CN104657611B - Naval vessel entirety elastoplasticity motion response Forecasting Methodology and system under underwater explosive damage - Google Patents

Abstract

The invention discloses naval vessel entirety elastoplasticity motion response Forecasting Methodology and system under a kind of underwater explosive damage, method includes step: S1, naval vessel is reduced to the free hull beam of uiform section；S2, hull beam load pressure free under underwater explosive damage is divided into five time phases；S3, solve free hull beam and explode under water the moving displacement in five time phases；S4, solve the moment of flexure of hull beam；S5, solve hull beam forward plasticity kinematic parameter；S6, solve hull beam reverse elasticity unloading kinematic parameter；S7, solve hull beam reverse plasticity kinematic parameter；S8, solve hull beam forward elastic movement parameter.Present invention also offers the system realizing said method.The present invention analyzes naval vessel plasticity motor process on the basis of considering shock wave and the bubble load percussion to Marine campaign system, can accurately and easily realize the engineering prediction of naval vessel entirety elastoplasticity motion deformation under underwater explosive damage.

Description

Ship overall elastic-plastic motion response prediction method and system under underwater explosion effect

Technical Field

The invention belongs to the technical field of resisting underwater explosion impact deformation of a ship overall structure, and particularly relates to a ship overall elastic-plastic motion response prediction method and system under the action of underwater explosion.

Background

In modern sea operations, underwater explosion attacks suffered by ships are mainly based on short-range effects, wherein impact effects formed by underwater short-range non-contact explosion are distributed on the whole ships, and large-range equipment damage and even ship whole breakage are often caused. The instant breaking and sinking event of the Korean 'Tianan number' protective ship which occurs at 26.3.2010 due to underwater explosion attack can be regarded as a typical example of the breaking and the breaking of the ship caused by underwater short-distance non-contact explosion.

For ships and warships under the action of underwater short-distance non-contact explosion, the ships and warships are firstly subjected to the action of shock waves, and then the structure is subsequently influenced by the pulsation of explosion bubbles and radiation pressure. Considering the precedence relationship between the shock wave and the bubble load, the existing research usually separately studies the shock wave and the bubble load or directly neglects the influence of the shock wave on the whole deformation of the structure, but in terms of the physical process, the structural response caused by the shock wave should be used as the initial input condition for calculating the structural response caused by the subsequent bubbles, and the influence should not be neglected. In addition, the underwater short-distance explosion action can cause obvious integral deformation of the ship, and sagging or mid-arch bending damage can occur according to different explosion working conditions, which is proved by actual ship explosion or model test research. At present, a set of theoretical calculation method for calculating the repeated arching or sagging plastic deformation of ships by completely considering the combined action of the blast shock wave and the air bubbles is lacked in China. In addition, based on the established ship overall response calculation method, the ship overall response characteristics and the damage mode are further researched, and the method has important significance for improving the design level of ship underwater explosion impact resistance protection, particularly for guiding optimization design of underwater weapon attack efficiency and attack mode selection.

Disclosure of Invention

The invention provides a response prediction method and a system for the ship bulk elastoplastic motion under the action of underwater explosion, which comprehensively consider the impact action of shock waves and bubble loads on the ship bulk, can reasonably reflect the response process of repeated loading and unloading of the ship, and can more simply, conveniently and accurately realize the engineering prediction of the ship elastoplastic motion deformation under the action of short-distance explosion.

In order to achieve the above object, according to the present invention, there is provided a method for predicting an overall elastoplastic motion response of a ship under the action of an underwater explosion, the method comprising the steps of:

s1, equating the ship to be a free ship beam with an equal section, so that the prediction of ship motion response is equivalent to the prediction of free ship beam motion response;

s2, dividing the load pressure process of the free hull beam under the underwater explosion into five time stages, and collecting the pressure peak value P of the shock wave_mNegative pressure peak value P in bubble pulsation stage_bThe first pulsating pressure peak value P of the air bubble_s(ii) a The five time periods are: t is more than or equal to 0 and less than t₁、t₁≤t＜t₂、t₂≤t＜t₃、t₃≤t＜t₄、t₄≤t＜t₅(ii) a Wherein t is₁θ is the shock wave attenuation constant; P₀is the hydrostatic pressure at the explosive, P_atmIs atmospheric pressure, c is the speed of sound in water, r₀The radius of charge is R is the detonation distance; m_efor charge equivalent, p_wIs the density of water, g is the acceleration of gravity; $t_5=3290\frac{r_0}{P_0^{0.71}}+t_4;$

s3, solving the motion displacement w (x, t) of the free hull beam in five time stages of underwater explosion:

wherein x is the abscissa value of the free hull beam at any point,k₂＝t₂-t₁， is the first-order natural vibration shape of the free hull beam,wherein ζ₁Amplitude of vibration, μ₁Is a beam motion frequency parameter, l is the free hull beam length;is an integration constant, ω, of five time phases₁The first-order mode shape natural frequency of the free beam; phi is a₁In order to be an integration constant, the first,m is the mass per unit beam length considering the mass of the attached water, p (x) is a pressure distribution characteristic function of the free hull beam in different time stages, and t is more than or equal to 0 and less than t₁、t₁≤t＜t₂The time period is a period of time,for the t-th₂≤t＜t₃、t₃≤t＜t₄、t₄≤t＜t₅The time period is a period of time, $p\left(x\right)=(1-\frac{2x-l}{l})\·\exp [-8{\left(\frac{2x-l}{l}\right)}^2+4{\left(\frac{2x-1}{l}\right)}^3]+{\left(\frac{2l-2x}{l}\right)}^{1.5}\·\left(\frac{2x-l}{l}\right);$

s4, solving the speed and bending moment values of the hull beam at different stages according to the elastic movement displacement w (x, t) of the hull beam at five time stages;

s5, when the absolute value of the midpoint bending moment of the hull beam exceeds the plastic limit bending moment M of the hull beam_sAnd (3) when the absolute value is obtained, the ship body beam enters forward plastic motion, and the motion amplitude H' (t) and the relative rotation angle α (t) when the ship body beam moves forward plastic are solved:

$\left\{\begin{array}{c}{\stackrel{\·\·}{H}}^{\′}\left(t\right)=\frac{3M_s-(l{\mathrm{\ξ}}_3+3{\mathrm{\ξ}}_4)p\left(t\right)}{(l{\mathrm{\ξ}}_1+3{\mathrm{\ξ}}_2)m}\\ \stackrel{\·\·}{\mathrm{\α}}\left(t\right)=\frac{24({\mathrm{\ξ}}_1{\mathrm{\ξ}}_4-{\mathrm{\ξ}}_2{\mathrm{\ξ}}_3)p\left(t\right)-24M_s{\mathrm{\ξ}}_1}{(l{\mathrm{\ξ}}_1+3{\mathrm{\ξ}}_2){\mathrm{ml}}^2}\end{array}\right.$

