CN112434428B - Ship transverse-swinging nonlinear dynamics analysis method in regular wave - Google Patents

Ship transverse-swinging nonlinear dynamics analysis method in regular wave Download PDF

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CN112434428B
CN112434428B CN202011347245.3A CN202011347245A CN112434428B CN 112434428 B CN112434428 B CN 112434428B CN 202011347245 A CN202011347245 A CN 202011347245A CN 112434428 B CN112434428 B CN 112434428B
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王丽元
刘峥
李焱
唐友刚
童波
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Abstract

The invention discloses a ship transverse throwing nonlinear dynamics analysis method in regular waves, which comprises the following steps: establishing a bow-rolling motion differential equation of the ship in a regular wave; determining the boundary of a stable domain and an unstable domain of the ship in the transverse throwing motion according to a nonlinear dynamics theory, obtaining a stable domain schematic diagram of a ship yawing motion differential equation solution, and obtaining a theoretical critical value of a parameter excitation term coefficient in the yawing motion differential equation according to the stable domain schematic diagram; calculating the bifurcation characteristic of the ship heading motion by adopting a numerical method based on a bifurcation nonlinear dynamics theory to obtain a bifurcation diagram of the ship heading angle along with the wave height so as to determine the reliability of a theoretical critical value of a parameter excitation term coefficient; and determining the critical condition of the ship for the transverse throwing according to the stability domain schematic diagram. The method adopts a nonlinear dynamics analysis method to analyze the stability region of the safe navigation of the ship for the transverse throwing motion of the ship in the regular wave, and provides an analysis method and a safe navigation basis for the safe navigation of the ship.

Description

Ship transverse-swinging nonlinear dynamics analysis method in regular wave
Technical Field
The invention relates to ship nonlinear motion analysis, in particular to a nonlinear dynamics calculation analysis method for transverse throwing of a ship under the excitation of regular waves.
Background
The vessel's wave/roll motion is one of the failure modes in the vessel's second generation of integrity stability. During the wave/transverse motion of the ship, the nonlinear dynamics of the ship are most remarkable in the unstable mode of the ship. The unstable mode of the transverse throwing motion of the ship means that when the ship sails in waves, the rudder loses the steering action on the ship, the ship moves with the large bow turning, and the ship may roll with the large bow turning in the process.
For the transverse throwing motion of the ship in the regular wave, a numerical analysis method is adopted in most researches. The method is direct in calculation, can consider more complex factors in the model, but only can calculate and consider one working condition every time, and is not easy to draw a general conclusion. Therefore, an efficient and accurate calculation research method is sought, and the understanding of the transverse throwing motion rule of the ship in the regular wave is improved. A plurality of researches show that the nonlinear dynamics method has the advantages of accurate, simple and rapid calculation and can intuitively obtain the stability domain of the transverse throwing motion of the ship in the regular wave. However, the application of nonlinear dynamics such as bifurcation to this problem is rare.
Disclosure of Invention
The invention aims to solve the limitation of a numerical method on the nonlinear transverse throwing motion of a ship in a regular wave, and provides a nonlinear dynamics analysis method.
The technical scheme adopted by the invention is as follows: a ship transverse-throwing nonlinear dynamics analysis method in regular waves comprises the following steps:
step 1, establishing a differential equation of the ship's yawing motion in a regular wave, and determining coefficients in the differential equation of the ship's yawing motion;
step 2, determining the boundary of a stable domain and an unstable domain of the ship for generating the transverse throwing motion according to a nonlinear dynamics theory, obtaining a stable domain schematic diagram of a ship yawing motion differential equation solution, and obtaining a theoretical critical value of a parameter excitation term coefficient in the yawing motion differential equation according to the stable domain schematic diagram of the ship yawing motion differential equation solution;
step 3, calculating the bifurcation characteristic of the ship heading motion by adopting a numerical method based on a bifurcation nonlinear dynamics theory to obtain a bifurcation diagram of the ship heading angle along with the wave height, and obtaining a numerical wave height critical value according to the bifurcation diagram of the ship heading angle along with the wave height so as to determine the reliability of the theoretical critical value of the parameter excitation item coefficient obtained in the step 2;
and 4, determining the critical condition of the ship for rolling according to the stability domain schematic diagram of the ship yawing motion differential equation solution obtained in the step 2, and reasonably forecasting the stability of the yawing motion in the ship waves.
