CN110955980A - Stability analysis method for time-lag underwater ultrahigh-speed navigation body - Google Patents
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Abstract
The invention belongs to the field of nonlinear hydrodynamic force, and particularly relates to a stability analysis method for a time-lag underwater ultrahigh-speed navigation body. The method comprises the following steps: step 1: obtaining a sliding force simplified model; step 2: according to the sliding force simplified model, a longitudinal movement time lag simplified model is obtained; and step 3: and judging the stability of the time-lag system according to the longitudinal motion time-lag simplified model. The stability analysis method of the time-lag underwater super-speed navigation body can provide a main reference calculation model for the stability problem of the time-lag super-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-lag effect.
Description
Technical Field
The invention belongs to the field of nonlinear hydrodynamic force, and particularly relates to a stability analysis method for a time-lag underwater ultrahigh-speed navigation body.
Background
According to the independent principle of Lobvinovich cavitation bubbles, the central line of the cavitation bubble always follows the trajectory of the cavitator, and the section radius of each cavitation bubble is determined by the state of the cavitator when the cavitation bubbles are generated. Thus, due to the cavitation deformation of the aft portion of the vehicle caused by the cavitators, there is a time lag depending on the speed of travel of the vehicle that affects the dynamic behavior of the vehicle by changing hydrodynamic and torque forces on the control surface. The time interval and the resulting effect thereof are generally referred to as skew effects. The time lag effect depends on the trajectory of the navigation body, the dimensions and geometry of the navigation body. Normally, the cavitator produces a time varying cavitation radius and a curved cavitation centerline that offsets the section of the cavitation along the centerline from the central section of the vehicle at the tail or tail portion of the vehicle. This time-varying offset and cavitation radius creates an asymmetric tail wetted area and glide force at the aft of the vehicle, and therefore has a time-lag term of influence in the expression for glide force.
The principle of independent expansion of the sections of the cavitation bubbles shows that the trajectory of each section of the cavitation bubble relative to the center of the cavitator expands independently of the state of motion of the cavitator before or after this moment, but determined by the instantaneous velocity of the cavitator through the plane of the section, the size of the cavitation bubble, the generated resistance and the difference between the ambient pressure and the internal pressure of the cavitation bubble. The arc along the central track of the cavitator defines the coordinate h, the position of which is determined relative to the static fluid. Drawing the plane Σ (h) perpendicular to the track at point h, the change in the cross section of the cavitation bubbles can be observed when t ═ 0 rises instantaneously to this plane.
Because the navigation body is when navigating, the change of navigation body gesture and the change of vacuole, navigation body afterbody can take place the interact with the vacuole, three kinds of situations can appear: the tail part is not contacted with the cavity wall in the cavity; the tail part slides on the inner wall of the cavity; the tail portion may flap against or even penetrate the cavity wall. The above three conditions can describe the stress condition of the tail part of the navigation body by using the sliding force. The gliding force is a special acting force of the navigation body different from other conventional underwater navigation bodies. The gliding force is generated by the interaction of the navigation body and the cavitation, so that the time lag effect of the cavitation can obviously influence the numerical value of the gliding force, and the time lag can influence the magnitude of the gliding force and further influence the stability of the navigation body.
Disclosure of Invention
The invention aims to provide a method for analyzing the stability of a time-lag underwater ultrahigh-speed navigation body.
The method for analyzing the stability of the time-lag underwater ultrahigh-speed navigation body comprises the following steps:
step 1: obtaining a sliding force simplified model according to the tail immersion depth and the tail immersion angle;
step 1.1: establishing an assumption of a horizontal straight-ahead state;
step 1.2: calculating the immersion depth of the tail part and simplifying;
step 1.3: calculating and simplifying the immersion angle of the tail part;
step 1.4: calculating a calculation item containing the immersion depth and the immersion angle;
step 1.5: and acquiring a sliding force simplified model.