ξ₁＝λ₁(l)-λ₁(l/2)，ξ₂＝λ₂(l)-λ₂(l/2)-λ₁(l)·l/2，

ξ₃＝η₁(l/2)-η₁(l)，ξ₄＝η₂(l/2)-η₂(l)+η₁(l)·l/2；

s6, when the ship body beam is in the positive directionWhen the plastic deformation reaches the maximum value α (t), the forward plastic deformation stage is ended, and the reverse elastic unloading is carried out according to the formula $\left\{\begin{array}{c}\mathrm{\α}\left(t\right)(x-l/2)+{\mathrm{\φ}}_1\left(x\right)H^{\′}\left(t\right)={\mathrm{\φ}}_1\left(x\right)H\left(t\right)\\ \stackrel{\·}{\mathrm{\α}}\left(t\right)(x-l/2)+{\mathrm{\φ}}_1\left(x\right){\stackrel{\·}{H}}^{\′}\left(t\right)={\mathrm{\φ}}_1\left(x\right)\stackrel{\·}{H}\left(t\right)\end{array}\right.$ Solving the motion amplitude H (t) of the reverse elastic unloading process of the hull beam;

s7, in the process of the reverse elastic deformation of the ship body beam, when the absolute value of the bending moment at the midpoint of the beam exceeds the plastic limit bending moment M_sWhen the absolute value is obtained, the hull beam enters reverse plastic motion, and the motion amplitude H of the beam after entering reverse plastic deformation is solved₁' (t) and angle of relative rotation α₁(t)：

$\left\{\begin{array}{c}{\stackrel{\·\·}{H}}_1^{\′}\left(t\right)=\frac{3(-M_s)-(l{\mathrm{\ξ}}_3+3{\mathrm{\ξ}}_4)p\left(t\right)}{(l{\mathrm{\ξ}}_1+3{\mathrm{\ξ}}_2)m}\\ {\stackrel{\·\·}{\mathrm{\α}}}_1\left(t\right)=\frac{24({\mathrm{\ξ}}_1{\mathrm{\ξ}}_4-{\mathrm{\ξ}}_2{\mathrm{\ξ}}_3)p\left(t\right)-24\left({-M}_s\right){\mathrm{\ξ}}_1}{(l{\mathrm{\ξ}}_1+3{\mathrm{\ξ}}_2){\mathrm{ml}}^2}\end{array}\right.;$

S8, α when the reverse plastic deformation of the hull beam reaches the maximum₁(t) reaching a maximum value, the reverse plastic deformation phase ending, entering a forward elastic unloading, according to $\left\{\begin{array}{c}{\mathrm{\α}}_1\left(t\right)(x-l/2)+{\mathrm{\φ}}_1\left(x\right)H_1^{\′}\left(t\right)={\mathrm{\φ}}_1\left(x\right)H_1\left(t\right)\\ {\stackrel{\·}{\mathrm{\α}}}_1\left(t\right)(x-l/2)+{\mathrm{\φ}}_1\left(x\right){\stackrel{\·}{H}}_1^{\′}\left(t\right)={\mathrm{\φ}}_1\left(x\right){\stackrel{\·}{H}}_1\left(t\right)\end{array}\right.$ Solving the motion amplitude H of the ship body beam in the forward elastic unloading stage₁(t)。

Correspondingly, the invention also provides a system for predicting the overall elastic-plastic motion response of a ship under the action of underwater explosion, which comprises:

the first module is used for enabling the ship to be equivalent to a free ship beam with an equal section, so that the prediction of ship motion response is equivalent to the prediction of free ship beam motion response;

the second module is used for dividing the load pressure process of the free hull beam under the underwater explosion action into five time stages and acquiring a shock wave pressure peak value P_mNegative pressure peak value P in bubble pulsation stage_bThe first pulsating pressure peak value P of the air bubble_s(ii) a The five time periods are: t is more than or equal to 0 and less than t₁、t₁≤t＜t₂、t₂≤t＜t₃、t₃≤t＜t₄、t₄≤t＜t₅(ii) a Wherein t is₁θ is the shock wave attenuation constant; P₀is the hydrostatic pressure at the explosive, P_atmIs atmospheric pressure, c is the speed of sound in water, r₀The radius of charge is R is the detonation distance; m_efor charge equivalent, p_wIs the density of water, g is the acceleration of gravity; $t_5=3290\frac{r_0}{P_0^{0.71}}+t_4;$

and the third module is used for solving the motion displacement w (x, t) of the free hull beam in five time phases of underwater explosion:

wherein x is the abscissa value of the free hull beam at any point,k₂＝t₂-t₁， is the first-order natural vibration shape of the free hull beam,wherein ζ₁Amplitude of vibration, μ₁Is a beam motion frequency parameter, l is the free hull beam length; is an integration constant, ω, of five time phases₁The first-order mode shape natural frequency of the free beam; phi is a₁In order to be an integration constant, the first,m is the mass per unit beam length considering the mass of the attached water, p (x) is a pressure distribution characteristic function of the free hull beam in different time stages, and t is more than or equal to 0 and less than t₁、t₁≤t＜t₂The time period is a period of time,for the t-th₂≤t＜t₃、t₃≤t＜t₄、t₄≤t＜t₅The time period is a period of time, $p\left(x\right)=(1-\frac{2x-l}{l})\·\exp [-8{\left(\frac{2x-l}{l}\right)}^2+4{\left(\frac{2x-1}{l}\right)}^3]+{\left(\frac{2l-2x}{l}\right)}^{1.5}\·\left(\frac{2x-l}{l}\right);$

the fourth module is used for solving the speed and bending moment values of the hull beam at different stages according to the elastic movement displacement w (x, t) of the hull beam at five time stages;

a fifth module for determining the absolute value of the bending moment at the midpoint of the hull beam exceeds the plastic limit bending moment M of the hull beam_sAnd in absolute value, judging that the ship body beam enters forward plastic motion, and solving the motion amplitude H' (t) and the relative rotation angle α (t) when the ship body beam moves forward plastic motion:

$\left\{\begin{array}{c}{\stackrel{\·\·}{H}}^{\′}\left(t\right)=\frac{3M_s-(l{\mathrm{\ξ}}_3+3{\mathrm{\ξ}}_4)p\left(t\right)}{(l{\mathrm{\ξ}}_1+3{\mathrm{\ξ}}_2)m}\\ \stackrel{\·\·}{\mathrm{\α}}\left(t\right)=\frac{24({\mathrm{\ξ}}_1{\mathrm{\ξ}}_4-{\mathrm{\ξ}}_2{\mathrm{\ξ}}_3)p\left(t\right)-24M_s{\mathrm{\ξ}}_1}{(l{\mathrm{\ξ}}_1+3{\mathrm{\ξ}}_2){\mathrm{ml}}^2}\end{array}\right.,$

ξ₁＝λ₁(l)-λ₁(l/2)，ξ₂＝λ₂(l)-λ₂(l/2)-λ₁(l)·l/2，

ξ₃＝η₁(l/2)-η₁(l)，ξ₄＝η₂(l/2)-η₂(l)+η₁(l)·l/2；

a sixth module, which is used for judging that the forward plastic deformation stage of the hull beam is finished and entering reverse elastic unloading when the forward plastic deformation of the hull beam reaches the maximum value α (t) is reached, and the forward plastic deformation stage of the hull beam enters the reverse elastic unloading according to a formula $\left\{\begin{array}{c}\mathrm{\α}\left(t\right)(x-l/2)+{\mathrm{\φ}}_1\left(x\right)H^{\′}\left(t\right)={\mathrm{\φ}}_1\left(x\right)H\left(t\right)\\ \stackrel{\·}{\mathrm{\α}}\left(t\right)(x-l/2)+{\mathrm{\φ}}_1\left(x\right){\stackrel{\·}{H}}^{\′}\left(t\right)={\mathrm{\φ}}_1\left(x\right)\stackrel{\·}{H}\left(t\right)\end{array}\right.$ Solving the motion amplitude H (t) of the reverse elastic unloading process of the hull beam;

a seventh module for exceeding the plastic limit bending moment M when the absolute value of the bending moment at the middle point of the beam exceeds the plastic limit bending moment M in the process of the reverse elastic deformation of the hull beam_sWhen the absolute value is obtained, the ship body beam is judged to enter the reverse plastic motion, and the motion amplitude H of the beam after the beam enters the reverse plastic deformation is solved₁' (t) and angle of relative rotation α₁(t)：

