CN112395722B - Method for acquiring motion response of hull beam under action of underwater explosion and wave load - Google Patents

Method for acquiring motion response of hull beam under action of underwater explosion and wave load Download PDF

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CN112395722B
CN112395722B CN201910567669.1A CN201910567669A CN112395722B CN 112395722 B CN112395722 B CN 112395722B CN 201910567669 A CN201910567669 A CN 201910567669A CN 112395722 B CN112395722 B CN 112395722B
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孔祥韶
周红昌
石干
吴卫国
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Wuhan University of Technology WUT
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Abstract

The invention relates to a method for acquiring the motion response of a hull beam under the action of underwater explosion and wave load, which comprises the following steps: 1) simplifying the calculation model into simply supported beams at two ends; 2) establishing a transverse vibration equation of a calculation model; 3) simplifying the solution of the wave load; 4) establishing a motion equation of a calculation model under the action of wave load; 5) solving the underwater explosion load; 6) establishing a motion equation of a calculation model under the underwater explosion load; 7) establishing a motion equation of a calculation model under the combined action of underwater explosion and wave load; 8) and solving the motion response of the calculation model under the combined action of the underwater explosion and the wave load by using the boundary conditions. The motion model is established based on an analytical method, the analytical method can effectively improve the calculation efficiency, and simultaneously can visually analyze the influence of each parameter on the calculation model, thereby realizing the mastering of the motion rule of the hull beam under the combined action of underwater explosion and wave load on the basic principle.

Description

Method for acquiring motion response of hull beam under action of underwater explosion and wave load
Technical Field
The invention relates to the technical field of ship and ocean engineering, in particular to a method for acquiring motion response of a hull beam under the action of underwater explosion and wave load.
Background
At present, students at home and abroad mainly develop motion response of ships under the action of single load (underwater explosion or wave load), a few of students research the motion response of the ships under the coupling action of the underwater explosion and the static wave load by a simplified approximate method, the research on the dynamic collapse process of the ships under the combined action of the underwater explosion and the waves is still in a starting stage, and the related experimental research developed by the students is especially digressive. However, in actual sea warfare, a ship cannot be in a calm sea area, and if the influence of the wave load on the ship is not considered or only the influence of the wave load and the underwater explosion load on the ship is simply superposed, the real coupling effect of the two loads cannot be effectively reflected, so that serious hidden danger is left for the safety evaluation of the ship.
However, as can be seen from the current research situation at home and abroad, although a large amount of basic theoretical research is conducted by scholars at home and abroad on the motion response of the hull beam under the action of a single load (underwater explosion or wave load), the research on the hull beam motion response calculation method under the combined action of underwater explosion and wave load still remains a blank at present. Therefore, a reasonable and effective simplified theoretical model and a calculation method are provided and established, and certain necessity and theoretical value are provided for researching the motion characteristics and rules of the hull beam under the coupling action of the two loads.
Disclosure of Invention
Technical problem to be solved
In order to solve the problems in the prior art, the invention provides a hull beam motion response obtaining method under the action of underwater explosion and wave load.
(II) technical scheme
In order to achieve the purpose, the invention adopts the main technical scheme that:
a ship body beam motion response obtaining method under the action of underwater explosion and wave load is designed, and the method comprises the following steps:
step 1): the hull beam is assumed to be a non-uniform simply supported beam. Let beam length be L, take the neutralization axis of beam as the Ox axle, the origin of coordinates is the left end of beam. In the coordinate system, the bending rigidity which changes along the length direction of the beam is EI (x), the mass per unit length is m (x), the width of the ship hull beam is b (x), the non-conservative force which acts on the beam per unit length is Q (x, t), and the transverse displacement z (x, t) on the neutral axis of the beam is a function which continuously changes along the coordinate x and the time t.
Step 2): and establishing a transverse vibration equation of the calculation model.
The micro-section of the beam is taken as a research object, the viscous external damping force is considered to be in direct proportion to the speed of the micro-section of the beam, and the external resistance of the x position on the beam in unit lengthThe damping coefficient is c (x), the beam micro-section is subjected to damping force
Figure GDA0003476899760000021
In the initial state, the gravity and the buoyancy are assumed to be balanced with each other, but in the motion process, the difference between the gravity and the buoyancy is assumed to be proportional to the ship width b (x) and the transverse displacement z (x, t), the differential load between the gravity and the buoyancy is ρ gb (x) z (x, t), and the non-conservative force Q (x, t) at any time satisfies the following relation:
Figure GDA0003476899760000022
the Hamilton principle is used for deducing the motion equation of the hull beam to obtain:
Figure GDA0003476899760000023
step 3): the solution of the wave load is simplified.
