CN109740211A - A Prediction Method for Inherent Characteristics of Fluid-Structure Interaction of Functional Pipelines - Google Patents

Abstract

The invention relates to the technical field of fluid-structure coupling dynamics of a flow transmission pipeline, and proposes a method for predicting the inherent characteristics of the fluid-structure coupling of a functional pipeline. The method includes: acquiring mechanical parameters, size parameters and installation parameters of a micron functional gradient pipeline; The coupled stress theory is used to establish the strain energy formula of the micron functionally graded pipeline structure; according to the strain energy formula, the differential equation of motion of the micron functionally graded pipeline is deduced by using Hamilton's principle and the boundary conditions are established; The mechanical parameters, size parameters and installation parameters of the micron functionally graded pipeline are used to solve the differential equation of motion, and then the natural frequency of the micron functionally graded pipeline is solved. The method for predicting the inherent characteristics of the fluid-structure interaction of functional pipelines proposed in the present disclosure can obtain the natural frequencies of micron functionally graded pipelines through theoretical research.

Description

A Prediction Method for Inherent Characteristics of Fluid-Structure Interaction of Functional Pipelines

technical field

The invention relates to the technical field of fluid-structure coupling dynamics of a flow pipeline, in particular to a method for predicting the inherent characteristics of the fluid-structure coupling of a functional pipeline.

Background technique

Micron-scale tubing can be used in microfluidic filtration devices, targeted drug delivery devices, fluid density, viscosity, and concentration detection. Micrometer-scale functionally graded materials can be applied to microelectromechanical systems, thin films, microsensors, and microactuators. Therefore, combining the advantages of the two structures can form micron functionally graded conduits composed of functionally graded materials.

At present, the method to obtain the natural frequency of micron functionally graded pipeline is mainly through experimental research. The experimental study is to obtain the natural frequency of the micron functionally graded pipe by means of physical experiments.

However, for experimental studies, due to the small feature size of the microstructures, the position control, pick-up, placement, fixture fabrication, loading and measurement of displacement deformation of these test pieces are very difficult, which leads to the difficulty of the microstructures. There are still great difficulties in experimental research.

It should be noted that the information disclosed in the above Background section is only for enhancing understanding of the background of the invention, and therefore may include information that does not form the prior art known to a person of ordinary skill in the art.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a method for predicting the inherent characteristics of the fluid-structure coupling of functional pipelines. The method for predicting the inherent characteristics of the fluid-structure interaction of functional pipelines can obtain the natural frequencies of micron functionally graded pipelines through theoretical research.

Other features and advantages of the present invention will become apparent from the following detailed description, or be learned in part by practice of the present invention.

According to one aspect of the present invention, there is provided a method for predicting the inherent characteristics of fluid-structure coupling of functional pipelines, the method comprising:

Obtain mechanical parameters, size parameters and installation parameters of the micron functionally graded pipeline, where the mechanical parameters include elastic modulus, density, and Poisson's ratio of two components in the micron functionally graded pipeline, and the size parameters include the micrometer functional gradient pipeline. the inner diameter, outer diameter and length of the functionally graded pipeline, the installation parameters including the number of nodes supporting the micron functionally graded pipeline and the distance between adjacent nodes;

According to the modified coupled stress theory, the strain energy formula of the micron functionally graded pipeline structure is established;

According to the strain energy formula, the differential equation of motion of the micron functionally graded pipeline is deduced by using Hamilton's principle, and the boundary conditions are established;

The hybridization method is used to solve the differential equation of motion according to the mechanical parameters, size parameters and installation parameters of the micron functionally graded pipeline, and then the natural frequency of the micron functionally graded pipeline is solved.

In an exemplary embodiment of the present invention, the formula for establishing the strain energy of the micron functionally graded pipeline structure includes:

build formula

in, U _m is the strain energy, E is the elastic modulus, I is the moment of inertia, μ is the shear modulus, A is the cross section of the micron functionally graded pipe, l is the material scale parameter, and w is the displacement of the target point in the z direction , L is the length of the micron functional gradient pipe, x is the coordinate of the target point in the x direction, z is the coordinate of the target point in the z direction, R _o and R _i represent the outer diameter and inner diameter of the micron functional gradient pipe, respectively, r is the radius of the target point, and θ is the rotation vector.

In an exemplary embodiment of the present invention, according to the strain energy formula, the differential equation of motion of the micron functionally graded pipeline is derived by using the Hamiltonian principle, including:

Establishing the Lagrangian function of the micron functional gradient pipeline: l _c =T _p +T _f -U _m , where T _p is the kinetic energy of the micron functional gradient pipeline, and T _f is the inner surface of the micron functional gradient pipeline the kinetic energy of the fluid;

According to Hamilton's principle, formulate the equation:

According to the equation Derive the differential equation of motion for the micron functionally graded pipeline:

Among them, m ^* is the mass per unit length of the micron functional gradient pipe, M _f is the mass of the fluid per unit length in the micron functional gradient pipe, U is the velocity of the fluid in the micron functional gradient pipe, and the mass m ^* per unit length of the micron functional gradient pipe can be Obtained according to the mechanical parameters and size parameters of the micron functional gradient pipeline.