$\left\{\begin{array}{c}{\stackrel{\·\·}{H}}_1^{\′}\left(t\right)=\frac{3(-M_s)-(l{\mathrm{\ξ}}_3+3{\mathrm{\ξ}}_4)p\left(t\right)}{(l{\mathrm{\ξ}}_1+3{\mathrm{\ξ}}_2)m}\\ {\stackrel{\·\·}{\mathrm{\α}}}_1\left(t\right)=\frac{24({\mathrm{\ξ}}_1{\mathrm{\ξ}}_4-{\mathrm{\ξ}}_2{\mathrm{\ξ}}_3)p\left(t\right)-24\left({-M}_s\right){\mathrm{\ξ}}_1}{(l{\mathrm{\ξ}}_1+3{\mathrm{\ξ}}_2){\mathrm{ml}}^2}\end{array}\right.;$

An eighth module for α when the reverse plastic deformation of the hull beam reaches a maximum₁(t) when the maximum value is reached, judging that the backward plastic deformation stage of the hull beam is finished, entering forward elastic unloading, and carrying out forward elastic unloading according to the result $\left\{\begin{array}{c}{\mathrm{\α}}_1\left(t\right)(x-l/2)+{\mathrm{\φ}}_1\left(x\right)H_1^{\′}\left(t\right)={\mathrm{\φ}}_1\left(x\right)H_1\left(t\right)\\ {\stackrel{\·}{\mathrm{\α}}}_1\left(t\right)(x-l/2)+{\mathrm{\φ}}_1\left(x\right){\stackrel{\·}{H}}_1^{\′}\left(t\right)={\mathrm{\φ}}_1\left(x\right){\stackrel{\·}{H}}_1\left(t\right)\end{array}\right.$ Solving the motion amplitude H of the ship body beam in the forward elastic unloading stage₁(t)。

Generally, compared with the prior art, the above technical solution conceived by the present invention mainly has the following technical advantages: the invention comprehensively considers the impact action of shock waves and bubble loads on the ship, simplifies the pressure of the underwater explosion load into five stages, calculates the movement displacement of the ship in different stages by establishing corresponding mathematical models, thereby obtaining the speed, the acceleration and the bending moment of the ship, and further analyzes the movement processes of forward plastic deformation, reverse elastic unloading, reverse plastic deformation and forward elastic unloading of the ship according to the parameters. Compared with the prior art, the method can accurately and simply realize the engineering prediction of the elastic-plastic motion deformation of the ship under the action of underwater short-distance explosion, and has guiding and reference significance for improving the design level of underwater explosion impact resistance protection of the ship, optimizing the attack efficiency and the attack mode of the underwater weapon and the like.

Drawings

FIG. 1 is a flow chart of the response prediction method of the ship overall elastic-plastic motion under the underwater explosion effect of the invention;

FIG. 2 is a schematic view of the hull beam explosion conditions of the present invention;

FIG. 3 is a schematic diagram of five stages of underwater explosive loading according to the present invention;

FIG. 4 is a schematic diagram of the stress-strain relationship during repeated loading and unloading of an ideal elastoplastic beam according to the present invention;

FIG. 5 is a schematic view of the beam pattern structure and dimensions according to an embodiment of the present invention;

FIG. 6 is a schematic view of a point bending moment time course curve of a hull beam under a full elastic motion condition according to an embodiment of the invention;

FIG. 7 is a schematic diagram of a midpoint displacement time course curve of a hull beam in elastic-plastic motion according to an embodiment of the invention;

FIG. 8 is a schematic diagram illustrating a comparison of deformation of the hull beam in the length direction at different times in accordance with an embodiment of the present invention;

fig. 9 is a schematic diagram of a time course curve of the change of the rotation angle alpha in the plastic movement process of the hull beam at different moments in one embodiment of the invention.

Detailed Description

In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

The invention provides a prediction method of ship integral elastic-plastic motion response under the action of underwater explosion, as shown in figure 1, the method firstly equates the ship motion to free ship beam motion, comprehensively considers the influence of shock wave and air bubble on the ship, divides the load pressure of the free ship beam under the action of underwater explosion into five stages, establishes corresponding mathematical models, further solves the elastic motion displacement of the ship in different stages, solves the motion displacement, speed, acceleration and bending moment of the ship beam in each stage according to the displacement, and further solves and analyzes the motion response of the beam forward plastic motion process, the beam reverse elastic unloading process, the beam reverse plastic deformation process and the beam forward elastic unloading process according to the parameters. The specific realization idea is as follows:

simplifying a ship into a free ship beam with an equal section according to the principle of total longitudinal strength equivalence and similarity;

simplifying the pressure curve of the underwater explosion load into five stages, wherein the pressure in the first stage and the second stage meets the linear change rule, the pressure in the third stage meets the sine function change relationship, and the pressure in the fourth stage and the pressure in the fifth stage meet the linear change rule;

determining the natural vibration shape function of the hull beam;

determining a pressure distribution function in the length direction of the hull beam in each stage according to the shock wave and the characteristics of the fluctuation of the pulsating pressure load of the air bubbles;

determining a main coordinate function of the motion deformation of the hull beam according to pressure distribution functions in the length direction of the hull beam at different pressure stages;

step six, determining a motion displacement function of the inner beam in the stage according to the shape function and the main coordinate function of the hull beam in different pressure stages;

step seven, solving the movement displacement, the speed, the acceleration and the bending moment of the hull beam in each stage according to the initial movement condition and the continuous movement condition of each stage;

step eight, determining the moment when the beam initially enters plastic deformation according to the distribution condition of the bending moment in the beam in the elastic movement stage, reasonably selecting a displacement function after the beam enters the plastic deformation, and solving a forward plastic movement process;

step nine, determining the moment when the beam generates reverse elastic unloading according to the forward plastic motion process of the beam, and obtaining the motion process of the beam in the elastic unloading stage by combining the motion equations and continuous conditions of the elastic and plastic stages;

step ten, determining the moment when the beam enters reverse plastic deformation according to the bending moment distribution of the beam in the reverse elastic unloading process, and determining the reverse plastic movement process by using a continuous boundary condition and an elastic-plastic movement control equation;

and step eleven, determining the moment when the beam enters the forward elastic unloading according to the maximum value of the reverse plastic deformation of the beam, and determining the forward plastic motion process by using a continuous boundary condition and an elastic-plastic motion control equation.

The specific implementation of the above steps is described in detail below.

When the explosive explodes under the middle part of the ship, the motion response of the explosive is relatively obvious. In one embodiment of the present invention, the typical working condition is used as a research object. As shown in fig. 2, under this condition, the method for predicting the overall elastoplastic motion response of a ship under the combined action of an underwater short-distance blast shock wave and a bubble comprises the following steps:

firstly, carrying out ship beam approximation of a ship structure.

The ship is simplified into the constant-section free ship beam, and compared with a real-scale ship beam and a scaled ship model, the simplified process meets the following two principles:

(a) keeping the total longitudinal moment of inertia of the prototype and the model similar geometrically;

(b) after the ship structure and the explosive are geometrically scaled according to the same scale, keeping the first-order wet frequency of the model consistent with the explosion bubble pulsation frequency of the scaled explosive.