Step 2: according to the sliding force simplified model, a longitudinal movement time lag simplified model is obtained;
and step 3: and judging the stability of a time-lag system according to a longitudinal motion time-lag simplified model, wherein the judgment basis is expressed as the following formula:
wherein A is a second parameter, AdAs a third parameter, if the first symmetric positive definite matrix P exists, the second symmetric positive definite matrix S ∈ Rn×nIf the above formula is satisfied, the longitudinal motion time lag simplified model described in step 2 is stable.
The sliding force in the horizontal straight flight state in step 1.1 is expressed as follows:
where h' is the depth of immersion, αpFor the immersion angle, V is the forward speed of the vehicle, L represents the length of the vehicle,
r represents the radius of the vehicle, RcIndicating the cavitation radius, m is the specific gravity of the navigation body and the cavitation radius RcRepresented by the formula:
wherein ,
σ is the cavitation number, RnIs the radius of the cavitator;
step 1.2 the depth of submersion h' is expressed as follows:
the method comprises the following steps that theta is a pitch angle of a navigation body, tau is a time delay time tau of the navigation body, wherein tau is L/V, and z represents the longitudinal displacement of the center of mass of the navigation body; when z (t) + θ L-z (t- τ) > R-RcWhen the tail of the navigation body is in contact with the lower wall of the vacuole, when z (t) + theta L-z (t-tau) < R-RcWhen z (t) + θ L-z (t- τ) ═ R-R, it means that the tail of the vehicle is in contact with the upper wall of the cavitycWhen the navigation body is in the cavity, the navigation body is not contacted with the cavity wall; the simplified depth of submersion h' is expressed as follows:
step 1.3 immersion angle αpRepresented by the formula:
wherein w is the longitudinal speed of the center of mass of the navigation body,
indicating the location of the sliding force, i.e. the shrinkage of the cavitation radius,
represented by the formula:
immersion angle αpSimplified to the following formula:
step 1.5 the simplified model of the sliding force is expressed as follows:
the dynamic model of the longitudinal motion described in step 2 is represented by the following equation:
wherein ,AIIs a first parameter matrix, BIIs a second parameter matrix, MIIs a third parameter matrix, FgravAs is the force of gravity and its moment,
first parameter matrix AIRepresented by the formula:
wherein the first parameter
Second parameter matrix BIRepresented by the formula:
third parameter matrix MIRepresented by the formula:
gravity and its moment FgravRepresented by the formula:
substituting the obtained simplified model of the sliding force containing the time-lag effect into a dynamic model of the longitudinal movement of the navigation body to obtain the following formula:
where | M | is a matrix of third parameters MIThe longitudinal motion time lag simplified model is expressed as the following formula:
wherein B is a fourth parameter.
The invention has the beneficial effects that:
the method for acquiring the longitudinal movement simplified model of the underwater super-speed navigation body and the method for judging the stability can provide a main reference calculation model for the stability problem of the time-lag super-speed navigation body, simplify the tail sliding force in the model and finally acquire the simplified model of the longitudinal movement of the navigation body with the time-lag effect. The time lag effect of the cavitation bubbles can obviously influence the numerical value of the sliding force, and the time lag can influence the magnitude of the sliding force and further influence the stability of the 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 representation of the cavitation lag effect.
FIG. 2 is a flow chart of the working process of the judgment method of the dynamics stability of the time-lag underwater super-speed vehicle.
FIG. 3 is a glide force simplification workflow.
