CN110955980B - Time-lag underwater ultra-high speed navigation body stability analysis method - Google Patents
Time-lag underwater ultra-high speed navigation body stability analysis method Download PDFInfo
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Abstract
The invention belongs to the field of nonlinear hydrodynamic force, and particularly relates to a stability analysis method of a time-lapse underwater ultra-high-speed navigation body. The method comprises the following steps: step 1: acquiring a sliding force simplified model; step 2: according to the sliding force simplified model, a longitudinal movement time lag simplified model is obtained; step 3: and judging the stability of the time lag system according to the longitudinal movement time lag simplified model. The stability analysis method of the time-lapse underwater ultra-high speed navigation body can provide a main reference calculation model for the stability problem of the time-lapse ultra-high speed navigation body, simplifies the tail sliding force in the model, and finally obtains a simplified model of the longitudinal movement of the navigation body with the time-lapse effect.
Description
Technical Field
The invention belongs to the field of nonlinear hydrodynamic force, and particularly relates to a stability analysis method of a time-lapse underwater ultra-high-speed navigation body.
Background
According to the Logvinovich cavitation independent principle, the cavitation center line always follows the trajectory of the cavitation device, and the section radius of each cavitation device is determined by the state of the cavitation device when the cavitation device generates cavitation bubbles. Thus, due to cavitation deformation of the tail of the vehicle by the cavitation device, there is a time lag depending on the speed of the vehicle, which affects the dynamics of the vehicle by changing the hydrodynamic forces and moments on the control surfaces. The usual time interval and the resulting effect is called the time lag effect. The time lag effect depends on the trajectory of the vehicle, the size and geometry of the vehicle. Normally, the cavitation device produces a time-varying cavitation radius and a curved cavitation centerline that causes the cross section of the cavitation bubble along the centerline to be offset from the central cross section of the vehicle at the tail or tail section of the vehicle. This time-varying offset and cavitation radius creates asymmetric tail wetting areas and taxiing forces at the tail of the vehicle, and thus has a time-lapse term in the expression of taxiing forces.
The principle of independent expansion of the cavitation cross-section indicates the ballistic expansion of each cross-section of the cavitation with respect to the centre of the cavitation device, the law of expansion of which is independent of the state of motion of the cavitation device before or after this instant, but is determined by the instant speed of the cavitation device through the plane of the cross-section, the size of the cavitation bubbles, the resistance produced and the difference between the ambient pressure and the internal pressure of the cavitation bubbles. The arc along the cavitation center trajectory determines the coordinate h, which is determined relative to the static fluid. At point h, the plane Σ (h) is drawn perpendicular to the trajectory, and the change in the cavitation cross section at the instant t=0 rising to this plane can be observed.
Due to the change of the attitude of the navigation body and the change of cavitation bubbles when the navigation body is in navigation, the tail of the navigation body can interact with the cavitation bubbles, and three situations can possibly occur: the tail part is not contacted with the cavity wall in the cavity; the tail part slides on the inner wall of the cavitation bubble; the tail appears to slap against the cavity wall and even penetrates the cavity wall. The three conditions can be used for describing the stress condition of the tail of the navigation body by the sliding force. The planing force is also a special force of this type of craft that is different from other conventional underwater craft. Because the sliding force is generated by interaction of the navigation body and the cavitation bubbles, the time lag effect of the cavitation bubbles has obvious influence on the numerical value of the sliding force, and the magnitude of the time lag can influence the magnitude of the sliding force so as to influence the stability of the navigation body, so that the stability analysis of the time lag system by utilizing the stability theory of the existing time lag system is significant for the design and stable navigation of the navigation body controller.
Disclosure of Invention
The invention aims to provide a stability analysis method for a time-lapse underwater ultra-high-speed navigation body.
