CN103500287A - Rotary blade-box rub-impact force determining method - Google Patents

Abstract

The invention discloses a rotary blade-box rub-impact force determining method and belongs to the technical field of machinery dynamics. The method includes the following steps of establishing a function equation Ue+Uc=W between a blade and a box at a certain moment according to the conservation law of mechanical energy; establishing the force equilibrium relation between the blade and the box and obtaining the expression of the radial force Fn in the position of a blade tip; decomposing the blade tip radial force Fn according to the normal direction and the tangential direction of the blade tip according to the bending deformation of the blade, wherein -FL=-Fncos[v'(L,t)], and -FT=Fnsin[v'(L,t)]; and finally, composing the stress on the blade tip.

Description

Rotating vane-casing touches definite method of the power of rubbing

Technical field

The invention belongs to the mechanical kinetics technical field, relate to a kind of definite method that rotating vane-casing touches the power of rubbing, the rotating vane-casing that particularly relates to a kind of rotation effect that comprises blade self, buckling effect and casing rigidity touches definite method of the power of rubbing.

Background technology

At present, definite method that existing blade-casing touches the power of rubbing mainly contains following several:

1. based on touching of the collision energy conservation power model that rubs

Blade is assumed to be to semi-girder, derived the normal direction contact force of blade and the relation between radial deformation, analyzed at single blade and two kinds of nonlinear dynamic characteristics that touch system in the situation of rubbing of multiple-blade, considered that single blade touches blade while rubbing-casing normal direction and touches the power of rubbing and be:

$F_n=\frac{{\mathrm{\&pi;}}^2}{4}\frac{\mathrm{<figure-callout\; id="2"\; label="EI\; L"\; filenames="HDA0000396415350000011.png,HDA0000396415350000021.png"\; state="\{\{state\}\}">EI</figure-callout>}}{L^{}}\frac{\frac{\mathrm{\&pi;}}{2}\sqrt{\frac{\mathrm{\&delta;}}{L}}}{\mathrm{\&mu;}+\frac{\mathrm{\&pi;}}{2}\sqrt{\frac{\mathrm{\&delta;}}{L}}}$

In formula: the bendind rigidity that EI is blade; L is length of blade; The radially intrusion amount that δ is blade tip; μ is friction factor.

On the basis of above-mentioned formula, by considering the impact of leaf dish and blade rotary centrifugal force, derived blade-casing normal direction and touched the power of rubbing, obtain its expression formula and be:

$F_n=\frac{{\mathrm{\&pi;}}^2}{4}\frac{\mathrm{EI}}{{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA0000396415350000011.png,HDA0000396415350000021.png"\; state="\{\{state\}\}">L</figure-callout>}}^2}\frac{\frac{\mathrm{\&pi;}}{2}\sqrt{\frac{\mathrm{\&delta;}}{L}}}{\mathrm{\&mu;}+\frac{\mathrm{\&pi;}}{2}\sqrt{\frac{\mathrm{\&delta;}}{L}}}+\frac{11}{56}\mathrm{\&rho;}{\mathrm{AL\&Omega;}}^2(\frac{5}{22}L+\frac{35}{22}R)\frac{\frac{\mathrm{\&pi;}}{2}\sqrt{\frac{\mathrm{\&delta;}}{L}}}{\mathrm{\&mu;}+\frac{\mathrm{\&pi;}}{2}\sqrt{\frac{\mathrm{\&delta;}}{L}}}---\left(1\right)$

In formula: the bendind rigidity that EI is blade; L is length of blade; The radially intrusion amount that δ is blade tip; μ is friction factor; ρ is density of material; A is area of blade section; Ω is angular velocity of rotation; R is leaf dish radius.

