CN110795783B - Method for forecasting natural frequency of liquid-filled cylindrical shell - Google Patents

Method for forecasting natural frequency of liquid-filled cylindrical shell Download PDF

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CN110795783B
CN110795783B CN201910924830.6A CN201910924830A CN110795783B CN 110795783 B CN110795783 B CN 110795783B CN 201910924830 A CN201910924830 A CN 201910924830A CN 110795783 B CN110795783 B CN 110795783B
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柳贡民
曹银行
张文平
张新玉
明平剑
曹贻鹏
国杰
赵晓臣
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Harbin Engineering University
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Abstract

The invention aims to provide a method for forecasting the natural frequency of a liquid-filled cylindrical shell, which comprises the following steps: establishing a free vibration equation of the cylindrical shell by adopting a Kennard thin shell theory; by solving a Helmholtz wave equation under a cylindrical coordinate system, acting fluid as a radial load on a cylindrical shell, and establishing a liquid-filled cylindrical shell dynamic model; selecting an improved state vector; the relationship between element displacement, corner, force and the like of the traditional state vector and the middle plane displacement is obtained by Kennard thin-shell theory, so that a conversion matrix between the improved state vector and the traditional state vector is established. And (4) introducing boundary conditions at two ends of the liquid filling cylinder shell to forecast the natural frequency of the liquid filling cylinder shell. The invention reduces the derivation workload of the first derivative equation and the transmission matrix and the possibility of the occurrence of derivation errors, has wide application range and simple and convenient realization process, and is very favorable for programming calculation; the forecasting precision is high, the variable degree of freedom can not be increased in the whole solving process, and the higher forecasting efficiency is ensured.

Description

Method for forecasting natural frequency of liquid-filled cylindrical shell
Technical Field
The invention relates to a natural frequency prediction method.
Background
Liquid-filled cylindrical shell structures are ubiquitous in building and ship pipeline systems, and a plurality of methods for forecasting the natural frequency of the structures also exist. The transfer matrix method has the advantages of few degrees of freedom related to variables, simplicity and quickness in solving and the like, is a common method for solving the problem of the dynamics of the chain structure, and is widely applied to solving the problems of beam or tube structure dynamics, sound propagation in a tube, fluid-solid coupling of a pipeline system and the like.
If the structure control differential equation is written into a first-order differential equation set form, a simple relation can be established between the state vectors of the starting point and the end point by using a transfer matrix, and the inherent frequency of the liquid-filled cylindrical shell can be forecasted by using boundary conditions. Therefore, the calculation accuracy of the transfer matrix method mainly depends on the selection of the state vector and the integral of the matrix equation. The dynamic equation integration method is widely researched, such as a fine time-course integration method, a Longge Kutta method and a homogeneous expansion integration method. With the development of computer technology, the integral calculation precision is continuously improved and basically meets the requirements. With the application of the transfer matrix method in a multi-degree-of-freedom system, the derivation of the state vector first-order equation becomes more difficult. The traditional state vector generally consists of displacement, speed, force, moment and the like, and the state vector cannot be directly obtained from a vibration equation, so that the derivation process of a first-order differential equation of the state vector is very complicated, the workload is increased, and errors often occur.
The dimensionless natural frequency of the liquid-filled cylindrical shell is solved by applying a transfer matrix method to the roman (the roman, research on fluid-solid coupling calculation of the liquid conveying pipeline, master academic paper 2014, university of harbin engineering), but the derivation process of a state vector first-order differential equation is complex, and the transfer matrix is a circumferential transfer matrix which is different from a common transfer matrix along the length direction of a chain structure.
Wanhaochuan et al (Wanhaochuan, Li Shang, Zheng, improved structure vibration transmission matrix method, vibration and impact, 2013,32(1): 173-.
Disclosure of Invention
The invention aims to provide a natural frequency forecasting method of a liquid-filled cylindrical shell, which improves the traditional transfer matrix method.