The method for analyzing the stability of the time-lapse underwater ultra-high-speed navigation body comprises the following steps of:
step 1: obtaining a sliding force simplified model according to the tail submerging depth and the tail submerging angle;
step 1.1: establishing a hypothesis of a horizontal direct-navigation state;
step 1.2: calculating the tail immersion depth and simplifying;
step 1.3: calculating and simplifying the tail submerging angle;
step 1.4: calculating a calculation item containing the immersion depth and the immersion angle;
step 1.5: and obtaining a sliding force simplified model.
Step 2: according to the sliding force simplified model, a longitudinal movement time lag simplified model is obtained;
step 3: according to the longitudinal movement time lag simplified model, judging the stability of the time lag system, wherein the judging basis is expressed as the following formula:
wherein A is a second parameter, A d For the third parameter, if there is a first symmetric positive definite matrix P, a second symmetric positive definite matrix S εR n×n So that the above equation holds, the longitudinal motion time lag simplified model described in step 2 is stable.
The sliding force in the horizontal straight navigation state described in step 1.1 is expressed as:
wherein h' is the immersion depth, α p For the immersion angle, V is the forward speed of the vehicle, L represents the length of the vehicle,r represents the radius of the navigation body, R c Represents the cavitation radius, m is the specific gravity of the navigation body, and the cavitation radius R c Expressed by the following formula:
wherein ,sigma is cavitation number, R n Is the radius of the cavitation device;
the depth of immersion h' in step 1.2 is expressed as:
wherein θ is the pitch angle of the vehicle, τ is the delay time τ=l/V of the vehicle, z represents the longitudinal displacement of the center of mass of the vehicle; when z (t) +thetaL-z (t-tau) > R-R c When z (t) +thetaL-z (t-tau) < R-R, the tail of the navigation body is in contact with the lower wall of the cavitation chamber c When z (t) +θL-z (t- τ) =R-R, the tail of the aircraft is in contact with the upper wall of the cavitation vessel c When the navigation body is positioned in the cavity, the navigation body is not contacted with the cavity wall; the simplified immersion depth h' is expressed as:
step 1.3 the immersion angle α p Expressed by the following formula:
where w is the longitudinal velocity of the centroid of the vehicle,representation ofThe sliding force position, i.e. the shrinkage of the cavitation radius, +.>Expressed by the following formula:
immersion angle alpha p Simplified to the following formula:
the simplified model of the sliding force in step 1.5 is expressed as follows:
the dynamics model of the longitudinal movement described in step 2 is expressed as:
wherein ,AI As the first parameter matrix, B I As a second parameter matrix, M I For a third parameter matrix, F grav The weight and the moment thereof are used for controlling the weight and the moment thereof,
first parameter matrix A I Expressed by the following formula:
wherein the first parameter
Second parameter matrix B I Expressed by the following formula:
third parameter matrix M I Expressed by the following formula:
gravity and moment F thereof grav Expressed by the following formula:
substituting the obtained sliding force simplified model containing the time-lag effect into a dynamics model of the longitudinal movement of the navigation body to obtain the following formula:
wherein |M| is the third parameter matrix M I Is represented by the following equation:
wherein B is a fourth parameter.
The invention has the beneficial effects that:
the acquisition method and the stability judging method of the longitudinal movement simplified model of the underwater ultra-high speed navigation body can provide a main reference calculation model for the stability problem of the time-lapse ultra-high speed navigation body, simplify the tail sliding force in the model, and finally obtain the longitudinal movement simplified model of the navigation body with the time-lapse effect. The time lag effect of cavitation can obviously influence the numerical value of the sliding force, and the magnitude of the time lag can influence the magnitude of the sliding force so as to influence the stability of a navigation body, so that the stability analysis of the time lag system is carried out by utilizing the stability theory of the existing time lag system, and the time lag system has important significance for the design and stable navigation of a navigation body controller.
Drawings
FIG. 1 is a schematic diagram of cavitation time lag effects.
FIG. 2 is a workflow diagram of a method for determining the dynamic stability of a time-lapse underwater ultra-high speed vehicle.
Fig. 3 is a sliding force reduction workflow diagram.
