CN109740211A - A kind of prediction technique of function pipeline fluid structurecoupling inherent characteristic - Google Patents
A kind of prediction technique of function pipeline fluid structurecoupling inherent characteristic Download PDFInfo
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Abstract
The present invention relates to fluid conveying pipe fluid structurecoupling dynamics technology fields, propose a kind of prediction technique of function pipeline fluid structurecoupling inherent characteristic, this method comprises: obtaining mechanics parameter, dimensional parameters and the installation parameter of micron functionally gradient pipeline；According to modified coupling stress theory, the strain energy formulation of the micron functionally gradient pipeline configuration is established；According to the strain energy formulation, the differential equation of motion of the micron functionally gradient pipeline is derived using Hamiton's principle and establishes boundary condition；The differential equation of motion is solved according to the mechanics parameter, dimensional parameters and installation parameter of the micron functionally gradient pipeline using hybrid method, and then solves the intrinsic frequency of the micron functionally gradient pipeline.The prediction technique for the function pipeline fluid structurecoupling inherent characteristic that the disclosure proposes can obtain a micron intrinsic frequency for functionally gradient pipeline by way of theoretical research.
Description
Technical field
The present invention relates to fluid conveying pipe fluid structurecoupling dynamics technology field more particularly to a kind of function pipeline fluid structurecouplings
The prediction technique of inherent characteristic.
Background technique
The pipeline of micro-meter scale can be used for microfluid filter plant, targeted drug delivery equipment, fluid density, viscosity and dense
The fields such as degree detection.The functionally graded material of micro-meter scale can be applied to microelectromechanical-systems, film, microsensor and micro- execution
Device.Therefore, the micron functionally gradient pipeline being made of functionally graded material can be formed by the advantage of two kinds of structures being combined.
Currently, the intrinsic frequency method for obtaining micron functionally gradient pipeline mainly passes through experimental study.Experimental study is logical
The method for crossing Physical Experiment obtains the intrinsic frequency of micron functionally gradient pipeline.
However, due to the characteristic size very little of micrometer structure, being controlled to the position of these testpieces for experimental study
System, pickup, placement, fixture manufacture, load and its measurement of displacement deformation are all very difficult, and which results in the realities of micrometer structure
Testing research, there is also great difficulties.
It should be noted that the information in the invention of above-mentioned background technology part is only used for reinforcing the reason to background of the invention
Solution, therefore may include the information not constituted to the prior art known to persons of ordinary skill in the art.
Summary of the invention
The purpose of the present invention is to provide a kind of prediction techniques of function pipeline fluid structurecoupling inherent characteristic.The function pipeline
The prediction technique of fluid structurecoupling inherent characteristic can obtain the intrinsic frequency of micron functionally gradient pipeline by way of theoretical research
Rate.
Other characteristics and advantages of the invention will be apparent from by the following detailed description, or partially by the present invention
Practice and acquistion.
According to an aspect of the present invention, a kind of prediction technique of function pipeline fluid structurecoupling inherent characteristic, the party are provided
Method includes:
Mechanics parameter, dimensional parameters and the installation parameter of micron functionally gradient pipeline are obtained, the mechanics parameter includes
The elasticity modulus of two kinds of components, density, Poisson's ratio in the micron functionally gradient pipeline, the dimensional parameters include the micron
Internal diameter, outer diameter and the length of functionally gradient pipeline, the installation parameter include the section for supporting the micron functionally gradient pipeline
The distance of point quantity and adjacent node；
According to modified coupling stress theory, the strain energy formulation of the micron functionally gradient pipeline configuration is established；
According to the strain energy formulation, the motion side of the micron functionally gradient pipeline is derived using Hamiton's principle
Journey and establish boundary condition；
It is solved using hybrid method according to the mechanics parameter, dimensional parameters and installation parameter of the micron functionally gradient pipeline
The differential equation of motion, and then solve the intrinsic frequency of the micron functionally gradient pipeline.
In a kind of exemplary embodiment of the invention, the strain energy formulation packet of the micron functionally gradient pipeline configuration is established
It includes:
Establish formula
Wherein,UmFor strain
Can, E is elasticity modulus, and I is the moment of inertia, and μ is modulus of shearing, and A is the cross section of micron functionally gradient pipeline, and l is test-material yardstick
Parameter, w are the displacement of target point in a z-direction, and L is the length of micron functionally gradient pipeline, and x is the seat of target point in the x direction
Mark, z are the coordinate of target point in a z-direction, RoAnd RiA micron outer diameter and inner diameter for functionally gradient pipeline is respectively indicated, r is target
The radius of point, θ is rotating vector.
In a kind of exemplary embodiment of the invention, according to the strain energy formulation, derived using Hamiton's principle described in
The differential equation of motion of micron functionally gradient pipeline, comprising:
Establish the Lagrangian of the micron functionally gradient pipeline: lc=Tp+Tf-Um, wherein TpFor the micron function
The kinetic energy of energy gradient pipeline, TfFor the kinetic energy of the micron functionally gradient fluids within pipes；
Equation is established according to Hamiton's principle:
According to equationDerive the differential equation of motion of the micron functionally gradient pipeline:
Wherein, m*For the quality of micron functionally gradient pipeline unit length, MfFor micron functionally gradient fluids within pipes unit
The quality of length, U are the speed of micron functionally gradient fluids within pipes, the quality m of micron functionally gradient pipeline unit length*It can
According to the mechanics parameter of the micron functionally gradient pipeline and dimensional parameters acquisition.