2. fusing adhesion rubbing model strikes off rubbing model with wearing away

The fusing adhesion model: supposition metal blade and the metal casing of obturaging touches while rubbing, and blade tip will melt, and crosses and touch while rubbing when blade pass, at cooling metal surface area, adhesive coating is arranged.Because blade moves with speed u in resisting medium, therefore can be in the top generation tangential force of blade.In addition, because blade tip and the interlayer of obturaging exist radially relative velocity, therefore also can produce normal force.When supposition blade tip width be far longer than its thickness (b>>a) time, the normal force F of fusing adhesion model _nwith tangential force F _texpression formula is as follows:

$\left\{\begin{array}{cc}{F}_n=\mathrm{cvb}{\left(\frac{a}{h}\right)}^3& F_n\&le;F_s\\ F_n=k(r-C)& F_n>F_s\end{array}\right.$

$F_t=\frac{\mathrm{cuab}}{h}$

In formula: the viscosity that c is deposite metal; The normal direction that v is blade tip is invaded speed; The tangential velocity that u is blade tip; A is blade tip thickness; B is the blade tip width; H is deposite metal thickness; F _ssupporting power for seal structure; K is the casing radial rigidity; The radial displacement that r is blade tip; The radial play that C is blade tip and sealing.

In the situation that blade melts and sticks on cooling seal structure, engine performance very soon can because of the blade wear pressure ratio reduce reduce, recover its efficiency and must change blade, this will shorten the overhaul phase, thereby increases maintenance cost.In order to reduce the wearing and tearing of blade tip, adopt at present mostly the easily low intensive layer of obturaging of mill, when metal blade and these materials touch while rubbing, all will ream some granules blade pass is out-of-date at every turn, remove the required energy U of unit volume material according to each blade, the expression formula that obtains its tangential force is:

F _t=(r-C)Ub

Its normal force is:

F _n=k(r-C)。

3. continuous elastic touches the power model that rubs

Normally used normal force model is the linear rigidity model, and expression formula is as follows:

F _n=k _nδ (2)

In formula: k _nfor contact stiffness; The radially intrusion amount that δ is blade tip.Due to k _nvalue more difficultly determine, therefore, concerning blade-casing rubbing model, its numerical value generally adopts the radial rigidity of blade.

Except linear model, nonlinear rigidity model also often is used, the modal Hertz contact theory that is based on wherein, be approximated to contacting between elastic cylinder and elastic half-space by blade tip and the collision of casing, then according to the local deformation between elastic body, determines normal force.Its normal force expression formula is as follows:

$F_n={\left(\frac{\mathrm{\&delta;}}{a}\right)}^{\frac{3}{2}}---\left(3\right)$

In formula: the radially intrusion amount that δ is blade tip; A is contact radius,

r ₁, R ₂be the radius-of-curvature of two contacts, the R here ₁and R ₂be respectively thickness radius h/2 and the casing radius R of blade _c; E ₁, υ ₁, E ₂, υ ₂be respectively elastic modulus and the Poisson ratio of two contacts.

4. surging force model

The pulse model is a kind of simplified model of normal force model, and it is mainly to simplify calculating by linearization, raises the efficiency.The pulse model can mean by the many kinds of function form:

(a) sine function pulse

$F_n=\left\{\begin{array}{cc}{F}_{\max }\sin \left(\frac{\mathrm{\&pi;}}{t_c}t\right)& 0\&le;t\&le;t_c\\ 0& t_c\&le;t\&le;t_p\end{array}\right.$

(b) cosine function pulse

$F_n=\left\{\begin{array}{cc}-\frac{F_{\max }}{2}+\frac{F_{\max }}{2}\cos \left(\frac{\mathrm{\&pi;}}{t_c}t\right)& 0\&le;t\&le;t\\ 0& t_c\&le;t\&le;t_p\end{array}\right.$

(c) rectangular function pulse

$F_n=\left\{\begin{array}{cc}{F}_{\max }& 0\&le;t\&le;t_c\\ 0& t_c\&le;t\&le;t_p\end{array}\right.$

In formula: F _maxfor maximum normal force; t _cfor duration of contact; t _pfor cycle length; T is the time.Duration of contact t _ccan determine (as shown in Figure 1) by the blade tip running orbit, expression formula is as follows:

