CN112149362A - Method for judging motion stability of supercavitation navigation body - Google Patents
Method for judging motion stability of supercavitation navigation body Download PDFInfo
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Abstract
The invention discloses a method for judging the motion stability of a supercavitation navigation body, which aims to solve the problem that the influence of initial conditions on the motion stability of the navigation body is not researched in the past. Step 1: establishing a supercavitation navigation body dynamic model; step 2: determining a parameter area with coexisting attractors in the movement of the navigation body through a dynamic map; and step 3: verifying the types of the coexisting attractors at different parameters by adopting a phase-track diagram; and 4, step 4: and judging the motion stability of the navigation body according to the change of the size and the shape of the attraction domain along with the parameters. And 5: and simulating and verifying the motion condition of the supercavitation navigation body at the corresponding parameters of different attraction domains. The dynamic map is adopted to clarify the parameter range of the coexisting attractors, and the strength of the motion stability of the navigation body is judged according to the change of the attraction domain, so that a new method is provided for judging the motion stability of the supercavitation navigation body under different initial launching conditions.
Description
Technical Field
The invention relates to a method for judging motion stability, in particular to a method for judging motion stability of a supercavitation navigation body.
Background
When the navigation body moves underwater at a high speed, the pressure near the surface of the navigation body is lower than the saturated steam pressure due to the Bernoulli effect, and the part of water area is vaporized to form a cavity, which is called as cavitation phenomenon. The bubble clusters formed when the cavitation phenomenon occurs are called cavitation bubbles, and the morphological dimension of the cavitation bubbles changes along with the change of the inflow conditions of the flow field. With further increase of the speed of the vehicle, the local pressure of the fluid near the surface decreases sharply, and when this cavity completely surrounds the vehicle, the phenomenon is called supercavitation, and the cavity is called supercavitation.
Although the supercavitation technology can greatly improve the speed and the range of the underwater vehicle, the supercavitation vehicle is different from a conventional underwater vehicle, and the unique navigation environment determines that the supercavitation vehicle is a multivariable system with the characteristics of multi-coupling and fluid dynamic parameter uncertainty. The fact that the system has initial condition sensitivity is that small changes of initial motion parameters make the motion of the system tend to different states and converge to different attractors, and finally the direction of the system is uncertain and often has very complex behaviors, and the phenomenon is called attractor coexistence. The phenomenon of coexistence of attractors causes the operation state of the system to have unforeseen factors which are completely inconsistent with the original preconceived by designers, and poses great threat to the motion stability of the system. However, the research on the motion stability of the supercavitation navigation body at present mainly aims at the research on the motion phenomenon generated by the change of system parameters such as cavitation number, cavitation device diameter, cavitation device deflection angle, empennage deflection angle and the like, and the influence of the change of initial motion parameters on the motion characteristics of the system is not reported.
In order to judge the influence of the initial motion parameters on the motion stability of the supercavitation navigation body, the unpredictable factors of the motion state of the supercavitation navigation body need to be studied in detail, so that some non-ideal destructive results are avoided. The method judges the range of the initial motion parameters corresponding to different motion states of the navigation body by using the attraction domain, analyzes the relationship between the motion stability of the navigation body and the size and the change of the attraction domain from the angle of the coexistence of attractors, and provides an initial motion parameter basis for launching the stably sailing supercavitation navigation body, thereby avoiding the instability of the motion of the navigation body.
Disclosure of Invention
The invention aims to provide a method for judging the motion stability of a supercavitation navigation body, which aims to disclose the initial value sensitivity of the supercavitation navigation body, judge the motion stability of the navigation body under different parameter conditions based on the change of an attraction domain and provide an initial condition basis for launching the supercavitation navigation body which stably moves. And a new way is provided for judging the motion stability of the supercavitation navigation body.
The invention provides a method for judging the motion stability of a supercavitation navigation body, which comprises the following steps:
firstly, establishing a dynamic model of a supercavitation navigation body
The original point of a body coordinate system for dynamic modeling of the supercavitation navigation body is positioned at the center of a circle of the top end face of a cavitation device at the head of the navigation body, the direction of an X axis is overlapped with the central axis of the navigation body and points forwards, the direction of a Z axis is vertical to the central axis and points downwards, and a ground system is taken as an inertial system. The modeling adopts the depth Z, the vertical speed w, the pitch angle theta and the pitch angle speed q of the navigation body as state variables to describe the dynamics of the navigation body, the direction of the vertical speed w is consistent with the direction of the Z axis, and V is the resultant speed of the navigation body in the longitudinal plane.
The control input of the supercavitation navigation body is the tail wing deflection angleeAnd cavitator deflection anglecThe invention is toe＝kθθ、c＝15z-30θ-0.3q，kθA feedback gain for the tail deflection angle control variable theta.
