CN113255241B - Method for establishing liquid shaking model in rectangular liquid tank with porous elastic baffle - Google Patents
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Abstract
The invention relates to the technical field of kinematics and dynamics modeling, and discloses a method for establishing a liquid shaking model in a rectangular liquid tank with a porous elastic baffle, wherein the porous elastic baffle is vertically arranged at the center of the rectangular liquid tank, the top end of the porous elastic baffle is connected with a pair of mooring lines with the rigidity of K, and the method is based on a hydrodynamic analysis method of a two-dimensional linear potential theory, and calculates the velocity potential of the liquid in the tank by using a Laplace equation and a linearization free liquid level boundary equation; and (3) according to a Green formula, matching a characteristic function expansion method and a Darcy theorem, solving to obtain bending moment and shear stress of liquid shaking on a baffle and liquid height and dynamic pressure in the liquid tank, thereby obtaining a specific analytical solution of important liquid shaking parameters in the tank. The mathematical model established by the invention can effectively solve the problem of dynamic analysis of the rectangular liquid tank with the porous elastic baffle, and applies smaller weight load to the tank body, thereby simplifying the calculation process and improving the accuracy of the model.
Description
Technical Field
The invention relates to the technical field of kinematics and dynamics modeling, in particular to a method for establishing a liquid shaking model in a rectangular liquid tank with a porous elastic baffle.
Background
When the container tank is externally excited, the liquid in the tank body is easy to generate intense free liquid level oscillation, and the phenomenon is called shaking, which is an important and worthy engineering problem. Previous researches show that impact load caused by shaking not only can damage the inner wall and the structure of the tank body, but also can cause traffic safety accidents caused by instability of the dangerous chemical tank truck under serious conditions, so that the introduction of the shaking prevention device is necessary for analyzing and researching liquid shaking in the storage tank.
Previously in anti-sloshing devices, vertical and horizontal rigid baffles have been considered as effective tools for inhibiting sloshing of liquids, but they also exert considerable gravity on the tank container, and so it would be of research value to introduce a lightweight porous elastic baffle. The liquid shaking analysis in the rectangular liquid tank with the porous elastic baffle is relatively less, and the kinematic and dynamic modeling technology is relatively deficient, so that a novel mathematical model for the liquid shaking in the rectangular liquid tank with the porous elastic baffle needs to be established, and the influence of the geometric shape and the structural parameters of the baffle on the liquid shaking can be further researched.
Disclosure of Invention
The invention aims to: aiming at the problems existing in the prior art, the invention provides a method for establishing a liquid shaking model in a rectangular liquid tank with a porous elastic baffle, which is a hydrodynamic analysis method based on a two-dimensional linear potential theory, and calculates the velocity potential of the liquid in the tank by using a Laplace equation and a linearization free liquid level boundary equationAccording to the Green formula matching characteristic function expansion method and Darcy theorem, the bending moment and shear stress of liquid shaking on the baffle plate and the liquid height and dynamic pressure in the liquid tank are obtained, so that the specific analysis of important liquid shaking parameters in the tank is obtained, and the mathematical model built by the invention canThe dynamic analysis problem of the rectangular liquid tank with the porous elastic baffle is effectively solved.
The technical scheme is as follows: the invention provides a method for establishing a liquid shaking model in a rectangular liquid tank with a porous elastic baffle, which comprises the following steps:
step 1: a porous elastic baffle is vertically arranged at the center of the rectangular liquid tank, the top end of the porous elastic baffle is connected with a pair of mooring lines with the spring rigidity of K, the other ends of the mooring lines are respectively fixed on the wall surfaces of tank bodies at two sides of the rectangular liquid tank, and the mooring lines and a y axis are distributed in an angle theta;
step 2: establishing a two-dimensional coordinate system of the rectangular liquid tank, taking the intersection point of the tank bottom of the rectangular liquid tank and the porous elastic baffle as the center o of the rectangular coordinate system, and vertically upwards along the y axis along one side of the bottom edge of the tank body in the positive direction of the x axis; the length of the rectangular liquid tank is 2a, the height l of the porous elastic baffle is equal to the height h of the liquid in the rectangular liquid tank;
step 3: obtaining the velocity potential of fluid movement in rectangular liquid tank by using green formula matching characteristic function expansion methodAn equation;
step 4: step expansion finishing is carried out on the velocity potential equation of the porous elastic baffle in the step 3 by utilizing a Galerkin method to obtain finished velocity potential;
step 5: when y is more than or equal to 0 and less than or equal to l, and x is more than or equal to 0, determining linearization dynamics and kinematic boundary condition equations which are required to be met by the fluid motion in the tank, and determining boundary condition equations at two ends of the baffle;
step 6: green formula G (y, y 0 ) Converting the Green formula by Laplace transformation method, and determining the complex displacement amplitude ζ of the baffle i (y)；
Step 7: amplitude ζ of complex displacement i (y) carrying out processing on the linear dynamics and kinematic boundary condition equation in the step 5, and carrying out integral processing on dynamic pressures of the wall surface and the bottom of the liquid tank to obtain an impact shaking load f generated by liquid shaking on the liquid tank S z ；
Step 8: and determining bending moment and shearing stress of the baffle plate according to the elastic structure displacement theory, and finally determining wave height and dynamic pressure in the rectangular liquid tank.