$t_c=\frac{{2\cos }^{-1}\left(\frac{{\mathrm{<figure-callout\; id="2"\; label="R\; c"\; filenames="HDA0000396415350000011.png,HDA0000396415350000021.png"\; state="\{\{state\}\}">R</figure-callout>}}_c^{}+{(\mathrm{\&Delta;}+{\mathrm{\&delta;}}_{\max })}^2-{\mathrm{<figure-callout\; id="2"\; label="r\; g"\; filenames="HDA0000396415350000011.png,HDA0000396415350000021.png"\; state="\{\{state\}\}">r</figure-callout>}}_g^{}}{{2R}_c(\mathrm{\&Delta;}+{\mathrm{\&delta;}}_{\max })}\right)}{\mathrm{\&Omega;}}$

In formula: o ₁, o ₂be respectively blade track center and casing center; R _cfor the casing radius; r _gfor tip radlus; Δ is the mean gap of casing when concentric with the blade track, Δ=R _c-r _g; δ _maxfor maximum blade tip-casing intrusion amount; Ω is angular velocity of rotation.

The above-mentioned applicable elements that each determines method is as follows:

1. based on touching of the collision energy conservation power model that rubs: the supposition casing is rigidity, has considered collision process Leaf elastic deformation, although, based on conservation of mechanical energy, do not consider energy dissipation.Therefore it mainly is applicable to take elastic collision as main, and the very large rotating machinery of casing rigidity, as gas turbine etc.

2. fusing adhesion rubbing model strikes off rubbing model with wearing away:

Fusing adhesion rubbing model: for spinner blade with touch the situation of rubbing without the coating metal seal structure.When this happens, may between blade and the basic unit of obturaging (as metal honeycomb structure) or seal structure (as the comb tooth of obturaging), produce one deck deposite metal.

Can wear away and strike off rubbing model: mainly based on the energy loss theory, describe touching that stator casing with the blade of abradable material and coating (as the pottery material of obturaging) occurs and rub.

3. continuous elastic touches the power model that rubs:

(1) spring-damper model, mainly be applicable to turn the stator elastic collision, as touching between the rotor seal part rubbed, not too is applicable to the situation of rubbing of touching that blade has moderate finite deformation; (2) Hertz contact model, both being applicable to elastic collision is that main touching rubs, as turn stator and slowly contact situation (contact relative speed be less than 0.5m/s), (lambda limiting process is to maximum intrusion amount can to pass through again to introduce the hysteresis contact force, provide set deformation volume), consider the impact of plastic yield.

4. surging force model: the power model that rubs is touched in the surging force part: by touching the form that the process simplification that rubs is surging force, complicated contact judgement and the solution procedure of nonlinear iteration have been avoided, greatly reduce calculated amount, but thereby the part of the comparatively complicated real blade-casing of simple analog touch and rub.But this model need to accurately be determined by Contact Dynamics emulation or experiment the duration of contact of blade-casing.

By above-mentioned introduce known: all there is significant limitation in use in definite method that existing blade-casing touches the power model that rubs, this will cause the characteristics of blade-casing collision not reflected fully, and differ larger with actual conditions, and can not consider the impact of each factor in blade-casing collision process.

The research of present stage focuses mostly on and touch the research of the fault of rubbing between rotor and stator, and also less for the research of blade and casing.Compare and turn stator and touch and rub, it is more outstanding that blade-casing touches the nonlinear characteristic of rubbing.Much smaller due to the bending stiffness of blade is compared casing rigidity, and, because be High Rotation Speed in the course of the work, so blade can bend when collision friction, thereby forced vibration can occur, whole system is exerted an influence.In order better to understand the fault signature that rubs that touches of blade-casing, avoid the generation of accident, reasonably touch the power model of rubbing and determine that the proposition of method just seems particularly important.

Summary of the invention

The problem existed for prior art, the invention provides a kind of definite method that rotating vane-casing touches the power of rubbing.Determine that from traditional power of rubbing of touching method is different, the method, for the characteristics of blade-casing collision friction, has been considered various factors, by setting up suitable model, can describe more accurately the collision friction phenomenon of blade-casing.