The dynamic model of the supercavitation navigation body can be obtained through the definition of the parameters:
in the formula: z (t), w (t), theta (t), q (t) respectively correspond to the depth, vertical speed, pitch angle and pitch angle speed of the navigation body at the time t. FplaningIs tail notLinear glide force. a is22(t,τ),a24(t,τ),a42(t,τ),a44(t,τ),b21(t,τ),b41(t,τ),b22,b42,c2,d2The coefficients of the system equation are as follows:
wherein the meanings and values of the variables are as follows: acceleration of gravity g is 9.81m/s2The radius R of the navigation body is 0.0508m, the length L of the navigation body is 1.8m, the density ratio m is 2, the tail efficiency n (t, tau), the lift coefficient Cx00.82, cavitator radius Rn0.0191m, cavitation number sigma epsilon [0.01980,0.03680 ∈]；
Secondly, determining a parameter area with coexisting attractors in the movement of the navigation body through a dynamic map, wherein the parameter area comprises the following steps:
step 2-1, obtaining a dynamic map of the navigation body based on the Lyapunov stability theory,
randomly selecting based on the kinetic model (1)Taking initial conditions, and calculating a stable solution R, a periodic solution G and a chaotic solution Y of the model according to a Lyapunov stability theory when sigma and k are equalθIs taken as sigma epsilon [0.01980,0.02738]，kθ∈[-0.27,23.37]The maximum Lyapunov index of the formula (1) is smaller than zero, the state variables z, w, theta and q are converged at a stable balance point, and the navigation body moves stably; when sigma e [0.01980,0.03680]，kθ∈[-50,24.78]The maximum Lyapunov index of the formula (1) is equal to zero, the state variables z, w, theta and q all periodically oscillate by taking a balance point as a center, and the navigation body periodically moves; when sigma e [0.03060,0.03352]，kθ∈[-3.88,7.85]The maximum Lyapunov index of the formula (1) is larger than zero, and z, w, theta and q all generate violent non-periodic oscillation to generate vibration and impact, so that the navigation body overturns; when sigma e [0.01980,0.03258]，kθ∈[23.82,50]The system is divergent, and the navigation body cannot navigate;
step 2-2, observing the dynamics map
Case 1: sigma e [0.02333,0.02838]，kθ∈[4.73,20.15]Then, the stable solution and the periodic solution are mutually alternated; sigma e [0.02737,0.02906]，kθ∈[20.15,36.57]The stable solution is interleaved with the periodic solution and the divergent state; the phenomenon that a balance point attractor and a periodic attractor coexist exists in the range;
case 2: sigma e [0.03175,0.03273]，kθ∈[-7.86,6.78]Within the range, the chaotic solution is periodically dispersed in the phenomenon that a periodic attractor and a chaotic attractor coexist;
case 3: sigma e [0.01980,0.02822]，kθ∈[17.16,32.59]Within the range, the stable solution is spread in the divergent part;
case 4: sigma e [0.02973,0.03680]，kθ∈[20.65,50.00]Within the enclosure, periodic points are dispersed in the diverging portion;
in the above range, a transition of the motion state is always caused by a slight change in the cavitation number σ, and there may be a phenomenon in which an attractor coexists;
thirdly, verifying the type of the coexisting attractors at different parameters by adopting a phase-orbit diagram, which comprises the following steps:
random selection of the parameter combination a (sigma, k) in the context of the first case of step 2-2θ) When the initial conditions are different, the solution of system equation (1) converges to two different results. When the initial condition is a2Then, converge on a balanced point attractor; on a phase track diagram corresponding to the attractor, converging state variables z, w, theta and q to a balance point, and enabling the navigation body to stably move; when the initial condition is a2In time, the solution of the equation (1) converges to a periodic attractor, a limit ring appears in the phase orbit diagram, and the navigation body oscillates periodically. This situation is known as the coexistence of the equilibrium point attractors with the periodic attractors;
random selection of the parameter combination b (sigma, k) in the context of the second case of step 2-2θ) When the initial conditions are different, the solution of system equation (1) converges to two different results, when the initial conditions are b1The solution of system equation (1) converges to a one-cycle attractor. The phase track diagram is a limit ring, and the navigation body periodically oscillates; when the initial condition is b2In time, a chaotic attractor is arranged in the phase orbit diagram, and the navigation body violently oscillates and even is unstable. This situation is known as the coexistence of periodic attractors with chaotic attractors;
random selection of the parameter combination c (σ, k) in the context of the third case of step 2-2θ) When the initial condition is c1When the solution of the system equation (1) converges to a balanced point attractor, the navigation body moves stably; when the initial condition is c2In time, the system equation (1) is not solved, the system is dispersed, and the navigation body overturns;
random selection of the parameter combination d (σ, k) in the context of the fourth case of step 2-2θ) When the initial condition is d1When the solution of the system equation (1) converges to a period 1 attractor, the navigation body oscillates periodically; when the initial condition is d2In time, the system equation (1) is not solved, the system is dispersed, and the navigation body overturns;
fourthly, judging the motion stability of the navigation body according to the change of the size and the shape of the attraction domain along with the parameters, as follows:
and (3) according to the four types of coexisting attractor phenomena in the step 3, visually representing whether the motion of the supercavitation navigation body under different initial conditions is stable or not through the attraction domain at the combination of the four parameters. In the supercavitation navigation system, the larger the stable solution and the periodic solution attraction domain are, the higher the probability that the navigation body keeps stable navigation under different launching conditions is, and the better the motion stability is;
1) attraction domain with stable attractors coexisting with periodic attractors
In case 1 of step 2-2, i.e., σ e [0.02333,0.02838]，kθ∈[4.73,20.15]、σ∈[0.02737,0.02906]，kθ∈[20.15,36.57]In the area, four groups of different parameter combinations are randomly selected to obtain an attraction area in z0-θ0The parameter combinations of the above sections are respectively: sigma1＝0.02468，kθ1＝3.42；σ2＝0.02603，kθ2＝5.90；σ3＝0.02721，kθ3＝7.47；σ4＝0.02855，kθ49.36; wherein σ1≈σ2≈σ3≈σ4，kθ1＜kθ2＜kθ3＜kθ4Its shape is approximately parallelogram; when initial emission depth and initial emission pitch angle z0、θ0When values are taken in a stable solution S area in the graph, the navigation body stably moves at a balance point; when z is0、θ0When the points in the R area are solved periodically correspondingly, the navigation body oscillates periodically; when the initial motion parameter value corresponds to the point in the divergence D area, the value is too large, so that the system diverges, and the navigation body cannot navigate.