The purpose of the invention is realized by the following steps:
the invention relates to a method for forecasting the natural frequency of a liquid filling cylindrical shell, which is characterized by comprising the following steps:
(1) dynamic modeling of the liquid-filled cylindrical shell:
firstly, establishing a free vibration equation of a cylindrical shell by adopting a Kennard thin shell theory, and simultaneously acting fluid as radial load on the cylindrical shell; and finally deducing to obtain a liquid-filled cylindrical shell dynamic model according to the boundary conditions of fluid and solid on the coupling surface by solving a Helmholtz wave equation under a cylindrical coordinate system:
Considering the influence of the inertia force of the shell and the action of fluid load, the Kennard thin-shell theoretical vibration differential equation is in the form as follows:
Figure BDA0002218621330000021
Figure BDA0002218621330000022
Figure BDA0002218621330000023
wherein
Figure BDA0002218621330000024
The propagation phase velocity of the telescopic wave in the flat plate is the same as that of the cylindrical shell material; p is the hydrodynamic pressure of the fluid in the pipe acting on the inner wall of the cylindrical shell;
for a vibration mode, ω, with an axial half wave number m and a circumferential wave number n mn For frequency, the solution of the above equation is written as:
Figure BDA0002218621330000025
Figure BDA0002218621330000026
Figure BDA0002218621330000027
the equation of the fluid inside the cylindrical shell satisfies:
Figure BDA0002218621330000028
assume that the solution of the internal fluid acoustic pressure is written as:
Figure BDA0002218621330000031
Figure BDA0002218621330000032
for radial wavenumbers, the following equation is satisfied:
Figure BDA0002218621330000033
wherein: k is a radical of f ＝ω/c f ，c f Is the wave velocity of free sound field of fluid in the pipe, k m Represents the axial wavenumber, and is related to the boundary conditions of the system.
At R ═ R, the continuity conditions are:
Figure BDA0002218621330000034
substituting the formula w (x, theta, t) and the formula p (theta, r, x, t) into the continuity condition to obtain:
Figure BDA0002218621330000035
(2) and (3) improving state vector selection:
u (x, theta, t), v (x, theta, t), w (x, theta, t) and p (x, theta, t) are expressed r＝R Substituting Kennard thin shell theory vibration differential equation to obtain:
Figure BDA0002218621330000036
Figure BDA0002218621330000037
Figure BDA0002218621330000038
the improved state vector is selected directly from the equation:
Figure BDA0002218621330000041
overwrite with the modified state vector:
Figure BDA0002218621330000042
c is an 8-order square matrix:
C(1,2)＝C(3,4)＝C(5,6)＝C(6,7)＝C(7,8)＝1；
Figure BDA0002218621330000043
Figure BDA0002218621330000044
Figure BDA0002218621330000045
Figure BDA0002218621330000046
(3) introducing a conversion matrix:
the state vector of the shell model is:
Figure BDA0002218621330000047
the Kennard thin-shell theory gives the relationship between element displacement, corner, force and mid-plane displacement of the traditional state vector, so that the improved state vector and the traditional state vector meet the following requirements:
Φ＝Dξ
D is a conversion matrix, and the non-0 elements of the conversion matrix satisfy:
D(1，1)＝D(2，3)＝D(3，5)＝D(4，6)＝1；
Figure BDA0002218621330000051
Figure BDA0002218621330000052
Figure BDA0002218621330000053
Figure BDA0002218621330000054
(4) forecasting natural frequency:
is composed of
Figure BDA0002218621330000055
And the transformation matrix can be used for solving a liquid-filled cylindrical shell transfer matrix T:
Φ(x)＝Dξ(χ)＝De Cs ξ(0)＝De Cs D -1 Dξ(0)＝De Cs D -1 Φ(0)＝TΦ(0)
introducing boundary conditions at two ends of the liquid filling cylindrical shell to forecast the natural frequency of the liquid filling cylindrical shell;
wherein x, r and theta are the directions of the cylindrical coordinates; mu is Poisson's ratio; omega is frequency variable, R is the radius of the middle surface of the cylindrical shell, E is Young's modulus, n is circumferential wave number, k m Is axial wavenumber, J n Is a Bessel function, N x Is middle equivalent film force, M x Is the middle equivalent bending moment, u, v, w is the displacement along the cylindrical coordinate direction, rho is the density, h is the thickness of the cylindrical shell, m is the axial half wave number,
Figure BDA0002218621330000056
Is radial wave number, t is time variable, c f Is fluid free acoustic field wave velocity, J' n As the first derivative of the Bessel function, F x Is a transverse shearing force, S x A thin film shear force.