Further, in step 1, the heading motion differential equation is as follows:
Figure GDA0003652325390000021
where ψ denotes a heading angle, F denotes a wave moment in a heading direction to which the ship is subjected, δ denotes a rudder angle, K, T denotes a ship steerability index, where,
Figure GDA0003652325390000022
i represents the ship body yawing moment of inertia, M represents the induced moment coefficient acting on the ship body, and N represents the damping coefficient of the ship body yawing motion;
each coefficient in the differential equation of the yawing motion is ship yawing moment inertia I, damping coefficient N of the ship yawing motion, induced moment coefficient M acting on the ship, wave moment F and rudder angle delta in the yawing motion direction received by the ship and ship maneuverability index K, T;
wherein the vessel maneuverability index K, T is obtained according to an empirical formula;
the rudder angle δ is shown in formula (2):
Figure GDA0003652325390000023
in the formula, k1、k2For rudder control parameters, #rRepresenting a target heading angle;
the wave moment F in the heading motion direction to which the vessel is subjected is determined according to the following method: calculating the heading wave moments under different wave directions by adopting hydrodynamic calculation software SESAM, giving the encounter frequency of the ship in actual sea conditions, interpolating the result of the wave moments to obtain first-order heading wave moments corresponding to a plurality of wave directions under the encounter frequency, arranging the wave moments from small to large according to wave angles, fitting the wave moments, and arranging the slope of the wave momentsIs the coefficient F0Corresponding values, the wave moment F is fitted to the form shown in equation (3):
F＝F0ψcos(ωet) (3)
in the formula, omegaeTo encounter frequency, t is time.
Further, step 2 further comprises:
substituting the formula (2) and the formula (3) into the formula (1) to obtain:
Figure GDA0003652325390000031
according to the nonlinear dynamics theory, introducing a yaw deviation angle psi△I.e. psi△＝ψ-ψrThen, formula (4) is converted to the following form:
Figure GDA0003652325390000032
in the formula (I), the compound is shown in the specification,
Figure GDA0003652325390000033
natural frequency of bow and roll motion;
Figure GDA0003652325390000034
is the coefficient of the damping term;
Figure GDA0003652325390000035
for a coefficient of the parametric excitation term, h is a quantity related to the wave height for a given vessel;
Figure GDA0003652325390000036
coefficients for the outer excitation term;
introducing a time scale tau-omega0t, changing τ to ω0Substitution of t for formula (5) yields the following expression:
Figure GDA0003652325390000037
in the formula (I), the compound is shown in the specification,
Figure GDA0003652325390000038
is a frequency ratio;
Figure GDA0003652325390000039
coefficients that are transformed damping terms;
Figure GDA00036523253900000310
coefficients that are transformed external excitation terms;
introducing a small parameter epsilon, and rearranging the formula (6) to obtain:
Figure GDA00036523253900000311
researching the sub-harmonic resonance of the heading motion, enabling the frequency ratio omega to satisfy omega ≈ 2, introducing a frequency ratio tuning factor sigma to describe the approximation degree between the frequency ratio omega and 2, and enabling:
Figure GDA00036523253900000312
substituting formula (8) for formula (7) to obtain:
Figure GDA00036523253900000313
assuming the solution of equation (9) is:
ψ△＝ψ0(T0,T1)+εψ1(T0,T1) (10)
in the formula, #nIs an n-order perturbation solution, n is 0,1,2 …; t is0τ, a variable representing a time fast scale; t is1ε τ, a variable representing the time slow scale;
introducing a calculation formula of a classical multi-scale method:
Figure GDA0003652325390000041
Figure GDA0003652325390000042
in the formula (I), the compound is shown in the specification,
Figure GDA0003652325390000043
all represent partial derivatives over a time scale;
substituting the equations (10), (11) and (12) into the equation (9), and expanding according to the zero-order term and the first-order term of epsilon, a partial differential equation set shown in the equations (13) and (14) is obtained:
Figure GDA0003652325390000044
Figure GDA0003652325390000045
let the expression of the solution of equation (13) be:
Figure GDA0003652325390000046
wherein A is a complex function; i is an imaginary unit;
substituting equation (15) into equation (14) yields:
Figure GDA0003652325390000047
in the formula (I), the compound is shown in the specification,
Figure GDA0003652325390000048
Figure GDA0003652325390000049
according to the elimination for a perpetual yearThe term (16) is set to zero for the perpetual term as shown in formula (17):
Figure GDA00036523253900000410
after eliminating the perpetual term, a first order approximation solution of equation (9) is obtained:
Figure GDA00036523253900000411
in the formula, θ is a variable representing the phase, and the variable θ and the variable r are determined by the formula (19) and the formula (20):
Figure GDA0003652325390000051
Figure GDA0003652325390000052
because formula (21) is considered:
Figure GDA0003652325390000053