According to the fact that when the actual ship body sails in waves, due to the fact that the relative positions of wave crests, wave troughs and the ship body are different, the ship body beams alternately generate sagging and arching motions. Taking the midspan motion of the hull beam as an example, assume that the wave load is distributed sinusoidally along the hull beam.
Assuming that wave force on each micro-segment of the hull beam is also distributed in a sine form at each moment, the specific load form is shown as formula (3):
Figure GDA0003476899760000024
in the formula (3), BwAmplitude of wave load, ωwFor the angular frequency at which the wave load acts, ψ is the initial phase angle.
Step 4): and establishing a motion equation of the calculation model under the wave load.
According to the knowledge of structural dynamics, the vibration mode function of the simply supported beam is as follows:
Figure GDA0003476899760000031
in the formula (4), the constant AmDetermined by the initial conditions. Under the action of wave load, Am＝Amw. Solving a generalized matrix of the simply supported beam vibration under the action of the wave load by adopting a modal superposition method, wherein a solving equation is as follows:
Figure GDA0003476899760000032
step 5): underwater explosions simplify the solution of the load. The basic assumptions are as follows:
1. it is assumed that the underwater blast shock wave loading does not cause local damage to the structure and that the hull beam structure is not plastically deformed. Therefore, the influence of the pulsating load of the underwater explosion bubbles is not considered.
2. Assuming that the fluid is incompressible and spin-free, the fluid domain satisfies the following Laplace equation:
Figure GDA0003476899760000033
wherein the velocity potential
Figure GDA0003476899760000034
From incident velocity potential
Figure GDA0003476899760000035
Diffraction velocity potential
Figure GDA0003476899760000036
And radiation velocity potential
Figure GDA0003476899760000037
And satisfies the following relation:
Figure GDA0003476899760000038
3. in the underwater explosion process of the far and middle fields, the influence of the hull beam on the bubbles is ignored, and only the influence of the bubbles on the hull beam is considered. Diffraction velocity potential in the fluid region around the bubble
Figure GDA0003476899760000039
And radiation velocity potential
Figure GDA00034768997600000310
Both are 0, and the fluid region near the hull beam satisfies equation (7).
4. The bubbles are assumed to be spherical and the transverse dimension of the hull beam is a tiny amount relative to the distance between the bubbles and the hull.
It can be seen from the longitudinal section and the transverse section that the coordinate system of the hull beam is xyz, the origin of the coordinate is established at the stern, the positive direction of the x axis is from the stern to the bow, the y axis is along the width direction of the ship, and the positive direction of the z axis is from the vertical direction. The bubble coordinate system is delta zeta gamma, the bubble center is taken as the coordinate origin, and the directions of the delta axis, the zeta axis and the gamma axis are respectively the same as the directions of the x axis, the y axis and the z axis. The distance from the center of the bubble to the hull is recorded as H0And the distance from the water surface is marked as H. The hull beam is a beam with equal section, and the height of the water part is marked as H1The underwater part circle radius is denoted as R.
For the fluid domain near the bubble, the velocity potential function satisfies the Laplace equation:
Figure GDA0003476899760000041
the lagrange integral of the unsteady non-rotational motion of the fluid on the surface of the bubble is:
Figure GDA0003476899760000042
in formula (9), PmWhere z is the pressure at infinity on the 0 plane, P is the fluid pressure at the outer surface of the bubble, and ρ is the fluid density.
The velocity component in the radial direction of the bubble is:
Figure GDA0003476899760000043
in the formula, vrIs the radial velocity of the surface of the bubble, rbIs the bubble radius.
The bubble load is regarded as a point source, the point source is positioned at the center of the bubble, and the intensity is recorded as Qb(t) of (d). The incident velocity potential caused by bubble loading in the flow field is:
Figure GDA0003476899760000044
in formula (11), raThe distance from the center of the bubble to any point.
