\\ 0,-1.38\&le;w\&le;1.38\\ -579w-796.2,-1.82<w\&le;-1.38\\ -36.9w+192.3,w\&le;-1.82\end{array}\right.$

(1) stability of balance point

Make system 4 $\stackrel{.}{w}=\stackrel{.}{q}=\stackrel{.}{\mathrm{\&theta;}}=\stackrel{.}{z}=0,$ Both

$\left\{\begin{array}{c}15w+78.27q-903.86{\mathrm{\&delta;}}_c+9.81-1.23F_P=0\\ \stackrel{.}{q}=-13w-5.72q+721.75{\mathrm{\&delta;}}_c+1.45F_P=0\\ q=0\\ w-73.4\mathrm{\&theta;}=0\end{array}\right.$

When σ=0.0313, get δ _c=k _zz-k _θθ-k _qq, δ _e=0,4 of systems have an equilibrium point in 1.38 < w < 1.82

S[1.3986,0,0.0190,(0.0190k _θ+0.0525)/k _z]

System 4, in equilibrium point S place's linearization, is obtained to Jacobi matrix and is

$J_S=\left[\begin{array}{c}w\\ q\\ \mathrm{\&theta;}\\ z\end{array}\right]=\left[\begin{array}{cccc}725.2636& 903.8622k_q+78.2711& 903.8622k_{\mathrm{\&theta;}}& -904k_z\\ -852.1539& -721.7495k_q-5.7243& -721.7495k_{\mathrm{\&theta;}}& 721.7495{\mathrm{<figure-callout\; id="0"\; label="k\; z"\; filenames="HDA0000407510310000014.png"\; state="\{\{state\}\}">k</figure-callout>}}_z\\ 0& 1& 0& 0\\ 1& 0& -73.4260& 0\end{array}\right]$

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 to linearization to system 4, obtain Jacobi matrix

$J_S=\left[\begin{array}{cccc}725.26& 349.43& 2711.587& -13560\\ -852.15& -222.25& -21652.49& 10826.24\\ 0& 1& 0& 0\\ 1& 0& -73.43& 0\end{array}\right]$

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, and index 2 equilibrium points, meet the necessary condition that forms chaotic attractor.

(2) chaotic attractor

Work as δ _c=15z – 30 θ – 0.3q, the running orbit of sail body be the projection of chaotic attractor in each plane as shown in Figure 4, corresponding Poincare mapping is as shown in Figure 5.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 phase rail figure, Poincare mapping and the Lyapunov index of system 4, system 4 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, kinetic description to supercavitating vehicle, adopted the force function that slides of a sectional linear fitting, build the kinetic model of supercavitating vehicle closed-loop control, thereby obtained, take the four dimensional chaos system that feedback control gain is variable element.Take feedback control gain as variable element, utilize conventional dynamic analysis instrument, 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:

$\begin{array}{c}{M}_0\left[\begin{array}{c}\stackrel{.}{w}\\ \stackrel{.}{q}\end{array}\right]=A_0\left[\begin{array}{c}w\\ q\end{array}\right]+B_0\left[\begin{array}{c}{\mathrm{\&delta;}}_e\\ {\mathrm{\&delta;}}_c\end{array}\right]+F_{\mathrm{gravity}}+\frac{1}{\mathrm{mL}}\left[\begin{array}{c}1\\ L\end{array}\right]F_{\mathrm{planing}}\\ \stackrel{.}{\mathrm{\&theta;}}=q\\ \stackrel{.}{z}\stackrel{}{=w-\mathrm{V\&theta;}}\end{array}$ Formula (1)

$F_{\mathrm{pianing}}=-V^2[1-{\left(\frac{R^{\&prime;}}{h+R^{\&prime;}}\right)}^2]\left(\frac{1+h^{\&prime;}}{1+2h^{\&prime;}}\right)\mathrm{\&alpha;}$ Formula (2)

Step 2: the non-linear force function that slides in supercavitating vehicle kinetic model is carried out to sectional linear fitting, obtain linearity and slide force function;

$F_p=\left\{\begin{array}{c}{c}_{2}w+b_2,w>-(b_1-b_2)/(c_1-c_2)\\ c_{1}w+b_1,w_{t0}<w\&le;-(b_1-b_2)/(c_1-c_2)\\ 0,-w_{t0}\&le;w\&le;w_{t0}\\ c_{1}w-b_1,(b_1-b_2)/(c_1-c_2)<w\&le;-w_{t0}\\ c_{2}w-b_2,w\&le;(b_1-b_2)/(c_1-c_2)\end{array}\right.$

Step 3: set supercavity kinetic parameters;

Step 4: adopt piecewise linearity to slide power F _pthe supercavity kinetic model of function obtains unique equilibrium point of supercavitating vehicle, and at equilibrium point place, system linearization is being processed, 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 points.