substituting equation (17) into equation (21) yields:
Figure GDA0003652325390000054
let the approximate solution of equation (22) be
A＝Br+iBiε (23)
In the formula, Br、BiAll are expressed as real numbers, and after equation (23) is substituted into equation (22), the real and imaginary parts of the equation are separated, so that:
Figure GDA0003652325390000055
let the solutions of equation (24) be:
Figure GDA0003652325390000056
in the formula, brAnd biAnd v is a constant, and is a boundary between the stable domain and the unstable domain, formula (25) is substituted into formula (24), and a small parameter epsilon is made to be 1, so that:
Figure GDA0003652325390000057
according to the formula (26), drawing the boundary of a stable domain and an unstable domain of the ship in the transverse swinging motion by taking the frequency ratio omega as a horizontal coordinate and taking the coefficient h of the parameter excitation term as a vertical coordinate, wherein the boundary of the stable domain and the unstable domain is a curve, the upper region of the curve and the upper region of the curve are unstable domains of the ship yawing motion differential equation solution, and the lower region of the curve is stable domains of the ship yawing motion differential equation solution;
for a determined sea state, the encounter frequency ω of the shipeIs determined while, for the vessel, the bow-like natural frequency omega0Also determined, then, in the determined sea state, the value of the frequency ratio Ω is determined; and (3) setting the coordinates of the lowest points of the curves shown by the boundaries of the stable regions and the unstable regions of the ship which are in the transverse throwing motion as (x, y), wherein x is the value of the frequency ratio omega under the sea condition, and y is the theoretical critical value of the parameter excitation term coefficient h.
Further, step 3 further comprises:
firstly, selecting wave height of waves as a bifurcation parameter, determining a calculation range of the wave height, and solving a heading motion differential equation of a ship in regular waves by sequentially adopting a Runge Kutta method for each wave height to obtain a time history chart of a heading angle and a heading angular velocity, wherein the time is a horizontal coordinate, and the heading angle and the heading angular velocity of the ship are vertical coordinates;
secondly, selecting a section of middle part data in a time history chart of the ship bow angle and the bow angular velocity, wherein the ordinate is the bow angular velocity of the ship, and the abscissa is the bow angle of the ship, so as to obtain a ship motion phase diagram;
then, according to the obtained ship motion phase diagram, selecting a section with zero ship bow angular velocity to obtain a bifurcation diagram of the ship bow angular velocity along with the change of the wave height, and in the drawing of the bifurcation diagram, when the wave height is lower than a certain value, the ship generates stable periodic bow motion, and a closed circle can be drawn on the bifurcation diagram under the wave height; when the wave height is equal to a certain value, the heading motion of the ship begins to diverge and is not stable reciprocating periodic heading motion any more, at the moment, a closed circle can not be drawn on the bifurcation diagram any more, and the value is a numerical wave height critical value;
and finally, obtaining a numerical critical value of the parameter excitation item coefficient corresponding to the numerical wave height critical value according to the relation between the coefficient h of the parameter excitation item and the wave height, and according to the calculation result, obtaining that the theoretical critical value of the parameter excitation item coefficient is equal to the numerical critical value of the parameter excitation item coefficient, thereby proving that the theoretical critical value of the parameter excitation item coefficient obtained in the step 2 is reliable.
Further, in step 4, the critical conditions for the ship to transverse throw are as follows: and when the parameter excitation term coefficient h reaches a value near a theoretical critical value of the parameter excitation term coefficient, the transverse throwing instability motion of the ship can occur.
The invention has the beneficial effects that:
1. the nonlinear dynamics calculation analysis method of the ship under the excitation of the regular wave can accurately forecast the motion instability condition of the ship transversely swinging in the regular wave, and solves the problem of solving the motion instability condition of the transversely swinging;
2. according to the method, the calculation result can directly obtain the transverse throwing motion stability domain of the ship in the regular wave according to modern nonlinear dynamics theory methods such as bifurcation and the like, the solving precision is high, and the efficiency is high;
3. the nonlinear dynamics calculation analysis method of the ship under the excitation of the regular wave is suitable for various ship types and has strong universality;
4. the invention belongs to an analysis theory and a method of movement in ship waves, and is used for two aspects of ship design and ship navigation.