In an exemplary embodiment of the present invention, establishing boundary conditions includes:

Build the formula:

In an exemplary embodiment of the present invention, the hybridization method is used to solve the differential equation of motion according to the mechanical parameters, size parameters and installation parameters of the micron functionally graded pipeline, and then to solve the natural frequency of the micron functionally graded pipeline, comprising: :

The Fourier transform pair governing the solution w(x,t) of the differential equation of motion is:

Will Bringing into the differential equation of motion we get:

set equation The solution in the frequency domain is where c is an undetermined constant, k is the wave number, and bring in Obtain the equation [(EI) ^* +(μA) ^* l ² ]k ⁴ -M _f U ² k ² -2ωM _f Uk-ω ² (M _f +m ^* )=0, set the equation [(EI) ^* + (μA) ^* l ² ]k ⁴ -M _f U ² k ² -2ωM _f Uk-ω ² (M _f +m ^* )=0 The solution in the frequency domain on the lateral displacement is

In an exemplary embodiment of the present invention, the hybridization method is used to solve the differential equation of motion according to the mechanical parameters, size parameters and installation parameters of the micrometer functionally graded pipeline, so as to solve the natural frequency of the micrometer functionally graded pipeline, and further include:

The formula d=Sa+s is established according to the theory of return rays, where S is the global scattering matrix; s is the global wave source matrix;

According to the theory of wave propagation method, the formula a=PUd is established, wherein,

T _qL is the q-th cross-unit left propagation matrix, T _qR is the q-th cross-unit right propagation matrix 0 _2×2 is a 2×2 zero matrix, and I _2×2 is a 2×2 unit matrix;

Combining the formula d=Sa+s and a=PUd, the formula (I-R)d=s is obtained, wherein, the return ray matrix R=SPU of the micron functionally graded pipeline;

According to the formula h(ω)=|I-R|=0, the natural frequency of the micron functionally graded pipeline is solved.

In an exemplary embodiment of the present invention, s=0.

In an exemplary embodiment of the present invention, the natural frequency of the micron functionally graded pipeline is solved according to the formula h(ω)=|I-R|=0, including:

Plot the curve of h(ω) as a function of ω;

When the real part and the imaginary part of h(ω) are both zero, the corresponding ω is the natural frequency of the micron functionally graded pipeline.

The present invention provides a method for predicting the inherent characteristics of the fluid-structure coupling of functional pipelines. The method includes: acquiring mechanical parameters, size parameters and installation parameters of micrometer functionally graded pipelines; The strain energy formula; according to the strain energy formula, use Hamilton's principle to deduce the differential equation of motion of the micron functionally graded pipeline and establish boundary conditions; use the hybrid method to calculate the mechanical parameters, size parameters and installation of the micron functionally graded pipeline according to the The parameter solves the differential equation of motion, and then solves the natural frequency of the micron functionally graded conduit. The method for predicting the inherent characteristics of the fluid-structure interaction of functional pipelines proposed in the present disclosure can obtain the natural frequencies of micron functionally graded pipelines through theoretical research.

It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the invention.

Description of drawings

The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the invention and together with the description serve to explain the principles of the invention. Obviously, the drawings in the following description are only some embodiments of the present invention, and for those of ordinary skill in the art, other drawings can also be obtained from these drawings without creative effort.

FIG. 1 is a flowchart of an exemplary embodiment of a method for predicting the inherent characteristics of fluid-structure coupling of functional pipelines of the present disclosure;

2 is a schematic structural diagram of a micron functional gradient pipeline in an exemplary embodiment of a method for predicting inherent properties of fluid-structure coupling of functional pipelines disclosed;

Fig. 3 is the variation of the inner layer material volume fraction along the thickness direction under the action of different volume fraction indices of the micron functionally graded pipeline in this exemplary embodiment;

Figure 4 is a schematic diagram of fluctuations in a multi-span micron functional gradient pipeline;

Figure 5 is a schematic diagram of the fluctuation in the m-th span pipeline;

6 is a schematic structural diagram of a micron functional gradient pipeline in this exemplary embodiment;

Fig. 7 is the variation of the first-order frequency of the pipeline with the fluid flow velocity u under the action of different dimensionless scale parameters Do/l of the micron functionally gradient pipeline in the exemplary embodiment;

FIG. 8 is the variation of the first three-order natural frequencies of the micron functional gradient pipeline with the flow velocity u when the micron functional gradient pipeline index n=0 in this exemplary embodiment (Do/l=10);

FIG. 9 is the variation of the first three-order natural frequencies of the micron functional gradient pipe with the flow velocity u when the micron functional gradient pipe index n=1 in the exemplary embodiment (Do/l=10);

FIG. 10 is the variation of the first three-order natural frequencies of the micron functional gradient pipeline with the flow velocity u when the micron functional gradient pipeline index n=10 in this exemplary embodiment (Do/l=10);

Fig. 11 is the variation of the first three-order natural frequencies of the micron functional gradient pipe with the flow velocity u when the micron functional gradient pipe index n=50 in the exemplary embodiment (Do/l=10);

Fig. 12 is the variation of the critical flow velocity with the parameter Do/l under the action of different exponents n of the micron functional gradient pipeline in this exemplary embodiment;

13 is the variation of the critical flow velocity with the index n under the action of different scale parameters Do/l of the micron functional gradient pipeline in the exemplary embodiment;

Fig. 14 is the variation of the critical flow velocity of the micron pipe with the position parameter L1/L under the action of different volume fraction indices n of the micron functional gradient pipe in the exemplary embodiment (Do/l=100);

Fig. 15 is the variation of the critical flow velocity of the micron pipe with the number of supports under the action of different volume fraction index n of the micron functionally graded pipe in the exemplary embodiment (Do/l=100).

Detailed ways

Example embodiments will now be described more fully with reference to the accompanying drawings. Example embodiments, however, can be embodied in various forms and should not be construed as limited to the embodiments set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the concept of example embodiments. conveyed to those skilled in the art. The same reference numerals in the drawings denote the same or similar parts, and thus their repeated descriptions will be omitted.