Step two, simplifying the pressure curve of the underwater explosion load into five stages

The underwater explosion load pressure curve is simplified into five stages I-V shown in figure 3. Wherein, the I, II th stage is a shock wave load attenuation stage which satisfies a linear change rule; the third stage is a flow field negative pressure change stage formed by the expansion and contraction motion of the bubbles under the boundary condition, and the flow field negative pressure change stage meets the sine function relationship; the IV and V stages are rising and falling stages of secondary pulsating pressure generated by bubble contraction, and satisfy the linear change rule, and the pressure load calculation formulas of all stages are listed as follows:

(1) stage I: p (t) ═ P_m·(1-t/k₁)， 0≤t＜t₁

(2) Stage II: $p\left(t\right)=P_m/e\·(1-\frac{t-t_1}{k_2}),$ t₁≤t＜t₂

(3) stage III: p (t) ═ P_b·sinβ(t-t₂) t₂≤t＜t₃

(4) And IV stage: $p\left(t\right)=P_s\·\left(\frac{t-t_3}{k_4}\right)$ t₃≤t＜t₄

(5) and (6) in stage V: $p\left(t\right)=P_s\·(1-\frac{t-t_4}{k_5})$ t₄≤t＜t₅

in the formula: $\begin{array}{cc}{P}_m=K_1\·{\left(\frac{{m_e}^{1/3}}{R}\right)}^{A_1}& k_1=\frac{\mathrm{e\θ}}{e-1},\end{array}$ t₁＝θ， $\mathrm{\θ}=K_2\·{m_e}^{1/3}\·{\left(\frac{{m_e}^{1/3}}{R}\right)}^{A_2},$ k₂＝t₂-t₁， $t_2=(\frac{850}{{\stackrel{\‾}{P}}_0^{0.18}}-\frac{20}{{\stackrel{\‾}{P}}_0^{1/3}}+n)\·\frac{r_0}{c},\stackrel{\‾}{P_0}=\frac{P_0}{P_{\mathrm{atm}}},$ P₀＝P_atm+ρ_wgH₀， $n=11.4-\frac{10.6}{{\stackrel{\‾}{r}}^{0.13}}+\frac{1.51}{{\stackrel{\‾}{r}}^{1.26}},$ $\stackrel{\‾}{r}=\frac{R}{r_0},\mathrm{\β}=\frac{\mathrm{\π}}{k_3},$ k₃＝t₃-t₂， $t_4=T=2.11\frac{{m_e}^{1/3}}{{(P_0/{\mathrm{\ρ}}_{w}g)}^{5/6}},P_s=\frac{39\×{10}^6+24P_0}{\stackrel{\‾}{r}},$ $P_b=\frac{-3.1\×{10}^4{m_e}^{1/3}{(H_0+10)}^{2/3}}{R},$ k₄＝t₄-t₃，k₅＝t₅-t₄， $k_4=k_5=3290\frac{r_0}{P_0^{0.71}}.$

wherein m is_eIs TNT charge equivalent, R is the detonation distance, P_mIs the peak shock wave pressure, K₁、K₂、A₁、A₂Is the constant of the shock wave, k₁、k₂、k₃、k₄、k₅Respectively, a parameter related to the duration of the five pressure phases, P₀Is the hydrostatic pressure at the explosive location,is a dimensionless pressure parameter, P_bIs the peak of negative pressure, P, during the pulsation of the bubble_sIn order to be the peak of the pulsating pressure,in order to characterize the dimensionless parameter of the detonation distance, θ is the shock wave attenuation constant, H₀Is the depth of charge, r₀Is the radius of charge, p_wIs the density of water, c is the speed of sound in water, P_atmThe pressure is atmospheric pressure, g is gravity acceleration, T is a bubble pulsation period, β is a pressure function angular frequency value of a bubble negative pressure stage, and physical quantities of all parameters adopt an international unit system.

Step three, determining the natural vibration mode function of the hull beam vibration under the underwater explosion action

When the underwater explosion load is small, the free hull beam elastically moves under the action of the explosion pressure P (x, t), and the motion control equation is as follows

$\mathrm{EI}\frac{{\∂}^{4}w}{\∂x^4}+m\frac{{\∂}^{2}w}{\∂t^2}=P(x,t)---\left(1\right)$

It is generally solved as

Wherein x is an abscissa value of the free hull beam at any point, w is a motion displacement function, E is an elastic modulus, I is a beam cross section inertia moment, and m is a unit beam length mass considering the attached water.

Wherein,is the i-th order natural mode of vibration of the beam, H_iAnd (t) is a principal coordinate corresponding to the ith order mode of the beam. In order to determine the motion displacement function of the beam, the eigenmode shape and the corresponding principal coordinates, respectively, should first be determined.

The general expression for the natural vibration shape of a beam is

Where μ is a variable related to the natural frequency of the beam, l is the length of the beam, ζ₁、ζ₂、ζ₃、ζ₄Is a constant.

For a free boundary beam, the bending moment and shear force at both ends are zero, so that the following boundary conditions are satisfied when x is 0 and x is l

Zeta is derived by substituting the above boundary condition with the formula (3)₁＝ζ₃,ζ₂＝ζ₄And is and

$\left\{\begin{array}{c}{\mathrm{\ζ}}_2(\mathrm{ch\μ}-\cos \mathrm{\μ})+{\mathrm{\ζ}}_1(\mathrm{sh\μ}-\sin \mathrm{\μ})=0\\ {\mathrm{\ζ}}_2(\mathrm{sh\μ}+\sin \mathrm{\μ})+{\mathrm{\ζ}}_1\left(\text{ch\μ-cos\μ}\right)=0\end{array}\right.---\left(5\right)$

for constant ζ₁And ζ₂In particular, a particular solution of the above system of equations requires that the determinant coefficients of the matrix be equal to zero, i.e. that

(chμ-cosμ)²-(sh²μ-sin²μ)＝0 (6)

The solution of the above equation determines the natural frequency of the free beam. In addition, the natural frequency of the beam is satisfied

$\mathrm{\ω}={\left(\frac{\mathrm{\μ}}{l}\right)}^2\sqrt{\frac{\mathrm{EI}}{m}}---\left(7\right)$

When mu takes the first non-zero solution mu₁4.730, the natural frequency of the first order mode of the beamAt this time ζ₂/ζ₁＝(sinμ₁-shμ₁)/(chμ₁-cosμ₁) 1, the first order shape function of the beam can be approximately expressed as

Step four, determining the pressure distribution function of the underwater explosive load in the length direction of the hull beam

For the underwater explosion pressure function P (x, t), the function satisfies P (x, t) ═ P (t) · P (x), wherein P (t) is a pressure time course attenuation curve at the middle point of the beam, and P (x) is an explosion load pressure distribution characteristic function relative to the middle point of the beam.

For the shock wave load attenuation phase (phase I, II), the characteristic function p of the pressure distribution on the beam_s(x) Can be characterized as

$p_s\left(x\right)=\frac{R^{1.13}}{{\sqrt{R^2+{(x-l/2)}^2}}^{1.13}}---\left(9\right)$

For the bubble pulsation phase (III-V phase), the characteristic function p of the pressure distribution on the beam_b(x) Can be characterized as

$p_b\left(x\right)=(1-\frac{2x-l}{l})\·\exp [-8{\left(\frac{2x-l}{l}\right)}^2+4{\left(\frac{2x-1}{l}\right)}^3]+{\left(\frac{2l-2x}{l}\right)}^{1.5}\·\left(\frac{2x-l}{l}\right)---\left(10\right)$

Respectively determining a characteristic function p of pressure distribution on a ship body beam in the shock wave stage and the bubble stage_s(x) And p_b(x) And combining the pressure time course change formula P (t) in the five stages to obtain the pressure function P (x, t).