Sa1、Sa2、Sa3、Sa4Respectively, the area of the attraction region where the stable attractor coexists with the periodic attractor, Sa1＝12.31、Sa2＝10.25、Sa3＝9.34、Sa4＝3.48，Sa1>Sa2>Sa3>Sa4The change of cavitation number can be ignored, and the feedback control gain k of the area of the attraction domain along with the deflection angle of the empennageθIs increased and decreased; within this class of parameters, σ ∈ [0.02333,0.02838]，kθ∈[4.73,20.15]、σ∈[0.02737,0.02906]，kθ∈[20.15,36.57]，kθThe larger the value of (A), the smaller the stable equilibrium point and the attraction domain of the periodic attractor are, and the worse the motion stability is;
2) attraction domain with coexisting periodic attractors and chaotic attractors
In case 2 of step 2-2 the range σ e [0.03175,0.03273 ]]，kθ∈[-7.86,6.78]Randomly selecting four groups of parameter combinations in the area to obtain an attraction area in z0-θ0The parameter combinations of the attraction domains in which the periodic attractors and the chaotic attractors coexist are respectively as follows: sigma1＝0.02838，kθ1＝-59.01；σ2＝0.03136，kθ2＝-67.44；σ3＝0.03238，kθ3＝-62.63；σ4＝0.03259，kθ4-56.04, the R region represents the initial motion parameters that eventually fall into the periodic orbit, the navigation body moving periodically; the area C represents the initial motion parameters which finally fall into a chaotic state, and the navigation body is unstable in motion and even overturns; region D represents the initial motion parameters that make the system divergent;
when the parameter combinations are different, the shapes of the attraction domains are approximately the same, the sizes of the attraction domains are different, and fractal boundaries exist. When the initial conditions slightly change, the motion of the system is transited from the stable periodic motion to the chaotic motion, which can cause the overturn of the navigation body and has poor motion stability. In order to ensure the motion stability of the navigation body, the selection of such parameter combinations should be avoided;
3) attraction domain of stable equilibrium point
In case 3 of step 2-2 the range σ e [0.01980,0.02822 ]]，kθ∈[17.16,32.59]Selecting four groups of parameter combinations in the area to obtain an attraction area in z0-θ0Wherein the parameter combinations of the attraction domains of the stable attractors are respectively: sigma1＝0.02788，kθ1＝8.87；σ2＝0.02838，kθ2＝9.37；σ3＝0.02754，kθ3＝11.06；σ4＝0.02636，kθ4-10.04. When the initial motion parameter z0、θ0When taking value in the stable S-solving area, a supercavity wrapping the whole navigation body can be formed, the navigation body moves stably, and the navigation body initially takes valueThe motion parameters exceed the stable solution S boundary, and the navigation body moves unstably;
the size of the attraction domain of the equilibrium point is dependent on the parameter combination c (σ, k)θ) Is changed; sc1、Sc2、Sc3、Sc4Suction area, S, of suction area of stable attractorc1＝6.78、Sc2＝3.27、Sc3＝2.16、Sc4＝0.65，Sc1>Sc2>Sc3>Sc4In the range, the change of cavitation number is negligible, and the feedback control gain k of the deflection angle of the tail wingθThe smaller the value is, the larger the area of the attraction area is, the better the motion stability is, and the navigation body is easier to stably navigate. 4) An attraction domain of a periodic attractor;
in case 4 of step 2-2, i.e., σ e [0.02973,0.03680 ∈]，kθ∈[20.65,50.00]Four sets of parameter combinations are selected from the area. Obtaining an attraction domain at z0-θ0Upper cross section. Wherein, the parameter combination of the attraction domain of the periodic attractor is respectively as follows: sigma1＝0.02434，kθ1＝-87.83；σ2＝0.03040，kθ2＝-91.93；σ3,＝0.02805，kθ3＝-90.90；σ4＝0.03428，kθ4-89.88. When the initial motion parameter z0、θ0When values are taken in the periodic R solving area, the tail of the navigation body collides with the vacuole from time to time, and the navigation body moves periodically; when z is0、θ0When the point in the divergent D area in the corresponding graph is divergent, the system diverges and the navigation body is unstable in motion;
attraction domain range of periodic attractors with control gain kθThe increase in value gradually decreases; in sigma e [0.02973,0.03680]，kθ∈[20.65,50.00]Within range, the stability of the motion of the vehicle as a function of kθAn increase in value and a decrease in value;
the motion stability of the navigation body is improved by adjusting the empennage deflection angle of the navigation body and selecting the initial motion parameters to be transmitted in the attraction domain.
In order to ensure stable movement of the navigation body, motion parameters and structure parameters are selected in the attraction domain.
Compared with the prior art, the invention has the following remarkable advantages: aiming at the dynamic model of the underwater supercavitation navigation body, a dynamic map between any two variable parameters can be obtained, and the type and the parameter range of the coexisting attractor are judged according to the dynamic map. And then obtaining an attraction domain at each type of coexisting attractors, and judging the motion stability of the navigation body according to the change of the attraction domain along with the deflection angle of the tail wing. And providing structural parameters and an initial motion parameter basis for launching the stably moving supercavity navigation body.
Drawings
FIG. 1 is a schematic diagram of longitudinal modeling of a supercavitation vehicle of the present invention.
FIG. 2 is a kinetic map of the present invention.
FIG. 3 shows the present invention at point a (σ, k)θ) Where the initial value is alpha1(σ1,kθ1) Phase trajectory plot of time.
FIG. 4 shows the present invention at point a (σ, k)θ) Where the initial value is a2(σ2,kθ2) Phase trajectory plot of time.
FIG. 5 shows the present invention at point a (σ, k)θ) And (3) stabilizing the phase trajectory of the equilibrium point and the periodic attractor.
FIG. 6 shows the present invention at point b (σ, k)θ) Where the initial value is beta1(σ1,kθ1) Phase trajectory plot of time.
FIG. 7 shows the present invention at point b (σ, k)θ) Where the initial value is beta2(σ2,kθ2) Phase trajectory plot of time.
FIG. 8 shows the present invention at point b (σ, k)θ) And (3) a phase trajectory diagram of the coexistence of the periodic attractors and the chaotic attractors.
FIG. 9 shows the present invention at point c (σ, k)θ) Where the initial value is gamma1(σ1,kθ1) Phase trajectory plot of time.
FIG. 10 shows the present invention at point d (σ, k)θ) Where the initial value is1(σ1,kθ1) Phase trajectory plot of time.
FIG. 11 is an attraction domain of the present invention in which a stable attractor coexists with a periodic attractor.
FIG. 12 is an attraction domain of the present invention in which a periodic attractor coexists with a chaotic attractor.
FIG. 13 is an attraction domain of the stable attractor of the present invention.
FIG. 14 is an attraction domain of a periodic attractor of the present invention.
FIG. 15 shows the present invention at point a (σ, k)θ) Time domain, frequency domain plot.
FIG. 16 shows the present invention at point b (σ, k)θ) Time domain, frequency domain plot.
Detailed Description
The present invention is described in further detail below with reference to the attached drawing figures.
The invention provides a method for judging the motion stability of a supercavitation navigation body, which comprises the following steps:
firstly, establishing a dynamic model of a supercavitation navigation body
The method comprises the following specific steps:
as shown in figure 1, the origin of a body coordinate system for dynamic modeling of a supercavitation navigation body is positioned at the center of a top end face of a cavitation device at the head of the navigation body, the direction of an X axis is overlapped with the central axis of the navigation body and points forwards, the direction of a Z axis is vertical to the central axis and points downwards, a ground system is taken as an inertial system, and the motion of the navigation body in a longitudinal plane is researched. The modeling adopts the depth Z, the vertical speed w, the pitch angle theta and the pitch angle speed q of the navigation body as state variables to describe the dynamics of the navigation body, the direction of the vertical speed w is consistent with the direction of the Z axis, and V is the resultant speed of the navigation body in the longitudinal plane.