Detailed Description
The flow chart of the invention is shown in fig. 2, because the gliding force contains the time lag information of the state of the navigation body, the model of the gliding force is researched, and the simplified model of the gliding 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 dynamic 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 a time-lag independent stability condition. The simplified flow of the non-linear acting force sliding force is shown in fig. 3, the sliding force is simplified when the navigation body is in a horizontal straight navigation state, the immersion depth of the tail of the navigation body is firstly calculated, namely the immersion depth is simply calculated, 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, a composite calculation item containing the immersion depth and the immersion angle is simplified, and finally the simplified result of the immersion depth, the immersion angle and the composite item containing the immersion depth and the immersion angle of the tail of the navigation body is brought into a sliding force calculation model to obtain a simplified sliding force model. The method for acquiring the longitudinal movement simplified model of the underwater super-speed navigation body and the method for judging the stability can provide a main reference calculation model for the stability problem of the time-lag super-speed navigation body, simplify the tail sliding force in the model and finally acquire the simplified model of the longitudinal movement of the navigation body with the time-lag effect.
The method consists of three parts, including simplification of a sliding force model; obtaining a longitudinal movement time lag model and judging the stability of the model.
A method for establishing a time-lag underwater super-speed navigation body dynamic model is characterized by researching and simplifying the tail sliding force. According to the content of the prior document, the expression of the sliding force can be written as
Wherein the depth of submersion h' and the glide angle αpThe calculation method when the time lag effect of the cavitation is considered is as follows:
wherein
Wherein theta is the pitch angle of the navigation body, tau is the time delay time tau of the navigation body, L/V, w is the longitudinal speed of the centroid of the navigation body, V is the forward speed of the navigation body,
the sliding force position and the shrinkage rate of the cavitation radius are shown. z represents the longitudinal displacement of the centre of mass of the vehicle, L represents the length of the vehicle, R represents the radius of the vehicle, RcDenotes the cavitation radius, RnIs the radius of the cavitator, and m is the specific gravity of the navigation body. In the expressions of the immersion depth and the glide angle of the aircraft, when z (t) + theta L-z (t-tau) > R-RcZ (t) + θ L-z (t- τ) ═ R-R, indicating that the tail of the vehicle is in contact with the lower wall of the cavity and vice versa, andcwhen in use, the navigation body is in the cavity and is not in contact with the cavity wall.
For a vehicle of known structure, the structural parameters of which are known and fixed, among which R, L and RnThe forward speed is constant for the horizontal straight voyage state, wherein the motion parameter V and the cavitation number sigma are constant. And because the longitudinal displacement of the center of mass of the navigation body is basically kept unchanged in the horizontal straight navigation state of the navigation body, namely z (t) -z (t-tau) ≈ 0, the immersion depth can be simplified to be 0 in the horizontal straight navigation state
In addition, in the horizontal straight navigation state, the pitch angle of the navigation body is changed within an angle range, so that the immersion depth of the tail part of the navigation body is [ h'min,h′max]An internal variation. For the immersion angle of the gliding force, w (t- τ) ≈ 0 when the vehicle is level straight, so that the immersion angle can be further simplified. The reduction of the final sliding force is as follows
Secondly, a method for establishing a time-lag underwater superspeed navigation body dynamic model, which is characterized in that on the basis of simplifying the treatment of the tail sliding force, a dynamic model of the longitudinal movement of the navigation body is established, and the obtained sliding force simplified model containing the time-lag effect is substituted into the dynamic model of the longitudinal movement of the navigation body
wherein AI，BI，MIFor corresponding parameter matrices, FgravFor gravity and its moment, the corresponding calculation formula is as follows:
obtaining a simplified time lag model of the longitudinal motion of the navigation body as
Where | M | is a parameter matrix MIThe above is modeled into a time-lag system form
And thirdly, a method for establishing a time-lag underwater superspeed navigation body dynamic model, which is characterized in that the stability of the system is judged on the basis of the time-lag dynamic model of the longitudinal movement of the navigation body. Based on the Lyapunov stability theory, the stability problem of a time-lag system is researched by adopting a linear matrix inequality. In the existing stability condition of a time-lag system, the stability condition is divided into two types of time-lag independence and time-lag dependence according to whether the time-lag in the system is depended on or not. When a system stability problem is detected, a time-lag independent stability condition is used for judging, and if the system stability problem is not successful, a time-lag dependent stability condition is used for detecting.