The present invention may further comprise:
1. the natural frequency forecast has two simply-supported boundaries as follows:
Figure BDA0002218621330000063
Figure BDA0002218621330000064
comprises the following steps:
Figure BDA0002218621330000061
order:
Figure BDA0002218621330000062
the non-zero solution condition is that T 'is 0, and T' is a frequency function, so that the modal natural frequency value of each order can be obtained; for different boundary constraints, different T's may be obtained, but all have T' equal to 0.
The invention has the advantages that: the invention reduces the derivation workload of the first derivative equation and the transmission matrix and the possibility of the occurrence of derivation errors, has wide application range and simple and convenient realization process, and is very favorable for programming calculation; the forecasting precision is high, the variable degree of freedom can not be increased in the whole solving process, and the higher forecasting efficiency is ensured.
Drawings
FIG. 1 is a flow chart of the present invention;
fig. 2 is a schematic diagram of a liquid-filled cylindrical shell structure based on a thin shell model.
Detailed Description
The invention will now be described in more detail by way of example with reference to the accompanying drawings in which:
with reference to fig. 1-2, the improved transfer matrix method for forecasting the natural frequency of the liquid-filled cylindrical shell is implemented based on the conventional transfer matrix method and the improved state vector.
The forecasting method of the invention comprises the following steps:
(1) and (4) dynamic modeling of the liquid-filled cylindrical shell.
Firstly, establishing a free vibration equation of a cylindrical shell by adopting a Kennard thin shell theory, and simultaneously acting fluid as radial load on the cylindrical shell; and finally deducing to obtain a liquid-filled cylindrical shell dynamic model according to the boundary conditions of the fluid and the solid on the coupling surface by solving a Helmholtz wave equation under a cylindrical coordinate system.
The Kennard thin shell theoretical differential equation for vibration, taking into account the effects of the inertial forces of the shell and the fluid loads, is of the form (where radial motion is positive away from the shell axis):
Figure BDA0002218621330000071
Figure BDA0002218621330000072
Figure BDA0002218621330000073
wherein beta is 2 ＝h 2 /12R 2 ，
Figure BDA0002218621330000074
The propagation phase velocity of the telescopic wave in the flat plate is the same as that of the cylindrical shell material; and p is the hydrodynamic pressure of the fluid in the pipe acting on the inner wall of the cylindrical shell, and reflects the influence of the fluid in the pipe.
For a vibration mode, ω, with an axial half wave number m and a circumferential wave number n mn For frequency, the solution of equation (1) can be written as:
Figure BDA0002218621330000075
Figure BDA0002218621330000076
Figure BDA0002218621330000077
the equation of the fluid inside the cylindrical shell satisfies:
Figure BDA0002218621330000078
assuming that the solution of the internal fluid acoustic pressure can be written as:
Figure BDA0002218621330000079
Figure BDA00022186213300000710
for radial wavenumbers, the following equation is satisfied:
Figure BDA00022186213300000711
wherein: k is a radical of f ＝ω/c f ，c f Is the wave velocity of free sound field of fluid in the pipe, k m Represents the axial wavenumber, and is related to the boundary conditions of the system.