The control input of the supercavitation navigation body is the tail wing deflection angleeAnd cavitator deflection anglecThe invention is toe＝kθθ、c＝15z-30θ-0.3q，kθA feedback gain for the tail deflection angle control variable theta.
The dynamic model of the supercavitation navigation body can be obtained through the definition of the parameters:
in the formula: z (t), w (t), theta (t), q (t) respectively correspond to the depth, vertical speed, pitch angle and pitch angle speed of the navigation body at the time t. FplaningIs the tail non-linear glide force. a is22(t,τ),a24(t,τ),a42(t,τ),a44(t,τ),b21(t,τ),b41(t,τ),b22,b42,c2,d2The coefficients of the system equation are as follows:
wherein, the meanings and values of the variables are shown in the following table 1:
TABLE 1 parameter values of supercavitation navigation body model
Secondly, determining a parameter area with coexisting attractors in the movement of the navigation body through a dynamic map, wherein the parameter area is as follows:
step 2-1, obtaining a dynamic map of the navigation body based on the Lyapunov stability theory
FIG. 2 shows that initial conditions are randomly selected based on a dynamic model (1), a stable solution S, a periodic solution R and a chaotic solution C of the model are calculated according to the Lyapunov stability theory, and a feedback control gain k is drawn according to a cavitation number sigma and a tail deflection angleθThe dynamic map of the supercavitation navigation body with bifurcation parameters describes the system dynamic behavior pairs sigma and kθThe dependency of (c). When sigma, kθWhen the value of (c) is a point corresponding to the stable solution S region, i.e., σ ∈ [0.01980,0.02738 ]]，kθ∈[-0.27,23.37]The maximum Lyapunov index of the formula (1) is smaller than zero, the state variables z, w, theta and q are converged at a stable balance point, and the navigation body moves stably; optional one point (σ, k) in the periodic R solution regionθ) I.e., σ e [0.01980,0.03680]，kθ∈[-50,24.78]The maximum Lyapunov index of the formula (1) is equal to zero, the state variables z, w, theta and q all periodically oscillate by taking a balance point as a center, and the navigation body periodically moves; when sigma, kθWhen the value is taken in the chaotic solution C area, the value is sigma-epsilon [0.03060,0.03352]，kθ∈[-3.88,7.85]The maximum Lyapunov index of the formula (1) is larger than zero, and z, w, theta and q all generate violent non-periodic oscillation to generate vibration and impact, so that the navigation body overturns; the divergent D region, σ e [0.01980,0.03258]，kθ∈[23.82,50]The system is shown to be divergent and the navigation body can not navigate.
Step 2-2, observing the dynamics map
1)σ∈[0.02333,0.02838]，kθ∈[4.73,20.15]Then, the stable solution S and the periodic solution R are mutually interspersed; sigma e [0.02737,0.02906]，kθ∈[20.15,36.57]The stable solution S is interleaved with the periodic solution R and the divergent state. The phenomenon that a balance point attractor and a periodic attractor coexist exists in the range.
2)σ∈[0.03175,0.03273]，kθ∈[-7.86,6.78]Within range, the chaotic solution C is spread in the periodic region R; the phenomenon that a periodic attractor and a chaotic attractor coexist exists in the range.
3)σ∈[0.01980,0.02822]，kθ∈[17.16,32.59]Within the range, the stable solution S is scattered in the divergent portion D.
4)σ∈[0.02973,0.03680]，kθ∈[20.65,50.00]Within the envelope, the periodic solution R is spread in the divergent portion D.
In the above range, a transition of the motion state is always caused by a slight change in the cavitation number σ, and there is a possibility that an attractor coexists.
Thirdly, verifying the types of the attractors coexisting at different parameters by adopting a phase-orbit diagram, which comprises the following steps:
in case 1 of step 2-2, the parameter combination a (σ, k) is chosen randomlyθ) When the initial condition is α, (0.02468,3.42)1(z0,w0,θ0,q0) (-0.1241, 1.4897, 1.4090, 1.4172), the solution of system equation (1) converges to one equilibrium point attractor. As shown in fig. 3, on the phase trajectory diagram corresponding to the attractor, the state variables z, w, θ, q converge to the equilibrium point, and the navigation body moves stably; when the initial condition is alpha2(z0,w0,θ0,q0) When (-0.1022, -0.2414, 0.3192, 0.3129), as shown in fig. 4, the solution of equation (1) converges to a periodic attractor, a limit cycle appears in the phase-orbit diagram, and the navigation body oscillates periodically. As shown in FIG. 5, when the two coexist, the trajectory is projected on the q-w-theta plane, the point represents the stable equilibrium point, and the ring is the periodic attractor. This situation is referred to as the coexistence of the equilibrium point attractors with the periodic attractors.
In case 2 of step 2-2 the combination of parameters b (σ, k) is chosen randomlyθ) (0.02838,59.01) under the initial condition of β1(z0，w0，θ0，q0) When (-0.1924, 0.8886, -0.7648, -1.4023), the solution of system equation (1) converges to one-cycle attractors as shown in fig. 6. The phase track diagram is a limit ring, and the navigation body periodically oscillates; when the initial condition is beta2(z0，w0，θ0，q0) When (-1.0616, 2.3505, -0.6156, -0.7481), as shown in fig. 7, the phase trajectory diagram is a chaotic attractor, and the navigation body oscillates violently and even becomes unstable. As shown in FIG. 8, when the two coexist, the projection of the phase orbit on the q-w-theta plane, the ring is a periodic attractor, and the other one is a chaotic attractor. This situation is known as the coexistence of periodic attractors with chaotic attractors.
In case 3 of step 2-2, the combination of parameters c (σ, k) is chosen randomlyθ) When the initial condition is γ, (0.02788,8.87)1(z0,w0,θ0,q0) When (-0.8314, -0.9792, -1.1564, -0.5336), as shown in fig. 9, the solution of system equation (1) converges to a balanced point attractor, and the vehicle moves steadily; when the initial condition is gamma2(z0,w0,θ0,q0) When the system equation (1) is solved (0.0229, -0.2620, -1.7502, -0.2857), the system diverges and the vehicle overturns.