Furthermore, the described features, structures, or characteristics may be combined in any suitable manner in one or more embodiments. In the following description, numerous specific details are provided in order to give a thorough understanding of the embodiments of the present disclosure. However, one skilled in the art will appreciate that the technical solutions of the present disclosure may be practiced without one or more of the specific details, or other methods, components, materials, devices, steps, etc. may be employed. In other instances, well-known structures, methods, devices, implementations, materials, or operations are not shown or described in detail to avoid obscuring aspects of the present disclosure.

The block diagrams shown in the figures are merely functional entities and do not necessarily necessarily correspond to physically separate entities. That is, these functional entities may be implemented in software, or in one or more software-hardened modules or parts of functional entities, or in different network and/or processor devices and/or microcontroller devices implement these functional entities.

Other features and advantages of the present invention will become apparent from the following detailed description, or be learned in part by practice of the present invention.

This exemplary embodiment first provides a method for predicting the inherent characteristics of fluid-structure coupling of functional pipelines. As shown in FIG. 1 , it is a flowchart of an exemplary embodiment of the method for predicting the inherent characteristics of fluid-structure coupling of functional pipelines of the present disclosure. Methods include:

Step S1: Obtain mechanical parameters, size parameters and installation parameters of the micron functionally graded pipeline, where the mechanical parameters include the elastic modulus, density, and Poisson's ratio of the two components in the micron functionally graded pipeline, and the size parameters include the inner diameter, outer diameter and length of the micron functionally graded pipeline, and the installation parameters include the number of nodes supporting the micron functionally graded pipeline and the distance between adjacent nodes;

Step S2: establishing a strain energy formula of the micron functionally graded pipeline structure according to the revised coupled stress theory;

Step S3: according to the strain energy formula, use the Hamiltonian principle to deduce the differential equation of motion of the micron functionally graded pipeline and establish boundary conditions;

Step S4 : using the hybridization method to solve the differential equation of motion according to the mechanical parameters, size parameters and installation parameters of the micron functionally graded pipeline, and then to solve the natural frequency of the micron functionally graded pipeline.

The present invention provides a method for predicting the inherent characteristics of the fluid-structure coupling of functional pipelines. The method includes: acquiring mechanical parameters, size parameters and installation parameters of micrometer functionally graded pipelines; The strain energy formula; according to the strain energy formula, use Hamilton's principle to deduce the differential equation of motion of the micron functionally graded pipeline and establish boundary conditions; use the hybrid method to calculate the mechanical parameters, size parameters and installation of the micron functionally graded pipeline according to the The parameter solves the differential equation of motion, and then solves the natural frequency of the micron functionally graded conduit. The method for predicting the inherent characteristics of the fluid-structure interaction of functional pipelines proposed in the present disclosure can obtain the natural frequencies of micron functionally graded pipelines through theoretical research.

The above steps are described in detail below:

First, this exemplary embodiment describes the material mechanical properties of the micron functionally graded pipeline, as shown in FIG. Schematic. Among them, the length of the micron functional gradient pipe is L, the average radius is R, the fluid velocity is U, and R _o and R _i represent the outer and inner diameters of the pipe, respectively. The displacement components in the x, y and z axis directions of the micron functionally graded pipe are denoted by u, v and w, respectively, and the micron functionally graded pipe can be composed of two materials. It should be pointed out that the plug flow model is still applicable at the micron scale for micron functionally graded pipes. Similarly, the material mechanical properties of the micron functionally graded pipe still change according to the power function law along the thickness direction, and its specific effective material properties can be expressed as:

E=V _i E _i +V _o E _o (1)

μ=V _i μ _i +V _o μ _o (2)

ρ=V _i ρ _i +V _o ρ _o (3)

where:

E is the elastic modulus of the pipeline;

μ is the shear modulus of the pipeline;

ρ——pipe density;

V—the volume fraction of the component material.

In micron functionally graded pipes, Poisson's ratio ν is assumed to be constant. The subscripts i and o denote the inner and outer material of the pipe, respectively. The volume fraction of component materials can be expressed as:

V _o =1-V _i (5)

where:

r - the radius of the target point;

n - volume fraction index.

As shown in Figure 3, it is the change of the volume fraction of the inner layer material along the thickness direction under the action of different volume fraction indices. Obviously, when the volume fraction index n=0, the micron functionally graded pipe degenerates into a micron functionally graded pipe of homogeneous material.

Step S1: Obtain mechanical parameters, size parameters and installation parameters of the micron functionally graded pipeline, where the mechanical parameters include the elastic modulus, density, and Poisson's ratio of the two components in the micron functionally graded pipeline, and the size parameters include The inner diameter, outer diameter and length of the micron functionally graded pipeline, and the installation parameters include the number of nodes supporting the micron functionally graded pipeline and the distance between adjacent nodes.

Through step S1, the elastic moduli E _i and E _o of the two components in the micron functionally graded pipe, the inner diameter Ri and the outer diameter R _o of the micron functionally graded pipe can be obtained, so _that according to formulas (1) and (4) , (5) Obtain the elastic modulus on the target point on the micron functional gradient pipeline.

Step S2: According to the modified coupled stress theory, the strain energy formula of the micron functionally graded pipeline structure is established.