Further, the linear shaking in the rectangular liquid tank with the elastic baffle is antisymmetric, namelyOnly the fluid movement in the half liquid tank is considered when x is less than or equal to 0, and the other half part can be reversely deduced.
Further, the velocity potential in the step 3The equation is:
wherein the characteristic function equation isFunction eigenvalue k n (n=1, 2.) satisfying the following discrete relation +.>Omega is vibration frequency, g is gravitational acceleration, delta i2 Is a Kronecker function, +.>Is a characteristic function f n Expansion coefficient of (y), horizontal velocity potential->
Further, the velocity potential after finishing in the step 4The method comprises the following steps:
wherein ,J 2p+1 as Bessel function, I 2p+1 To correct the Bessel function, the order of the two functions is 2p+1; a, a ip Is an unknown coefficient; a is half of the total length of the tank body,n-degree Chebyshev polynomial->ψ p (y) is a square root singular point satisfying a boundary condition; k is the spring rate of the mooring line.
Further, the linearization dynamics and kinematics boundary condition equations to be satisfied by the fluid motion in the tank in the step 5 are:
wherein ,ρ is the fluid density, σ is the composite parameter, m is the mass per unit length of the baffle, ω is the vibration frequency, and EI is the bending stiffness of the baffle; j=1, 2, the excitation mode is 1 when swinging, and 2 when rolling;
the boundary condition equation at the two ends of the baffle is:
wherein ,is χ i The composite displacement amplitude of the baffle after normalization is K, which is the spring rigidity of the mooring line and χ i For the movement amplitude, subscript i is the excited movement mode of the liquid tank, 1 is swinging, and 2 is rolling; EI is the bending stiffness of the baffle.
Further, the complex displacement amplitude ζ of the baffle in the step 6 i (y) is:
wherein ,y0 Is a singular point.
Further, in the step 7, the impact shaking load generated by the liquid shaking on the liquid tankThe method comprises the following steps:
wherein ,
further, in the step 8, the bending moment and the shear stress of the baffle are respectively:
wherein ,mi (y) is the bending moment of the baffle plate, q i And (y) is a shear stress.
Further, the wave height and dynamic pressure in the rectangular liquid tank are as follows:
where i=1, 2.
The beneficial effects are that:
1. the method for establishing the liquid shaking model in the rectangular liquid tank with the porous elastic baffle provided by the invention can be used for an actual liquid shaking model in the rectangular liquid tank, is a fluid dynamics analysis method based on a two-dimensional linear potential theory, and calculates the velocity potential of the liquid in the tank by utilizing a Laplace equation and a linear free liquid level boundary equationAccording to the Green formula matching characteristic function expansion method and Darcy theorem, the bending moment and shear stress of liquid shaking on the baffle and the liquid height and dynamic pressure in the liquid tank are obtained, so that the specific analytical solution of important liquid shaking parameters in the tank is obtained.
2. Compared with the added absolute rigid baffle, the invention has the advantages that the density and Young modulus of the added elastic baffle are small, the mass is also small, the density and Young modulus of the former rigid baffle are large, the bending resistance is poor, the larger weight load is applied to the tank body, the vibration suppressing effect is not good, and the comparison of the provided ANSYS simulation diagrams shows that the modeling method of the baffle can effectively solve the problem. And the liquid is rocked and analyzed in the rectangular liquid tank by using the elastic baffle plate with holes, so that a smaller weight load is applied to the tank body, and the sloshing inhibition effect is better than that of the absolute rigid baffle plate.