In the 4 th case of step 2-2, the parameter combination d (σ, k) is randomly chosenθ) When the initial conditions are (0.02434, -87.83)1(z0，w0，θ0，q0) When (-1.0891, 0.0326, 0.5525, 1.1006), as shown in fig. 10, the solution of system equation (1) converges to one cycle 1 attractor, the navigation body oscillates periodically; when the initial conditions are2(z0，w0，θ0，q0) When (-1.0891, 0.0326, 0.5525, 1.1006), the system equation (1) is solved, the system diverges, and the vehicle overturns.
And fourthly, judging the motion stability of the navigation body according to the variation of the size and the shape of the attraction domain along with the parameters. The method comprises the following specific steps:
and (3) according to the four types of coexisting attractor phenomena in the step 3, visually representing whether the motion of the supercavitation navigation body under different initial conditions is stable or not through the attraction domain at the combination of the four parameters. In the supercavitation navigation system, the larger the stable solution and the periodic solution attraction domain are, the higher the probability that the navigation body keeps stable navigation under different launching conditions is, and the better the motion stability is.
1) Attraction domain with stable attractors coexisting with periodic attractors
In case 1 of step 2-2, i.e., σ e [0.02333,0.02838]，kθ∈[4.73,20.15]、σ∈[0.02737,0.02906]，kθ∈[20.15,36.57]In the region, four different parameter combinations are randomly selected, as shown in FIG. 11, to obtain the attraction domain at z0-θ0The cross-section of fig. 11(a) -11(d), wherein the parameter combinations of fig. 11(a) -11(d) are: sigma1＝0.02468，kθ1＝3.42；σ2＝0.02603，kθ2＝5.90；σ3＝0.02721，kθ3＝7.47；σ4＝0.02855，kθ49.36. Wherein σ1≈σ2≈σ3≈σ4，kθ1＜kθ2＜kθ3＜kθ4. The shape of which is approximately parallelogram. When initial emission depth and initial emission pitch angle z0、θ0When values are taken in a stable solution S area in the graph, the navigation body stably moves at a balance point; when z is0、θ0When the points in the R area are solved periodically correspondingly, the navigation body oscillates periodically; when the initial motion parameter value corresponds to the point in the divergent solution D area, the value is too large, so that the system diverges, and the navigation body cannot navigate.
Sa1、Sa2、Sa3、Sa4FIG. 11(a), FIG. 11(b), FIG. 11(c), and FIG. 11(d) show the areas of the attraction regions where the stable attractors coexist with the periodic attractors, Sa1＝12.31、Sa2＝10.25、Sa3＝9.34、Sa4＝3.48，Sa1>Sa2>Sa3>Sa4The change of cavitation number can be ignored, and the feedback control gain k of the area of the attraction domain along with the deflection angle of the empennageθIs increased and decreased. Within such parameters, i.e., σ ∈ [0.02333,0.02838]，kθ∈[4.73,20.15]、σ∈[0.02737,0.02906]，kθ∈[20.15,36.57]，kθThe larger the value of (A), the smaller the stable equilibrium point and the attraction domain of the periodic attractor, and the worse the motion stability. In practical engineering application, the tail wing deflection angle of the navigation body can be adjusted to suckAnd selecting the initial motion parameters of the launching in the navigation area to improve the motion stability of the navigation body.
2) Attraction domain with coexisting periodic attractors and chaotic attractors
In the range of case 2 of step 2-2, i.e., σ e [0.03175,0.03273]，kθ∈[-7.86,6.78]Randomly selecting four groups of parameter combinations in the region, as shown in FIG. 12, to obtain the attraction domain in z0-θ012(a) -12(d), wherein the parameter combinations of fig. 12(a) -12(d) are respectively: sigma1＝0.02838，kθ1＝-59.01；σ2＝0.03136，kθ2＝-67.44；σ3＝0.03238，kθ3＝-62.63；σ4＝0.03259，kθ4-56.04. The R area represents the initial motion parameters finally falling into the periodic orbit, and the navigation body moves periodically; the area C represents the initial motion parameters which finally fall into a chaotic state, and the navigation body is unstable in motion and even overturns; the D-region represents the initial motion parameters that make the system divergent.
When the parameter combinations are different, the shapes of the attraction domains are approximately the same, the sizes of the attraction domains are different, and fractal boundaries exist. When the initial conditions slightly change, the motion of the system is transited from the stable periodic motion to the chaotic motion, which often causes the overturn of the navigation body and has poor motion stability. To ensure the stability of the movement of the vehicle, the selection of such a combination of parameters should be avoided.
3) Attraction domain of stable equilibrium point
In the case of case 3 of step 2-2, i.e., σ e [0.01980,0.02822]，kθ∈[17.16,32.59]Selecting four groups of parameter combinations in the area, as shown in FIG. 13, to obtain the attraction domain in z0-θ013(a) -13(d), wherein the parameter combinations of fig. 13(a) -13(d) are respectively: sigma1＝0.02788，kθ1＝8.87；σ2＝0.02838，kθ2＝9.37；σ3＝0.02754，kθ3＝11.06；σ4＝0.02636，kθ4-10.04. When the initial motion parameter z0、θ0When the value is taken in the stable solution S area, a supercavity wrapping the whole navigation body can be formed, the navigation body stably moves, and when the initial motion parameters exceedAnd if a stable solution S boundary is formed, the navigation body moves unstably.
The size of the attraction domain of the equilibrium point is dependent on the parameter combination c (σ, k)θ) May vary. Sc1、Sc2、Sc3、Sc4The areas of the suction regions S in FIGS. 13(a), 13(b), 13(c) and 13(d), respectivelyc1＝6.78、Sc2＝3.27、Sc3＝2.16、Sc4＝0.65，Sc1>Sc2>Sc3>Sc4In the range, the change of cavitation number is negligible, and the feedback control gain k of the deflection angle of the tail wingθThe smaller the value is, the larger the area of the attraction area is, the better the motion stability is, and the navigation body is easier to stably navigate. In practical application, in order to ensure the stable movement of the navigation body, the motion parameters and the structure parameters are selected in the attraction area.