According to the coupled stress theory, the strain energy density of a material is related to both the strain tensor and the rotational gradient tensor. In its constitutive relation, two material intrinsic characteristic scale parameters are additionally introduced. Based on the coupled stress theory, the present exemplary embodiment is based on the modified coupled stress theory. By introducing a new equilibrium condition, the strain energy density of the material is only related to the symmetric components of the strain tensor and the rotational gradient tensor, but has nothing to do with the antisymmetric component of the rotational gradient tensor. The advantage of the modified coupled stress theory is that only one additional material scale parameter needs to be introduced into the constitutive relation of the microstructure, which makes the analysis of the whole problem simpler.

According to the modified coupled stress theory, the strain energy of the structure can be expressed as:

where:

Um——structural strain energy;

Ω——structure volume;

σ——stress tensor;

ε——strain tensor;

m——the offset part of the symmetric coupling stress;

χ——symmetric curvature tensor;

These tensors can be specifically represented as:

σ=λtr(ε)I+2με (7)

m=2l ² μχ (9)

where:

u——displacement vector;

λ——elastic constant;

μ——shear modulus;

θ——rotation vector;

l——Material scale parameter.

In functionally graded materials, the scale parameter of the material is variable. Here, in order to simplify the analysis, the scale parameter l of the material is assumed to be a constant. Alternatively the rotation vector can be written as:

According to the Euler-Bernoulli beam theory, the displacements of any point in Fig. 2 along the x, y and z directions can be written as:

u=-zψ(x,t),v=0,w=w(x,t) (12)

where:

ψ——the angle of the pipe cross section;

Under the condition of small deformation, the turning angle of the pipe can be expressed as:

Substituting equations (12) and (13) into (8) yields the only non-zero strain component:

Substituting equations (12) and (13) into (11) gives:

Substituting equation (15) into (10) gives:

The differential equations of motion and boundary conditions of micrometer functionally graded pipelines are derived.

To simplify the analysis, the effect of Poisson's ratio is ignored when writing the stress components. Substituting equation (14) into (7), the stress component of the micron functionally graded pipe can be expressed as:

Substituting equation (16) into (9) gives:

The strain energy of a micron functionally graded pipe can be written as:

The above formula can be further abbreviated as:

in:

where U _m is the strain energy, E is the elastic modulus, I is the moment of inertia, μ is the shear modulus, A is the cross section of the micron functionally graded pipe, l is the material scale parameter, and w is the target point in the z direction displacement, L is the length of the micron functional gradient pipe, x is the coordinate of the target point in the x direction, z is the coordinate of the target point in the z direction, R _o and R _i represent the outer diameter and inner diameter of the micron functional gradient pipe, respectively , r is the radius of the target point, and θ is the rotation vector. The above parameters l, R _o and _Ri can be obtained directly through step S1; the elastic modulus E can be obtained through the above formula (1); the shear modulus μ can be obtained through the above formulas (2), (4) and (5) , the shear moduli μ _i and μ _o of the two components in the micron functionally graded pipeline can be obtained from the elastic modulus of each component using the formula μ=E/2(1+ν), where ν is Poisson’s ratio , the Poisson's ratio can be obtained directly through step S1; the moment of inertia can be obtained through the formula Obtain, the rotation vector θ can be obtained by formula (11).

Step S3: According to the strain energy formula, use Hamilton's principle to deduce the differential equation of motion of the micron functionally graded pipeline and establish boundary conditions.

The kinetic energy of the micron functionally graded pipe can be written as:

The mass per unit length m ^* of a micron functional gradient pipe can be defined as:

The kinetic energy of the flow in the pipe can be written as:

According to the Hamiltonian principle of flow pipeline, for the case of flow pipeline simply supported at both ends, it can be written as:

where l _c =T _p +T _f -U _m is the Lagrangian function of the micron functionally graded pipe. Substitute the above quantities into equation (26), and through variational operations, the differential equation of motion of the micron functional gradient pipeline can be obtained:

It should be noted that the effects of gravity, damping force, and fluid pressure are ignored in the above formula. The simply supported boundary conditions at both ends of the pipe can be expressed as:

Compared with the differential equation of motion of the classical flow pipeline, it can be found that the effective bending stiffness of the micron functionally graded pipeline is (EI) ^* +(μA) ^* l ² , the mass per unit length of the pipeline is m ^* , and the mass per unit length of the fluid is M _f .

Among them, the density ρ of the micron functionally graded pipe can be obtained according to formulas (3), (4) and (5), and the densities ρ _i and ρ _o of the two materials can be obtained directly through step S1. The mass m ^* per unit length of the pipe can be obtained from the density ρ of the micrometer functionally graded pipe.

Step S4 : using the hybridization method to solve the differential equation of motion according to the mechanical parameters, size parameters and installation parameters of the micron functionally graded pipeline, and then to solve the natural frequency of the micron functionally graded pipeline.

When solving the differential equation of motion, it is first necessary to determine some parameters in the differential equation of motion, for example, the elastic modulus E, the moment of inertia I, the shear modulus μ, the mass per unit length of the pipe m ^* , the material scale parameter l, etc. The above content How to obtain the above parameters has been described in detail. Additionally, solving the differential equation of motion by the hybridization method can include:

The Fourier transform pair governing the solution w(x,t) of differential equation (27) can be written as:

Substituting equation (30) into equation (27) gives:

The displacement solution in the frequency domain can be set as:

where:

c——undetermined constant;

k - wave number.