In a kind of exemplary embodiment of the invention, establishing boundary condition includes:
Establish formula:
In a kind of exemplary embodiment of the invention, joined using hybrid method according to the mechanics of the micron functionally gradient pipeline
Number, dimensional parameters and installation parameter solve the differential equation of motion, and then solve consolidating for the micron functionally gradient pipeline
There is frequency, comprising:
Control the Fourier transform pairs of the solution w (x, t) of the differential equation of motion are as follows:
It willThe differential equation of motion is brought into obtain:
Set equationFrequently
Solution in domain isWherein, c is undetermined constant, and k is wave number, and willIt brings intoObtain equation [(EI)*+(μ
A)*l2]k4-MfU2k2-2ωMfUk-ω2(Mf+m*)=0 is set equation [(EI)*+(μA)*l2]k4-MfU2k2-2ωMfUk-ω2
(Mf+m*Solution on the lateral displacement of)=0 in frequency domain is
In a kind of exemplary embodiment of the invention, joined using hybrid method according to the mechanics of the micron functionally gradient pipeline
Number, dimensional parameters and installation parameter solve the differential equation of motion, and then solve consolidating for the micron functionally gradient pipeline
There is frequency, further includes:
Formula d=Sa+s is established according to passback ray theory, wherein S is global collision matrix；S is global wave source matrix；
Formula a=PUd is established according to wave propagation method theory, wherein
TqLIt is q across the left propogator matrix of unit, TqRFor q across
The right propogator matrix 0 of unit2×2For 2 × 2 null matrix, I2×2For 2 × 2 unit matrix；
Formula (I-R) d=s is obtained in conjunction with formula d=Sa+s and a=PUd, wherein the passback of micron functionally gradient pipeline
Ray matrix R=SPU；
According to formula h (ω)=| I-R |=0 solves the intrinsic frequency of the micron functionally gradient pipeline.
In a kind of exemplary embodiment of the invention, s=0.
In a kind of exemplary embodiment of the invention, according to formula h (ω)=| I-R |=0 solves micron function ladder
Spend the intrinsic frequency of pipeline, comprising:
Draw the curve that h (ω) changes with ω；
When the real and imaginary parts of h (ω) are simultaneously zero, corresponding ω is consolidating for the micron functionally gradient pipeline
There is frequency.
The present invention proposes a kind of prediction technique of function pipeline fluid structurecoupling inherent characteristic, this method comprises: obtaining micron
Mechanics parameter, dimensional parameters and the installation parameter of functionally gradient pipeline；According to modified coupling stress theory, the micron is established
The strain energy formulation of functionally gradient pipeline configuration；According to the strain energy formulation, the micron function is derived using Hamiton's principle
Can gradient pipeline differential equation of motion and establish boundary condition；Using hybrid method according to the micron functionally gradient pipeline
Mechanics parameter, dimensional parameters and installation parameter solve the differential equation of motion, and then solve the micron functionally gradient pipe
The intrinsic frequency in road.The prediction technique for the function pipeline fluid structurecoupling inherent characteristic that the disclosure proposes can pass through theoretical research
Mode obtains a micron intrinsic frequency for functionally gradient pipeline.
It should be understood that above general description and following detailed description be only it is exemplary and explanatory, not
It can the limitation present invention.
Detailed description of the invention
The drawings herein are incorporated into the specification and forms part of this specification, and shows and meets implementation of the invention
Example, and be used to explain the principle of the present invention together with specification.It should be evident that the accompanying drawings in the following description is only the present invention
Some embodiments for those of ordinary skill in the art without creative efforts, can also basis
These attached drawings obtain other attached drawings.
Fig. 1 is a kind of process of exemplary embodiment of prediction technique of disclosure function pipeline fluid structurecoupling inherent characteristic
Figure；
Fig. 2 is micron function in a kind of exemplary embodiment of prediction technique of disclosure function pipeline fluid structurecoupling inherent characteristic
The structural schematic diagram of energy gradient pipeline；
Fig. 3 is that micron functionally gradient pipeline different volumes fractional exponent acts on lower inner layer material body in the present exemplary embodiment
The variation of fraction through-thickness；
Fig. 4 is the fluctuation schematic diagram in multispan micron functionally gradient pipeline；
Fig. 5 is m across the fluctuation schematic diagram in pipeline；
Fig. 6 is the structural schematic diagram of micron functionally gradient pipeline in the present exemplary embodiment；
Fig. 7 is that micron functionally gradient pipeline difference dimensionless scale parameter Do/l acts on lower pipeline in the present exemplary embodiment
Fundamental frequency with fluid flow rate u variation；
First three rank of micron functionally gradient pipeline when Fig. 8 is micron functionally gradient pipeline index n=0 in the present exemplary embodiment
Intrinsic frequency with flow velocity u variation (Do/l=10)；
First three rank of micron functionally gradient pipeline when Fig. 9 is micron functionally gradient pipeline index n=1 in the present exemplary embodiment
Intrinsic frequency with flow velocity u variation (Do/l=10)；
When Figure 10 is micron functionally gradient pipeline index n=10 in the present exemplary embodiment micron functionally gradient pipeline first three
Rank intrinsic frequency with flow velocity u variation (Do/l=10)；
When Figure 11 is micron functionally gradient pipeline index n=50 in the present exemplary embodiment micron functionally gradient pipeline first three
Rank intrinsic frequency with flow velocity u variation (Do/l=10)；
Figure 12 is that micron functionally gradient pipeline different index n acts on lower critical flow velocity with parameter in the present exemplary embodiment
The variation of Do/l；
Figure 13 is that micron functionally gradient pipeline different scale parameter Do/l acts on lower critical flow velocity in the present exemplary embodiment
With the variation of index n；
Figure 14 is that micron functionally gradient pipeline different volumes fractional exponent n acts on lower micron pipeline in the present exemplary embodiment
Critical flow velocity with location parameter L1/L variation (Do/l=100)；
Figure 15 is that micron functionally gradient pipeline different volumes fractional exponent n acts on lower micron pipeline in the present exemplary embodiment
Critical flow velocity with supported amount variation (Do/l=100).
Specific embodiment
Example embodiment is described more fully with reference to the drawings.However, example embodiment can be real in a variety of forms
It applies, and is not understood as limited to embodiment set forth herein；On the contrary, provide these embodiments be intended to so that the disclosure comprehensively and
Completely, and by the design of example embodiment comprehensively it is communicated to those skilled in the art.The identical appended drawing reference table in figure
Show same or similar part, thus repetition thereof will be omitted.