To achieve these goals, the present invention adopts following technical scheme, and a kind of rotating vane-casing touches definite method of the power of rubbing, and comprises the steps:

Step 1: physical dimension and service condition to blade and casing are measured, comprising the length L of blade, the width b of blade, the thickness h of blade, the radius R of blade disk _d, the casing radius R _c, the minor increment c between blade tip and casing inwall _minand rotating speed Ω;

Step 2: according to the law of conservation of mechanical energy, set up the functional equation that blade and casing are carved at a time:

U _e+U _c=W

In formula, U _efor the flexural deformation energy of blade, U _cfor centrifugal potential energy, W is blade tip radial force and transverse force work;

Step 3: set up the dynamic balance relation of blade and casing, obtain the radial force F at blade tip place _nexpression formula:

$F_n=\frac{5}{3}{\mathrm{L\&Gamma;k}}_{\mathrm{ca}\sin g}\frac{\frac{\mathrm{\&delta;}}{L}+\frac{5\mathrm{\&Gamma;}}{6}-\frac{\sqrt{15}}{6}\sqrt{\frac{15}{9}{\mathrm{\&Gamma;}}^2+4({\mathrm{\&mu;}}^2-\mathrm{\&Gamma;})\frac{\mathrm{\&delta;}}{\mathrm{<figure-callout\; id="10"\; label="L"\; filenames="HDA0000396415350000041.png,HDA0000396415350000042.png"\; state="\{\{state\}\}">L</figure-callout>}}}}{\frac{10\mathrm{\&Gamma;}}{3}-\frac{5}{3}{\mathrm{\&mu;}}^2+\frac{\mathrm{\&delta;}}{L}}$

In formula, Γ is the stiffness term coefficient,

k _casingfor casing rigidity; The elastic modulus that E is blade, the cross sectional moment of inertia that I is blade, the density that ρ is blade, the area of section that A is blade, the length that L is blade, R _dfor the radius of blade disk, the friction factor that μ is surface of contact, δ is the intrusion amount, the angular velocity of rotation that Ω is blade;

Step 4: according to the flexural deformation of blade, by blade tip radial force F _naccording to the normal direction of blade tip with tangentially decomposed:

-F _L=-F _ncos[v′(L,t)]

-F _T=-F _nsin[v′(L,t)]

In formula, F _lfor radial force F _ndecompose the power on the blade tip normal direction, F _tfor radial force F _ndecompose the power of blade tip on tangential, the bending displacement angle that v ' (L, t) is blade tip, the length that L is blade, t is the time;

Step 5: finally the stressed of blade tip place synthesized:

${\stackrel{\&OverBar;}{F}}_n=-F_L=-F_n\cos \left[v^{\&prime;}(L,t)\right]$

${\stackrel{\&OverBar;}{F}}_t=-\mathrm{\&mu;}{\stackrel{\&OverBar;}{F}}_n-F_T=-\mathrm{\&mu;}{\stackrel{\&OverBar;}{F}}_n-F_n\sin \left[v^{\&prime;}(L,t)\right]$

In formula,

normal force for the blade tip place after synthetic,

tangential force for the blade tip place after synthetic, F _lfor radial force F _ndecompose the power on the blade tip normal direction, F _tfor radial force F _ndecompose the power of blade tip on tangential, the bending displacement angle that v ' (L, t) is blade tip, the length that L is blade, t is the time, the friction factor that μ is surface of contact.

The mathematical model adopted for the intrusion amount δ described in determining step three is:

δ=u _L(t)-c _rub(t)

In formula, c _rub(t) be the t gap of blade-casing constantly, its expression formula is:

In formula, R _cfor the casing radius; r _gfor blade tip orbital radius, r _g=L+R _d, the length that L is blade, R _dradius for the blade disk; Δ is the mean gap of casing when concentric with the blade track, Δ=R _c-r _g; c _minfor the minor increment between blade tip and casing inwall, wherein, c _minthe initial minimum clearance of 0 expression, c _min<0 means initial maximum invasion depth; for phasing degree; n _pfor the pitch diameter number; T is the time; The angular velocity of rotation that Ω is blade.