Substituting equation (32) into equation (31) gives:

[(EI) ^* +(μA) ^* l ² ]k ⁴ -M _f U ² k ² -2ωM _f Uk-ω ² (M _f +m ^* )=0 (33)

The k in equation (33) corresponds to four solutions k1, k2, k3 and k4, which correspond to the four wave modes respectively. According to the method of returning rays, these waves can be divided into incoming and outgoing waves. In the wave propagation method, it can be divided into left traveling waves and right traveling waves. Since k in Equation (33) has four solutions, the assumed solution of the lateral displacement in the frequency domain can be rewritten as:

Correspondingly, the turning angle of the micron functionally graded pipeline in the frequency domain bending moment and shear force It can be expressed as:

According to the return ray theory, as shown in Fig. 4, it is a schematic diagram of the fluctuation in the multi-span micron functional gradient pipeline. Double superscript will be introduced here to describe the incoming and outgoing waves. The superscripts J and K denote adjacent nodes. It should be pointed out that in the incident wave, represents the movement of the wave from node K to node J, while in the outgoing wave, Indicates that the wave moves from node J to node K.

A schematic diagram of the fluctuation of a typical multi-span micron functional gradient pipeline is shown in Figure 4. The pipe is supported by n simply supported supports. According to the return ray theory, the incoming and outgoing waves at each node can be expressed as:

in the formula

a——incident wave;

d - outgoing wave.

The superscripts 1, 2, ... n represent the pipeline nodes, and the subscripts 1, 2, 3, and 4 correspond to k1, k2, k3, and k4, respectively.

According to the theory of returning rays, the incident wave and the outgoing wave have the following relationship:

d=Sa+s (40)

where:

S - global scattering matrix;

s - global source matrix.

The wave source matrix s is related to the external load. In the analysis of this chapter, the free vibration and stability of the pipeline are mainly considered, and the external load is not considered, so s=0 in equation (40). The global scattering matrix S is assembled from the local scattering matrices at each node. The local scattering matrix of the node is established by the displacement continuity condition and the force balance condition at the node.

For example, in Figure 4, the boundary conditions for node 1 can be written as:

Substituting equations (34) and (36) into (41) gives:

in

The incoming and outgoing waves at node 1 are:

Substituting the incoming and outgoing waves at node 1 into equations (42) and (43) yields:

The incoming and outgoing waves at node 1 have the following relationship:

d ¹ =S ¹ a ¹ (47)

where:

S ¹ - Local scattering matrix for node 1.

Combining equations (46) and (47), the scattering matrix at node 1 can be obtained as:

The local scattering matrix at node 2 in Fig. 4 can be obtained by the continuity condition and equilibrium condition at node 2, which can be written as:

where:

"-" - the left end of the node;

"+" - the right end of the node;

and is the corner of the pipe cross-section.

The incoming and outgoing waves at node 2 are:

After substituting the expressions of displacement, rotation angle and bending moment into the above formula, and after finishing, the local scattering matrix at node 2 can be obtained:

where λ _j =ik _j .

Other intermediate nodes have the same continuity conditions and equilibrium conditions as those of node 2, and their local scattering matrices can also be established by the same method, which will not be repeated here.

The boundary conditions for node n can be written as:

The incoming and outgoing waves at node n are:

Combining equations (34) and (36), and sorting out, the scattering matrix Sn at node ⁿ can be obtained as:

After obtaining the local scattering matrices of all nodes, according to the wave positions of the nodes, the global scattering matrix of the structure can be obtained by assembling:

Next, the wave propagation method will be used to establish the wave propagation matrix of the sub-span unit of the multi-span pipeline according to the propagation relationship of the waves in the sub-span unit.

Figure 5 depicts the propagation of waves in the mth spanning pipe unit. As mentioned earlier, in the wave propagation method, the waves in the pipeline are divided into right-traveling waves and left-traveling waves. Without loss of generality, the subscripts 1 and 2 are in The middle indicates the left traveling wave, while the subscripts 3 and 4 indicate the right traveling wave.

The left traveling wave of the mth spanning pipeline unit has the following propagation relationship:

where:

T _mL ——m cross-unit left propagation matrix, m can be obtained according to the installation parameters of the micron functional gradient pipeline.

The specific expressions of the matrix in the above formula are:

lm is the distance from the _mth node to the _m +1th node, and lm can be obtained from the installation parameters of the micron functional gradient pipeline.

The right traveling wave of the mth spanning pipeline unit has the following propagation relationship:

where:

T _{mR ——the} m-th cross-unit right propagation matrix.

The specific expressions of the matrix in the above formula are:

The wave propagation matrices of other sub-span cells can also be established in the same way.

In the following analysis, the return ray matrix method combined with the wave propagation method will be used to establish the characteristic equation of the multi-span pipeline. Introducing the incoming and outgoing waves into the wave propagation matrix, and combining equations (58) and (60), the following matrix can be obtained:

where the incoming and outgoing waves at node m and node m+1 are:

By assembling the local wave propagation matrices of all sub-span cells, the second relation for incoming and outgoing waves can be obtained:

a=PUd (65)

in:

By combining equations (40) and (65), we get:

(I-R)d=s (68)

where R=SPU is called the return ray matrix of the micron functional gradient pipeline. The natural frequencies of micron functionally graded pipes can be obtained by equalizing the coefficient matrix to zero:

h(ω)=|I-R|=0 (69)

Using the above formula, the natural frequency of the micron functionally graded pipeline can be obtained. Obtained by plotting h(ω) as a function of ω. When the real and imaginary parts of h(ω) are zero at the same time, the corresponding ω is the natural frequency of the pipeline.

The present exemplary embodiment also provides a natural frequency solution process of a specific micron functional gradient pipeline and an analysis method for the solution.

The micron functionally graded pipeline is composed of aluminum and aluminum oxide. The mechanical parameters and dimensional parameters are shown in Table 1. The density of the fluid ρ _f =1000 kg/m ³ . In order to study the scale effect of micron functionally graded pipelines, in the following example, the scale parameter l of micron functionally graded pipelines is selected as 15 μm.