3. The method simplifies the calculation process of deducing the complex displacement amplitude of the baffle by using the Green formula, thereby obtaining the bending moment and the shear stress of the baffle as key parameters, wherein the former deduction process is to establish the baffle boundary condition equation before deduction, the calculation process is long, and the method simplifies the calculation process.
4. Compared with the traditional modeling method, the method has the advantages that the equivalent mechanical model is mostly used, the liquid height is a numerical value obtained by observing rough measurement, the method is not accurate enough, the calculated numerical value is more accurate due to the fact that the method has specific formulas of liquid height and dynamic pressure, and the built model for researching liquid shaking is more accurate. According to an ANSYS simulation diagram, the simulation can be performed by the modeling method, so that the condition of impact between fluid in the reaction tank and the baffle can be accurately achieved, and the problem of analysis of the anti-shaking effect of the elastic baffle in the rectangular liquid tank can be effectively solved.
Drawings
FIG. 1 is a schematic diagram of two-dimensional fluid movement within a partially-baffled rectangular-section tank of the present invention;
FIG. 2 is a three-dimensional schematic view of a porous elastomeric baffle of the present invention;
fig. 3 is a graph showing the change in fluid level when the rigid baffle t=2s is installed;
fig. 4 is a graph showing the change of the fluid level when the elastic baffle t=2s is installed;
fig. 5 is a graph showing the change in fluid level when the rigid baffle t=4s is installed;
fig. 6 is a graph showing the change in fluid level when the elastic shutter t=4s is attached.
Detailed Description
The invention is further described below with reference to the accompanying drawings. The following examples are only for more clearly illustrating the technical aspects of the present invention, and are not intended to limit the scope of the present invention.
The invention discloses a method for establishing a liquid shaking model in a rectangular liquid tank with a porous elastic baffle, which comprises the following steps:
step 1: the multi-hole elastic baffle is vertically arranged at the center of the rectangular liquid tank, the top end of the multi-hole elastic baffle is connected with a pair of mooring lines with the spring rigidity of K, the other ends of the mooring lines are respectively fixed on the wall surfaces of tank bodies at two sides of the rectangular liquid tank, and the mooring lines are distributed with a y axis in an angle theta. And establishing a two-dimensional coordinate system of the rectangular liquid tank, taking the intersection point of the tank bottom and the baffle plate as the center o of the rectangular coordinate system, and enabling the positive direction of the x axis to be right along the bottom edge of the rectangular tank body and the y axis to be vertically upward. Wherein the length of the rectangular liquid tank is 2a, the height l of the baffle plate and the height h of the liquid filled in the tank are equal.
Step 2: under the conditions that the fluid in the rectangular liquid tank is incompressible, non-viscous and has small liquid level fluctuation, the velocity potential phi is established according to the linear potential flow theory i Is defined by the equation:
wherein the rectangular tank body is subjected to a sine function X i (t)＝χ i sin ωt excites motion, g is gravitational acceleration, y R As the rotation center χ i For the motion amplitude, ω is the vibration frequency, subscript i is the tank excited motion mode, 1 is wobble, 2 is roll.
Step 3: setting the deflection of the baffle to be xi i (y, t), when x=0, 0.ltoreq.y.ltoreq.l, the flexibility satisfies the linearized kinematic and dynamic boundary equations according to the Darcy theorem as
Wherein EI is the bending rigidity of the baffle, ρ is the fluid density, m is the mass per unit length of the baffle, U ri (y, t) is the horizontal velocity of the fluid relative to the tank, σ is the composite parameter.
Step 4: assuming that the mooring line at the upper end of the baffle in the rectangular tank is taut, i.e. when y=l, the boundary condition is satisfied:
the velocity potential and deflection of the baffle can be written as:
wherein ,is χ i And the Re is a Reynolds equation.
It can be seen that the linear sloshing in the rectangular tank with the elastic baffle is antisymmetric, i.eTherefore, only the fluid movement in the half liquid tank is needed to be considered when x is less than or equal to 0, and the other half part can be reversely deduced.
Step 5: from the above estimation, the velocity potential of the fluid movement in the rectangular tankLaplace equation and boundary conditions thereof:
when x is less than or equal to 0 and is less than or equal to 0, the formulas 5 and 6 can be combined in a tidying way, and the velocity potential is the same as thatSatisfy the equation
Step 6: matching a characteristic function expansion method according to the green formula, wherein the characteristic function equation is as follows:
its function characteristic value k n (n=1, 2.) the following discrete relationship is satisfied:
from the definition of 8, the characteristic equation f can be known n (y) is orthogonal.
wherein ,δmn Is a Kronecker function.