The pressure inside the bubble can be determined by the following equation of state:
Figure GDA0003476899760000045
in the formula (12), PbIs the internal pressure of the bubble, PcAnd P0Respectively the saturated vapor pressure of the gas and the initial pressure of the bubbles, V0And V is the initial volume of the bubble and the volume of the bubble at an arbitrary time, γ is the adiabatic index of the gas, and γ is 1.4 for an ideal gas. Irrespective of the tension on the bubble surface and the gas saturation vapor pressure, the fluid pressure P at the bubble outer surface is expressed as:
Figure GDA0003476899760000051
the simultaneous equations (9) to (13) can be obtained:
Figure GDA0003476899760000052
Figure GDA0003476899760000053
the velocity potential of the flow field near the hull beam is:
Figure GDA0003476899760000054
based on the 4 th assumption, it can be considered that: y is<<H，z<<H. Therefore, the second order trace y in equation (16)2、z2Negligible, on the basis of which the taylor expansion of equation (16) is performed,
Figure GDA0003476899760000055
omitting the higher order infinite term, equation (17) can be transformed into:
Figure GDA0003476899760000056
because the invention is suitable for middle and far field underwater explosion, the incident velocity of the fluid around the hull beam can only consider a vertical component, and the expression is as follows:
Figure GDA0003476899760000057
in the middle and far field explosion, the flow field velocity near the hull beam caused by the underwater explosion bubble load can be considered to be mainly along the y-axis and z-axis directions based on the 4 th hypothesis,
Figure GDA0003476899760000061
the flow field near the hull beam can be converted into a two-dimensional flow field, and the velocity potential function of the two-dimensional flow field meets the Laplace equation:
Figure GDA0003476899760000062
the velocity potential near the hull beam meets the surface impenetrable condition:
Figure GDA0003476899760000063
in the formula (22), raThe' is the distance between any point on the surface of the ship body beam and the ship body beam in the two-dimensional flow field, and theta is the included angle between the z axis and the normal vector of the surface of the ship body beam.
According to the Laplace equation of the two-dimensional flow field, the diffraction velocity potential and the incident velocity potential satisfy the following relationship on the surface of the ship body:
Figure GDA0003476899760000064
diffraction velocity potential
Figure GDA0003476899760000065
The solution is in the form:
Figure GDA0003476899760000066
radiation velocity potential
Figure GDA0003476899760000067
The ship surface impenetrable condition is met:
Figure GDA0003476899760000068
diffraction velocity potential
Figure GDA0003476899760000069
The solution is in the form:
Figure GDA00034768997600000610
considering that the explosion distance is longer, the fluid velocity potential gradient near the ship body is smaller, namely the requirement of meeting
Figure GDA00034768997600000611
And in the total pressure composition of the hull surface, the position water head z (x, t) is negligible, and the pressure of the hull surface meets the following linearized Bernoulli equation:
Figure GDA0003476899760000071
the fluid load on any section of the hull beam can be found by integrating P over the hull surface:
Figure GDA0003476899760000072
substituting the formula (18), the formula (24) and the formula (26) into the formula (28) yields:
Figure GDA0003476899760000073
in the formula, ma(x)＝πρR2Is the added mass of the hull beam profile in the water.
Step 6): and establishing a motion equation of the calculation model under the underwater explosion load.
Substituting the formula (29) into the damped vibration differential equation of the simply supported beam under the action of the external load,
Figure GDA0003476899760000074
in the formula (30), the first and second groups,
Figure GDA0003476899760000075
solving by adopting a modal superposition method to obtain:
Figure GDA0003476899760000076
in the formula, lambda is the proportionality coefficient of the additional mass and the mass of the hull beam section in water,
Figure GDA0003476899760000077
B'm(t) is the vibration mode force corresponding to B' (x, t), and the expression is as follows:
Figure GDA0003476899760000078
in formula (33), AmbDetermined by the initial conditions of underwater explosion bubble loading.
Step 7): and step 5) and step 6) are combined to obtain a motion equation of the calculation model under the combined action of the underwater explosion and the wave load, and then the motion response of the calculation model under the combined action of the underwater explosion and the wave load can be solved according to the boundary conditions.
(III) advantageous effects
The invention has the beneficial effects that: the invention combines the underwater explosion load with the wave load, provides a hull beam motion response calculation method under the combined action of the underwater explosion and the wave load, has simple operation and is easy to realize; the motion model is established based on an analytical method, the analytical method can effectively improve the calculation efficiency, and simultaneously can visually analyze the influence of each parameter on the calculation model, thereby realizing the understanding of the motion rule of the hull beam under the combined action of underwater explosion and wave load on the basic principle.