Table 1 Mechanical properties of micron functionally graded pipeline components

As shown in FIG. 6 , it is a schematic structural diagram of the micron functional gradient pipeline in this exemplary embodiment. The entire micron functional gradient pipeline is supported by four hinged supports. The geometrical parameters of the micron functional gradient pipeline are _{Di/D o =} 0.9, L ₁ =L ₂ =L/3, and L/D _o =20. In order to study the effect of material scale parameters on the vibration and stability of the micron functionally graded pipe, here the diameter of the micron functionally graded pipe is set as a variable, and the dimensionless scale parameter is defined as D _o /l:

In order to simplify the analysis and make the results general, the following dimensionless parameters are introduced in the following analysis:

where:

ξ——dimensionless length;

l ^* ——dimensionless scale parameter;

u — dimensionless fluid velocity;

- the natural frequency of the dimensionless pipeline;

In addition, EI and m respectively represent the flexural rigidity and the mass per unit length of the micron functionally graded pipe when the volume fraction index n=0.

In order to study the effect of material scale parameters on the free vibration and stability of the micron functionally graded pipeline, as shown in Figure 7, it is the change of the first-order frequency of the pipeline with the fluid velocity u under the action of different dimensionless scale parameters D _o /l. Table 2 gives the critical flow velocity ud under the action of different dimensionless scale parameters _{D o} /l. It has been stated earlier that in the stable phase of the pipeline, the real part of the frequency It decreases with the increase of the fluid velocity u. When the velocity exceeds a certain value, the real part of the first-order frequency drops to zero, which indicates that the pipeline system is statically unstable, and the corresponding fluid velocity is called the critical velocity u. _d , the imaginary part of the frequency is

Table 2 Critical flow velocity under the action of different dimensionless scale parameters D _o /l

It can be seen from Fig. 7 and Table 2 that the natural frequency and critical flow velocity obtained by the modified coupled stress theory are larger than those calculated by the classical theory. At the same time, it is also found that with the gradual increase of the dimensionless scale parameter D _o /l, the results calculated by the modified coupled stress theory gradually converge to the results calculated by the classical theory. For example, when the dimensionless scale parameter D _o /l=1, the critical flow velocity calculated by the modified coupled stress theory is 2.14 times that of the classical beam theory calculation. When the dimensionless scaling parameter D _o /l=10, this ratio is reduced to 1.02. Therefore, it can be concluded that the scale parameters of the material have an important influence on the vibration and stability of the micron functionally graded piping system. when approaching. This is because the scale parameter of the material can increase the effective bending stiffness (EI) ^* +(μA) ^* l ² of the micron functionally graded pipe, but as the dimensionless scale parameter D _o /l gradually increases, this effect slowing shrieking. When the diameter of the pipe is larger than the scale parameter of the material (D _o /l>10), the natural frequency and critical flow velocity obtained by the modified coupled stress theory converge to the results obtained by the classical theory.

In order to study the effect of the volume fraction index n on the free vibration and stability of the micron functionally graded pipeline, Figures 8, 9, 10, and 11 show the micron functional gradient pipeline at different volume fraction index n (n=0, 1, 10, 50 ), when the dimensionless scale parameter D _o /l=10, the first three-order natural frequencies change with the fluid velocity u. Among them, 1st-mode represents the first-order mode, 2st-mode represents the second-order mode, and 3st-mode represents the third-order mode. It can be seen from Fig. 8 that when the index n=0, the micron functionally graded pipeline presents a more complex dynamic phenomenon. The specific performance is that static instability occurs when the first-order frequency of the pipeline is at the dimensionless flow velocity u=9.59. When the second-order frequency is u=11.78, static instability occurs, and when the flow velocity u=15.69, the static instability occurs at the third-order natural frequency of the pipeline. Then, when the flow velocity u continues to increase to 17.60, the second-order frequency of the pipeline is directly coupled with the third-order frequency of the pipeline, and the dynamic instability of the coupling occurs. It should be pointed out that in a single-span flow pipeline, the coupled dynamic instability of the pipeline first occurs at the first and second order frequencies of the pipeline. The coupled dynamic instability of this example occurs directly in the second and third order modes, which is different from the single-span flow pipeline and is also related to the support.

It can also be found from Figures 8, 9, 10, and 11 that the real part of the natural frequency of the micron functionally graded pipe and the critical flow velocity increase with the increase of the volume fraction index n. For example, when the exponent n=0, the static instability of the first mode occurs at u=9.59. When the index n=1, the critical flow velocity of the micron functional gradient pipeline _ud =17.50, when the index n=10 and 50, the micron functional gradient pipeline does not have static instability in the range of flow velocity u<18. Therefore, it can be concluded that the stability of the micron functionally graded pipeline increases with the increase of the volume fraction index n. This is mainly due to the fact that with the increase of the index n, the composition content of alumina in the micron functionally graded pipe increases while the composition of aluminum decreases, and the Young's modulus of alumina is much larger than that of aluminum. quantity. These conclusions are the same as for functionally graded macroscopic flow conduits.

In order to further study the effects of volume fraction index n and dimensionless scale parameter D _o /l on the stability of micron functionally graded pipelines, under the action of different scale parameters D _o /l (D _o /l=1, 2, 5, 10) , the variation of the dimensionless critical flow velocity u _d with the volume fraction exponent n is shown in Fig. 12, and the result calculated by the classical beam theory (l=0) is also given in the figure. Figure 13 shows the variation of the dimensionless critical flow velocity _ud with the scale parameter D _o /l under the action of different volume fraction exponents n (n=0, 1, 10, 50).