In addition, described feature, structure or characteristic can be incorporated in one or more implementations in any suitable manner
In example.In the following description, many details are provided to provide and fully understand to embodiment of the disclosure.However,
It will be appreciated by persons skilled in the art that can be with technical solution of the disclosure without one in the specific detail or more
It is more, or can adopt with other methods, constituent element, material, device, step etc..In other cases, it is not shown in detail or describes
Known features, method, apparatus, realization, material or operation are to avoid fuzzy all aspects of this disclosure.
Block diagram shown in the drawings is only functional entity, not necessarily must be corresponding with physically separate entity.
I.e., it is possible to realize these functional entitys using software form, or these are realized in the module of one or more softwares hardening
A part of functional entity or functional entity, or realized in heterogeneous networks and/or processor device and/or microcontroller device
These functional entitys.
Other characteristics and advantages of the invention will be apparent from by the following detailed description, or partially by the present invention
Practice and acquistion.
The present exemplary embodiment provides a kind of prediction technique of function pipeline fluid structurecoupling inherent characteristic first, such as Fig. 1 institute
Show, is a kind of flow chart of exemplary embodiment of prediction technique of disclosure function pipeline fluid structurecoupling inherent characteristic, this method
Include:
Step S1: mechanics parameter, dimensional parameters and the installation parameter of micron functionally gradient pipeline, the mechanics ginseng are obtained
Number includes the elasticity modulus of two kinds of components, density, Poisson's ratio in the micron functionally gradient pipeline, and the dimensional parameters include institute
Micron internal diameter of functionally gradient pipeline, outer diameter and a length are stated, the installation parameter includes supporting the micron functionally gradient pipe
The number of nodes in road and the distance of adjacent node；
Step S2: according to modified coupling stress theory, the strain energy formulation of the micron functionally gradient pipeline configuration is established；
Step S3: according to the strain energy formulation, the fortune of the micron functionally gradient pipeline is derived using Hamiton's principle
It moves the differential equation and establishes boundary condition；
Step S4: using hybrid method according to mechanics parameter, dimensional parameters and the installation of the micron functionally gradient pipeline
Differential equation of motion described in parametric solution, and then solve the intrinsic frequency of the micron functionally gradient pipeline.
The present invention proposes a kind of prediction technique of function pipeline fluid structurecoupling inherent characteristic, this method comprises: obtaining micron
Mechanics parameter, dimensional parameters and the installation parameter of functionally gradient pipeline；According to modified coupling stress theory, the micron is established
The strain energy formulation of functionally gradient pipeline configuration；According to the strain energy formulation, the micron function is derived using Hamiton's principle
Can gradient pipeline differential equation of motion and establish boundary condition；Using hybrid method according to the micron functionally gradient pipeline
Mechanics parameter, dimensional parameters and installation parameter solve the differential equation of motion, and then solve the micron functionally gradient pipe
The intrinsic frequency in road.The prediction technique for the function pipeline fluid structurecoupling inherent characteristic that the disclosure proposes can pass through theoretical research
Mode obtains a micron intrinsic frequency for functionally gradient pipeline.
Above-mentioned steps are described in detail below:
Firstly, the present exemplary embodiment is illustrated the material mechanical performance of micron functionally gradient pipeline, such as Fig. 2 institute
Show, is micron functionally gradient pipe in a kind of exemplary embodiment of prediction technique of disclosure function pipeline fluid structurecoupling inherent characteristic
The structural schematic diagram in road.Wherein, the length of micron functionally gradient pipeline is L, mean radius R, fluid flow rate U, RoAnd RiPoint
Not Biao Shi pipeline outer diameter and inner diameter.For micron functionally gradient pipeline in x, the displacement component in y and z-axis direction uses u, v and w table respectively
Show, this micron of functionally gradient pipeline can be made of two kinds of materials.It should be pointed out that micron functionally gradient pipeline is in micron meter
Under degree, plug flow model is still applicable in.Equally, the material mechanical performance of micron functionally gradient pipeline still through-thickness here
Change by Bending influence, specific effectively material property can indicate are as follows:
E=ViEi+VoEo (1)
μ=Viμi+Voμo (2)
ρ=Viρi+Voρo (3)
In formula:
E --- pipeline elasticity modulus；
μ --- pipeline modulus of shearing；
ρ --- channel density；
V --- component material volume fraction.
In micron functionally gradient pipeline, Poisson's ratio ν is assumed to constant.Subscript i and o respectively indicate the inner layer material of pipeline
And cladding material.The volume fraction of component material can indicate are as follows:
Vo=1-Vi (5)
In formula:
The radius of r --- target point；
N --- volume fraction index.
As shown in figure 3, acting on the variation of lower inner layer material volume fraction through-thickness for different volumes fractional exponent.It is aobvious
So as volume fraction index n=0, micron functionally gradient pipeline is degenerated for the micron functionally gradient pipeline of homogeneous material.
Step S1: mechanics parameter, dimensional parameters and the installation parameter of micron functionally gradient pipeline, the mechanics ginseng are obtained
Number includes the elasticity modulus of two kinds of components, density, Poisson's ratio in the micron functionally gradient pipeline, and the dimensional parameters include institute
Micron internal diameter of functionally gradient pipeline, outer diameter and a length are stated, the installation parameter includes supporting the micron functionally gradient pipe
The number of nodes in road and the distance of adjacent node.
By step S1, the elastic modulus E of two kinds of components in available micron functionally gradient pipelineiAnd Eo, micron function
The internal diameter R of gradient pipelineiWith outer diameter Ro, so as to obtain target on micron functionally gradient pipeline according to formula (1), (4), (5)
Elasticity modulus on point.
Step S2: according to modified coupling stress theory, the strain energy formulation of the micron functionally gradient pipeline configuration is established.