4) Attraction domain of periodic attractors
In case 4 of step 2-2, i.e., σ e [0.02973,0.03680 ∈]，kθ∈[20.65,50.00]Four sets of parameter combinations are selected from the area. As shown in FIG. 14, the attraction domain is found at z0-θ0Upper cross section. In fig. 14(a) to 14(d), the parameter combinations are: sigma1＝0.02434，kθ1＝-87.83；σ2＝0.03040，kθ2＝-91.93；σ3,＝0.02805，kθ3＝-90.90；σ4＝0.03428，kθ4-89.88. When the initial motion parameter z0、θ0When values are taken in the periodic R solving area, the tail of the navigation body collides with the vacuole from time to time, and the navigation body moves periodically; when z is0、θ0When points in the divergent solution D area in the corresponding graph are dispersed, the system diverges and the navigation body is unstable in motion.
Attraction domain range of periodic attractors with control gain kθThe increase in value gradually decreases. In sigma e [0.02973,0.03680]，kθ∈[20.65,50.00]Within range, the stability of the motion of the vehicle as a function of kθThe value increases and decreases.
And fifthly, simulating the motion condition of the supercavitation navigation body at the corresponding parameters of different attraction domains.
The method comprises the following specific steps:
and (3) programming by using matlab software, setting different initial conditions at the parameters of the four attractor subtypes, and simulating the time domain response of each variable of the navigation body, thereby verifying the relationship between the motion stability and the initial conditions in the step 4. As shown in fig. 15, a is 0.02468, kθTaking the initial motion parameter value α in the stable solution S region and the periodic solution R region in fig. 11(a), for example, 3.421(z0,w0,θ0,q0) (-0.1241, 1.4897, 1.4090, 1.4172) and α2(z0,w0,θ0,q0) (-0.1022, -0.2414, 0.3192, 0.3129). The dotted line represents the initial motion parameter α1The motion conditions of the four state variables z, w, theta and q of the system changing along with time, and the solid line represents the initial motion parameter alpha2Time domain and frequency domain response conditions of each variable of the system. It can be found that the initial motion parameter is alpha1In the process, under the action of a control law, state variables z, w, theta and q are quickly stabilized on balance points (0.0772,3.0198,0.0365 and 0) to enable the navigation body to stably move; initial motion parameter of alpha2When the four state variables are all periodically oscillated around the equilibrium point (-0.0282, -1.2007, -0.0145, 0); the tail of the navigation body continuously collides with the wall surface of the cavity, and the system has long-time obvious periodic oscillation.
At σ 0.03259, kθTaking the initial motion parameter value β in the periodic solution R region and the chaotic solution C region in fig. 12(a), for example, -56.041,，β2. The simulation result of the navigation body motion characteristic at this time is shown in fig. 16, and the solid line indicates that the initial motion parameter is β1(z0，w0，θ0，q0) (-0.1924, 0.8886, -0.7648, -1.4023), the system's four state variables z, w, θ, q, motion over time and corresponding frequency spectrum; dotted line is the initial motion parameter beta2(z0，w0，θ0，q0) (-1.0616, 2.3505, -0.6156, -0.7481) time domain response of each variable of the system. When the initial motion parameter is beta1Four state variables z, w, theta, qThe oscillations are periodically centered on the equilibrium points (0.0019, -0.0138, -0.0002, 0), respectively, thereby indicating that the vehicle is periodically vibrating within the vacuole. When the initial motion parameter is beta2In the process, the state variables of the navigation body oscillate non-periodically along with the change of time near a balance point, and because each state variable of the system is in a non-periodic unstable state, the motion of the navigation body is also easily interfered by the outside, thereby causing instability.
Claims (3)
1. A method for judging the motion stability of a supercavitation navigation body is characterized by comprising the following steps:
step 1, establishing a dynamic model of the supercavitation navigation body:
the body coordinate system origin of the dynamic model of the supercavitation navigation body is positioned at the center of a circle of the top end face of a cavitation device at the head of the navigation body, the direction of an X axis is overlapped with the central axis of the navigation body and points forwards, the direction of a Z axis is vertical to the central axis and points downwards, and a ground system is taken as an inertial system to study the motion of the navigation body in a longitudinal plane; the modeling adopts the depth Z, the vertical speed w, the pitch angle theta and the pitch angle speed q of the navigation body as state variables to describe the dynamics of the navigation body, the direction of the vertical speed w is consistent with the direction of the Z axis, and V is the resultant speed of the navigation body in the longitudinal plane;
the control input of the supercavitation navigation body is the tail wing deflection angleeAnd cavitator deflection anglecWhereine＝kθθ、c＝15z-30θ-0.3q，kθFeedback gain of a control variable theta for the deflection angle of the tail wing;
and (3) obtaining a dynamic model of the supercavitation navigation body by defining the parameters:
in the formula: z (t), w (t), theta (t), q (t) respectively correspond to the depth, vertical speed, pitch angle and pitch angle speed of the navigation body at the time t, FplaningTail nonlinear sliding force; a is22(t,τ),a24(t,τ),a42(t,τ),a44(t,τ),b21(t,τ),b41(t,τ),b22,b42,c2,d2For the coefficients of the system equation:
wherein the meanings and values of the variables are as follows: acceleration of gravity g is 9.81m/s2The radius R of the navigation body is 0.0508m, the length L of the navigation body is 1.8m, the density ratio m is 2, the tail efficiency n (t, tau), the lift coefficient Cx00.82, cavitator radius Rn0.0191m, cavitation number sigma epsilon [0.01980,0.03680 ∈]；
Step 2, determining a parameter area with coexisting attractors in the movement of the navigation body through a dynamic map, wherein the parameter area comprises the following steps:
step 2-1, obtaining a dynamic map of the navigation body based on the Lyapunov stability theory,