Detailed Description
The flow chart of the invention is shown in figure 2, because the sliding force contains time lag information of the state of the navigation body, the model of the sliding force is researched, and a simplified model of the sliding force is calculated according to the actual navigation state of the navigation body; then, the simplified model of the sliding force is brought into a longitudinal motion dynamics model of the underwater supercavitation navigation body, and a time lag simplified model of the longitudinal motion of the navigation body is calculated; and finally judging the stability of the system by using independent time-lag stability conditions. The simplified flow of the nonlinear acting force sliding force is shown in fig. 3, the sliding force is simplified under the horizontal straight navigation state of the navigation body, the immersion depth of the tail of the navigation body is calculated first, namely, the immersion depth is simplified, then the immersion angle of the tail of the navigation body is calculated, a simplified calculation mode is obtained, after the maximum range of the immersion depth and the immersion angle is determined, the composite calculation item containing the immersion depth and the immersion angle is simplified, and finally, the immersion depth, the immersion angle and the simplified result containing the composite item of the immersion depth and the immersion angle of the tail of the navigation body are brought into a calculation model of the sliding force, and a simplified sliding force model is obtained. The acquisition method and the stability judging method of the longitudinal movement simplified model of the underwater ultra-high speed navigation body can provide a main reference calculation model for the stability problem of the time-lapse ultra-high speed navigation body, simplify the tail sliding force in the model, and finally obtain the longitudinal movement simplified model of the navigation body with the time-lapse effect.
The method consists of three parts, including simplification of a sliding force model; and (5) acquiring a longitudinal movement time lag model and judging the stability of the model.
1. A method for establishing a time-lapse underwater ultra-high speed navigation body dynamics model is characterized in that the research and the simplification of the tail sliding force are performed. According to the prior literature, the expression of the sliding force can be written as
Wherein the depth of immersion h' and the glide angle alpha p The calculation method when considering the time-lag effect of cavitation is as follows:
wherein
Where θ is the pitch angle of the vehicle, τ is the time delay of the vehicle τ=l/V, w is the longitudinal velocity of the vehicle centroid, V is the forward velocity of the vehicle,the sliding force position and the shrinkage of the cavitation radius are indicated. z represents the longitudinal displacement of the centroid of the vehicle, L represents the length of the vehicle, R represents the radius of the vehicle, R c Represents the radius of cavitation, R n The radius of the cavitation device is the specific gravity of the navigation body. In the expression of the submerging depth and the planing angle of the navigation body, when z (t) +thetaL-z (t-tau) > R-R c Represents the contact of the tail of the navigation body with the lower wall of the cavitation bubble and vice versa with the upper wall of the cavitation bubble, z (t) +thetal-z (t-tau) =r-R c When the navigation body is inside the cavitation bubbles, the navigation body is not contacted with the cavitation bubble wall.
For structurally known aircraft, the structural parameters are known and fixed, R, L and R being among the parameters n Is constant, for a horizontal straight-through condition, forward velocity is constant for a vehicle where the motion parameter V, and cavitation number σ are both constant. In addition, because the longitudinal displacement of the mass center of the navigation body is basically unchanged in the horizontal direct navigation state of the navigation body, and the z (t) -z (t-tau) is approximately equal to 0, the immersion depth can be simplified into
In addition, the pitch angle of the navigation body is changed within an angle range under the horizontal straight navigation state, so that the submerging depth of the tail part of the navigation body is [ h ]' min ,h′ max ]Internal variation. For the submerging angle of the sliding force, w (t- τ) ≡0 when the navigation body is horizontally and directly navigated, so the submerging angle can be further simplified. The final sliding force reduction results are as follows
2. A method for establishing a time-lapse underwater ultra-high-speed navigation body dynamics model is characterized in that on the basis of simplifying the tail sliding force, a dynamics model of the longitudinal movement of the navigation body is established, and the obtained sliding force simplified model containing the time-lapse effect is substituted into the dynamics model of the longitudinal movement of the navigation body
wherein AI ，B I ，M I For a corresponding parameter matrix, F grav The corresponding calculation formulas for the gravity and the moment thereof are as follows:
the simplified time lag model of the longitudinal movement of the navigation body is obtained as
Where |M| is the parameter matrix M I Modeling the above into a time-lapse system form
3. A method for establishing a time-lapse underwater ultra-high-speed navigation body dynamics model is characterized in that the stability of a system is judged on the basis of the time-lapse dynamics model of longitudinal movement of a navigation body. Based on Lyapunov stability theory, linear matrix inequality is adopted to study stability problem of time-lapse system. In the existing time lag system stability conditions, the stability conditions are divided into independent time lag and time lag dependence according to whether the time lag in the system is dependent or not. When a system stability problem is detected, a time-lag independent stability condition is firstly applied to judge, and if the system stability problem is unsuccessful, a time-lag dependent stability condition is applied to detect.