Drawings
FIG. 1: the invention relates to a flow chart of a ship transverse throwing nonlinear dynamics analysis method in a regular wave;
FIG. 2: a bow-shake 1-order wave moment transfer function;
FIG. 3: the stability domain schematic diagram of the ship transverse throwing motion equation solution is obtained;
FIG. 4: the invention obtains a bifurcation diagram of the ship bow angle along with the change of the wave height.
Detailed Description
In order to further understand the contents, features and effects of the present invention, the following embodiments are illustrated and described in detail with reference to the accompanying drawings:
the invention combines the ship motion theory and the nonlinear dynamics theory, provides a new measure index and a new concept of ship motion instability, and establishes an analysis method and an analysis process of nonlinear dynamics.
As shown in fig. 1, a method for analyzing the nonlinear dynamics of a ship transverse throwing in a regular wave comprises the following steps:
1. firstly, establishing a bow motion differential equation of a ship in a regular wave:
Figure GDA0003652325390000071
where ψ denotes a heading angle, F denotes a wave moment in a heading direction to which the ship is subjected, δ denotes a rudder angle, K, T denotes a ship steerability index, where,
Figure GDA0003652325390000072
i represents the ship body yawing moment of inertia, M represents the induced moment coefficient acting on the ship body, and N represents the damping coefficient of the ship body yawing motion.
Secondly, determining coefficients of each item in a ship yawing motion differential equation: the ship body yawing moment I, the damping coefficient N of the ship body yawing motion, the induced moment coefficient M acting on the ship body, the wave moment F in the yawing motion direction received by the ship, the rudder angle delta and the ship maneuverability index K, T.
Wherein the ship maneuverability index K, T is obtained according to an empirical formula[1]。
The rudder angle δ is as shown in formula (2):
Figure GDA0003652325390000073
in the formula, k1、k2For rudder control parameters, #rIndicating the target heading angle.
The wave moment F in the heading direction of the ship is calculated by using hydrodynamic calculation software SESAM, as shown in fig. 2. Giving the encounter frequency of the ship in the actual sea condition, interpolating the results of the wave moments to obtain first-order yawing wave moments corresponding to a plurality of waves under the encounter frequency, arranging the wave moments from small to large according to wave angles, fitting the wave moments, wherein the slope is a coefficient F0Corresponding values, the wave moment F is fitted to the form shown in equation (3):
F＝F0ψcos(ωet) (3)
in the formula, ωeTo encounter frequency, t is time.
2. According to a nonlinear dynamics theory, determining the boundary of a stable domain and an unstable domain of the ship for generating the transverse swinging motion, obtaining a stable domain schematic diagram of a ship yawing motion differential equation solution, and obtaining a theoretical critical value of a parameter excitation term coefficient in the yawing motion differential equation according to the stable domain schematic diagram of the ship yawing motion differential equation solution. The method specifically comprises the following steps:
substituting the formula (2) and the formula (3) into the formula (1) to obtain:
Figure GDA0003652325390000081
according to the nonlinear dynamics theory, introducing a yaw deviation angle psi△I.e. psi△＝ψ-ψrThen, the formula (4) can be converted into the following form:
Figure GDA0003652325390000082
in the formula (I), the compound is shown in the specification,
Figure GDA0003652325390000083
natural frequency of bow and roll motion;
Figure GDA0003652325390000084
is the coefficient of the damping term;
Figure GDA0003652325390000085
for a coefficient of the parametric excitation term, h is a quantity related to the wave height for a given vessel;
Figure GDA0003652325390000086
are coefficients of the external excitation term.
Introducing a time scale tau-omega0t, changing τ to ω0Substitution of t for formula (5) yields the following expression:
Figure GDA0003652325390000087
in the formula (I), the compound is shown in the specification,
Figure GDA0003652325390000088
is a frequency ratio;
Figure GDA0003652325390000089
coefficients that are transformed damping terms;
Figure GDA00036523253900000810
to change overCoefficients of the latter external excitation term.