Step 7: obtainable according to boundary condition 6
Wherein (1)>Is a characteristic function f n Expansion coefficient of (y).
When x=0, the horizontal velocity potential u i (y) can be defined by the characteristic function f n (y) represents;
step 8: substituting formula 7 into formula 12 to combine formula 11, the product u can be obtained in A of the representation in ,B in The following are provided:
substituting equation 13 into equation 7, velocity potentialCan be arranged and written into
Step 9: the step expansion of the velocity potential equation of the baffle plate is realized by adopting a Galerkin method
wherein ,aip For unknown coefficients (the solution 24 can be a numerical value),n-degree Chebyshev polynomial->ψ p (y) is a square root singular point satisfying the boundary condition.
Then substituting equation 14 into equation 15, multiplying f by both sides simultaneously m (y) and integrating y to obtain
wherein ,J 2p+1 as Bessel function, I 2p+1 To correct the Bessel function, the order of the two functions is 2p+1.
Then the velocity potential equation of equation 14 can be written as
Step 10: when y is equal to or less than 0 and equal to or less than l, and x is equal to or less than 0, the motion of the fluid in the tank needs to meet the following linearization dynamics and kinematics boundary condition equation:
wherein ,
step 11: the boundary condition equation of the two ends of the baffle can be deduced from 3 as follows
To solve the boundary condition value problem, green formula G (y, y 0 )：
wherein ,y0 Is a singular point.
Then converting the Green formula by Laplace transformation method to obtain
wherein ,
the complex displacement amplitude ζ of the baffle can be obtained by the formula 19 and the Green formula 22 i (y) is
Step 12: then, formula 23 is substituted into formula 18, both sides are multiplied by ψ q (y) and is integrated over the range 0.ltoreq.y.ltoreq.l
Wherein q is taken
Step 13: then integrating dynamic pressures of the wall surface and the bottom of the liquid tank to obtain impact shaking load of liquid shaking on the liquid tankThe formula of (2) is:
wherein ,
step 14: the baffle bending moment formula is M according to the reference i (y,t)＝Re[m i (y)e -jωt ]The shear stress formula is Q i (y,t)＝Re[q i (y)e -jωt ]And then, obtaining the bending moment and the shear stress of the baffle by utilizing the elastic structure displacement theory:
the wave height and dynamic pressure in a rectangular tank can then be expressed as:
where i=1, 2.
According to the thought principle, fluid simulation is carried out in ANSYS software, a rectangular tank body with the length of 580mm, the width of 500mm and the height of 360mm and the installed baffle thickness of 12mm and the height of 80mm is selected as a study object, the fluid in the tank is set to be water, and the rigid baffle and the elastic baffle are respectively impacted by a water column so as to verify that the shaking prevention effect of adding the porous elastic baffle is better under the modeling method, so that the mathematical model established by the invention can effectively solve the kinetic analysis problem of the rectangular liquid tank. Simulation setting parameters: the density of the rigid baffle is 7800kg/m 3 Young's modulus of 2X 10 8 Pa; the density of the elastic baffle is 2700kg/m 3 Young's modulus of 10 6 Pa, fig. 3 and 5 show changes in the fluid free surface when the rigid shutters t=2s and t=4s are installed, and fig. 4 and 6 show changes in the fluid free surface when the elastic shutters t=2s and t=4s are installed. Through observation and comparison, the rise value of the free liquid level of fluid in the liquid tank provided with the rigid baffle is faster than that of the liquid tank provided with the elastic baffle, obvious liquid level crushing and liquid splashing phenomena occur, and the liquid level change in the liquid tank provided with the elastic baffle is more gentle.
The foregoing embodiments are merely illustrative of the technical concept and features of the present invention, and are intended to enable those skilled in the art to understand the present invention and to implement the same, not to limit the scope of the present invention. All equivalent changes or modifications made according to the spirit of the present invention should be included in the scope of the present invention.