According to coupling stress theory, the strain energy density of material is related with strain tensor and rotation gradient tensor simultaneously.At it
It is additional to introduce the intrinsic characteristic dimension parameter of two materials in constitutive relation.On the basis of coupling stress theory, this is exemplary
Embodiment is according to modified coupling stress theory.Make the strain energy density of material by introducing a new equilibrium condition and answer
The symmetrical components for becoming tensor sum rotation gradient tensor are related, and unrelated with the rotation antisymmetric component of gradient tensor.Modified coupling
The advantage of stress theory is, one test-material yardstick parameter of additional introducing is only needed in the constitutive relation of micrometer structure, so that
Entire case study becomes simpler.
According to modified coupling stress theory, the strain energy of structure can be indicated are as follows:
In formula:
Um --- structural strain energy；
Ω --- structural volume；
σ --- stress tensor；
ε --- strain tensor；
M --- the deviator part of symmetrical coupling stress；
χ --- symmetrical contravariant tensor；
These tensors can be embodied as:
σ=λ tr (ε) I+2 μ ε (7)
M=2l2μχ (9)
In formula:
U --- motion vector；
λ --- elastic constant；
μ --- modulus of shearing；
θ --- rotating vector；
L --- test-material yardstick parameter.
In functionally graded material, the scale parameter of material is variable.Here for simplifying the analysis, the scale of material is joined
Number l is assumed to be a constant.In addition rotating vector can be written as:
According to Euler-Bernoulli beam theory, arbitrary point can be written as respectively along the displacement of x, y and z directionss in Fig. 2:
U=-z ψ (x, t), v=0, w=w (x, t) (12)
In formula:
ψ --- cross-section of pipeline corner；
Under the conditions of small deformation, the corner of pipeline can be indicated are as follows:
Equation (12) and (13) are substituted into (8), available unique non-zero components of strain:
Equation (12) and (13), which are substituted into (11), to be obtained:
Equation (15) substitution (10) can be obtained:
Derive the differential equation of motion and boundary condition of micron functionally gradient pipeline.
For simplifying the analysis, when writing the components of stress, the influence of Poisson's ratio will be ignored.Equation (14) are substituted into (7), micron
The components of stress of functionally gradient pipeline can indicate are as follows:
Equation (16) substitution (9) can be obtained:
The strain energy of micron functionally gradient pipeline can be written as:
Above formula can further be write a Chinese character in simplified form are as follows:
Wherein:
Wherein, UmFor strain energy, E is elasticity modulus, and I is the moment of inertia, and μ is modulus of shearing, and A is micron functionally gradient pipeline
Cross section, l is test-material yardstick parameter, and w is the displacement of target point in a z-direction, and L is the length of micron functionally gradient pipeline, and x is
The coordinate of target point in the x direction, z are the coordinate of target point in a z-direction, RoAnd RiRespectively indicate a micron functionally gradient pipeline
Outer diameter and inner diameter, r be target point radius, θ is rotating vector.Above-mentioned parameter l, Ro、RiIt can directly be obtained by step S1
It takes；Elastic modulus E can be obtained by above-mentioned formula (1)；Modulus of shearing μ can be obtained by above-mentioned formula (2), (4), (5),
The modulus of shearing μ of two kinds of components in micron functionally gradient pipelineiAnd μoFormula μ=E/2 (1+ ν) be can use according to each component
Elasticity modulus obtains, wherein ν is Poisson's ratio, and Poisson's ratio can be obtained directly by step S1；The moment of inertia can pass through formulaIt obtains, rotating vector θ can be obtained by formula (11).
Step S3: according to the strain energy formulation, the fortune of the micron functionally gradient pipeline is derived using Hamiton's principle
It moves the differential equation and establishes boundary condition.
The kinetic energy of micron functionally gradient pipeline can be written as:
Micron functionally gradient pipeline linear mass m*It can be with is defined as:
The kinetic energy flowed in pipeline can be written as:
According to the Hamiton's principle of fluid conveying pipe, the case where to the fluid conveying pipe of both ends freely-supported, can be written as:
Wherein lc=Tp+Tf-UmIt is a micron Lagrangian for functionally gradient pipeline.Above-mentioned each amount is substituted into equation
(26), and pass through variation operation, the differential equation of motion of available micron functionally gradient pipeline:
It should be noted that having ignored the influence of gravity, damping force, Fluid pressure in above formula.The freely-supported of pipe ends
Boundary condition can indicate are as follows:
Compared to classical fluid conveying pipe differential equation of motion, it can be found that effective bending stiffness of micron functionally gradient pipeline
For (EI)*+(μA)*l2, pipeline linear mass is m*, fluid units linear mass is Mf。
Wherein, the density p of micron functionally gradient pipeline can be obtained according to formula (3), (4) (5), the density of two kinds of materials
ρiAnd ρoIt can be directly acquired by step S1.Pipeline linear mass m*The density p of micron functionally gradient pipeline can be passed through
It obtains.
Step S4: using hybrid method according to mechanics parameter, dimensional parameters and the installation of the micron functionally gradient pipeline
Differential equation of motion described in parametric solution, and then solve the intrinsic frequency of the micron functionally gradient pipeline.
When solving differential equation of motion, it is necessary first to some parameters in differential equation of motion are determined, for example, springform
Measure E, the moment of inertia I, modulus of shearing μ, pipeline linear mass m*, test-material yardstick parameter l etc., how above content to obtaining
Above-mentioned parameter is taken to be described in detail.In addition, solving the differential equation of motion by hybrid method and may include:
The Fourier transform pairs of the solution w (x, t) of control differential equation (27) can be written as:
Equation (30) substitution equation (27) can be obtained:
Displacement solution in frequency domain can be set to:
In formula:
C --- undetermined constant；
K --- wave number.