And (3) introducing the small parameter epsilon, and rearranging the formula (6) to obtain the small parameter epsilon, wherein the small parameter epsilon is 1 after the calculation is finished and before the formula (26) is obtained:
Figure GDA0003652325390000091
researching the subharmonic resonance of the heading motion, enabling the ratio of the encounter frequency to the inherent frequency of the heading motion to satisfy omega ≈ 2, introducing a frequency ratio tuning factor sigma to describe the approximation degree between the frequency ratio omega and 2, and enabling:
Figure GDA0003652325390000092
substituting formula (8) for formula (7) to obtain:
Figure GDA0003652325390000093
assuming the solution of equation (9) is:
ψ△＝ψ0(T0,T1)+εψ1(T0,T1) (10)
in the formula, #nIs an n-order perturbation solution, n is 0,1,2 …; t is0τ, a variable representing a time fast scale; t is1ε τ, a variable representing the time slow scale;
introducing a calculation formula of a classical multi-scale method:
Figure GDA0003652325390000094
Figure GDA0003652325390000095
in the formula (I), the compound is shown in the specification,
Figure GDA0003652325390000096
both represent partial derivatives over the time scale.
By substituting equations (10), (11) and (12) into equation (9) and expanding according to the zero-order term and the first-order term of ∈, a partial differential equation set as shown in equations (13) and (14) can be obtained:
Figure GDA0003652325390000097
Figure GDA0003652325390000098
let the expression of the solution of equation (13) be:
Figure GDA0003652325390000099
wherein A is a complex function; i is an imaginary unit.
Substituting equation (15) into equation (14) yields:
Figure GDA0003652325390000101
where cc represents the sum of the conjugate terms of the preceding terms, i.e.,
Figure GDA0003652325390000102
Figure GDA0003652325390000103
according to the condition of eliminating the perpetual term, the perpetual term in the formula (16) is made to be zero, as shown in the formula (17):
Figure GDA0003652325390000104
after eliminating the perpetual term, a first order approximation solution of formula (9) can be obtained[2]：
Figure GDA0003652325390000105
In the formula, θ is a variable representing the phase, and the variable θ and the variable r are determined by the formula (19) and the formula (20):
Figure GDA0003652325390000106
Figure GDA0003652325390000107
because formula (21) is considered:
Figure GDA0003652325390000108
substituting equation (17) into equation (21) yields:
Figure GDA0003652325390000109
let the approximate solution of equation (22) be
A＝Br+iBiε (23)
In the formula, Br、BiAll are expressed as real numbers, and after equation (23) is substituted into equation (22), the real part and imaginary part of the equation are separated, and the following can be obtained:
Figure GDA0003652325390000111
let the solutions of equation (24) be:
Figure GDA0003652325390000112
in the formula, brAnd biAnd v is a constant, and is a boundary between the stable domain and the unstable domain, formula (25) is substituted into formula (24), and a small parameter epsilon is made to be 1, so that:
Figure GDA0003652325390000113
the formula (26) is to solve the heading motion equation of the ship by adopting a multi-scale method, determine the stable domain and the unstable domain of the ship in the transverse throwing motion and obtain the boundary of the stable domain and the unstable domain.
According to the formula (26), the boundary between the stable domain and the unstable domain of the ship in the transverse throwing motion is drawn by taking the frequency ratio omega as the abscissa and the coefficient h of the parameter excitation term as the ordinate, so as to obtain a stable domain schematic diagram of the ship transverse throwing motion equation solution, as shown in fig. 3. The boundary of the stable region and the unstable region is a curve, the upper region of the curve and the curve is the unstable region of the ship yawing motion differential equation solution, namely the unstable condition of the ship yawing motion, and the lower region of the curve is the stable region of the ship yawing motion differential equation solution.
As can be seen in fig. 3, the coordinates of the lowest point of the curve are (x, y). For a determined sea state, the encounter frequency ω of the shipeIs determined while, for the vessel, the bow-like natural frequency omega0Also determined, then, under the determined sea condition, the value of the frequency ratio Ω is determined, then, x is the value of the frequency ratio Ω under the sea condition, and y is the theoretical critical value of the parameter excitation term coefficient h.
3. Based on the bifurcation nonlinear dynamics theory, calculating the bifurcation characteristic of the ship heading motion by adopting a numerical method to obtain a bifurcation diagram of the ship heading angle along with the wave height, and obtaining a numerical wave height critical value according to the bifurcation diagram of the ship heading angle along with the wave height so as to determine the reliability of the theoretical critical value of the parameter excitation item coefficient obtained in the step 2. The method specifically comprises the following steps:
firstly, selecting wave height of waves as a bifurcation parameter, determining a calculation range of the wave height, and solving a heading motion differential equation of the ship in regular waves by sequentially adopting a Runge Kutta method for each wave height to obtain a time history chart of a heading angle and a heading angular velocity, wherein the time is a horizontal coordinate, and the heading angle and the heading angular velocity of the ship are vertical coordinates.