Claims (8)
1. The method for establishing the liquid shaking model in the rectangular liquid tank with the porous elastic baffle is characterized by comprising the following steps of:
step 1: a porous elastic baffle is vertically arranged at the center of the rectangular liquid tank, the top end of the porous elastic baffle is connected with a pair of mooring lines with the spring rigidity of K, the other ends of the mooring lines are respectively fixed on the wall surfaces of tank bodies at two sides of the rectangular liquid tank, and the mooring lines and a y axis are distributed in an angle theta;
step 2: establishing a two-dimensional coordinate system of the rectangular liquid tank, taking the intersection point of the tank bottom of the rectangular liquid tank and the porous elastic baffle as the center o of the rectangular coordinate system, and vertically upwards along the y axis along one side of the bottom edge of the tank body in the positive direction of the x axis; the length of the rectangular liquid tank is 2a, the height l of the porous elastic baffle is equal to the height h of the liquid in the rectangular liquid tank;
step 3: obtaining the velocity potential of fluid movement in rectangular liquid tank by using green formula matching characteristic function expansion methodAn equation; speed potential->The equation is:
wherein the characteristic function equation isFunction eigenvalue k n (n=1, 2.) satisfying the following discrete relation +.>Omega is vibration frequency, g is gravitational acceleration, delta i2 Is a Kronecker function, +.>Is a characteristic function f n Expansion coefficient of (y), horizontal velocity potential->
Step 4: step expansion finishing is carried out on the velocity potential equation of the porous elastic baffle in the step 3 by utilizing a Galerkin method to obtain finished velocity potential;
step 5: when y is more than or equal to 0 and less than or equal to l, and x is more than or equal to 0, determining linearization dynamics and kinematic boundary condition equations which are required to be met by the fluid motion in the tank, and determining boundary condition equations at two ends of the baffle;
step 6: green formula G (y, y 0 ) Converting the Green formula by Laplace transformation method, and determining the complex displacement amplitude ζ of the baffle i (y)；
Step 7: amplitude ζ of complex displacement i (y) carrying out processing on the linear dynamics and kinematic boundary condition equation in the step 5, and carrying out integral processing on dynamic pressures of the wall surface and the bottom of the liquid tank to obtain the impact shaking load of liquid shaking on the liquid tank
Step 8: and determining bending moment and shearing stress of the baffle plate according to the elastic structure displacement theory, and finally determining wave height and dynamic pressure in the rectangular liquid tank.
2. The method for creating a model of liquid sloshing in a rectangular liquid tank with a porous elastic baffle according to claim 1, wherein the linear sloshing in the rectangular liquid tank with a porous elastic baffle is antisymmetric, namely
3. The method for establishing a liquid sloshing model in a rectangular liquid tank with a porous elastic baffle according to claim 1, wherein the velocity profile after finishing in step 4 is characterized in thatThe method comprises the following steps:
wherein ,J 2p+1 as Bessel function, I 2p+1 To correct the Bessel function, the order of the two functions is 2p+1; a, a ip Is an unknown coefficient; a is half of the total length of the tank body,n-degree Chebyshev polynomial->ψ p (y) is a square root singular point satisfying a boundary condition; k is the spring rate of the mooring line.
4. The method for creating a liquid sloshing model in a rectangular liquid tank with elastic porous baffles according to claim 1, wherein the linearization dynamics and kinematic boundary condition equations to be satisfied by the movement of the liquid in the tank in step 5 are:
wherein ,ρ is the fluid density, σ is the composite parameter, m is the mass per unit length of the baffle, ω is the vibration frequency, and EI is the bending stiffness of the baffle; j=1, 2, the excitation mode is 1 when swinging, and 2 when rolling;
the boundary condition equation at the two ends of the baffle is:
wherein ,is χ i The composite displacement amplitude of the baffle after normalization is K, which is the spring rigidity of the mooring line and χ i For the movement amplitude, subscript i is the excited movement mode of the liquid tank, 1 is swinging, and 2 is rolling; EI is the bending stiffness of the baffle.
5. The method for creating a sloshing model of liquid in a rectangular liquid tank with elastic porous baffles according to claim 4, wherein the complex displacement amplitude ζ of the baffles in step 6 i (y) is:
wherein ,y0 Is a singular point.
6. The method for creating a liquid sloshing model in a rectangular liquid tank with elastic porous baffle according to claim 1, wherein the impact sloshing load of the liquid sloshing on the liquid tank in step 7The method comprises the following steps:
wherein ,
7. the method for establishing a liquid sloshing model in a rectangular liquid tank with a porous elastic baffle according to claim 1, wherein the bending moment and the shearing stress of the baffle in the step 8 are respectively:
wherein ,mi (y) is the bending moment of the baffle plate, q i And (y) is a shear stress.
8. The method for creating a sloshing model of liquid in a rectangular liquid tank with elastic porous baffles according to claim 7, wherein the wave height and dynamic pressure in the rectangular liquid tank are:
where i=1, 2.
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