Equation (32) substitution equation (31) can be obtained:
[(EI)*+(μA)*l2]k4-MfU2k2-2ωMfUk-ω2(Mf+m*)=0 (33)
K in equation (33) is corresponding there are four k1, k2, k3 and k4 is solved, and has respectively corresponded four wave modes.It is penetrated according to passback
Collimation method, these waves can be divided into incidence wave and outgoing wave.And in wave propagation method, then it can be divided into left lateral wave and right lateral wave.By
K in equation (33) is there are four solution, therefore lateral displacement setting solution and can be re-written as in frequency domain:
Correspondingly, corner of the micron functionally gradient pipeline in frequency domainMoment of flexureAnd shearingIt can indicate are as follows:
According to passback ray theory, as shown in figure 4, for the fluctuation schematic diagram in multispan micron functionally gradient pipeline.Here
Double subscripts will be introducedTo describe incidence wave and outgoing wave.Subscript J and K indicate adjacent node.It needs
, it is noted that in incidence wave,Indicate that wave is moved from node K to node J, and in outgoing wave,Indicate wave from node
J is moved to node K.
The fluctuation schematic diagram of one section of typical multispan micron functionally gradient pipeline is as shown in Figure 4.Pipeline is by n freely-supported support branch
Support.According to passback ray theory, incidence wave and outgoing wave at each node can indicate respectively are as follows:
In formula
A --- incidence wave；
D --- outgoing wave.
Wherein subscript 1,2, n expression pipeline nodes, and subscript 1,2,3,4 respectively correspond k1, k2, k3 and k4.
By passback ray theory, incidence wave and outgoing wave have following relationship:
D=Sa+s (40)
In formula:
S --- global collision matrix；
S --- global wave source matrix.
Wave source matrix s is related to external applied load, and the free vibration and stability of pipeline are mainly considered in this chapter analysis, is not examined
Consider external applied load, therefore the s=0 in equation (40).Global collision matrix S it is assembled by the local collision matrix of each node and
?.And the local collision matrix of node is established by the displacement continuity condition and dynamic balance condition of node.
Such as in Fig. 4, the boundary condition of node 1 be can be written as:
Equation (34) and (36), which are substituted into (41), to be obtained:
Wherein
Incidence wave and outgoing wave at node 1 are as follows:
By at node 1 incidence wave and outgoing wave substitute into equation (42) and (43) it is available:
Incidence wave and outgoing wave at node 1 have following relationship:
d1=S1a1 (47)
In formula:
S1--- the local collision matrix of node 1.
In conjunction with the collision matrix at equation (46) and (47) available node 1 are as follows:
Local collision matrix at Fig. 4 interior joint 2 can be obtained by the condition of continuity at node 2 and equilibrium condition,
It specifically can be written as:
In formula:
"-" --- node left end；
"+" --- node right end；
WithFor cross-section of pipeline corner.
Incidence wave and outgoing wave at node 2 are respectively as follows:
It will be displaced, after corner and Expression of Moment formula substitution above formula, and be scattered by arranging the part at available node 2
Matrix:
Wherein λj=ikj。
Other intermediate nodes have the condition of continuity and equilibrium condition as node 2, and local collision matrix can also
To establish by the same method, which is not described herein again.
The boundary condition of node n can be written as:
Incidence wave and outgoing wave at node n are respectively as follows:
The collision matrix S at node n can be obtained by equation (34) and (36), and by arrangementnAre as follows:
After the local collision matrix for obtaining all nodes, according to the wave position of node, it can be tied by assembly
The global collision matrix of structure:
Next wave propagation method will be utilized, according to the sub propagation relationship across unit medium wave, establishes multispan pipeline neutron across list
The wave propogator matrix of member, as shown in figure 5, being m across the fluctuation schematic diagram in pipeline.
Fig. 5 describes propagation of the m across piping unit medium wave.As previously mentioned, the wave in pipeline is divided in wave propagation method
For right lateral wave and left lateral wave.Herein without loss of generality it is assumed that subscript 1 and 2 existsMiddle expression left lateral wave, and the table of subscript 3 and 4
Show right lateral wave.
M has following propagation relationship across the left lateral wave of piping unit:
In formula:
TmL--- m can be obtained across the left propogator matrix of unit, m according to the installation parameter of micron functionally gradient pipeline.
The expression of matrix is respectively as follows: in above formula
lmFor the distance of m node to m+1 node, lmIt can be obtained by the installation parameter of micron functionally gradient pipeline.
M has following propagation relationship across the right lateral wave of piping unit:
In formula:
TmR--- m is across the right propogator matrix of unit.
The expression of matrix is respectively as follows: in above formula
Other wave propogator matrixes of the son across unit, can also establish in the same way.
In next analysis, reverberation-ray matrix combination wave propagation method will be used, and establish the feature side of multispan pipeline
Journey.Incidence wave and outgoing wave are introduced into wave propogator matrix, and combine equation (58) and (60), available following square
Battle array:
Wherein it is respectively as follows: in the incidence wave and outgoing wave of m node and m+1 node
By assembling all sub local wave propogator matrixes across unit, second pass of available incidence wave and outgoing wave
It is formula:
A=PUd (65)
Wherein:
It is available using combination equation (40) and (65):
(I-R) (68) d=s
Wherein R=SPU is known as the passback ray matrix of micron functionally gradient pipeline.The intrinsic frequency of micron functionally gradient pipeline
Rate can be equal to zero by coefficient matrix and obtain:
H (ω)=| I-R |=0 (69)
A micron intrinsic frequency for functionally gradient pipeline can be found out using above formula.The song changed by drawing h (ω) with ω
Line obtains.When the real and imaginary parts of h (ω) are simultaneously zero, corresponding ω is the intrinsic frequency of fluid conveying pipe.
The present exemplary embodiment also provides the intrinsic frequency solution procedure of a kind of specific micron functionally gradient pipeline and right
The analysis method of solution.