based on a kinetic model (1), randomlySelecting initial conditions, and calculating a stable solution (S), a periodic solution (R) and a chaotic solution (C) of the model according to a Lyapunov stability theory when sigma and k are equalθIs taken as sigma epsilon [0.01980,0.02738]，kθ∈[-0.27,23.37]The maximum Lyapunov index of the formula (1) is smaller than zero, the state variables z, w, theta and q are converged at a stable balance point, and the navigation body moves stably; when sigma e [0.01980,0.03680]，kθ∈[-50,24.78]The maximum Lyapunov index of the formula (1) is equal to zero, the state variables z, w, theta and q all periodically oscillate by taking a balance point as a center, and the navigation body periodically moves; when sigma e [0.03060,0.03352]，kθ∈[-3.88,7.85]The maximum Lyapunov index of the formula (1) is larger than zero, and z, w, theta and q all generate violent non-periodic oscillation to generate vibration and impact, so that the navigation body overturns; when sigma e [0.01980,0.03258]，kθ∈[23.82,50]The system is divergent, and the navigation body cannot navigate;
step 2-2, analyzing the dynamics map
Case 1: sigma e [0.02333,0.02838]，kθ∈[4.73,20.15]The stable solution (S) and the periodic solution (R) are interleaved; sigma e [0.02737,0.02906]，kθ∈[20.15,36.57]The stable solution (S) is interleaved with the periodic solution (R) and the divergent state; the phenomenon that a balance point attractor and a periodic attractor coexist exists in the range;
case 2: sigma e [0.03175,0.03273]，kθ∈[-7.86,6.78]Within the scope, the chaotic solution (C) is periodically dispersed in the periodic solution (R), a phenomenon in which periodic attractors coexist with chaotic attractors;
case 3: sigma e [0.01980,0.02822]，kθ∈[17.16,32.59]Within the range, the stable solution (S) is dispersed in the divergent portion (D);
case 4: sigma e [0.02973,0.03680]，kθ∈[20.65,50.00]Within the enclosure, the periodic solution (R) is dispersed in a divergent portion (D);
in the above range, the transition of the motion state always occurs with a slight change in the cavitation number σ, and the phenomenon of coexistence of attractors occurs;
step 3, verifying the types of the coexisting attractors at different parameters by adopting a phase-orbit diagram, which comprises the following steps:
in case 1 of step 2-2, the parameter combination a (σ, k) is chosen randomlyθ) When the initial condition is α, (0.02468,3.42)1(z0,w0,θ0,q0) (-0.1241, 1.4897, 1.4090, 1.4172), the solution of system equation (1) converges to a balanced point attractor navigation body stable motion; when the initial condition is alpha2(z0,w0,θ0,q0) (-0.1022, -0.2414, 0.3192, 0.3129), the solution of equation (1) converges to one periodic attractor, the navigation volume oscillates periodically;
in case 2 of step 2-2 the combination of parameters b (σ, k) is chosen randomlyθ) (0.02838,59.01) under the initial condition of β1(z0，w0，θ0，q0) (-0.1924, 0.8886, -0.7648, -1.4023), the solution of system equation (1) converges to one period attractor, the navigation volume oscillates periodically; when the initial condition is beta2(z0，w0，θ0，q0) When (-1.0616, 2.3505, -0.6156, -0.7481), the vehicle oscillates violently and even becomes unstable;
in case 3 of step 2-2, the combination of parameters c (σ, k) is chosen randomlyθ) When the initial condition is γ, (0.02788,8.87)1(z0,w0,θ0,q0) When (-0.8314, -0.9792, -1.1564, -0.5336), the solution of system equation (1) converges to a balanced point attractor and the vehicle moves steadily; when the initial condition is gamma2(z0,w0,θ0,q0) (0.0229, -0.2620, -1.7502, -0.2857), system equation (1) is solved, system divergence occurs, and the vehicle overturns;
in the 4 th case of step 2-2, the parameter combination d (σ, k) is randomly chosenθ) When the initial conditions are (0.02434, -87.83)1(z0，w0，θ0，q0) (-1.0891, 0.0326, 0.5525, 1.1006), the solution of system equation (1) converges to one periodic attractor, the navigation volume oscillates periodically; when the initial conditions are2(z0，w0，θ0，q0) When (-1.0891, 0.0326, 0.5525, 1.1006), the system equation (1) is solved, the system diverges, and the vehicle overturns;
step 4, judging the motion stability of the navigation body according to the change of the size and the shape of the attraction domain along with the parameters, as follows:
according to the phenomenon of the four types of coexisting attractors in the step 3, whether the motion of the supercavitation navigation body under different initial conditions is stable or not is visually represented through an attraction domain at the combination of the four parameters; in the supercavitation navigation system, the larger the stable solution and the periodic solution attraction domain are, the higher the probability that the navigation body keeps stable navigation under different launching conditions is, and the better the motion stability is;
1) attraction domain with stable attractors coexisting with periodic attractors
In case 1 of step 2-2 the range σ e [0.02333,0.02838 ]]，kθ∈[4.73,20.15]、σ∈[0.02737,0.02906]，kθ∈[20.15,36.57]In the area, four groups of different parameter combinations are randomly selected to obtain an attraction area in z0-θ0The parameter combinations of the above sections are respectively: sigma1＝0.02468，kθ1＝3.42；σ2＝0.02603，kθ2＝5.90；σ3＝0.02721，kθ3＝7.47；σ4＝0.02855，kθ49.36; wherein σ1≈σ2≈σ3≈σ4，kθ1＜kθ2＜kθ3＜kθ4Its shape is approximately parallelogram; when initial emission depth and initial emission pitch angle z0、θ0When the value is taken in the stable solution (S) area in the graph, the navigation body stably moves at the balance point; when z is0、θ0When corresponding to the point in the periodic solution (R) area, the navigation body oscillates periodically; when the initial motion parameter value corresponds to a point in the divergent solution (D) area, the system diverges and the navigation body cannot navigate;
Sa1、Sa2、Sa3、Sa4respectively, the area of the attraction region where the stable attractor coexists with the periodic attractor, Sa1＝12.31、Sa2＝10.25、Sa3＝9.34、Sa4＝3.48，Sa1>Sa2>Sa3>Sa4The change of cavitation number can be ignored, and the feedback control gain k of the area of the attraction domain along with the deflection angle of the empennageθIs increased and decreased; within this class of parameters, σ ∈ [0.02333,0.02838]，kθ∈[4.73,20.15]、σ∈[0.02737,0.02906]，kθ∈[20.15,36.57]，kθThe larger the value of (A), the smaller the stable equilibrium point and the attraction domain of the periodic attractor are, and the worse the motion stability is; 2) an attraction domain in which the periodic attractors and the chaotic attractors coexist;