Step five, determining main coordinate functions of the movement of the hull beam in different pressure stages

For principal coordinate H under forced vibration condition_i(t), can be represented as

$H_i\left(t\right)=a_i\cos {\mathrm{\ω}}_{i}t+b_i\sin {\mathrm{\ω}}_{i}t+\frac{1}{{\mathrm{\ω}}_i}{\∫}_0^t{f}_i\left(\mathrm{\τ}\right)\sin {\mathrm{\ω}}_i(t-\mathrm{\τ})\mathrm{d\τ}---\left(11\right)$

In the formulaIs a generalized excitation force corresponding to a generalized mass, a_i、b_iThe integral constant is determined by the initial condition (displacement and velocity are zero) and the motion continuation condition.

Defining pressure distribution characteristic function p of each stage of underwater explosion_s(x) Then, parameters are introduced to simplify the parameter expression form of subsequent formula derivation

Step six, determining the motion displacement function of the inner beam in different pressure stages

Considering that the explosive charge when exploded under the middle of the beam mainly provokes the low order motion response of the beam, for simplicity, it is assumed that the hull beam now exhibits mainly first order motion modes. Determining the mode shape function according to the equations (8) and (11)(x) And its corresponding primary coordinate function H₁After (t), an approximate displacement function of the beam can be obtained as

The underwater explosion load is divided into 5 different stages, and the vibration shape function of each stageKeeping the same, and corresponding forced vibration principal coordinate function H₁(t) according to the pressure calculation formulas at different stages, the corresponding motion displacement functions are obtained as (sequentially and respectively formula (13) to formula (17))

Stage I:

stage II:

stage III:

and IV stage:

and (6) in stage V:

and seventhly, determining the motion parameters of the inner beams in different pressure stages, and predicting the whip motion response condition of the hull beam according to the motion parameters.

The time t in the equations (13) to (17) is zero at the start time of each pressure phase, and the coefficient Determined by initial motion conditions (zero displacement and speed) and continuous motion conditions of each stage, and the parameter phi₁And determining according to the pressure distribution characteristic functions of different stages.

After the displacement function w (x, t) of each stage is determined, the speed, the acceleration and the bending moment of the beam can be obtained by solving partial derivatives of the displacement function on time t or a variable x.

Step eight, enabling the beam to enter a positive plastic movement process after plastic deformation from elastic deformation

With the increase of the underwater explosion intensity, the hull beam converts the full elastic motion response into the elastic plastic motion response, and the deformation process comprises the following steps: elastic movement → forward plastic deformation → reverse elastic unloading → reverse plastic deformation → forward elastic unloading. For an ideal elastoplastic hull beam, the kinematic deformation process and stress-strain relationship is shown in fig. 4.

At a certain stage of underwater explosion impact load, if the bending moment of the middle part of the hull beam exceeds the plastic limit bending moment, assuming that a fixed plastic hinge is formed at the middle part of the hull beam, the whole beam rotates relatively around the plastic hinge while the beam continues to maintain first-order motion deformation. Considering the symmetry of the hull beam, taking the right half hull beam as a research object, the displacement function is listed as follows

In the formula, l/2<x≤l，In the first-order mode shape function, H' (t) is the corresponding motion amplitude function, and α (t) is the rotation angle function when relative rotation occurs.

In the initial elastic movement process of the beam, the bending moment in the middle part is the largest, and when the maximum bending moment value exceeds the plastic limit bending moment M_sWhen the absolute value of the bending moment is larger than the preset value, the middle part of the beam is provided with a fixed plastic hinge, the beam enters a plastic deformation stage from an elastic stage, and the midpoint bending moment value of the beam is kept to be M_sAnd is not changed. At this time, the motion control equation of the beam satisfies

$\frac{{\&PartialD;}^{2}M}{\&PartialD;x^2}+m\frac{{\&PartialD;}^{2}w}{\&PartialD;t^2}=P(x,t)---\left(19\right)$

Substituting equation (18) into the above equation, and performing first and second order integration, using boundary conditions: when x is l/2, Q is 0, M is M_sQ is the shear force in the beam, M is the bending moment of the beam, and when x is l, Q is 0 and M is 0, the calculation equations for functions H' (t) and α (t) can be obtained, respectively:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{H}}^{\&prime;}\left(t\right)=\frac{3M_s-(l{\mathrm{\&xi;}}_3+3{\mathrm{\&xi;}}_4)p\left(t\right)}{(l{\mathrm{\&xi;}}_1+3{\mathrm{\&xi;}}_2)m}\\ \stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&alpha;}}\left(t\right)=\frac{24({\mathrm{\&xi;}}_1{\mathrm{\&xi;}}_4-{\mathrm{\&xi;}}_2{\mathrm{\&xi;}}_3)p\left(t\right)-24M_s{\mathrm{\&xi;}}_1}{(l{\mathrm{\&xi;}}_1+3{\mathrm{\&xi;}}_2){\mathrm{ml}}^2}\end{array}\right.---\left(20\right)$

wherein

ξ₁＝λ₁(l)-λ₁(l/2)，

ξ₂＝λ₂(l)-λ₂(l/2)-λ₁(l)·l/2，

ξ₃＝η₁(l/2)-η₁(l)，

ξ₄＝η₂(l/2)-η₂(l)+η₁(l)·l/2。

The above formula incorporates the following integral function

${\&Integral;}_0^{x}p\left(x\right)\mathrm{dx}={\mathrm{\&eta;}}_1\left(x\right),$

${\&Integral;}_0^x{\&Integral;}_0^{x}p\left(x\right)\mathrm{dxdx}={\mathrm{\&eta;}}_2\left(x\right),$

The beam meets the momentum and energy balance condition in the elastic-plastic motion conversion process, wherein the momentum balance condition is

$2{\&Integral;}_{l/2}^{l}m[{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_0(x-l/2)+{\stackrel{\&CenterDot;}{H}}_0^{\&prime;}{\mathrm{\&phi;}}_1\left(x\right)]\mathrm{dx}={\&Integral;}_0^{l}m{\mathrm{\&phi;}}_1\left(x\right)\stackrel{\&CenterDot;}{H}\left(t_s\right)\mathrm{dx}---\left(21\right)$

The energy balance condition is

$2{\&Integral;}_{l/2}^{l}0.5m{[{\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_0(x-l/2)+{\stackrel{\&CenterDot;}{H}}_0^{\&prime;}{\mathrm{\&phi;}}_1\left(x\right)]}^2\mathrm{dx}={\&Integral;}_0^{l}0.5m{\mathrm{\&phi;}}_1^2\left(x\right){\stackrel{\&CenterDot;}{H}}^2\left(t_s\right)\mathrm{dx}---\left(22\right)$

In the formula t_sIs the initial moment at which the plastic movement occurs,is the initial parameter value of the displacement function when the plastic motion occurs.

And (3) performing first and second integrations on the time t by using the equation system (20), and combining momentum and energy balance conditions of the beam during elastic-plastic motion conversion to obtain the displacement and relative rotation angle of the beam in the plastic deformation stage in any pressure time period.

Ninth, beam reverse elastic unloading process

After the hull beam enters plastic movement, when the middle plastic deformation of the hull beam reaches the maximum value, the rotation angle alpha (t) of the two ends of the beam rotating relatively around the fixed plastic hinge also reaches the maximum value, then the beam starts to move reversely, the structural stress is released, and the middle point of the beam drives the two ends to enter elastic unloading.