This micron of functionally gradient pipeline is formed by aluminium and alumina composite, the cladding material of micron functionally gradient pipeline by
100% aluminium oxide is constituted, and the inner layer material of pipeline is made of 100% aluminium, the power of micron functionally gradient pipeline component material
It learns parameter and dimensional parameters is as shown in table 1.The density p of fluidf=1000kg/m3.In order to study a micron ruler for functionally gradient pipeline
Degree influences, and in following example, micron functionally gradient tubular dimensions parameter l is selected as 15 μm.
1 micron of functionally gradient pipeline component material mechanical performance of table
As shown in fig. 6, for the structural schematic diagram of micron functionally gradient pipeline in the present exemplary embodiment.Entire micron function
Gradient pipeline is by four hinged seat supports.The geometric parameter of micron functionally gradient pipeline is Di/Do=0.9, L1=L2=L/3, L/
Do=20.Influence for research material scale parameter to micron functionally gradient pipe vibration and stability, here by micron
The diameter of functionally gradient pipeline is set as variable, and defining nondimensional scale parameter is Do/ l:
In order to analyze simplicity, and make result that there is generality, following dimensionless group introduced in analysis below:
In formula:
ξ --- nondimensional length；
l*--- dimensionless scale parameter；
U --- dimensionless fluid flow rate；
--- dimensionless pipeline intrinsic frequency；
In addition when EI and m respectively indicates volume fraction index n=0, the bending stiffness and unit of micron functionally gradient pipeline
Linear mass.
In order to which research material scale parameter is to micron functionally gradient pipeline free vibration and the influence of stability, such as Fig. 7
It is shown, it is different dimensionless scale parameter Do/ l acts on lower pipeline fundamental frequency with the variation of fluid flow rate u.Table 2 gives not
With dimensionless scale parameter DoCritical flow velocity u under/l effectd.Front is it is stated that in the stabilization sub stage of fluid conveying pipe, frequency
Real partReduce with the increase of fluid flow rate u, when flow velocity is more than a certain determining value, the real part drop of fundamental frequency
It is zero, this indicates that static instability occurs for pipe-line system, and corresponding fluid flow rate is known as critical flow velocity ud, the imaginary part of frequency is
The different dimensionless scale parameter D of table 2oCritical flow velocity under/l effect
From Fig. 7 and table 2 as can be seen that the intrinsic frequency and critical flow velocity ratio that are obtained by modified coupling stress theory are by classics
The value that theoretical calculation obtains is bigger.Simultaneously, it was also found that with dimensionless scale parameter Do/ l's is gradually increased, and is answered by modified coupling
The result that power theoretical calculation obtains gradually converges on the result calculated by classical theory.For example, working as nondimensional scale parameter Do/
It is 2.14 times of Classical Beam the calculated results by the critical flow velocity that modified coupling stress theory is calculated when l=1.Work as nothing
The scale parameter D of dimensionoWhen/l=10, this ratio is reduced to 1.02.Therefore it can be concluded that, the scale parameter of material is to micron
The vibration of functionally gradient pipe-line system and stability have a major impact, and the steady of micron functionally gradient pipe-line system can be improved in it
It is qualitative, especially when the diameter of micron functionally gradient pipeline and the close scale parameter of material.This is because material
Scale parameter can increase effective bending stiffness (EI) of micron functionally gradient pipeline*+(μA)*l2, but with nondimensional ruler
Spend parameter Do/ l is gradually increased, and this influence is gradually reduced.When the diameter of pipeline compares the larger (D of scale parameter of materialo/l>
10) when, the intrinsic frequency and critical flow velocity obtained by modified coupling stress theory converges on the knot being calculated by classical theory
Fruit.
In order to study volume fraction index n to micron functionally gradient pipeline free vibration and the influence of stability, Fig. 8,
9,10,11 micron functionally gradient pipeline dimensionless ruler in the case where different volumes fractional exponent n (n=0,1,10,50) is acted on is given
Spend parameter DoWhen/l=10, first three rank intrinsic frequency with fluid flow rate u variation.Wherein, 1st-mode indicates first-order modal,
2st-mode indicates that second-order modal, 3st-mode indicate three rank mode.From figure 8, it is seen that as index n=0, micron function
Gradient pipeline presents more complicated dynamic phenomena.It is embodied in, the first order frequency of pipeline is in velocity of flow of zero dimension u=
When 9.59, static instability occurs.In u=11.78 static instability, as flow velocity u=15.69, pipeline occur for the second order frequency
Static instability occurs for third rank intrinsic frequency.Then, when flow velocity u continues to increase to 17.60, the second order frequency of pipeline is direct
It is coupled with the third order frequency of pipeline, and the dynamic buckling coupled.It should be pointed out that in single span fluid conveying pipe,
The Coupled Dynamic unstability of pipeline first occurs at the first rank and the second order frequency of pipeline.And the Coupled Dynamic unstability of this example is straight
Sending and receiving are raw in second-order and third rank mode, this is different with single span fluid conveying pipe, while also related with support.
From Fig. 8,9,10,11 it is also found that micron functionally gradient pipeline intrinsic frequency real part and critical flow velocity with
The increase of volume fraction index n and increase.For example, the static instability of first step mode occurs in u=as index n=0
9.59.As index n=1, the critical flow velocity u of micron functionally gradient pipelined=17.50, as index n=10 and 50, micron
There is no static instabilities in the range of flow velocity u < 18 for functionally gradient pipeline.Therefore it can be concluded that, micron functionally gradient pipeline
Stability improved with the increase of volume fraction index n.This is mainly due to the increases with index n, and aluminium oxide is micro-
Component content in rice functionally gradient pipeline in increase and the component of aluminium is being reduced, and the Young's modulus of aluminium oxide is far longer than aluminium
Young's modulus.These conclusions are identical as functionally gradient macroscopic view fluid conveying pipe.