Beneficial effect of the present invention:

Definite method of the present invention is for the characteristics of blade-casing collision friction, considered various factors, by setting up suitable model, the collision friction phenomenon of blade-casing can be described more exactly, to different operating mode lower blades-the casing fault is simulated, and its accuracy also can improve greatly; Simultaneously, can react more really the fault signature of blade-casing collision friction part; The adjustment that the method not only can be blade-casing gap provides theoretical foundation, also can be the early diagnosis that blade touches the fault of rubbing technical support is provided, and to improving overall performance, also has important directive significance.

The accompanying drawing explanation

Fig. 1 is blade tip running orbit schematic diagram;

Fig. 2 is the dynamic balance schematic diagram in blade-casing collision process;

The schematic diagram that rubs that touches that Fig. 3 is single blade-elasticity casing;

Fig. 4 is blade-stator System schematic diagram;

Fig. 5 is that the blade-casing adopted in the present invention touches the motion schematic diagram of determining rotating vane under rotating speed in the example that rubs;

Fig. 6 is the change curve (μ=0.3) of the normal force of blade tip place after synthetic with the intrusion amount;

Normal force after blade tip place when Fig. 6 (a) is rigidity for casing is synthetic is with the change curve (μ=0.3) of intrusion amount;

Fig. 6 (b) is for working as k _casingbe 5 * 10 ⁸n/m, c _casing=1 * 10 ³during Ns/m, the normal force after the blade tip place under different rotating speeds is synthetic is with the change curve (μ=0.3) of intrusion amount;

Fig. 7 is the change curve (k of the normal force of blade tip place after synthetic with rotating speed _casing=5 * 10 ⁸n/m, c _casing=1 * 10 ³ns/m, δ=20 μ m, μ=0.3);

Fig. 8 is the change curve of the normal force of blade tip place after synthetic with the stiffness term coefficient;

Fig. 9 is the change curve (k of the normal force of blade tip place after synthetic with friction factor _casing=5 * 10 ⁸n/m, c _casing=1 * 10 ³ns/m, δ=20 μ m, Ω=5000r/min);

Figure 10 is the change curve (δ=20 μ ms, μ=0.3) of casing radial deflection distance with casing rigidity;

In figure, 1-blade, 2-casing.

Embodiment

Below in conjunction with the drawings and specific embodiments, the present invention is described in further detail:

A kind of rotating vane-casing touches definite method of the power of rubbing, and comprises the steps:

Step 1: physical dimension and service condition to blade and casing are measured, comprising the length L of blade, the width b of blade, the thickness h of blade, the radius R of blade disk _d, the casing radius R _c, the minor increment c between blade tip and casing inwall _minand rotating speed Ω;

Step 2: according to the law of conservation of mechanical energy, set up the functional equation that blade and casing are carved at a time:

In certain time of supposing to contact with casing at blade, blade meets conservation of mechanical energy:

U _e+U _c=W (4)

Wherein:

U _efor the flexural deformation energy of blade, its expression formula is:

$U_e=\frac{1}{2}{\&Integral;}_0^L\mathrm{EI}{\left(\frac{{\&PartialD;}^{2}v}{{\&PartialD;x}^2}\right)}^2\mathrm{dx}---\left(5.1\right)$

In formula, the elastic modulus that E is blade, the cross sectional moment of inertia that I is blade, the length that L is blade, the bending displacement that v is blade;

U _cfor centrifugal potential energy, its expression formula is:

$U_c=\frac{1}{2}{\&Integral;}_0^{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA0000396415350000052.png"\; state="\{\{state\}\}">L</figure-callout>}}\frac{1}{2}{\mathrm{\&rho;A\&Omega;}}^2({\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="HDA0000396415350000011.png,HDA0000396415350000021.png"\; state="\{\{state\}\}">L</figure-callout>}}^2+{2R}_{d}L-2R_{d}x-x^2){\left(\frac{\&PartialD;v}{\&PartialD;x}\right)}^2\mathrm{dx}---\left(5.2\right)$

In formula, the length that L is blade, the bending displacement that v is blade, the density that ρ is blade, the area of section that A is blade, R _dfor the radius of blade disk, the angular velocity of rotation that Ω is blade;