Drawings
FIG. 1 is a schematic flow chart of a hull beam motion response acquisition method under the action of underwater explosion and wave load;
FIG. 2 is a schematic diagram of a simplified model of a hull beam that is a non-uniform simply supported beam;
FIG. 3 is a result of solving a simplified model of a hull beam in an embodiment of the invention;
FIG. 4 is a schematic illustration of the sinusoidal distribution of wave loads along a hull beam;
FIG. 5 is a schematic diagram of a displacement time course curve of the middle part of a hull beam under the action of wave load in the embodiment of the invention;
FIG. 6 is a simplified model schematic view (longitudinal cross-sectional view) of a hull beam under underwater explosive bubble loading;
FIG. 7 is a simplified model schematic (cross-sectional) view of a hull beam under underwater explosive bubble loading;
FIG. 8 is a schematic representation of a time history of a point source intensity of an underwater explosive load in an embodiment of the present invention;
FIG. 9 is a schematic graph of the time history of the bubble radius of an underwater explosive load in an embodiment of the present invention;
FIG. 10 is a schematic diagram of a displacement time course curve of the middle part of a hull beam under the action of underwater explosive load in the embodiment of the invention;
fig. 11 is a schematic diagram of a displacement response time course curve of the middle part of the hull beam under the combined action of underwater explosion and wave load in the embodiment of the invention.
Detailed Description
For the purpose of better explaining the present invention and to facilitate understanding, the present invention will be described in detail by way of specific embodiments with reference to the accompanying drawings.
The invention provides a method for acquiring the motion response of a hull beam under the action of underwater explosion and wave load, which comprises the following steps of:
step 1): assuming that the hull beam is an uneven simply supported beam, as shown in fig. 2, let the beam length be L, take the neutralization axis of the beam as the Ox axis, and the origin of coordinates be the left end of the beam. In the coordinate system, the bending rigidity which changes along the length direction of the beam is EI (x), the mass per unit length is m (x), the width of the ship hull beam is b (x), the non-conservative force which acts on the beam per unit length is Q (x, t), and the transverse displacement z (x, t) on the neutral axis of the beam is a function which continuously changes along the coordinate x and the time t.
Step 2): and establishing a transverse vibration equation of the calculation model.
The beam micro-section is taken as a research object, the viscous external damping force is considered to be in direct proportion to the speed of the beam micro-section, the external damping coefficient of x on the beam in unit length is set as c (x), and the damping force borne by the beam micro-section is
Figure GDA0003476899760000091
In the initial state, the gravity and the buoyancy are assumed to be balanced with each other, but in the motion process, the difference between the gravity and the buoyancy is assumed to be proportional to the ship width b (x) and the transverse displacement z (x, t), the differential load between the gravity and the buoyancy is ρ gb (x) z (x, t), and the non-conservative force Q (x, t) at any time satisfies the following relation:
Figure GDA0003476899760000092
the Hamilton principle is used for deducing the motion equation of the hull beam to obtain:
Figure GDA0003476899760000093
step 3): the solution of the wave load is simplified.
According to the fact that when the actual ship body sails in waves, due to the fact that the relative positions of wave crests, wave troughs and the ship body are different, the ship body beams alternately generate sagging and arching motions. Taking the midspan motion of the hull beam as an example, assume that the wave load is distributed sinusoidally along the hull beam, as shown in fig. 4.
Assuming that wave force on each micro-segment of the hull beam is also distributed in a sine form at each moment, the specific load form is shown as formula (3):
Figure GDA0003476899760000101
in the formula (3), BwIs the wave load amplitude, omegawFor the angular frequency of the wave loading action ψ is the initial phase angle.
Step 4): and establishing a motion equation of the calculation model under the wave load.
According to the knowledge of structural dynamics, the vibration mode function of the simply supported beam is as follows:
Figure GDA0003476899760000102
in the formula (4), the constant AmDetermined by the initial conditions. Under the action of wave load, Am＝Amw. Solving a generalized matrix of the simply supported beam vibration under the action of the wave load by adopting a modal superposition method, wherein a solving equation is as follows:
Figure GDA0003476899760000103
step 5): the underwater explosion simplifies the solving of the load. The basic assumptions are as follows:
1. it is assumed that the underwater blast shock wave load does not cause local damage to the structure and the hull beam structure does not plastically deform. Therefore, the influence of the pulsating load of the underwater explosion bubbles is not considered.
2. Assuming that the fluid is incompressible and spin-free, the fluid domain satisfies the following Laplace equation:
Figure GDA0003476899760000104
wherein the velocity potential
Figure GDA0003476899760000105
From incident velocity potential
Figure GDA0003476899760000106
Diffraction velocity potential
Figure GDA0003476899760000107
And radiation velocity potential
Figure GDA0003476899760000108
And satisfies the following relation:
Figure GDA0003476899760000109
3. in the underwater explosion process of the far and middle fields, the influence of the hull beam on the bubbles is ignored, and only the influence of the bubbles on the hull beam is considered. Diffraction velocity potential in the fluid region around the bubble
Figure GDA00034768997600001010
And radiation velocity potential
Figure GDA00034768997600001011
All are 0, and the fluid area near the hull beam satisfies equation (7).