in case 2 of step 2-2 the range σ e [0.03175,0.03273 ]]，kθ∈[-7.86,6.78]Randomly selecting four groups of parameter combinations in the area to obtain an attraction area in z0-θ0The parameter combinations of the attraction domains in which the periodic attractors and the chaotic attractors coexist are respectively as follows: sigma1＝0.02838，kθ1＝-59.01；σ2＝0.03136，kθ2＝-67.44；σ3＝0.03238，kθ3＝-62.63；σ4＝0.03259，kθ4-56.04, the stable solution (S) region represents the initial motion parameters that eventually fall into the periodic orbit, the periodic motion of the navigation body; the chaotic solution (C) area represents the initial motion parameters which finally fall into a chaotic state, and the navigation body is unstable in motion and even overturns; the divergent solution (D) region represents the initial motion parameters that make the system divergent;
in order to ensure the motion stability of the navigation body, the selection of parameter combination of fractal boundaries is avoided;
3) attraction domain of stable equilibrium point
In case 3 of step 2-2 the range σ e [0.01980,0.02822 ]]，kθ∈[17.16,32.59]Selecting four groups of parameter combinations in the area to obtain an attraction area in z0-θ0Wherein the parameter combinations of the attraction domains of the stable attractors are respectively: sigma1＝0.02788，kθ1＝8.87；σ2＝0.02838，kθ2＝9.37；σ3＝0.02754，kθ3＝11.06；σ4＝0.02636，kθ4-10.04; when the initial motion parameter z0、θ0In the stabilization ofWhen the value is taken in the solution (S) area, a supercavity wrapping the whole navigation body is formed, the navigation body moves stably, and when the initial motion parameter exceeds the stable solution (S) boundary, the navigation body moves unstably;
the size of the attraction domain of the equilibrium point is dependent on the parameter combination c (σ, k)θ) Is changed; sc1、Sc2、Sc3、Sc4Suction area, S, of suction area of stable attractorc1＝6.78、Sc2＝3.27、Sc3＝2.16、Sc4＝0.65，Sc1>Sc2>Sc3>Sc4In the range, the change of cavitation number is negligible, and the feedback control gain k of the deflection angle of the tail wingθThe smaller the value is, the larger the area of the attraction area is, the better the motion stability is, and the navigation body can navigate stably;
4) attraction domain of periodic attractors
In case 4 of step 2-2, i.e., σ e [0.02973,0.03680 ∈]，kθ∈[20.65,50.00]Four sets of parameter combinations are selected from the area. Obtaining an attraction domain at z0-θ0Cross section of (a); wherein, the parameter combination of the attraction domain of the periodic attractor is respectively as follows: sigma1＝0.02434，kθ1＝-87.83；σ2＝0.03040，kθ2＝-91.93；σ3,＝0.02805，kθ3＝-90.90；σ4＝0.03428，kθ4-89.88; when the initial motion parameter z0、θ0When values are taken in the periodic solution (R) area, the tail of the navigation body collides with the vacuole from time to time, and the navigation body moves periodically; when z is0、θ0For points in the divergence (D) region, the system diverges and the navigation body motion is unstable;
attraction domain range of periodic attractors with control gain kθThe increase in value gradually decreases. In sigma e [0.02973,0.03680]，kθ∈[20.65,50.00]Within range, the stability of the motion of the vehicle as a function of kθThe value increases and decreases.
2. Method according to claim 1, characterized in that the stability of the movement of the navigation body is increased by selecting the initial movement parameters of the launch within the attraction domain by adjusting the tail deflection angle of the navigation body.
3. Method according to claim 1, characterized in that the motion parameters and the structure parameters are selected within the attraction domain in order to guarantee a stable movement of the navigation body.
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113821996A (en) * | 2021-07-14 | 2021-12-21 | 南京理工大学 | Novel method for quickly calculating high-speed water-entering trajectory of projectile |
| CN114323552A (en) * | 2021-11-18 | 2022-04-12 | 厦门大学 | Method for judging stability of water entering and exiting from cross-medium navigation body |
Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN103512438A (en) * | 2013-10-25 | 2014-01-15 | 北京电子工程总体研究所 | Adjustable control method for orbit control direct force on air-defense missile |
| CN103558009A (en) * | 2013-11-04 | 2014-02-05 | 南京理工大学 | Piecewise linear method for analyzing supercavitation navigation body kinetic characteristics |
| CN106644382A (en) * | 2016-12-27 | 2017-05-10 | 南京理工大学 | Method for planning motion track of supercavitating vehicle based on multi-steady-state analysis |
-
2020
- 2020-07-08 CN CN202010650128.8A patent/CN112149362A/en active Pending
Patent Citations (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN103512438A (en) * | 2013-10-25 | 2014-01-15 | 北京电子工程总体研究所 | Adjustable control method for orbit control direct force on air-defense missile |
| CN103558009A (en) * | 2013-11-04 | 2014-02-05 | 南京理工大学 | Piecewise linear method for analyzing supercavitation navigation body kinetic characteristics |
| CN106644382A (en) * | 2016-12-27 | 2017-05-10 | 南京理工大学 | Method for planning motion track of supercavitating vehicle based on multi-steady-state analysis |
Non-Patent Citations (4)
| Title |
|---|
| YIPIN LV等: "Robustness of Supercavitating Vehicles Based on Multistability Analysis", 《ADVANCES IN MATHEMATICAL PHYSICS》, pages 2 - 4 * |
| 吕一品;熊天红;易文俊;: "基于分叉理论的水下超空泡航行体运动特性研究", 舰船科学技术, no. 23, pages 24 - 29 * |
| 吕一品;熊天红;易文俊;管军;: "系统参数对超空泡航行体非线性动力学特性的影响", 振动与冲击, no. 24, pages 161 - 169 * |
| 张重先;李向林;刘玉秋;: "复杂外形导弹带空泡出水动力学特性研究", 现代防御技术, no. 01, pages 48 - 53 * |
Cited By (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN113821996A (en) * | 2021-07-14 | 2021-12-21 | 南京理工大学 | Novel method for quickly calculating high-speed water-entering trajectory of projectile |
| CN113821996B (en) * | 2021-07-14 | 2024-01-30 | 南京理工大学 | Novel rapid calculation method for high-speed entry trajectory of projectile |
| CN114323552A (en) * | 2021-11-18 | 2022-04-12 | 厦门大学 | Method for judging stability of water entering and exiting from cross-medium navigation body |
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