It can be seen from Fig. 12 that the critical flow velocity calculated by the modified coupled stress theory is larger than the value calculated by the classical beam theory. This is due to the fact that the scale parameter can increase the effective stiffness of micrometer functionally graded pipes. As the dimensionless scale parameter D _o /l increases from 1 to 10, the dimensionless critical flow rate decreases sharply and finally converges to the result calculated by classical theory. It can be seen from Fig. 12 that the critical flow velocity decreases with the increase of the scale parameter D _o /l. When the scale parameter D _o /l>10, the critical flow velocity calculated by the modified coupled stress theory gradually approaches a certain value. a fixed value. From the above results, it can be seen that the scale effect caused by the coupled stress has a significant effect only when the pipe size is similar to the material scale parameter. When the dimensionless scale parameter is large (D _o /l>10), that is, when the pipe diameter is much larger than the material scale parameter, the results calculated by the modified coupled stress theory converge to the results calculated by the classical theory.

On the other hand, it can also be seen from Figure 12 that the critical flow rate increases with the increase of the volume fraction index n, especially when n<10, the critical flow rate increases rapidly with the increase of the index n. With the further increase of the exponent n, its influence on the critical flow rate gradually decreases. When the exponent n=50, the critical flow rate has approached a constant. This is similar to the results for the macro-micron functional gradient pipeline.

In engineering practice, the support position of the pipe clamp is usually limited by the surrounding environment, and cannot be evenly distributed in the entire pipe system. Therefore, it is necessary to study the influence of the position of the pipe support clamp on the stability of the pipeline. In this calculation example, it is assumed that the support 2 and the support 3 in Fig. 6 move from the two ends of the pipeline to the midpoint of the pipeline respectively (0≤L ₁ =L ₂ ≤L/2). Under the action of different volume fraction exponents n (n=0, 1, 10, 50), the critical flow velocity of the micron functional gradient pipeline (D _o /l=100) varies with the _ud support position L ₁ /L as shown in Fig. 14 Show.

It can be seen from Fig. 14 that when L ₁ /L=0 and the volume fraction exponent n=0, the critical velocity of the pipeline is 3.142≈π (π is the analytical solution of the critical velocity of the simply branched pipeline at both ends). In fact, when L ₁ /L=0, supports 2 and 3 in Fig. 6 coincide with supports 1 and 4 respectively, indicating that the three-span flow transmission system at this time degenerates into a single-span flow transmission pipeline system, and the index n= 0 means that the micron functionally graded pipe also degenerates into a uniform material pipe, and the dimensionless scale parameter D _o /l=100 means that the result calculated by the modified coupled stress theory has converged to the result calculated by the classical theory at this time. Therefore, the critical flow velocity 3.142 calculated by the method in this paper is in good agreement with the analytical solution of the critical flow velocity of a single-span uniform material pipeline. This also proves the correctness of the analysis method in this paper. It can also be seen from Fig. 14 that when the position parameter L ₁ /L is approximately equal to 0.33, the entire pipeline system has the maximum critical flow velocity. This is because for a three-span flow pipeline, when the intermediate supports are evenly arranged on the pipeline, the stiffness of the entire system is the largest. It can also be found from Figure 14 that when the position parameter L ₁ /L increases from zero to a small value, the critical flow velocity of the pipeline system also increases sharply. It can be seen that for a single-span pipeline, even in its The addition of support clamps near both ends will greatly improve the stability of the flow pipeline system, and it also shows that multi-span support can significantly improve the stability of the flow pipeline system compared to single-span support.

In order to study the effect of the number of clamp supports on the stability of the micron functionally graded pipeline, Fig. 15 shows the variation of the dimensionless critical flow velocity _ud with the number of pipeline supports (D _o /l=100). In this study, the support clamps are evenly distributed over the entire piping system. It can be seen from Fig. 14 that when the number of supports is in the range of 10, with the increase of the number of supports, the critical flow velocity of the flow pipeline shows a linear increase as a whole. It should be pointed out that when the number of supports is 2, it means that the micron functional gradient pipeline system is a single-span pipeline at this time, and when the number of supports is 3, the pipeline is a two-span pipeline at this time. It can be found from Figure 15 that when the pipeline transitions from a single-span pipeline to a two-span pipeline, the critical flow velocity of the pipeline increases sharply, and when the pipeline system transitions from a two-span pipeline to a three-span pipeline, the change of the critical velocity does not obvious. After that, with the increase of the number of pipe supports, the critical flow velocity of the pipe system basically shows a linear increase. Overall, increasing the number of supports of the flow conduits can significantly improve the stiffness of the conduit system and expand the stable region of the micron functionally graded conduits. These results have implications for designing micron-scale piping systems.

Additionally, although the various steps of the methods of the present disclosure are depicted in the figures in a particular order, this does not require or imply that the steps must be performed in the particular order or that all illustrated steps must be performed to achieve the desired result. Additionally or alternatively, certain steps may be omitted, multiple steps may be combined into one step for execution, and/or one step may be decomposed into multiple steps for execution, and the like.

Other embodiments of the present disclosure will readily suggest themselves to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of the present disclosure that follow the general principles of the present disclosure and include common knowledge or techniques in the technical field not disclosed by the present disclosure . The specification and examples are to be regarded as exemplary only, with the true scope and spirit of the disclosure being indicated by the claims.