4. The bubbles are assumed to be spherical and the transverse dimension of the hull beams is a minute amount relative to the distance between the bubbles and the hull.
As shown in fig. 6 and 7, it can be seen from the combination of the longitudinal section and the transverse section that the coordinate system of the hull beam is xyz, the origin of the coordinate is established at the stern, the positive direction of the x-axis is from the stern to the bow, the y-axis is along the width direction of the ship, and the positive direction of the z-axis is from the vertical direction. The bubble coordinate system is delta zeta gamma, the bubble center is taken as the coordinate origin, and the directions of the delta axis, the zeta axis and the gamma axis are respectively the same as the directions of the x axis, the y axis and the z axis. The distance from the center of the bubble to the hull is recorded as H0And the distance from the water surface is marked as H. The hull beam is a beam with equal section, and the height of the overwater part is recorded as H1The underwater part circle radius is denoted as R.
For the fluid domain near the bubble, the velocity potential function satisfies the Laplace equation:
Figure GDA0003476899760000111
the lagrange integral of the unsteady non-rotational motion of the fluid on the surface of the bubble is:
Figure GDA0003476899760000112
in formula (9), PmWhere z is the pressure at infinity on the 0 plane, P is the fluid pressure at the outer surface of the bubble, and ρ is the fluid density.
The velocity component in the radial direction of the bubble is:
Figure GDA0003476899760000113
in the formula, vrIs the radial velocity of the surface of the bubble, rbIs the bubble radius.
The bubble load is regarded as a point source, the point source is positioned at the center of the bubble, and the intensity is recorded as Qb(t) of (d). The incident velocity potential caused by bubble loading in the flow field is:
Figure GDA0003476899760000114
in formula (11), raThe distance from the center of the bubble to any point.
The pressure inside the bubble can be determined by the following equation of state:
Figure GDA0003476899760000121
in the formula (12), PbIs the internal pressure of the bubble, PcAnd P0Respectively the saturated vapor pressure of the gas and the initial pressure of the bubbles, V0And V is the initial volume of the bubble and the volume of the bubble at an arbitrary time, γ is the adiabatic index of the gas, and γ is 1.4 for an ideal gas. Irrespective of the tension on the bubble surface and the gas saturation vapor pressure, the fluid pressure P at the bubble outer surface is expressed as:
Figure GDA0003476899760000122
the simultaneous equations (9) to (13) can be obtained:
Figure GDA0003476899760000123
Figure GDA0003476899760000124
the velocity potential of the flow field near the hull beam is:
Figure GDA0003476899760000125
based on the 4 th assumption, it can be considered that: y is<<H，z<<H. Therefore, the second order trace y in equation (16)2、z2Negligible, on the basis of which the Taylor expansion of equation (16) is performed,
Figure GDA0003476899760000126
omitting the higher order infinite term, equation (17) can be transformed into:
Figure GDA0003476899760000127
because the invention is suitable for underwater explosion in middle and far fields, the incident velocity of the fluid around the hull beam can only consider the vertical component, and the expression is as follows:
Figure GDA0003476899760000131
in the middle and far field explosion, the flow field velocity near the hull beam caused by the underwater explosion bubble load can be considered to be mainly along the y-axis and z-axis directions based on the 4 th hypothesis,
Figure GDA0003476899760000132
the flow field near the hull beam can be converted into a two-dimensional flow field, and the velocity potential function of the two-dimensional flow field meets the Laplace equation:
Figure GDA0003476899760000133
the velocity potential near the hull beam meets the surface impenetrable condition:
Figure GDA0003476899760000134
in the formula (22), raThe' is the distance between any point on the surface of the ship body beam and the ship body beam in the two-dimensional flow field, and theta is the included angle between the z axis and the normal vector of the surface of the ship body beam.