Assuming that the beam displacement function still meets the requirement of the formula (12) in the elastic unloading process, solving the motion amplitude H (t) of the ship hull beam in the reverse elastic unloading process according to the displacement and speed continuous conditions at the moment of plastic and elastic conversion, wherein the equation is as follows:

$\left\{\begin{array}{c}\mathrm{\&alpha;}\left(t\right)(x-l/2)+{\mathrm{\&phi;}}_1\left(x\right)H^{\&prime;}\left(t\right)={\mathrm{\&phi;}}_1\left(x\right)H\left(t\right)\\ \stackrel{\&CenterDot;}{\mathrm{\&alpha;}}\left(t\right)(x-l/2)+{\mathrm{\&phi;}}_1\left(x\right){\stackrel{\&CenterDot;}{H}}^{\&prime;}\left(t\right)={\mathrm{\&phi;}}_1\left(x\right)\stackrel{\&CenterDot;}{H}\left(t\right)\end{array}\right.---\left(23\right)$

the motion equation of the beam in the elastic unloading stage can be obtained by taking the beam midpoint as a survey point and combining the motion equations obtained in the elastic and plastic stages and simultaneously utilizing the continuous conditions. It should be noted that, besides the beam midpoint, the motion displacement of other parts needs to be superimposed with a relative rotation displacement on the basis of elastic deformation.

Step ten, the reverse plastic movement process of the beam

In the elastic unloading and reverse movement processes, when the absolute value of the midpoint bending moment of the beam exceeds the plastic limit bending moment M_sWhen the absolute value is larger, the beam enters the reverse plastic movement process, the fixed plastic hinge appears in the middle, the beam continues to keep the first-order movement deformation, and the two ends of the beam rotate relatively around the plastic hinge. Assuming that the displacement function and the motion control equation of the beam still meet the requirements of the equations (18) and (19) in the process of reverse plastic motion, and keeping the midpoint bending moment value of the beam to be-M_sThe motion amplitude H of the beam in the reverse plastic deformation can be obtained by using continuous boundary conditions without change₁' (t) and relative angle of rotation α₁(t)：

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{H}}_1^{\&prime;}\left(t\right)=\frac{3(-M_s)-(l{\mathrm{\&xi;}}_3+3{\mathrm{\&xi;}}_4)p\left(t\right)}{(l{\mathrm{\&xi;}}_1+3{\mathrm{\&xi;}}_2)m}\\ {\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&alpha;}}}_1\left(t\right)=\frac{24({\mathrm{\&xi;}}_1{\mathrm{\&xi;}}_4-{\mathrm{\&xi;}}_2{\mathrm{\&xi;}}_3)p\left(t\right)-24\left({-M}_s\right){\mathrm{\&xi;}}_1}{(l{\mathrm{\&xi;}}_1+3{\mathrm{\&xi;}}_2){\mathrm{ml}}^2}\end{array}\right.---\left(24\right)$

It is noted here that the pressure load p (x, t) changes during the reverse plastic deformation phase, i.e., whether p (x, t) spans multiple load phases is determined. The motion control equation is determined in stages according to the actual condition of the explosive load p (x, t).

Eleven, beam positive elastic unloading process

When the reverse plastic deformation of the middle part of the hull beam reaches the maximum value, the two ends of the beam rotate relative to each other around the fixed plastic hinge at the rotating angle α₁(t) will also reach a maximum value after which the beam starts to move in a forward direction, the structural stresses are relieved and the beam midpoint carries the other parts to elastically unload again.

Assuming that the beam displacement function still satisfies equation (12) during the beam forward spring unloading process) According to the continuous conditions of displacement and speed at the moment of plastic and elastic conversion, the calculation equation of the formula (23) can still be obtained, namely the motion amplitude H of the ship body beam in the forward elastic unloading stage₁(t) satisfies the following formula:

$\left\{\begin{array}{c}{\mathrm{\&alpha;}}_1\left(t\right)(x-l/2)+{\mathrm{\&phi;}}_1\left(x\right)H_1^{\&prime;}\left(t\right)={\mathrm{\&phi;}}_1\left(x\right)H_1\left(t\right)\\ {\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1\left(t\right)(x-l/2)+{\mathrm{\&phi;}}_1\left(x\right){\stackrel{\&CenterDot;}{H}}_1^{\&prime;}\left(t\right)={\mathrm{\&phi;}}_1\left(x\right){\stackrel{\&CenterDot;}{H}}_1\left(t\right)\end{array}\right.---\left(25\right).$

the motion equation of the beam in the forward elastic unloading stage can be obtained by taking the beam midpoint as a survey point, combining the motion equations obtained in the elastic and plastic stages and simultaneously utilizing the continuous conditions. It should be noted that, besides the beam midpoint, the motion displacement of other parts needs to be superimposed with a relative rotation displacement on the basis of elastic deformation.

In a specific embodiment of the present invention, a certain beam model is selected as an analysis object, and the relevant dimension parameters are as follows: the beam length is 2.8m, the width is 0.3m, the height is 0.08m, the plate thickness is 1mm, the beam plastic limit bending moment is 1.8e4Nm, and the specific structural form is shown in FIG. 5. And (3) selecting an explosion working condition of 5g of TNT (trinitrotoluene) dose and 0.7m of explosion distance to analyze the overall movement response process of the beam.

Fig. 6 shows the time course curve of the mid-point bending moment of the hull beam under the assumed condition of full elastic motion. It can be seen that the beam mid-point bending moment value has exceeded the plastic limit bending moment M during the bubble expansion process_sIt is shown that the beam has entered the plastic movement process at the beginning of the bubble expansion. After the beam enters plastic deformation, the follow-up motion response needs to further determine whether reverse plastic deformation and unloading process exist according to the conditions of elastic unloading and reverse loading.

Fig. 7 shows the middle point displacement time course curve when the ship body beam generates elastic-plastic movement. It can be seen that the hull beam still shows obvious heave motion under the working condition, the deformation of the beam is rapidly increased in the bubble collapse stage, the middle arch deformation is obvious and exceeds the initial middle arch deformation (3.8cm) and the middle sag bending deformation value (5.1cm), and whip motion is shown; compared with the working condition of 1m of explosion distance, the sagging deformation of the beam caused by the movement of the bubbles under 0.7m of explosion distance is relatively obvious, which is related to the distribution and change characteristics of a low-pressure flow field; the kinematic response of the beam undergoes a complex process of intrados plastic deformation → elastic unloading and reverse loading → midperpendicular plastic deformation → elastic unloading again and intrados plastic deformation. As can be seen from comparison with fig. 6, the plastic deformation causes the reciprocating period of the beam to increase, and the first order response frequency to decrease.

Fig. 8 shows a comparison of the length-wise deformation of the hull beams at typical times. It can be seen that the beam as a whole exhibits first order deformation, but the presence of plastic deformation causes the beam not to move in forward and reverse directions strictly around a fixed stagnation point; at the moment of 37.1ms, the beam generates maximum sagging plastic deformation, and a relatively obvious plastic hinge is visible in the middle; for a certain moment, the movement displacement of the two ends of the beam exceeds the displacement value of the middle point.

And (5) taking the plastic relative rotation angle alpha as a research object, and further analyzing the plastic deformation process of the beam. Fig. 9 shows the time course of the change of the rotation angle α. It can be seen that in the initial arch-centering deformation stage, the rotation speed of the beam plastic hinge is relatively gentle, and the maximum rotation angle is-1.7 e-3; from the later expansion stage of the bubbles to the contraction stage, the sagging plastic deformation of the beam is rapidly increased, and the plastic hinge of the beam is changed from-1.7 e-3 to 1 e-2; after the bubbles are collapsed, the pulsating pressure causes the beam to generate rapid mid-arch plastic deformation, and when the calculation is finished, the plastic corner is-8 e-3.