2. the subsection-linear method of supercavitating vehicle dynamical property analysis according to claim 1, is characterized in that: in the formula of described step 1 (1), m is density ratio (ρ m/ ρ), and 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

$M_0=\left[\begin{array}{cc}\frac{7}{9}& \frac{17L}{36}\\ \frac{17L}{36}& \frac{11}{60}{R}^2+\frac{133}{405}{L}^2\end{array}\right]$

$A_0=\mathrm{CV}\left[\begin{array}{cc}-\frac{(1+n)}{\mathrm{mL}}& -\frac{n}{m}\\ -\frac{n}{m}& -\frac{\mathrm{nL}}{m}\end{array}\right]+V\left[\begin{array}{cc}0& \frac{7}{9}\\ 0& \frac{17L}{36}\end{array}\right]$

$B_0={\mathrm{CV}}^2\left[\begin{array}{cc}-\frac{n}{\mathrm{mL}}& -\frac{n}{\mathrm{mL}}\\ -\frac{n}{m}& 0\end{array}\right]$

$F_{\mathrm{gravity}}=\left[\begin{array}{c}\frac{7}{9}\\ \frac{17L}{36}\end{array}\right]g$

Defined parameters C is

$C=0.5C_{x0}(1+\mathrm{\&sigma;}){\left(\frac{R_n}{R}\right)}^2$

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 of described step 1 (2), the expression formula of R ' is:

$R^{\&prime;}=\frac{R_c-R}{R}$

Supercavitating vehicle tail end submergence represents with parameter h ', can be expressed as

$h^{\&prime;}=\tanh \left(\mathrm{kw}\right)\frac{L}{2\mathrm{RV}}f\left(w\right)$

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) VL is the positive w value that is positioned at transition point, k be one for selecting to control the constant of approximate error.

α is supercavitating vehicle submergence angle, and expression formula is:

$\mathrm{\&alpha;}=\frac{w}{V}-\tanh \left(\mathrm{kw}\right)\frac{{\stackrel{.}{R}}_c}{V}$

In formula, apart from the cavity radius R at cavitation device distance L place _cand expansion and contraction

have

$R_c=R_n\sqrt{0.82\frac{1+\mathrm{\&sigma;}}{\mathrm{\&sigma;}}}{K}_2$

${\stackrel{.}{R}}_c=\frac{-\frac{20}{17}{\left(0.82\frac{1+\mathrm{\&sigma;}}{\mathrm{\&sigma;}}\right)}^{0.5}V(1-\frac{4.5\mathrm{\&sigma;}}{1+\mathrm{\&sigma;}})K_1^{23/17}}{K_2(\frac{1.92}{\mathrm{\&sigma;}}-3)}$

In formula,

$K_1=\frac{L}{R_n(\frac{1.92}{\mathrm{\&sigma;}}-3)}-1$

$K_2=\sqrt{1-(1-\frac{4.5\mathrm{\&sigma;}}{1+\mathrm{\&sigma;}})K_1^{40/17}}.$

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 $F_{\mathrm{pianing}}=-V^2[1-{\left(\frac{R^{\&prime;}}{h+R^{\&prime;}}\right)}^2]\left(\frac{1+h^{\&prime;}}{1+2h^{\&prime;}}\right)\mathrm{\&alpha;}$

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 and slide power F _p.

5. piecewise linearity 15 methods of supercavitating vehicle dynamical property analysis according to claim 1, is characterized in that: the supercavity kinetic parameters in described step 3, the feedback control gain of supercavitating vehicle is respectively k _z=15, k _θ=30 and k _q=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, 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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