W is blade tip radial force F _nwith transverse force F _twork, blade tip transverse force F _tthe friction force produced while contacting for blade-casing, friction force adopts Coulomb friction model, i.e. F here _t=μ F _n, the expression formula of merit is:

$W=\frac{1}{2}{F}_n{u}_L+\frac{1}{2}{\mathrm{\&mu;F}}_n{v}_L---\left(5.3\right)$

Wherein: v _lfor the bending displacement at blade tip place, the friction factor that μ is surface of contact, u _lfor the radial displacement at blade tip place, its expression formula is:

$u_L=\frac{1}{2}{\&Integral;}_0^L{\left(\frac{\&PartialD;v}{\&PartialD;x}\right)}^2\mathrm{dx}---\left(6\right)$

In formula, the length that L is blade, the bending displacement that v is blade;

Step 3: set up the dynamic balance relation of blade and casing, obtain the radial force F at blade tip place _nexpression formula:

Can be obtained the dynamic balance relation of radial impact by Fig. 2, and the amount δ of intrusion shown in Fig. 3 and casing radial deflection are apart from the radial displacement u at d and blade tip place _lrelation:

F _n=k _casingd (7)

δ=u _L+d (8)

In formula, k _casingfor casing rigidity;

In Fig. 3, o ₁for disc centre, c _casingfor the casing damping;

Obtaining blade semi-girder line of deflection formula by integral method is:

$v=\frac{F_t{x}^2}{6\mathrm{EI}}(3L-x)---\left(9\right)$

In formula, the elastic modulus that E is blade, the cross sectional moment of inertia that I is blade, the length that L is blade, F _tfor the blade tip transverse force, the bending displacement that v is blade;

According to blade semi-girder line of deflection formula, the bending displacement v by the bending displacement v of rotating vane with the blade tip place _lmean:

$v={\mathrm{<figure-callout\; id="1"\; label="v\; L"\; filenames="HDA0000396415350000052.png"\; state="\{\{state\}\}">v</figure-callout>}}_L\frac{1}{2}(\frac{{3x}^2}{L^2}-\frac{x^3}{L^3})---\left(10\right)$

In formula, the length that L is blade, v _lbending displacement for the blade tip place;

By bending displacement v expression formula (10) the substitution formula (5.1) of rotating vane and formula (5.2), and convolution (5.3), finally by conservation of mechanical energy equation (4), obtain the bending displacement v at blade tip place _lfor:

$v_L=\frac{{\mathrm{\&mu;F}}_n}{\frac{3\mathrm{EI}}{L^3}-\frac{{3F}_n}{5L}+3{\mathrm{\&rho;A\&Omega;}}^2(\frac{27}{280}L+\frac{1}{8}{R}_d)}---\left(11\right)$

In formula, the length that L is blade, the density that ρ is blade, the area of section that A is blade, R _dfor the radius of blade disk, the angular velocity of rotation that Ω is blade, the friction factor that μ is surface of contact, F _nfor blade tip radial force, the elastic modulus that E is blade, the cross sectional moment of inertia that I is blade;

Obtain v _lafter, can through type (6) and formula (10) try to achieve u _l, then by u _lexpression formula with d

be updated in formula (8):

$\mathrm{\&delta;}=\frac{5L{\mathrm{\&mu;}}^2{{\mathrm{<figure-callout\; id="2"\; label="F\; n"\; filenames="HDA0000396415350000011.png,HDA0000396415350000021.png"\; state="\{\{state\}\}">F</figure-callout>}}_n}^2}{3{[F_n-5L(\frac{\mathrm{EI}}{L^3}+{\mathrm{\&rho;A\&Omega;}}^2(\frac{27}{280}L+\frac{1}{8}{R}_d))]}^2}+\frac{F_n}{k_{\mathrm{ca}\sin g}}---\left(12\right)$

In formula, the length that L is blade, the density that ρ is blade, the area of section that A is blade, R _dfor the radius of blade disk, the angular velocity of rotation that Ω is blade, the friction factor that μ is surface of contact, F _nfor blade tip radial force, the elastic modulus that E is blade, the cross sectional moment of inertia that I is blade, k _casingfor casing rigidity, δ is the intrusion amount;