In order to further study volume fraction index n and dimensionless scale parameter Do/ l is to micron functionally gradient pipeline stabilization
The influence of property, in different scale parameter Do/l(Do/ l=1,2,5,10) under effect, dimensionless critical flow velocity udRefer to volume fraction
The variation of number n is as shown in figure 12, also gives the result calculated by classical beam theory (l=0) in figure simultaneously.Figure 13 gives
Under different volumes fractional exponent n (n=0,1,10,50) effect, nondimensional critical flow velocity udWith scale parameter DoThe variation of/l.
It can be recognized from fig. 12 that the critical flow velocity ratio being calculated by modified coupling stress theory is by Classical Beam theoretical calculation
Obtained value is big.This is because a micron effective rigidity for functionally gradient pipeline can be improved in scale parameter.With dimensionless scale
Parameter Do/ l increases to 10 from 1, and nondimensional critical flow velocity strongly reduces, and finally converges to the knot calculated by classical theory
Fruit.It can be recognized from fig. 12 that critical flow velocity is with scale parameter DoThe increase of/l and reduce, as scale parameter DoBehind/l > 10, by
The critical flow velocity that modified coupling stress theory calculates gradually levels off to a certain fixed value.It can be seen from the results above that being answered by coupling
Scale effect caused by power only just has a significant impact when line size is with test-material yardstick parameter similar.When dimensionless scale
Larger (the D of parametero/ l > 10) when, i.e., when pipe diameter is much larger than test-material yardstick parameter, it is calculated by amendment coupling stress theory
Result converge on the result being calculated by classical theory.
On the other hand, from Figure 12 it can also be seen that critical flow velocity increases with the increase of volume fraction index n, especially
It is as n < 10, critical flow velocity is increased rapidly with the increase of index n.As index n is further increased, to critical flow velocity
Influence be also gradually reduced, as index n=50, critical flow velocity has leveled off to a constant.This and macroscopical micron function are terraced
The result for spending pipeline is similar.
Engineering in practice, the Support Position of pipeline clamp is usually limited by ambient enviroment, and cannot equably be divided
Cloth is in entire pipe-line system.Therefore need to study influence of the position of pipeline support clip to fluid conveying pipe stability.At this
In example, it is assumed that support 2 and support 3 in Fig. 6 move (0≤L from the both ends of pipeline toward pipeline midpoint respectively1=L2≤L/2)。
Under different volumes fractional exponent n (n=0,1,10,50) effect, micron functionally gradient pipeline (Do/ l=100) critical flow velocity
With udSupport Position L1The variation of/L is as shown in figure 14.
As seen from Figure 14, work as L1/ L=0 and when volume fraction index n=0, the critical flow velocity of pipeline is 3.142 ≈ π
(analytic solutions that π is both ends freely-supported fluid conveying pipe critical flow velocity).In fact, working as L1When/L=0, the support 2 and 3 in Fig. 6 is distinguished
With support 1 and 4 be overlapped, indicate at this time three across defeated streaming system degenerate be single span Pipe Conveying Fluid, and index n=0 indicate it is micro-
It is homogeneous material pipeline, while nondimensional scale parameter D that rice functionally gradient pipeline, which is also degenerated,o/ l=100 is indicated at this time by repairing
The result that positive coupling stress theory calculates has converged on the result calculated by classical theory.Therefore faced by what context of methods calculated
Boundary's flow velocity 3.142 and the analytic solutions of single span homogeneous material fluid conveying pipe critical flow velocity are very identical.This is also demonstrated again herein
The correctness of analysis method.From Figure 14 it can also be seen that working as location parameter L1When/L is approximately equal to 0.33, entire pipe-line system has most
Big critical flow velocity.This is because for three across fluid conveying pipe, when intermediate supports are uniformly distributed on pipeline, entirely
The rigidity of system is maximum.From Figure 14 it is also found that working as location parameter L1/ L from zero increase to a small value when, pipe-line system
Critical flow velocity also has an increase sharply, it can be seen that, for single span pipeline, even if increasing in its both ends neighbouring position
Add support clip that can also greatly improve the stability of Pipe Conveying Fluid, while also illustrating that multispan support can compared to single span support
To significantly improve the stability of Pipe Conveying Fluid.
In order to study influence of the clip support number to micron functionally gradient pipeline stability, Figure 15 gives nondimensional
Critical flow velocity udWith the variation (D of pipeline support numbero/ l=100).In this example, support clip is distributed evenly at whole
In a pipe-line system.It is seen from figure 14 that when support number is in the range of 10, as support sizes purpose increases, pipe conveying fluid
The critical flow velocity in road shows linear rising on the whole.It should be pointed out that being indicated micro- at this time when supporting number is 2
Rice functionally gradient pipe-line system is single span pipeline, and when supporting number is 3, pipeline is two across pipeline at this time.It can be sent out from Figure 15
It is existing, when pipeline by single span pipeline be transitioned into two it is across pipeline when, the critical flow velocity of pipeline increased dramatically at this time, and when pipe-line system by
Two across pipeline be transitioned into three it is across pipeline when, the variation of critical flow velocity is not obvious.Hereafter with the increase of pipeline support number, pipe
The critical flow velocity of road system substantially shows linear increase.On the whole, the support number for increasing fluid conveying pipe can be significant
The rigidity of pipe-line system is improved, and expands a micron stability region for functionally gradient pipeline.These results are to design micron order pipeline
System has reference.
In addition, although describing each step of method in the disclosure in the accompanying drawings with particular order, this does not really want
These steps must be executed in this particular order by asking or implying, or having to carry out step shown in whole could realize
Desired result.Additional or alternative, it is convenient to omit multiple steps are merged into a step and executed by certain steps, and/
Or a step is decomposed into execution of multiple steps etc..
Those skilled in the art after considering the specification and implementing the invention disclosed here, will readily occur to its of the disclosure
His embodiment.This application is intended to cover any variations, uses, or adaptations of the disclosure, these modifications, purposes or
Adaptive change follow the general principles of this disclosure and including the undocumented common knowledge in the art of the disclosure or
Conventional techniques.The description and examples are only to be considered as illustrative, and the true scope and spirit of the disclosure are by claim
It points out.