According to the Laplace equation of the two-dimensional flow field, the diffraction velocity potential and the incident velocity potential satisfy the following relationship on the surface of the ship body:
Figure GDA0003476899760000135
diffraction velocity potential
Figure GDA0003476899760000136
The solution is in the form:
Figure GDA0003476899760000137
radiation velocity potential
Figure GDA0003476899760000138
The ship surface impenetrable condition is met:
Figure GDA0003476899760000139
diffraction velocity potential
Figure GDA00034768997600001310
The solution is in the form:
Figure GDA0003476899760000141
considering that the explosion distance is longer, the fluid velocity potential gradient near the ship body is smaller, namely the requirement of meeting
Figure GDA0003476899760000142
And in the total pressure composition of the hull surface, the position water head z (x, t) is negligible, and the pressure of the hull surface meets the following linearized Bernoulli equation:
Figure GDA0003476899760000143
the fluid load on any section of the hull beam can be found by integrating P over the hull surface:
Figure GDA0003476899760000144
substituting the formula (18), the formula (24) and the formula (26) into the formula (28) yields:
Figure GDA0003476899760000145
in the formula, ma(x)＝πρR2Is the added mass of the hull beam profile in the water.
Step 6): and establishing a motion equation of the calculation model under the underwater explosion load.
Substituting the formula (29) into the damped vibration differential equation of the simply supported beam under the action of the external load,
Figure GDA0003476899760000146
in the formula (30), the first and second groups,
Figure GDA0003476899760000147
solving by adopting a modal superposition method to obtain:
Figure GDA0003476899760000148
in the formula, lambda is the proportionality coefficient of the additional mass and the mass of the hull beam section in water,
Figure GDA0003476899760000149
B'm(t) is the mode shape force corresponding to B' (x, t), and the expression is as follows:
Figure GDA0003476899760000151
in formula (33), AmbDetermined by the initial conditions of underwater explosion bubble loading.
Step 7): and 5) and 6) simultaneously, obtaining a motion equation of the calculation model under the combined action of the underwater explosion and the wave load, and further solving the motion response of the calculation model under the combined action of the underwater explosion and the wave load according to the boundary conditions.
The invention is applied in the following examples: step 1): as shown in fig. 2, the hull beam is simplified into an eulerian bernoulli beam with two simply-supported ends, and the structural parameters include section inertia moment I of 2.8m4Taking the length L of the ship as 40m and the elastic modulus E as 2 x 1011N · m, the external load F (x, t) is 0 when freely vibrating. The calculation model is obtained by solving through a numerical method, and a specific solving result is shown in fig. 3. Step 2): as shown in fig. 4, the wave load is reduced to a sinusoidal load form. Assuming that the displacement of the middle part of the hull beam does not exceed 0.01m under the action of the wave load, the hull beam is a straight beam with an equal section
Figure GDA0003476899760000152
In addition, let
Figure GDA0003476899760000153
And solving the motion equation of the calculation model under the wave load by adopting a numerical calculation method to obtain a motion equation curve when the middle part of the hull beam is displaced, wherein the motion equation curve is shown in figure 5. And step 3): the simplified model of the hull beam under the underwater explosion bubble load is shown in fig. 6 and 7, and the working conditions of the underwater explosion load are as follows: 450kgTNT in an explosion of 60m under a midship, the initial radius r01.2m, initial pressure P0＝4.98×107Pa, solving the simplified load of the underwater explosion to obtain time history curves of the point source intensity and the bubble radius of the underwater explosion load, wherein the time history curves are respectively shown in fig. 8 and fig. 9. Step 4): substituting the underwater explosive load in the step 3) into the external load F (x, t) in the step 1) to obtain a motion equation of the calculation model under the action of the underwater explosive load:
Figure GDA0003476899760000154
in the formula (35), the first and second groups,
Figure GDA0003476899760000161
and solving by adopting a modal superposition method to obtain a displacement time-course curve of the middle part of the hull beam under the action of the underwater explosive load, as shown in FIG. 10. Step 5): combining the step 2) and the step 4) to obtain a motion equation of the calculation model under the combined action of the underwater explosion and the wave load:
Figure GDA0003476899760000162
initial conditions in the step 1), the step 2) and the step 3) are taken, a modal superposition method is adopted for solving, and a hull beam middle displacement response time course curve under each working condition can be obtained, as shown in fig. 11.
While the present invention has been described with reference to the particular embodiments illustrated in the drawings, which are meant to be illustrative only and not limiting, it will be apparent to those of ordinary skill in the art in light of the teachings of the present invention that numerous modifications can be made without departing from the spirit and scope of the invention as defined by the appended claims.