Secondly, selecting a section of middle part data in a time history chart of the ship's heading angle and the ship's heading angle speed, wherein the ordinate is the ship's heading angle speed, and the abscissa is the ship's heading angle, and obtaining a ship motion phase diagram.
Then, according to the obtained ship motion phase diagram, selecting a section with zero ship bow angular velocity to obtain a bifurcation diagram of the ship bow angular velocity along with the change of the wave height, as shown in fig. 4. In the drawing of the bifurcation diagram, when the wave height is lower than a certain value, the ship generates stable periodic yawing motion, and a closed circle can be drawn on the bifurcation diagram under the wave height; when the wave height is equal to a certain value, the heading motion of the ship begins to diverge and is not stable reciprocating periodic heading motion any more, at the moment, a closed circle can not be drawn on the bifurcation diagram any more, and the value is the critical value of the numerical wave height.
And finally, obtaining a numerical critical value of the parameter excitation item coefficient corresponding to the numerical wave height critical value according to the relation between the coefficient h of the parameter excitation item and the wave height, and according to the calculation result, obtaining that the theoretical critical value of the parameter excitation item coefficient is equal to the numerical critical value of the parameter excitation item coefficient, thereby proving that the theoretical critical value of the parameter excitation item coefficient obtained in the step 2 is reliable.
4. And (3) determining the critical condition of the ship for generating the transverse swinging according to the stability domain schematic diagram of the ship yawing motion differential equation solution obtained in the step (2), and reasonably forecasting the stability of the yawing motion in the ship waves.
As shown in fig. 3, the coordinates (x, y) of the lowest point of the curve are expressed that when the frequency ratio Ω is x, the parameter excitation term h in the ship yawing motion equation will generate the rolling and unstable motion of the ship when reaching the value near y (y is the theoretical critical value of the parameter excitation term coefficient as shown above), and the rolling and unstable motion of the ship is not easy to occur under other frequency ratios. By stability analysis of the solution, the critical conditions under which the cross-throw occurs can be determined. The critical condition of the ship with a certain frequency ratio for transverse swinging can be clearly seen through the stability domain schematic diagram of the ship transverse swinging motion equation solution, the result is proved to be convenient and easy to implement, and the method is a feasible ship transverse swinging motion nonlinear dynamics calculation analysis method.
By the method, the stability of the bow rolling motion in the ship waves and whether the ship is flapped can be reasonably forecasted.
In summary, the invention provides a ship transverse throwing nonlinear dynamics analysis method in regular waves based on ship and ocean engineering hydrodynamics and nonlinear dynamics theories, and the purpose of rapidly solving unstable motion conditions of ships through nonlinear dynamics is achieved. Aiming at the nonlinear problem of the transverse throwing motion of the ship in the regular wave, the unstable dynamic condition of the ship in transverse throwing is determined in a mode of combining numerical analysis and a nonlinear dynamics theory, so that the possible motion risk of the ship can be forecasted. At present, no such analysis method exists at home and abroad.
Reference:
[1] plum waves, zhangobviously bank, zhangyang. vessel maneuverability index K, T based on SPSS technology forecasts [ J ]. marine technology, 2007 (5): 2-5.
[2] Liu towns, ship riding wave transverse throwing and pure stability loss state nonlinear dynamics characteristic research [ D ]. Tianjin university 2020.
Although the preferred embodiments of the present invention have been described above with reference to the accompanying drawings, the present invention is not limited to the above-described embodiments, which are merely illustrative and not restrictive, and those skilled in the art can make many modifications without departing from the spirit and scope of the present invention as defined in the appended claims.

Claims (5)

1. A ship transverse throwing nonlinear dynamics analysis method in regular waves is characterized by comprising the following steps:
step 1, establishing a differential equation of the ship's yawing motion in a regular wave, and determining coefficients in the differential equation of the ship's yawing motion;
step 2, determining the boundary of a stable domain and an unstable domain of the ship in the transverse swinging motion according to a nonlinear dynamics theory, obtaining a stable domain schematic diagram of a ship yawing motion differential equation solution, and obtaining a theoretical critical value of a parameter excitation term coefficient in the yawing motion differential equation according to the stable domain schematic diagram of the ship yawing motion differential equation solution;
step 3, calculating the bifurcation characteristic of the ship heading motion by adopting a numerical method based on a bifurcation nonlinear dynamics theory to obtain a bifurcation diagram of the ship heading angle along with the wave height, and obtaining a numerical wave height critical value according to the bifurcation diagram of the ship heading angle along with the wave height so as to determine the reliability of the theoretical critical value of the parameter excitation item coefficient obtained in the step 2;
and 4, determining a critical condition for the transverse throwing of the ship according to the stability domain schematic diagram of the differential equation solution of the ship heading motion obtained in the step 2, and reasonably forecasting the stability of the heading motion in the waves of the ship.