It is to be understood that the present disclosure is not limited to the precise structures described above and illustrated in the accompanying drawings, and that various modifications and changes may be made without departing from the scope thereof. The scope of the present disclosure is limited only by the appended claims.

Claims (10)

1. a prediction method for the inherent characteristics of functional pipeline fluid-structure coupling, is characterized in that, comprises: Obtain mechanical parameters, size parameters and installation parameters of the micron functionally graded pipeline, where the mechanical parameters include elastic modulus, density, and Poisson's ratio of two components in the micron functionally graded pipeline, and the size parameters include the micrometer functional gradient pipeline. the inner diameter, outer diameter and length of the functionally graded pipeline, the installation parameters including the number of nodes supporting the micron functionally graded pipeline and the distance between adjacent nodes; According to the modified coupled stress theory, the strain energy formula of the micron functionally graded pipeline structure is established; According to the strain energy formula, the differential equation of motion of the micron functionally graded pipeline is deduced by using Hamilton's principle, and the boundary conditions are established; The hybridization method is used to solve the differential equation of motion according to the mechanical parameters, size parameters and installation parameters of the micron functionally graded pipeline, and then the natural frequency of the micron functionally graded pipeline is solved. 2 . The method for predicting the inherent characteristics of fluid-structure coupling of functional pipelines according to claim 1 , wherein establishing the strain energy formula of the micron functionally graded pipeline structure comprises: 3 . build formula in, U _m is the strain energy, E is the elastic modulus, I is the moment of inertia, μ is the shear modulus, A is the cross section of the micron functionally graded pipe, l is the material scale parameter, and w is the displacement of the target point in the z direction , L is the length of the micron functional gradient pipe, x is the coordinate of the target point in the x direction, z is the coordinate of the target point in the z direction, R _o and R _i represent the outer diameter and inner diameter of the micron functional gradient pipe, respectively, r is the radius of the target point, and θ is the rotation vector. 3. The method for predicting the inherent characteristics of the fluid-structure coupling of functional pipelines according to claim 2, wherein the moment of inertia is 4. The method for predicting the inherent characteristics of functional pipeline fluid-structure coupling according to claim 2, wherein, according to the strain energy formula, the differential equation of motion of the micron functionally graded pipeline is deduced by using the Hamiltonian principle, comprising: Establishing the Lagrangian function of the micron functional gradient pipeline: l _c =T _p +T _f -U _m , where T _p is the kinetic energy of the micron functional gradient pipeline, and T _f is the inner surface of the micron functional gradient pipeline the kinetic energy of the fluid; According to Hamilton's principle, formulate the equation: According to the equation Derive the differential equation of motion for the micron functionally graded pipeline: Among them, m ^* is the mass per unit length of the micron functional gradient pipe, M _f is the mass of the fluid per unit length in the micron functional gradient pipe, and U is the velocity of the fluid in the micron functional gradient pipe. 5. The method for predicting the inherent characteristics of functional pipeline fluid-structure coupling according to claim 4, wherein establishing boundary conditions comprises: Build the formula: 6. the prediction method of the inherent characteristic of functional pipeline fluid-structure coupling according to claim 5, is characterized in that, utilizes hybridization method to solve described differential equation of motion according to the mechanical parameter, size parameter and installation parameter of described micron functional gradient pipeline, Then solve the natural frequency of the micron functionally graded pipeline, including: The Fourier transform pair governing the solution w(x,t) of the differential equation of motion is: Will Bringing into the differential equation of motion we get: set equation The solution in the frequency domain is where c is an undetermined constant, k is the wave number, and bring in Obtain the equation [(EI) ^* +(μA) ^* l ² ]k ⁴ -M _f U ² k ² -2ωM _f Uk-ω ² (M _f +m ^* )=0, set the equation [(EI) ^* + (μA) ^* l ² ]k ⁴ -M _f U ² k ² -2ωM _f Uk-ω ² (M _f +m ^* )=0 The solution in the frequency domain on the lateral displacement is 7. the prediction method of the inherent characteristic of functional pipeline fluid-structure coupling according to claim 6, is characterized in that, utilizes hybridization method to solve described differential equation of motion according to the mechanical parameter, size parameter and installation parameter of described micron functional gradient pipeline, and then solving the natural frequency of the micron functional gradient pipeline, further comprising: The formula d=Sa+s is established according to the theory of return rays, where S is the global scattering matrix; s is the global wave source matrix; According to the theory of wave propagation method, the formula a=PUd is established, wherein, T _qL is the q-th cross-unit left propagation matrix, T _qR is the q-th cross-unit right propagation matrix 0 _2×2 is a 2×2 zero matrix, and I _2×2 is a 2×2 unit matrix; Combining the formula d=Sa+s and a=PUd, the formula (I-R)d=s is obtained, wherein, the return ray matrix R=SPU of the micron functionally graded pipeline; According to the formula h(ω)=|I-R|=0, the natural frequency of the micron functionally graded pipeline is solved. 8 . The method for predicting the inherent characteristics of fluid-structure coupling of functional pipelines according to claim 7 , wherein s=0. 9 . 9. The method for predicting the inherent characteristics of functional pipeline fluid-structure interaction according to claim 7, wherein the global scattering matrix where Sn is the scattering matrix at node ⁿ . 10 . The method for predicting the inherent characteristics of the fluid-structure interaction of functional pipelines according to claim 7 , wherein, calculating the natural frequency of the micron functionally graded pipeline according to the formula h(ω)=|I-R|=0, comprising: 10 . Plot the curve of h(ω) as a function of ω; When the real part and the imaginary part of h(ω) are both zero, the corresponding ω is the natural frequency of the micron functionally graded pipeline.