Theorem for the system (4), if a symmetric positive definite matrix P exists,
so that
The system (4) is asymptotically stable.
The matrix inequality (5) is a linear matrix inequality related to the matrixes P and S, theorem 1 provides a sufficient condition for asymptotically stabilizing the system by using the feasibility of the linear matrix inequality, and an LMI tool box can be used for solving the linear matrix inequality to judge whether the system meets the sufficient condition.
Claims (3)
1. The method for analyzing the stability of the time-lag underwater ultrahigh-speed navigation body is characterized by comprising the following steps of:
step 1: obtaining a sliding force simplified model according to the tail immersion depth and the tail immersion angle;
step 1.1: establishing an assumption of a horizontal straight-ahead state;
step 1.2: calculating the immersion depth of the tail part and simplifying;
step 1.3: calculating and simplifying the immersion angle of the tail part;
step 1.4: calculating a calculation item containing the immersion depth and the immersion angle;
step 1.5: and acquiring a sliding force simplified model.
Step 2: according to the sliding force simplified model, a longitudinal movement time lag simplified model is obtained;
and step 3: and judging the stability of a time-lag system according to a longitudinal motion time-lag simplified model, wherein the judgment basis is expressed as the following formula:
wherein A is a second parameter, AdAs a third parameter, if the first symmetric positive definite matrix P exists, the second symmetric positive definite matrix S ∈ Rn×nIf the above formula is satisfied, the longitudinal motion time lag simplified model described in step 2 is stable.
2. The method for analyzing the stability of a time-lag underwater ultra-high speed vehicle according to claim 1, wherein the sliding force in the horizontal straight-ahead state in step 1.1 is expressed by the following formula:
where h' is the depth of immersion, αpFor the immersion angle, V is the forward speed of the vehicle, L represents the length of the vehicle,
r represents the radius of the vehicle, RcIndicating the cavitation radius, m is the specific gravity of the navigation body and the cavitation radius RcRepresented by the formula:
wherein ,
σ is the cavitation number, RnIs the radius of the cavitator;
step 1.2 the depth of submersion h' is expressed as follows:
wherein, theta is the pitch angle of the navigation body, tau is the time delay time tau of the navigation body which is L/VZ represents the longitudinal displacement of the centre of mass of the navigation body; when z (t) + θ L-z (t- τ) > R-RcWhen the tail of the navigation body is in contact with the lower wall of the vacuole, when z (t) + theta L-z (t-tau) < R-RcWhen z (t) + θ L-z (t- τ) ═ R-R, it means that the tail of the vehicle is in contact with the upper wall of the cavitycWhen the navigation body is in the cavity, the navigation body is not contacted with the cavity wall; the simplified depth of submersion h' is expressed as follows:
step 1.3 immersion angle αpRepresented by the formula:
wherein w is the longitudinal speed of the center of mass of the navigation body,
indicating the location of the sliding force, i.e. the shrinkage of the cavitation radius,
represented by the formula:
immersion angle αpSimplified to the following formula:
step 1.5 the simplified model of the sliding force is expressed as follows:
3. the method for analyzing the stability of a time-lapse underwater superspeed vehicle according to claim 1, wherein the dynamic model of the longitudinal movement in step 2 is represented by the following formula:
wherein ,AIIs a first parameter matrix, BIIs a second parameter matrix, MIIs a third parameter matrix, FgravAs is the force of gravity and its moment,
first parameter matrix AIRepresented by the formula:
wherein the first parameter
Second parameter matrix BIRepresented by the formula:
third parameter matrix MIRepresented by the formula:
gravity and its moment FgravRepresented by the formula:
substituting the obtained simplified model of the sliding force containing the time-lag effect into a dynamic model of the longitudinal movement of the navigation body to obtain the following formula:
where | M | is a matrix of third parameters MIThe longitudinal motion time lag simplified model is expressed as the following formula:
wherein B is a fourth parameter.
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| 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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