It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (2)

1. A ship overall elastic-plastic motion response prediction method under the action of underwater explosion is characterized by comprising the following steps:

s1, equating the ship to be a free ship beam with an equal section, so that the prediction of ship motion response is equivalent to the prediction of free ship beam motion response;

s2, dividing the load pressure process of the free hull beam under the underwater explosion into five time stages, and collecting the pressure peak value P of the shock wave_mNegative pressure peak value P in bubble pulsation stage_bFirst pulsating pressure of bubblesPeak value of force P_s(ii) a The five time periods are: t is more than or equal to 0 and less than t₁、t₁≤t＜t₂、t₂≤t＜t₃、t₃≤t＜t₄、t₄≤t＜t₅(ii) a Wherein t is₁θ is the shock wave attenuation constant; P₀is the hydrostatic pressure at the explosive, P_atmIs atmospheric pressure, c is the speed of sound in water, r₀The radius of charge is R is the detonation distance; m_efor charge equivalent, p_wIs the density of water, g is the acceleration of gravity; $t_5=3290\frac{r_0}{P_0^{0.71}}+t_4;$

s3, solving the motion displacement w (x, t) of the free hull beam in five time stages of underwater explosion:

wherein x is the abscissa value of the free hull beam at any point,k₂＝t₂-t₁， is the first-order natural vibration shape of the free hull beam,wherein ζ₁Amplitude of vibration, μ₁Is a beam motion frequency parameter, l is the free hull beam length;is an integration constant, ω, of five time phases₁The first-order mode shape natural frequency of the free beam; phi is a₁In order to be an integration constant, the first,m is the mass per unit beam length considering the mass of the attached water, p (x) is a pressure distribution characteristic function of the free hull beam in different time stages, and t is more than or equal to 0 and less than t₁、t₁≤t＜t₂The time period is a period of time,for the t-th₂≤t＜t₃、t₃≤t＜t₄、t₄≤t＜t₅The time period is a period of time, $p\left(x\right)=(1-\frac{2x-l}{l})\&CenterDot;\exp [-8{\left(\frac{2x-l}{l}\right)}^2+4{\left(\frac{2x-l}{l}\right)}^3]+{\left(\frac{2l-2x}{l}\right)}^{1.5}\&CenterDot;\left(\frac{2x-l}{l}\right);$

s4, solving the speed and bending moment values of the hull beam at different stages according to the elastic movement displacement w (x, t) of the hull beam at five time stages;

s5, when the absolute value of the bending moment of the midpoint of the hull beam exceeds the absolute value of the plastic limit bending moment Ms of the hull beam, the hull beam enters forward plastic motion, and the motion amplitude H' (t) and the relative rotation corner alpha (t) of the hull beam during the forward plastic motion are solved:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{H}}^{\&prime;}\left(t\right)=\frac{3M_s-(l{\mathrm{\&xi;}}_3+3{\mathrm{\&xi;}}_4)p\left(t\right)}{(l{\mathrm{\&xi;}}_1+3{\mathrm{\&xi;}}_2)m}\\ \stackrel{}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&alpha;}}}\left(t\right)=\frac{24({\mathrm{\&xi;}}_1{\mathrm{\&xi;}}_4-{\mathrm{\&xi;}}_2{\mathrm{\&xi;}}_3)p\left(t\right)-24M_s{\mathrm{\&xi;}}_1}{(l{\mathrm{\&xi;}}_1+3{\mathrm{\&xi;}}_2)m{l}^2}\end{array}\right.,$

ξ₁＝λ₁(l)-λ₁(l/2)，ξ₂＝λ₂(l)-λ₂(l/2)-λ₁(l)·l/2，

ξ₃＝η₁(l/2)-η₁(l)，ξ₄＝η₂(l/2)-η₂(l)+η₁(l)·l/2；

${\&Integral;}_0^{x}p\left(x\right)\mathrm{dx}={\mathrm{\&eta;}}_1\left(x\right),{\&Integral;}_0^x{\&Integral;}_0^{x}p\left(x\right)\mathrm{dxdx}={\mathrm{\&eta;}}_2\left(x\right);$

s6, when the forward plastic deformation of the ship body beam reaches the maximum value, namely α (t) reaches the maximum value, the forward plastic deformation stage is ended, reverse elastic unloading is carried out, and the ship body beam is subjected to elastic unloading according to a formula $\left\{\begin{array}{c}\mathrm{\&alpha;}\left(t\right)(x-l/2)+{\mathrm{\&phi;}}_1\left(x\right)H^{\&prime;}\left(t\right)={\mathrm{\&phi;}}_1\left(x\right)H\left(t\right)\\ \stackrel{\&CenterDot;}{\mathrm{\&alpha;}}\left(t\right)(x-l/2)+{\mathrm{\&phi;}}_1\left(x\right){\stackrel{\&CenterDot;}{H}}^{\&prime;}\left(t\right)={\mathrm{\&phi;}}_1\left(x\right)\stackrel{\&CenterDot;}{H}\left(t\right)\end{array}\right.$ Solving the motion amplitude H (t) of the reverse elastic unloading process of the hull beam;

s7, in the process of the reverse elastic deformation of the ship body beam, when the absolute value of the bending moment at the midpoint of the beam exceeds the plastic limit bending moment M_sWhen the absolute value is obtained, the hull beam enters reverse plastic motion, and the motion amplitude H of the beam after entering reverse plastic deformation is solved₁' (t) and angle of relative rotation α₁(t)：

$\left\{\begin{array}{c}{{\stackrel{\&CenterDot;\&CenterDot;}{H}}_1}^{\&prime;}\left(t\right)=\frac{3(-M_s)-(l{\mathrm{\&xi;}}_3+3{\mathrm{\&xi;}}_4)p\left(t\right)}{(l{\mathrm{\&xi;}}_1+3{\mathrm{\&xi;}}_2)m}\\ {\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&alpha;}}}_1\left(t\right)=\frac{24({\mathrm{\&xi;}}_1{\mathrm{\&xi;}}_4-{\mathrm{\&xi;}}_2{\mathrm{\&xi;}}_3)p\left(t\right)-24(-M_s){\mathrm{\&xi;}}_1}{(l{\mathrm{\&xi;}}_1+3{\mathrm{\&xi;}}_2)m{l}^2}\end{array}\right.;$

S8, α when the reverse plastic deformation of the hull beam reaches the maximum₁(t) reaching a maximum value, the reverse plastic deformation phase ending, entering a forward elastic unloading, according to $\left\{\begin{array}{c}{\mathrm{\&alpha;}}_1\left(t\right)(x-l/2)+{\mathrm{\&phi;}}_1\left(x\right){H_1}^{\&prime;}\left(t\right)={\mathrm{\&phi;}}_1\left(x\right)H_1\left(t\right)\\ {\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1\left(t\right)(x-l/2)+{\mathrm{\&phi;}}_1\left(x\right){{\stackrel{\&CenterDot;}{H}}_1}^{\&prime;}\left(t\right)={\mathrm{\&phi;}}_1\left(x\right){\stackrel{\&CenterDot;}{H}}_1\left(t\right)\end{array}\right.$ Solving the motion amplitude H of the ship body beam in the forward elastic unloading stage₁(t)。