It should be understood that the present disclosure is not limited to the precise structures that have been described above and shown in the drawings, and
And various modifications and changes may be made without departing from the scope thereof.The scope of the present disclosure is only limited by the attached claims.
Claims (10)
1. a kind of prediction technique of function pipeline fluid structurecoupling inherent characteristic characterized by comprising
Mechanics parameter, dimensional parameters and the installation parameter of micron functionally gradient pipeline are obtained, the mechanics parameter includes described
The elasticity modulus of two kinds of components, density, Poisson's ratio in micron functionally gradient pipeline, the dimensional parameters include the micron function
Internal diameter, outer diameter and the length of gradient pipeline, the installation parameter include the number of nodes for supporting the micron functionally gradient pipeline
The distance of amount and adjacent node；
According to modified coupling stress theory, the strain energy formulation of the micron functionally gradient pipeline configuration is established；
According to the strain energy formulation, using Hamiton's principle derive the differential equation of motion of the micron functionally gradient pipeline with
And establish boundary condition；
Using hybrid method according to the solution of the mechanics parameter of the micron functionally gradient pipeline, dimensional parameters and installation parameter
Differential equation of motion, and then solve the intrinsic frequency of the micron functionally gradient pipeline.
2. the prediction technique of function pipeline fluid structurecoupling inherent characteristic according to claim 1, which is characterized in that establish institute
Stating a micron strain energy formulation for functionally gradient pipeline configuration includes:
Establish formula
Wherein,UmFor strain
Can, E is elasticity modulus, and I is the moment of inertia, and μ is modulus of shearing, and A is the cross section of micron functionally gradient pipeline, and l is test-material yardstick
Parameter, w are the displacement of target point in a z-direction, and L is the length of micron functionally gradient pipeline, and x is the seat of target point in the x direction
Mark, z are the coordinate of target point in a z-direction, RoAnd RiA micron outer diameter and inner diameter for functionally gradient pipeline is respectively indicated, r is target
The radius of point, θ is rotating vector.
3. the prediction technique of function pipeline fluid structurecoupling inherent characteristic according to claim 2, which is characterized in that the moment of inertia
4. the prediction technique of function pipeline fluid structurecoupling inherent characteristic according to claim 2, which is characterized in that according to institute
Strain energy formulation is stated, the differential equation of motion of the micron functionally gradient pipeline is derived using Hamiton's principle, comprising:
Establish the Lagrangian of the micron functionally gradient pipeline: lc=Tp+Tf-Um, wherein TpFor the micron function ladder
Spend the kinetic energy of pipeline, TfFor the kinetic energy of the micron functionally gradient fluids within pipes；
Equation is established according to Hamiton's principle:
According to equationDerive the differential equation of motion of the micron functionally gradient pipeline:
Wherein, m*For the quality of micron functionally gradient pipeline unit length, MfFor micron functionally gradient fluids within pipes unit length
Quality, U be micron functionally gradient fluids within pipes speed.
5. the prediction technique of function pipeline fluid structurecoupling inherent characteristic according to claim 4, which is characterized in that establish side
Boundary's condition includes:
Establish formula:
6. the prediction technique of function pipeline fluid structurecoupling inherent characteristic according to claim 5, which is characterized in that utilize miscellaneous
Friendship method solves the motion side according to the mechanics parameter, dimensional parameters and installation parameter of the micron functionally gradient pipeline
Journey, and then solve the intrinsic frequency of the micron functionally gradient pipeline, comprising:
Control the Fourier transform pairs of the solution w (x, t) of the differential equation of motion are as follows:
It willThe differential equation of motion is brought into obtain:
Set equationIn frequency domain
Solution beWherein, c is undetermined constant, and k is wave number, and willIt brings intoObtain equation [(EI)*+(μ
A)*l2]k4-MfU2k2-2ωMfUk-ω2(Mf+m*)=0 is set equation [(EI)*+(μA)*l2]k4-MfU2k2-2ωMfUk-ω2
(Mf+m*Solution on the lateral displacement of)=0 in frequency domain is
7. the prediction technique of function pipeline fluid structurecoupling inherent characteristic according to claim 6, which is characterized in that utilize miscellaneous
Friendship method solves the motion side according to the mechanics parameter, dimensional parameters and installation parameter of the micron functionally gradient pipeline
Journey, and then solve the intrinsic frequency of the micron functionally gradient pipeline, further includes:
Formula d=Sa+s is established according to passback ray theory, wherein S is global collision matrix；S is global wave source matrix；
Formula a=PUd is established according to wave propagation method theory, wherein
TqLIt is q across the left propogator matrix of unit, TqRFor across the unit right side q
Propogator matrix 02×2For 2 × 2 null matrix, I2×2For 2 × 2 unit matrix；
Formula (I-R) d=s is obtained in conjunction with formula d=Sa+s and a=PUd, wherein the passback ray of micron functionally gradient pipeline
Matrix R=SPU；
According to formula h (ω)=| I-R |=0 solves the intrinsic frequency of the micron functionally gradient pipeline.
8. the prediction technique of function pipeline fluid structurecoupling inherent characteristic according to claim 7, which is characterized in that s=0.
9. the prediction technique of function pipeline fluid structurecoupling inherent characteristic according to claim 7, which is characterized in that the overall situation dissipates
Penetrate matrixWherein, SnFor the collision matrix at node n.
10. the prediction technique of function pipeline fluid structurecoupling inherent characteristic according to claim 7, which is characterized in that according to
Formula h (ω)=| I-R |=0 solves the intrinsic frequency of the micron functionally gradient pipeline, comprising:
Draw the curve that h (ω) changes with ω；
When the real and imaginary parts of h (ω) are simultaneously zero, corresponding ω is the intrinsic frequency of the micron functionally gradient pipeline
Rate.
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