Theorem for system (4), if there is a symmetric positive definite matrix P,so that
The system (4) is asymptotically stable.
The matrix inequality (5) is a linear matrix inequality with respect to the matrices P and S, and theorem 1 gives a sufficient condition for asymptotically stabilization of the system with the feasibility of the linear matrix inequality, which can be solved by using the LMI toolbox to determine whether the system satisfies.
Claims (1)
1. The method for analyzing the stability of the time-lapse underwater ultra-high-speed navigation body is characterized by comprising the following steps of:
step 1: establishing a simplified model of the sliding force of the time-lapse underwater ultra-high-speed navigation body;
the expression of the sliding force is:
wherein V represents the forward speed of the vehicle; m represents the specific gravity of the navigation body; l represents the length of the vehicle;r represents the radius of the vehicle; r is R c Representing the radius of the cavitation bubbles; h' represents the depth of immersion, α p The calculation method for representing the sliding angle when considering the time lag effect of cavitation is as follows:
wherein θ represents a pitch angle of the vehicle;shrinkage, which represents the radius of cavitation; τ represents the delay time of the craft, τ=l/V;
wherein ,Rn Representing the cavitation radius; sigma represents cavitation number;
for structurally known aircraft, the structural parameters are known and fixed, R, L and R being among the parameters n Is a constant; for the followingThe horizontal direct-navigation state, the forward speed of the navigation body with constant forward speed, and the motion parameter V and the cavitation number sigma are constants; in addition, because the longitudinal displacement of the mass center of the navigation body is basically unchanged in the horizontal direct navigation state of the navigation body, namely z (t) -z (t-tau) approximately equal to 0 is present, the immersion depth is simplified into:
since the pitch angle of the vehicle varies over a range of angles in a horizontal straight-through condition, the depth of immersion of the tail of the vehicle is [ h ]' min ,h′ max ]An internal variation; for the immersion angle of the sliding force, when the navigation body is in horizontal direct navigation, w (t-tau) is approximately equal to 0, so that the immersion angle is further simplified, and the final reduction result of the sliding force is as follows:
step 2: establishing a longitudinal movement time lag simplified model of the time lag underwater ultrahigh-speed navigation body;
on the basis of simplified tail sliding force treatment, a dynamics model of longitudinal movement of the navigation body is established
wherein ,
substituting the sliding force simplified model containing the time-lag effect into a dynamics model of the longitudinal movement of the navigation body to obtain a simplified time-lag model of the longitudinal movement of the navigation body, wherein the simplified time-lag model comprises the following steps:
wherein ,x(t)＝[z θ w q] T ；/>i M I is a parameter matrix M I Determinant value of (2); />u(t)＝[δ f δ c ] T ；
Step 3: judging the stability of the time-lapse underwater ultrahigh-speed navigation body according to the longitudinal movement time-lapse simplified model;
if there is a symmetric positive definite matrix P, S ε R n×n So thatAnd judging that the time-lapse underwater ultra-high-speed navigation body meets the stability.
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