Continuity conditions are (at R ═ R):
Figure BDA0002218621330000081
substituting the formula (2c) and the formula (4) into the continuity condition to obtain:
Figure BDA0002218621330000082
(2) improved state vector selection
Substituting the formula (2) and the formula (7) into the formula (1) to obtain:
Figure BDA0002218621330000083
Figure BDA0002218621330000084
Figure BDA0002218621330000085
the improved state vector is selected directly from equation (8):
Figure BDA0002218621330000086
rewriting equation (8) with the improved state vector:
Figure BDA0002218621330000087
c is an 8-order square matrix, and the non-0 elements are directly obtained by the formula (8):
C(1,2)＝C(3,4)＝C(5,6)＝C(6,7)＝C(7,8)＝1；
Figure BDA0002218621330000091
Figure BDA0002218621330000092
Figure BDA0002218621330000093
Figure BDA0002218621330000094
(3) introducing a transformation matrix
The traditional state vector generally consists of displacement, velocity, force, moment, etc., and the state vector of the shell model generally is:
Figure BDA0002218621330000099
the Kennard thin-shell theory gives the relationship between element displacement, corner, force and the like of the traditional state vector and the displacement of the middle plane, so that the improved state vector and the traditional state vector meet the following requirements:
Φ＝Dξ
d is a conversion matrix, and the non-0 elements of the conversion matrix meet the following conditions:
D(1,1)＝D(2,3)＝D(3,5)＝D(4,6)＝1；
Figure BDA0002218621330000095
Figure BDA0002218621330000096
Figure BDA0002218621330000097
Figure BDA0002218621330000098
(4) natural frequency prediction
The liquid-filled cylindrical shell transfer matrix T can be found from equation (9) and the transformation matrix:
Φ(x)＝Dξ(x)＝De Cs ξ(0)＝De Cs D -1 Dξ(0)＝De Cs D -1 Φ(0)
＝TΦ(0)
therefore, the problem that the derivation process of the first-order derivative of the state vector of the traditional transfer matrix method is complex is solved by improving the selection of the state vector, and finally, the natural frequency of the liquid-filled cylindrical shell can be forecasted by introducing boundary conditions at two ends of the liquid-filled cylindrical shell. Take the two-end simply-supported boundary as an example:
Figure BDA0002218621330000101
Figure BDA0002218621330000102
Comprises the following steps:
Figure BDA0002218621330000103
order:
Figure BDA0002218621330000104
the non-zero solution condition of equation (10) is T 'is 0 and T' is a frequency function, so that the modal natural frequency value of each order can be obtained. For different boundary constraints, different T's may be obtained, but all have T' equal to 0.
Example (c):
the material and fluid parameters of the thin shell model are shown in Table 1, the two ends of the pipeline are fixedly supported, and the axial wave number k corresponding to several common boundaries m The values are shown in Table 2. The results of the calculations are shown in table 3 in comparison to the reference.
TABLE 1 thin shell model materials and fluid parameters
Figure BDA0002218621330000105
TABLE 2 axial wavenumbers for common boundary conditions
Figure BDA0002218621330000111
TABLE 3 inherent circular frequency of fluid-solid coupling of two-end solid branch fluid transmission pipeline
Figure BDA0002218621330000112
As can be seen from the results of the implementation of the present invention in the above examples, the present invention can be used for the liquid-filled cylindrical shell natural frequency prediction (example 1). The result of solving the natural frequency of the liquid-filled cylindrical shell system by the novel forecasting method is well matched with the result of the reference document, and the accuracy of the method is further verified.

Claims (2)

1. The method for forecasting the natural frequency of the liquid-filled cylindrical shell is characterized by comprising the following steps:
(1) dynamic modeling of the liquid-filled cylindrical shell:
firstly, establishing a free vibration equation of a cylindrical shell by adopting a Kennard thin shell theory, and simultaneously acting fluid as radial load on the cylindrical shell; and finally deducing to obtain a liquid-filled cylindrical shell dynamic model according to the boundary conditions of fluid and solid on the coupling surface by solving a Helmholtz wave equation under a cylindrical coordinate system:
Considering the influence of the inertia force of the shell and the action of fluid load, the Kennard thin-shell theoretical vibration differential equation is in the form as follows:
Figure FDA0003640834450000011
Figure FDA0003640834450000012
Figure FDA0003640834450000013
wherein
Figure FDA0003640834450000014