Claims (1)

1. A hull beam motion response obtaining method under the action of underwater explosion and wave load is characterized by comprising the following steps: the method comprises the following steps:
step 1): simplifying the calculation model into simply supported beams at two ends;
step 2): establishing a transverse vibration equation of a calculation model;
step 3): simplifying the solution of the wave load;
step 4): establishing a motion equation of a calculation model under the action of wave load;
step 5): solving the underwater explosion load;
step 6): establishing a motion equation of a calculation model under the underwater explosion load;
step 7): establishing a motion equation of a calculation model under the combined action of underwater explosion and wave load;
step 8): solving the motion response of the calculation model under the joint action of the underwater explosion and the wave load by utilizing the boundary conditions;
in the step 1), assuming that the length of the beam is L, taking a neutralization axis of the beam as an Ox axis and taking a coordinate origin as the left end part of the beam, and establishing a coordinate system; in the coordinate system, the bending rigidity which changes along the length direction of the beam is EI (x), the mass per unit length is m (x), the width of the ship body beam is b (x), the non-conservative force which acts on the beam per unit length is Q (x, t), and the transverse displacement z (x, t) on the beam neutral axis is a function which continuously changes along the coordinate x and the time t;
in the step 2), the beam micro-section is taken as a research object, the viscous external damping force is in direct proportion to the speed of the beam micro-section, the external damping coefficient of x on the beam in unit length is set as c (x), and the damping force borne by the beam micro-section is
Figure FDA0003476899750000011
In the initial state, assuming that the gravity and the buoyancy are balanced mutually, and assuming that the difference between the gravity and the buoyancy is proportional to the ship width b (x) and the transverse displacement z (x, t) in the motion process, the differential load of the gravity and the buoyancy is rhogb (x) z (x, t), rho is the fluid density, and the non-conservative force Q (x, t) at any time satisfies the following conditions:
Figure FDA0003476899750000012
the Hamilton principle is used for deducing the motion equation of the hull beam to obtain:
Figure FDA0003476899750000021
in step 3), when the ship body navigates in the waves, the ship body beams alternately generate the motions of sagging and arching, the wave load is assumed to be distributed in a sine form along the ship body beams, the wave force on each micro-segment of the ship body beams is also assumed to be distributed in a sine form at each moment, and the specific load form is as follows:
Figure FDA0003476899750000022
in the formula (3), BwIs the wave load amplitude, omegawThe angular frequency of the wave load action is shown, and psi is the initial phase angle;
in the step 4), the vibration mode function of the simply supported beam is as follows:
Figure FDA0003476899750000023
in the formula (4), the constant AmDetermined by initial conditions; suppose A is under wave loadm＝Amw(ii) a Solving a generalized matrix of the simply supported beam vibration under the action of the wave load by adopting a modal superposition method,
Figure FDA0003476899750000024
in said step 5), the solution is performed based on the following assumptions:
1) the influence of the underwater explosion bubble pulsation load is not considered;
2) if the fluid is incompressible and not spin-free, the fluid domain satisfies the following Laplace equation:
Figure FDA0003476899750000025
wherein the velocity potential
Figure FDA0003476899750000026
From incident velocity potential
Figure FDA0003476899750000027
Diffraction velocity potential
Figure FDA0003476899750000028
And radiation velocity potential
Figure FDA0003476899750000029
And satisfies the following relation:
Figure FDA00034768997500000210
3) in the underwater explosion process of the medium and far field, the influence of the hull beam on the bubbles is ignored, and only the influence of the bubbles on the hull beam, namely the diffraction velocity potential in the fluid area around the bubbles is considered
Figure FDA0003476899750000031
And radiation velocity potential
Figure FDA0003476899750000032
Are all 0, the fluid area near the hull beam satisfies the formula (7);
4) the bubbles are assumed to be spherical, and the transverse size of the hull beam is a tiny amount relative to the distance between the bubbles and the hull;
the coordinate system of the hull beam is xyz, the origin of the coordinate is established at the stern, the positive direction of the x axis is from the stern to the bow, the y axis is along the width direction of the ship, and the z axis is in the vertical direction as the positive direction; the bubble coordinate system is delta zeta gamma, the bubble center is taken as the coordinate origin, and the directions of a delta axis, a zeta axis and a gamma axis are the same as the directions of an x axis, a y axis and a z axis respectively; the distance from the center of the bubble to the hull is recorded as H0And the distance from the water surface is marked as H; the hull beam is a beam with equal section, and the height of the water part is marked as H1The radius of the underwater part circle is marked as R;