2. The method for analyzing nonlinear dynamics of ship transverse throwing in regular waves according to claim 1, wherein in step 1, the differential equation of heading motion is as follows:
Figure FDA0003652325380000011
where ψ denotes a heading angle, F denotes a wave moment in a heading direction to which the ship is subjected, δ denotes a rudder angle, K, T denotes a ship steerability index, where,
Figure FDA0003652325380000012
i represents the ship body yawing moment of inertia, M represents the induced moment coefficient acting on the ship body, and N represents the damping coefficient of the ship body yawing motion;
each coefficient in the differential equation of the yawing motion is ship yawing moment inertia I, damping coefficient N of the ship yawing motion, induced moment coefficient M acting on the ship, wave moment F and rudder angle delta in the yawing motion direction received by the ship and ship maneuverability index K, T;
wherein the vessel maneuverability index K, T is obtained according to an empirical formula;
the rudder angle δ is as shown in formula (2):
Figure FDA0003652325380000013
in the formula, k1、k2For rudder control parameters, #rRepresenting a target heading angle;
the wave moment F in the heading motion direction to which the vessel is subjected is determined according to the following method: calculating the heading wave moments under different wave directions by adopting hydrodynamic calculation software SESAM, interpolating the result of the wave moments by giving the encounter frequency of the ship in actual sea conditions to obtain first-order heading wave moments corresponding to a plurality of wave directions under the encounter frequency, arranging the wave moments from small to large according to wave angles, fitting the wave moments, wherein the slope is a coefficient F0Corresponding values, the wave moment F is fitted to the form shown in equation (3):
F＝F0ψcos(ωet) (3)
in the formula, ωeTo encounter frequency, t is time.
3. The method for analyzing nonlinear dynamics of ship cross-throwing in regular waves according to claim 2, wherein the step 2 further comprises:
substituting the formula (2) and the formula (3) into the formula (1) to obtain:
Figure FDA0003652325380000021
according to the nonlinear dynamics theory, introducing a yaw deviation angle psi△I.e. psi△＝ψ-ψrThen, formula (4) is converted to the following form:
Figure FDA0003652325380000022
in the formula (I), the compound is shown in the specification,
Figure FDA0003652325380000023
natural frequency of bow and roll motion;
Figure FDA0003652325380000024
is the coefficient of the damping term;
Figure FDA0003652325380000025
is the coefficient of the parametric excitation term, h is a quantity related to the wave height for a given vessel;
Figure FDA0003652325380000026
coefficients for the outer excitation term;
introducing a time scale tau-omega0t, changing τ to ω0Substitution of t for formula (5) yields the following expression:
Figure FDA0003652325380000027
in the formula (I), the compound is shown in the specification,
Figure FDA0003652325380000028
is a frequency ratio;
Figure FDA0003652325380000029
coefficients that are transformed damping terms;
Figure FDA00036523253800000210
coefficients that are transformed external excitation terms;
introducing a small parameter epsilon, and rearranging the formula (6) to obtain:
Figure FDA00036523253800000211
researching the sub-harmonic resonance of the yawing motion, enabling the frequency ratio omega to meet omega ≈ 2, introducing a frequency ratio tuning factor sigma to describe the approximation degree between the frequency ratio omega and 2, and enabling:
Figure FDA0003652325380000031
substituting formula (8) for formula (7) to obtain:
Figure FDA0003652325380000032
assuming the solution of equation (9) is:
ψ△＝ψ0(T0,T1)+εψ1(T0,T1) (10)
in the formula, #nIs an n-order perturbation solution, n is 0,1,2 …; t is0τ, a variable representing a time fast scale; t is1ε τ, a variable representing the time slow scale;
introducing a calculation formula of a classical multi-scale method:
Figure FDA0003652325380000033
Figure FDA0003652325380000034
in the formula (I), the compound is shown in the specification,
Figure FDA0003652325380000035
all represent partial derivatives over a time scale;
substituting the equations (10), (11) and (12) into the equation (9), and expanding according to the zero-order term and the first-order term of epsilon, a partial differential equation set shown in the equations (13) and (14) is obtained:
Figure FDA0003652325380000036
Figure FDA0003652325380000037
let the expression of the solution of equation (13) be:
Figure FDA0003652325380000038
wherein A is a complex function; i is an imaginary unit;
substituting equation (15) into equation (14) yields:
Figure FDA0003652325380000039
in the formula (I), the compound is shown in the specification,
Figure FDA0003652325380000041
Figure FDA0003652325380000042
according to the condition of eliminating the perpetual term, the perpetual term in the formula (16) is made to be zero, as shown in the formula (17):
