CN106294927A - A kind of computational methods of liquid-filled pipe dynamic trait - Google Patents

Abstract

The present invention relates to engineering calculation field, particularly relate to the computational methods of a kind of liquid-filled pipe dynamic trait.What the present invention was correct describes the stream flow velocity impact on pipeline transverse bending vibration natural frequency in liquid-filled pipe, it is proposed that correct computational methods, reduces the security risk that pipeline in use exists.

Description

A kind of computational methods of liquid-filled pipe dynamic trait

Technical field

The present invention relates to engineering calculation field, particularly relate to the computational methods of a kind of liquid-filled pipe dynamic trait.

Background technology

A kind of special equipment that pipeline carries as material plays critically important in modernization commercial production and people's lives Effect, pipeline accident happens occasionally, and drastically influence the safety of life and property of the people.Safety in production is caused by pipe vibration The biggest threat.Strong pipe vibration can make the connecting portion of pipeline fittings, particularly pipeline and the connecting portion of conduit fittings Position etc. occurs to loosen and rupture, and causes major accident.

Present stage, the transverse curvature dynamic trait natural frequency that the fluid flowing in pipeline can affect pipeline is carried out During calculating, the result of calculation drawn demonstrates, along with flow velocity increase pipeline natural frequencies reduce, but with reality The situation on border does not corresponds, and in test, the flexible pipe of a bending can stretch along with the increase of flow rate of water flow, and i.e. pipeline is intrinsic Frequency improves with the increase of flow velocity.

Summary of the invention

The technical problem to be solved in the present invention is how to overcome the deficiencies in the prior art, it is provided that a kind of liquid-filled pipe power is special The computational methods of property.

The present invention the technical scheme is that the computational methods of a kind of liquid-filled pipe dynamic trait for achieving the above object, Including step calculated as below:

(1) pipeline infinitesimal section is carried out dynamic analysis, draw the kinetics equation of pipeline infinitesimal section, i.e. formula 1

$(m+\mathrm{\ρ}A)\frac{{\∂}^{2}y}{\∂t^2}+EI\frac{{\∂}^{4}y}{\∂x^4}-{\mathrm{\ρAv}}^2\frac{{\∂}^{2}y}{\∂x^2}-2\mathrm{\ρ}Av\frac{{\∂}^{2}y}{\∂x\∂t}=0$

In formula:

The quality of m pipeline unit length；

The transversely deforming (amount of deflection) of y pipeline；

The coordinate of x pipe lengths；

ρ fluids within pipes density；

A inner-walls of duct cross-sectional area；

Stream flow velocity in v pipeline；

EI cross-section of pipeline composite bending modulus；

The t time；

(2) formula 1 is solved, on the basis of being built upon small deformation supposition due to formula 1, the angle speed that cross section rotates Degree is high-order a small amount of, therefore, ignores Coriolis accelerationAfter known Mode Equation, i.e. formula 2:

φ^Ⅳ(x)-α φ " (x)-β φ (x)=0

In formula:

φ (x) model function of vibration,

$\mathrm{\α}=\frac{{\mathrm{\ρAv}}^2}{EI},\mathrm{\β}=\frac{{\mathrm{\ω}}^2(m+\mathrm{\ρ}A)}{EI},$

ω liquid-filled pipe transverse bending vibration natural frequency；

(3) formula 2 is solved, if φ (x)=e^px, substituting into formula 2 must be about 4 equation of n th order n of p:

p⁴-αp²-β=0

Make p²=s, then above formula is the quadratic equation with one unknown of s

s²-α s-β=0

Tried to achieve by above formula

$s_1=\frac{\mathrm{\α}+\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}}{2},s_2=\frac{\mathrm{\α}-\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}}{2}$

Then 4 solutions of formula 2 are:

$p_{1,2}=\±\sqrt{\frac{\mathrm{\α}+\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}}{2}},p_{3,4}=\±i\sqrt{\frac{\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}-\mathrm{\α}}{2}}$

Order

$\mathrm{\δ}=\sqrt{\frac{\mathrm{\α}+\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}}{2}},\mathrm{\ϵ}=\sqrt{\frac{\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}-\mathrm{\α}}{2}}$

Thus

φ (x)=C₁e^δx+C₂e^-δx+C₃e^iεx+C₄e^-iεx

By Euler's formula

$\cosh x=\frac{e^x+e^{-x}}{2},\sinh x=\frac{e^x-e^{-x}}{2}$

$\cos x=\frac{e^{ix}+e^{-ix}}{2},\sin x=\frac{e^{ix}-e^{-ix}}{2i}$

Obtain model function of vibration, i.e. formula 3:

φ (x)=D₁chδx+D₂shδx+D₃cosεx+D₄sinεx

In formula:

D₁、D₂、D₃、D₄Undetermined coefficient；

The parameter that δ, ε pipe ends constraints is relevant,

The constraints of pipe ends includes: two ends freely-supported, two ends are fixed, one end is fixed, one end freely-supported is fixed in one end and One end is fixed freely in one end.