for the fluid domain near the bubble, the velocity potential function of the bubble satisfies the Laplace equation:
Figure FDA0003476899750000033
the lagrange integral of the unsteady non-rotational motion of the fluid on the surface of the bubble is:
Figure FDA0003476899750000034
in formula (9), P∞The pressure at an infinite distance on a plane z is 0, P is the fluid pressure on the outer surface of the bubble, and rho is the fluid density;
the velocity component in the radial direction of the bubble is:
Figure FDA0003476899750000035
in the formula (10), vrIs the radial velocity of the surface of the bubble, rbIs the bubble radius;
the bubble load is regarded as a point source, the point source is positioned at the center of the bubble, and the intensity is recorded as Qb(t) gas in the flow fieldThe incident velocity potential due to bubble loading is:
Figure FDA0003476899750000036
in formula (11), raThe distance from the center of the bubble to any point;
the pressure inside the bubble is determined by equation (12):
Figure FDA0003476899750000041
in the formula (12), PbIs the internal pressure of the bubble, PcAnd P0Respectively the saturated vapor pressure of the gas and the initial pressure of the bubbles, V0V is the initial volume of the bubble and the volume of the bubble at any moment, and gamma is the adiabatic index of the gas; regardless of the tension on the bubble surface and the gas saturation vapor pressure, the fluid pressure P at the bubble outer surface is expressed as:
Figure FDA0003476899750000042
the following equations (9) to (13) yield:
Figure FDA0003476899750000043
Figure FDA0003476899750000044
velocity potential of flow field near hull beam
Figure FDA0003476899750000045
Comprises the following steps:
Figure FDA0003476899750000046
based on the fourth assumption, y<<H，z<<H; therefore, the second order trace y in equation (16)2、z2Neglecting, the Taylor expansion is performed on the basis of the formula (16),
Figure FDA0003476899750000047
omitting the higher order infinite terms, equation (17) translates to:
Figure FDA0003476899750000051
for medium and far field underwater explosion, only the vertical component is considered for the incident velocity of the fluid around the hull beam, and the expression is as follows:
Figure FDA0003476899750000052
in the middle and far field explosion, based on the 4 th hypothesis, the flow field velocity near the ship body beam caused by the underwater explosion bubble load is mainly along the y-axis and z-axis directions,
Figure FDA0003476899750000053
then the flow field near the hull beam is converted into a two-dimensional flow field, and the velocity potential function of the two-dimensional flow field meets the Laplace equation:
Figure FDA0003476899750000054
the velocity potential near the hull beam meets the surface impenetrable condition:
Figure FDA0003476899750000055
in the formula (22), r'aThe distance between any point on the surface of the ship body beam and the ship body beam in the two-dimensional flow field is shown, and theta is an included angle between a z-axis and a normal vector of the surface of the ship body beam;
obtaining a diffraction velocity potential and an incidence velocity potential by a two-dimensional flow field Laplace equation, wherein the diffraction velocity potential and the incidence velocity potential meet the requirements on the surface of the ship body:
Figure FDA0003476899750000056
diffraction velocity potential
Figure FDA0003476899750000057
The solution is in the form:
Figure FDA0003476899750000058
radiation velocity potential
Figure FDA0003476899750000059
The ship surface impenetrable condition is met:
Figure FDA00034768997500000510
radiation velocity potential
Figure FDA00034768997500000511
The solution is in the form:
Figure FDA0003476899750000061
due to the far explosion distance, the fluid velocity potential gradient near the ship body is small, namely the requirement is met
Figure FDA0003476899750000062
And in the total pressure composition of the hull surface, the position water head z (x, t) is ignored, and then the pressure of the hull surface meets the following linearized Bernoulli equation:
Figure FDA0003476899750000063
the fluid load on any section of the hull beam is then found by the integral of P over the hull surface:
Figure FDA0003476899750000064
substituting the formula (18), the formula (24) and the formula (26) into the formula (28) yields:
Figure FDA0003476899750000065
in the step 6), the formula (29) is substituted into a damped vibration differential equation of the simply supported beam under the action of the external load,
Figure FDA0003476899750000066
in the formula (30), the first and second groups,
Figure FDA0003476899750000067
solving by adopting a modal superposition method to obtain:
Figure FDA0003476899750000068
in the formula (31), lambda is the proportionality coefficient of the additional mass and the mass of the hull beam section in water,
Figure FDA0003476899750000069
B'm(t) is the vibration mode force corresponding to B' (x, t), and the expression is as follows:
Figure FDA0003476899750000071
in formula (33), AmbDetermining the initial conditions of underwater explosion bubble load;
in the step 7) and the step 8), the step 5) and the step 6) are combined to obtain a motion equation of the calculation model under the combined action of the underwater explosion and the wave load, and then the motion response of the calculation model under the combined action of the underwater explosion and the wave load is solved according to the boundary conditions.
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