Figure FDA0003652325380000043
after eliminating the perpetual term, a first order approximation solution of equation (9) is obtained:
Figure FDA0003652325380000044
in the formula, θ is a variable representing the phase, and the variable θ and the variable r are determined by the formula (19) and the formula (20):
Figure FDA0003652325380000045
Figure FDA0003652325380000046
because formula (21) is considered:
Figure FDA0003652325380000047
substituting equation (17) into equation (21) yields:
Figure FDA0003652325380000048
let the approximate solution of equation (22) be
A＝Br+iBiε (23)
In the formula, Br、BiAll are expressed as real numbers, and after equation (23) is substituted into equation (22), the real and imaginary parts of the equation are separated, and are given as:
Figure FDA0003652325380000049
let the solutions of equation (24) be:
Figure FDA00036523253800000410
in the formula, brAnd biAnd v is a constant, and is a boundary between the stable domain and the unstable domain, formula (25) is substituted into formula (24), and a small parameter epsilon is made to be 1, so that:
Figure FDA0003652325380000051
according to the formula (26), drawing the boundary of a stable domain and an unstable domain of the ship in the transverse swinging motion by taking the frequency ratio omega as a horizontal coordinate and taking the coefficient h of the parameter excitation term as a vertical coordinate, wherein the boundary of the stable domain and the unstable domain is a curve, the upper region of the curve and the upper region of the curve are unstable domains of the ship yawing motion differential equation solution, and the lower region of the curve is stable domains of the ship yawing motion differential equation solution;
for a determined sea state, the encounter frequency ω of the shipeIs determined while, for the vessel, the bow-like natural frequency omega0Also determined, then, in the determined sea state, the value of the frequency ratio Ω is determined; and (3) setting the coordinates of the lowest points of the curves shown by the boundaries of the stable regions and the unstable regions of the ship which are in the transverse throwing motion as (x, y), wherein x is the value of the frequency ratio omega under the sea condition, and y is the theoretical critical value of the parameter excitation term coefficient h.
4. The method for analyzing nonlinear dynamics of ship cross-throwing in regular waves according to claim 1, wherein the step 3 further comprises:
firstly, selecting wave height of waves as a bifurcation parameter, determining a calculation range of the wave height, and solving a heading motion differential equation of a ship in regular waves by sequentially adopting a Runge Kutta method for each wave height to obtain a time history chart of a heading angle and a heading angular velocity, wherein the time is a horizontal coordinate, and the heading angle and the heading angular velocity of the ship are vertical coordinates;
secondly, selecting a section of middle part data in a time history chart of the ship bow angle and the bow angular velocity, wherein the ordinate is the bow angular velocity of the ship, and the abscissa is the bow angle of the ship, so as to obtain a ship motion phase diagram;
then, according to the obtained ship motion phase diagram, selecting a section with zero ship bow angular velocity to obtain a bifurcation diagram of the ship bow angular velocity along with the change of the wave height, and in the drawing of the bifurcation diagram, when the wave height is lower than a certain value, the ship generates stable periodic bow motion, and a closed circle can be drawn on the bifurcation diagram under the wave height; when the wave height is equal to a certain value, the heading motion of the ship begins to diverge and is not stable reciprocating periodic heading motion any more, at the moment, a closed circle can not be drawn on the bifurcation diagram any more, and the value is a numerical wave height critical value;
and finally, obtaining a numerical critical value of the parameter excitation item coefficient corresponding to the numerical wave height critical value according to the relation between the coefficient h of the parameter excitation item and the wave height, and according to the calculation result, obtaining that the theoretical critical value of the parameter excitation item coefficient is equal to the numerical critical value of the parameter excitation item coefficient, thereby proving that the theoretical critical value of the parameter excitation item coefficient obtained in the step 2 is reliable.
5. The method for analyzing nonlinear dynamics of ship transverse throwing in regular waves according to claim 1, wherein in step 4, the critical conditions for ship transverse throwing are as follows: and when the parameter excitation term coefficient h reaches a value near a theoretical critical value of the parameter excitation term coefficient, the transverse throwing instability motion of the ship can occur.
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