Such as: two ends freely-supported condition be amount of deflection be zero-sum moment of flexure be zero, i.e. φ (0)=φ (l)=0, φ " (0)=φ " L ()=0, substitutes them in above formula and can obtain δ, ε, and D₁、D₂、D₃、D₄These known skills being in theory of structural dynamics Art obtains in above-mentioned solution procedure simultaneously；

(4) according to pipe ends physical constraint condition, ε parameter value is calculated；

(5) formula 3 is had to try to achieve the formula about ω, i.e. formula 4 according to the constraints of pipe ends:

$\mathrm{\ω}=\frac{{\mathrm{\ϵ}}^{2}EI}{(m+\mathrm{\ρ}A)}\sqrt{1+\frac{{\mathrm{\ρAv}}^2}{{\mathrm{\ϵ}}^{2}EI}}$

(6) carry out the measuring and calculating of necessity for EI, m, A of liquid-filled pipe in formula 4, and calculate ginseng according to practical situation The value of number ρ, v, substitutes in formula 4 and draws liquid-filled pipe transverse bending vibration natural frequency, i.e. ω value.

What the present invention was correct describes the stream flow velocity impact on pipeline transverse bending vibration natural frequency in liquid-filled pipe, carries Go out correct computational methods, reduce the security risk that pipeline in use exists.

Accompanying drawing explanation

The result of the test diagram of Fig. 1 test method of the present invention.

The stress of Fig. 2 inventive pipeline infinitesimal section and motion schematic diagram,

Wherein, T tension force, N, dN shearing and shearing increment, M, dM moment of flexure and moment of flexure increment,

D θ pipeline infinitesimal section sectional twisting angle,The outlet of pipeline infinitesimal section and the fluid momentum in exit,Pipeline infinitesimal section transverse acceleration, dx pipeline infinitesimal segment length

Detailed description of the invention

Set up correct liquid-filled pipe dynamic characteristic calculation method, it is possible to the correct fluid flowing in reaction liquid-filled pipe Impact on pipeline transverse curvature dynamic trait (natural frequency).

Current liquid-filled pipe dynamic trait calculates based on following equations:

In formula:

The quality of m pipeline unit length；

The transversely deforming (amount of deflection) of y pipeline；

The coordinate of x pipe lengths；

ρ fluids within pipes density；

A inner-walls of duct cross-sectional area；

Stream flow velocity in v pipeline；

Owing to the vibration of pipeline belongs to small deformation, the angular velocity that its cross section rotates is the least, therefore, and Coriolis acceleration itemNow, equation 1 is reduced to:

$(m+\mathrm{\ρ}A)\frac{{\∂}^{2}y}{\∂t^2}+EI\frac{{\∂}^{4}y}{\∂x^4}+{\mathrm{\ρAv}}^2\frac{{\∂}^{2}y}{\∂x^2}=0$

And calculate the natural frequency computing formula of liquid-filled pipe:

$\mathrm{\ω}=\frac{{\mathrm{\ϵ}}^{2}EI}{(m+\mathrm{\ρ}A)}\sqrt{1-\frac{{\mathrm{\ρAv}}^2}{{\mathrm{\ϵ}}^{2}EI}}$

In formula: the parameter that ε is relevant with pipe ends constraints, the constraints of pipe ends is

During simple boundary condition,

From above-mentioned natural frequency computing formula it can be seen that along with flow velocity increase pipeline natural frequencies Reduce, but this is not inconsistent with natural phenomena, is not inconsistent with experimental phenomena yet；The flexible pipe of a piece bending can increase along with flow rate of water flow and stretch Directly, such as fire hose.Fig. 1 is the result of the test of test method of the present invention, which characterizes the natural frequency of liquid-filled pipe with The increase of flow velocity and improve.

To sum up, the computational methods of current liquid-filled pipe transverse bending vibration natural frequency are incorrect, and reason is in " side Journey 1 " can not be correct description fact phenomenon.