The propagation phase velocity of the telescopic wave in the flat plate is the same as that of the cylindrical shell material; p is the hydrodynamic pressure of the fluid in the pipe acting on the inner wall of the cylindrical shell;
for a vibration mode, ω, with an axial half wave number m and a circumferential wave number n mn For frequency, the solution of the above equation is written as:
Figure FDA0003640834450000015
Figure FDA0003640834450000016
Figure FDA0003640834450000017
the equation of the fluid inside the cylindrical shell satisfies:
Figure FDA0003640834450000018
assume that the solution of the internal fluid acoustic pressure is written as:
Figure FDA0003640834450000019
Figure FDA0003640834450000021
for radial wavenumbers, the following equation is satisfied:
Figure FDA0003640834450000022
wherein: k is a radical of f ＝ω/c f ，c f Is the wave velocity of the free sound field of the fluid in the pipe, k m Represents the axial wavenumber, and is related to the boundary condition of the system;
at R ═ R, the continuity conditions are:
Figure FDA0003640834450000023
substituting the formula w (x, theta, t) and the formula p (theta, r, x, t) into the continuity condition to obtain:
Figure FDA0003640834450000024
(2) and (3) improving state vector selection:
using formula u (x, theta, t), v (x, theta, t), w (x, theta, t) and formula p (x, theta, t) r＝R Substituting Kennard thin shell theory vibration differential equation to obtain:
Figure FDA0003640834450000025
Figure FDA0003640834450000026
Figure FDA0003640834450000027
the improved state vector is selected directly from the equation:
Figure FDA0003640834450000028
overwrite with the modified state vector:
Figure FDA0003640834450000029
c is an 8-order square matrix:
C(1,2)＝C(3,4)＝C(5,6)＝C(6,7)＝C(7,8)＝1；
Figure FDA0003640834450000031
Figure FDA0003640834450000032
Figure FDA0003640834450000033
Figure FDA0003640834450000034
Figure FDA0003640834450000035
(3) introducing a conversion matrix:
the state vector of the shell model is:
Figure FDA0003640834450000036
the Kennard thin-shell theory gives the relationship between element displacement, corner, force and mid-plane displacement of the traditional state vector, so that the improved state vector and the traditional state vector meet the following requirements:
Φ＝Dξ
D is a conversion matrix, and the non-0 elements of the conversion matrix meet the following conditions:
D(1,1)＝D(2,3)＝D(3,5)＝D(4,6)＝1；
Figure FDA0003640834450000037
Figure FDA0003640834450000038
Figure FDA0003640834450000039
Figure FDA00036408344500000310
(4) forecasting natural frequency:
is composed of
Figure FDA00036408344500000311
And the transformation matrix can be used for solving a liquid-filled cylindrical shell transfer matrix T:
Φ(x)＝Dξ(x)＝De Cs ξ(0)＝De Cs D -1 Dξ(0)＝De Cs D -1 Φ(0)
＝TΦ(0)
introducing boundary conditions at two ends of the liquid filling cylindrical shell to forecast the natural frequency of the liquid filling cylindrical shell;
wherein x, r and theta are the directions of the cylindrical coordinates; mu is Poisson's ratio; omega is frequency variable, R is the radius of the middle surface of the cylindrical shell, E is Young's modulus, n is circumferential wave number, k m Is axial wavenumber, J n Is a Bessel function, N x Is middle equivalent film force, M x Is the middle equivalent bending moment, u, v, w is the displacement along the cylindrical coordinate direction, rho is the density, h is the thickness of the cylindrical shell, m is the axial half wave number,
Figure FDA0003640834450000041
Is radial wave number, t is time variable, c f Is fluid free acoustic field wave velocity, J' n As the first derivative of the Bessel function, F x Is a transverse shearing force, S x A thin film shear force.
2. The method of claim 1 for forecasting the natural frequency of a liquid filled cylinder shell, comprising: the natural frequency forecast has two simply-supported boundaries as follows:
Figure FDA0003640834450000042
Figure FDA0003640834450000043
comprises the following steps:
Figure FDA0003640834450000044
order:
Figure FDA0003640834450000045
the non-zero solution condition is that T 'is 0, and T' is a frequency function, so that the modal natural frequency value of each order can be obtained; for different boundary constraints, different T's may be obtained, but all have T' equal to 0.
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Publication number Priority date Publication date Assignee Title
US7560632B1 (en) * 2006-07-17 2009-07-14 Lanzel Kenneth W Bass drum with compliant resonant head
CN109800512A (en) * 2019-01-23 2019-05-24 东北大学 Rotating cylindrical shell-variable cross-section disk-pretwist blade system dynamic modeling method
CN109948180A (en) * 2019-01-25 2019-06-28 北京航空航天大学 A kind of orthotropy opposite side freely-supported rectangular thin plate vibration analysis method

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