2. A system for predicting the response of the ship to the overall elastoplastic motion under the action of underwater explosion, which is characterized by comprising:

the first module is used for enabling the ship to be equivalent to a free ship beam with an equal section, so that the prediction of ship motion response is equivalent to the prediction of free ship beam motion response;

the second module is used for dividing the load pressure process of the free hull beam under the underwater explosion action into five time stages and acquiring a shock wave pressure peak value P_mNegative pressure peak value P in bubble pulsation stage_bThe first pulsating pressure peak value P of the air bubble_s(ii) a The five time periods are: t is more than or equal to 0 and less than t₁、t₁≤t＜t₂、t₂≤t＜t₃、t₃≤t＜t₄、t₄≤t＜t₅(ii) a Wherein t is₁θ is the shock wave attenuation constant; $\stackrel{\&OverBar;}{P_0}=\frac{P_0}{P_{\mathrm{atm}}},n=11.4-\frac{10.6}{{\stackrel{\&OverBar;}{r}}^{0.13}}+\frac{1.51}{{\stackrel{\&OverBar;}{r}}^{1.26}},\stackrel{\&OverBar;}{r}=\frac{R}{r_0},$ P₀is the hydrostatic pressure at the explosive, P_atmIs atmospheric pressure, c is the speed of sound in water, r₀The radius of charge is R is the detonation distance; m_efor charge equivalent, p_wIs the density of water, g is the acceleration of gravity; $t_5=3290\frac{r_0}{P_0^{0.71}}+t_4;$

and the third module is used for solving the motion displacement w (x, t) of the free hull beam in five time phases of underwater explosion:

wherein x is the abscissa value of the free hull beam at any point,k₂＝t₂-t₁， is the first-order natural vibration shape of the free hull beam,wherein ζ₁Amplitude of vibration, μ₁Is a beam motion frequency parameter, l is the free hull beam length;is an integration constant, ω, of five time phases₁The first-order mode shape natural frequency of the free beam; phi is a₁In order to be an integration constant, the first,m is the mass per unit beam length considering the mass of the attached water, p (x) is a pressure distribution characteristic function of the free hull beam in different time stages, and t is more than or equal to 0 and less than t₁、t₁≤t＜t₂The time period is a period of time,for the t-th₂≤t＜t₃、t₃≤t＜t₄、t₄≤t＜t₅The time period is a period of time, $p\left(x\right)=(1-\frac{2x-l}{l})\&CenterDot;\exp [-8{\left(\frac{2x-l}{l}\right)}^2+4{\left(\frac{2x-l}{l}\right)}^3]+{\left(\frac{2l-2x}{l}\right)}^{1.5}\&CenterDot;\left(\frac{2x-l}{l}\right);$

the fourth module is used for solving the speed and bending moment values of the hull beam at different stages according to the elastic movement displacement w (x, t) of the hull beam at five time stages;

the fifth module is used for judging that the ship body beam enters forward plastic motion when the absolute value of the bending moment of the midpoint of the ship body beam exceeds the absolute value of the plastic limit bending moment Ms of the ship body beam, and solving the motion amplitude H' (t) and the relative rotation corner alpha (t) of the ship body beam during the forward plastic motion:

$\left\{\begin{array}{c}{\stackrel{\&CenterDot;\&CenterDot;}{H}}^{\&prime;}\left(t\right)=\frac{3M_s-(l{\mathrm{\&xi;}}_3+3{\mathrm{\&xi;}}_4)p\left(t\right)}{(l{\mathrm{\&xi;}}_1+3{\mathrm{\&xi;}}_2)m}\\ \stackrel{}{\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&alpha;}}}\left(t\right)=\frac{24({\mathrm{\&xi;}}_1{\mathrm{\&xi;}}_4-{\mathrm{\&xi;}}_2{\mathrm{\&xi;}}_3)p\left(t\right)-24M_s{\mathrm{\&xi;}}_1}{(l{\mathrm{\&xi;}}_1+3{\mathrm{\&xi;}}_2)m{l}^2}\end{array}\right.,$

ξ₁＝λ₁(l)-λ₁(l/2)，ξ₂＝λ₂(l)-λ₂(l/2)-λ₁(l)·l/2，

ξ₃＝η₁(l/2)-η₁(l)，ξ₄＝η₂(l/2)-η₂(l)+η₁(l)·l/2；

${\&Integral;}_0^{x}p\left(x\right)\mathrm{dx}={\mathrm{\&eta;}}_1\left(x\right),{\&Integral;}_0^x{\&Integral;}_0^{x}p\left(x\right)\mathrm{dxdx}={\mathrm{\&eta;}}_2\left(x\right);$

a sixth module for plastically deforming the hull beam in the forward directionWhen the maximum value is reached, namely α (t) reaches the maximum value, judging that the forward plastic deformation stage of the ship body beam is ended, entering reverse elastic unloading, and according to a formula $\left\{\begin{array}{c}\mathrm{\&alpha;}\left(t\right)(x-l/2)+{\mathrm{\&phi;}}_1\left(x\right)H^{\&prime;}\left(t\right)={\mathrm{\&phi;}}_1\left(x\right)H\left(t\right)\\ \stackrel{\&CenterDot;}{\mathrm{\&alpha;}}\left(t\right)(x-l/2)+{\mathrm{\&phi;}}_1\left(x\right){\stackrel{\&CenterDot;}{H}}^{\&prime;}\left(t\right)={\mathrm{\&phi;}}_1\left(x\right)\stackrel{\&CenterDot;}{H}\left(t\right)\end{array}\right.$ Solving the motion amplitude H (t) of the reverse elastic unloading process of the hull beam;

a seventh module for exceeding the plastic limit bending moment M when the absolute value of the bending moment at the middle point of the beam exceeds the plastic limit bending moment M in the process of the reverse elastic deformation of the hull beam_sWhen the absolute value is obtained, the ship body beam is judged to enter the reverse plastic motion, and the motion amplitude H of the beam after the beam enters the reverse plastic deformation is solved₁' (t) and angle of relative rotation α₁(t)：

$\left\{\begin{array}{c}{{\stackrel{\&CenterDot;\&CenterDot;}{H}}_1}^{\&prime;}\left(t\right)=\frac{3(-M_s)-(l{\mathrm{\&xi;}}_3+3{\mathrm{\&xi;}}_4)p\left(t\right)}{(l{\mathrm{\&xi;}}_1+3{\mathrm{\&xi;}}_2)m}\\ {\stackrel{\&CenterDot;\&CenterDot;}{\mathrm{\&alpha;}}}_1\left(t\right)=\frac{24({\mathrm{\&xi;}}_1{\mathrm{\&xi;}}_4-{\mathrm{\&xi;}}_2{\mathrm{\&xi;}}_3)p\left(t\right)-24(-M_s){\mathrm{\&xi;}}_1}{(l{\mathrm{\&xi;}}_1+3{\mathrm{\&xi;}}_2)m{l}^2}\end{array}\right.;$

An eighth module for α when the reverse plastic deformation of the hull beam reaches a maximum₁(t) when the maximum value is reached, judging that the backward plastic deformation stage of the hull beam is finished, entering forward elastic unloading, and carrying out forward elastic unloading according to the result $\left\{\begin{array}{c}{\mathrm{\&alpha;}}_1\left(t\right)(x-l/2)+{\mathrm{\&phi;}}_1\left(x\right){H_1}^{\&prime;}\left(t\right)={\mathrm{\&phi;}}_1\left(x\right)H_1\left(t\right)\\ {\stackrel{\&CenterDot;}{\mathrm{\&alpha;}}}_1\left(t\right)(x-l/2)+{\mathrm{\&phi;}}_1\left(x\right){{\stackrel{\&CenterDot;}{H}}_1}^{\&prime;}\left(t\right)={\mathrm{\&phi;}}_1\left(x\right){\stackrel{\&CenterDot;}{H}}_1\left(t\right)\end{array}\right.$ Solving the motion amplitude H of the ship body beam in the forward elastic unloading stage₁(t)。