Liquid-filled pipe bends when vibrating, due to the flexural deformation of pipeline so that the momentum of tube fluid changes, For straight tube, although change in flow causes the change of momentum, but owing to the direction of momentum is parallel with conduit axis, therefore, will not Pipeline is made to produce horizontal force.But during for the pipeline bent or pipeline generation transversely deforming, even if flow velocity is constant (fixed length Flow velocity), owing to the change in flow velocity direction also will cause fluid momentum to change, and its direction of momentum of this part change is vertical In flow velocity (the flow velocity direction momentum increment that causes of change is perpendicular to flow velocity), i.e. it is perpendicular to pipeline, thus the transversely acting produced Power.

Increment due to momentum is equal to the momentum of power, and therefore, the increment of this part momentum is that the tension force momentum of pipeline is dynamic The projection in amount increment direction.Fig. 2 shows stress and the motion schematic diagram of pipeline infinitesimal section, it is analyzed, thus may be used Obtain the kinetics equation of pipeline infinitesimal section, i.e. formula 1

$(m+\mathrm{\ρ}A)\frac{{\∂}^{2}y}{\∂t^2}+EI\frac{{\∂}^{4}y}{\∂x^4}-{\mathrm{\ρAv}}^2\frac{{\∂}^{2}y}{\∂x^2}-2\mathrm{\ρ}Av\frac{{\∂}^{2}y}{\∂x\∂t}=0$

In formula:

The quality of m pipeline unit length；

The transversely deforming (amount of deflection) of y pipeline；

The coordinate of x pipe lengths；

ρ fluids within pipes density；

A inner-walls of duct cross-sectional area；

Stream flow velocity in v pipeline；

EI cross-section of pipeline composite bending modulus；

The t time；

(2) formula 1 is solved, on the basis of being built upon small deformation supposition due to formula 1, the angle speed that cross section rotates Degree is high-order a small amount of, therefore, ignores Coriolis accelerationAfter known Mode Equation, i.e. formula 2:

φ^Ⅳ(x)-α φ " (x)-β φ (x)=0

In formula:

φ (x) model function of vibration,

$\mathrm{\α}=\frac{{\mathrm{\ρAv}}^2}{EI},\mathrm{\β}=\frac{{\mathrm{\ω}}^2(m+\mathrm{\ρ}A)}{EI},$

ω liquid-filled pipe transverse bending vibration natural frequency；

(3) formula 2 is solved, if φ (x)=e^px, substituting into formula 2 must be about 4 equation of n th order n of p:

p⁴-αp²-β=0

Make p²=s, then above formula is the quadratic equation with one unknown of s

s²-α s-β=0

Tried to achieve by above formula

$s_1=\frac{\mathrm{\α}+\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}}{2},s_2=\frac{\mathrm{\α}-\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}}{2}$

Then 4 solutions of formula 2 are:

$p_{1,2}=\±\sqrt{\frac{\mathrm{\α}+\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}}{2}},p_{3,4}=\±i\sqrt{\frac{\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}-\mathrm{\α}}{2}}$

Order

$\mathrm{\δ}=\sqrt{\frac{\mathrm{\α}+\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}}{2}},\mathrm{\ϵ}=\sqrt{\frac{\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}-\mathrm{\α}}{2}}$

Thus

φ (x)=C₁e^δx+C₂e^-δx+C₃e^iεx+C₄e^-iεx

By Euler's formula

$\cosh x=\frac{e^x+e^{-x}}{2},\sinh x=\frac{e^x-e^{-x}}{2}$

$\cos x=\frac{e^{ix}+e^{-ix}}{2},\sin x=\frac{e^{ix}-e^{-ix}}{2i}$

Obtain model function of vibration, i.e. formula 3:

φ (x)=D₁chδx+D₂shδx+D₃cosεx+D₄sinεx

In formula:

D₁、D₂、D₃、D₄Undetermined coefficient；

The parameter that δ, ε pipe ends constraints is relevant,

The constraints of pipe ends includes: two ends freely-supported, two ends are fixed, one end is fixed, one end freely-supported is fixed in one end and One end is fixed freely in one end.

One of which constraints: two ends freely-supported condition be amount of deflection be zero-sum moment of flexure be zero, i.e. φ (0)=φ (l)=0, φ " (0)=φ " (l)=0, substitutes them in above formula and can obtain δ, ε, and D₁、D₂、D₃、D₄These are structural dynamic scientific principle Known technology in Lun obtains in above-mentioned solution procedure simultaneously；

(4) according to pipe ends physical constraint condition, ε parameter value is calculated；

(5) formula 3 is had to try to achieve the formula about ω, i.e. formula 4 according to the constraints of pipe ends:

$\mathrm{\ω}=\frac{{\mathrm{\ϵ}}^{2}EI}{(m+\mathrm{\ρ}A)}\sqrt{1+\frac{{\mathrm{\ρAv}}^2}{{\mathrm{\ϵ}}^{2}EI}}$

(6) carry out the measuring and calculating of necessity for EI, m, A of liquid-filled pipe in formula 4, and calculate ginseng according to practical situation The value of number ρ, v, substitutes in formula 4 and draws liquid-filled pipe transverse bending vibration natural frequency, i.e. ω value.

Above-described embodiment simply to illustrate that the technology design of the present invention and feature, its objective is to be to allow in this area Those of ordinary skill will appreciate that present disclosure and implements according to this, can not limit the scope of the invention with this.All It is the change according to the equivalence done by the essence of present invention or modification, all should contain within the scope of the present invention.

Claims (1)

1. the computational methods of a liquid-filled pipe dynamic trait, it is characterised in that include step calculated as below:

(1) pipeline infinitesimal section is carried out dynamic analysis, draw the kinetics equation of pipeline infinitesimal section, i.e. formula 1

$(m+\mathrm{\ρ}A)\frac{{\∂}^{2}y}{\∂t^2}+EI\frac{{\∂}^{4}y}{\∂x^4}-{\mathrm{\ρAv}}^2\frac{{\∂}^{2}y}{\∂x^2}-2\mathrm{\ρ}Av\frac{{\∂}^{2}y}{\∂x\∂t}=0$

In formula:

The quality of m pipeline unit length；

The transversely deforming (amount of deflection) of y pipeline；

The coordinate of x pipe lengths；

ρ fluids within pipes density；

A inner-walls of duct cross-sectional area；

Stream flow velocity in v pipeline；

EI cross-section of pipeline composite bending modulus；

The t time；

(2) solving formula 1, on the basis of being built upon small deformation supposition due to formula 1, the angular velocity that cross section rotates is High-order in a small amount, therefore, ignores Coriolis accelerationAfter known Mode Equation, i.e. formula 2:

φ^Ⅳ(x)-α φ " (x)-β φ (x)=0

In formula:

φ (x) model function of vibration,

$\mathrm{\α}=\frac{{\mathrm{\ρAv}}^2}{EI},\mathrm{\β}=\frac{{\mathrm{\ω}}^2(m+\mathrm{\ρ}A)}{EI},$

ω liquid-filled pipe transverse bending vibration natural frequency；

(3) formula 2 is solved, if φ (x)=e^px, substituting into formula 2 must be about 4 equation of n th order n of p:

p⁴-αp²-β=0

Make p²=s, then above formula is the quadratic equation with one unknown of s

s²-α s-β=0

Tried to achieve by above formula

$s_1=\frac{\mathrm{\α}+\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}}{2},s_2=\frac{\mathrm{\α}-\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}}{2}$

Then 4 solutions of formula 2 are:

$p_{1,2}=\±\sqrt{\frac{\mathrm{\α}+\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}}{2}},p_{3,4}=\±i\sqrt{\frac{\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}-\mathrm{\α}}{2}}$

Order

$\mathrm{\δ}=\sqrt{\frac{\mathrm{\α}+\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}}{2}},\mathrm{\ϵ}=\sqrt{\frac{\sqrt{{\mathrm{\α}}^2+4\mathrm{\β}}-\mathrm{\α}}{2}}$

Thus

φ (x)=C₁e^δx+C₂e^-δx+C₃e^iεx+C₄e^-iεx

By Euler's formula

$\cosh x=\frac{e^x+e^{-x}}{2},\sinh x=\frac{e^x-e^{-x}}{2}$

$\cos x=\frac{e^{ix}+e^{-ix}}{2},\sin x=\frac{e^{ix}-e^{-ix}}{2i}$

Obtain model function of vibration, i.e. formula 3:

φ (x)=D₁chδx+D₂shδx+D₃cosεx+D₄sinεx

In formula:

D₁、D₂、D₃、D₄Undetermined coefficient；

The parameter that δ, ε pipe ends constraints is relevant,

(4) according to pipe ends physical constraint condition, ε parameter value is calculated.

(5) formula about ω, i.e. formula 4 are tried to achieve according to the constraints of pipe ends by formula 3:

$\mathrm{\ω}=\frac{{\mathrm{\ϵ}}^{2}EI}{(m+\mathrm{\ρ}A)}\sqrt{1+\frac{{\mathrm{\ρAv}}^2}{{\mathrm{\ϵ}}^{2}EI}}$

(6) for EI, m, A of liquid-filled pipe in formula 4 carry out necessity measuring and calculating, and according to practical situation calculate parameter ρ, v、Value, substitute in formula 4 and draw liquid-filled pipe transverse bending vibration natural frequency, i.e. ω value.