Omit high-order term, finally arrange and obtain F _nanalytical expression;

$F_n=\frac{5}{3}\mathrm{L\&Gamma;}{k}_{\mathrm{ca}\sin g}\frac{\frac{\mathrm{\&delta;}}{L}+\frac{5\mathrm{\&Gamma;}}{6}-\frac{\sqrt{15}}{6}\sqrt{\frac{15}{9}{\mathrm{\&Gamma;}}^2+4({\mathrm{\&mu;}}^2-\mathrm{\&Gamma;})\frac{\mathrm{\&delta;}}{\mathrm{<figure-callout\; id="10"\; label="L"\; filenames="HDA0000396415350000041.png,HDA0000396415350000042.png"\; state="\{\{state\}\}">L</figure-callout>}}}}{\frac{10\mathrm{\&Gamma;}}{3}-\frac{5}{3}{\mathrm{\&mu;}}^2+\frac{\mathrm{\&delta;}}{L}}---\left(13\right)$

In formula, Γ is the stiffness term coefficient, k _casingfor casing rigidity, the elastic modulus that E is blade, the cross sectional moment of inertia that I is blade, the density that ρ is blade, the area of section that A is blade, the length that L is blade, R _dfor the radius of blade disk, the friction factor that μ is surface of contact, δ is the intrusion amount, the angular velocity of rotation that Ω is blade, F _nfor the blade tip radial force;

Step 4: according to the flexural deformation of blade, by blade tip radial force F _naccording to the normal direction of blade tip with tangentially decomposed:

Because rotating vane may produce the situation that flexural deformation is larger in touching the process of rubbing, so the radial force of blade should be revised along the bending direction of blade.When radially invasion depth is larger, it is necessary that this revision just becomes.Blade tip radial force F _ndecompose the normal direction of blade tip and tangential with the form of the bending displacement angle v ' (L, t) of blade tip:

-F _L=-F _ncos[v′(L,t)]

-F _T=-F _nsin[v′(L,t)]

In formula, F _lfor blade tip radial force F _ndecompose the power on the blade tip normal direction, F _tfor blade tip radial force F _ndecompose the power of blade tip on tangential, the bending displacement angle that v ' (L, t) is blade tip, the length that L is blade, t is the time;

Step 5: finally to the normal force at blade tip place

and tangential force

synthesized:

${\stackrel{\&OverBar;}{F}}_n=-F_L=-F_n\cos \left[v^{\&prime;}(L,t)\right]$

${\stackrel{\&OverBar;}{F}}_t=-\mathrm{\&mu;}{\stackrel{\&OverBar;}{F}}_n-F_T=-\mathrm{\&mu;}{\stackrel{\&OverBar;}{F}}_n-F_n\sin \left[v^{\&prime;}(L,t)\right]$

In formula,

normal force for the blade tip place after synthetic,

tangential force for the blade tip place after synthetic, F _lfor blade tip radial force F _ndecompose the power on the blade tip normal direction, F _tfor blade tip radial force F _ndecompose the power of blade tip on tangential, the bending displacement angle that v ' (L, t) is blade tip, the length that L is blade, t is the time, the friction factor that μ is surface of contact.

The definite method of intrusion amount δ described in step 3 is as follows:

Blade-stator System as shown in Figure 4, in figure, o, o ₁be respectively casing center and disc centre; R _cfor the casing radius; r _gfor blade tip orbital radius, r _g=L+R _d, the length that L is blade, R _dradius for the blade disk; Δ is the mean gap of casing when concentric with the blade track, Δ=R _c-r _g; c _minfor the minor increment between blade tip and casing inwall, if c _min0, c _minmean initial minimum clearance, if c _min<0, c _minmean initial maximum invasion depth; The angular velocity of rotation that Ω is blade; c _rub(t) be the t gap of blade-casing constantly;

According to the geometric relationship in figure:

AB ²+oB ²=oA ²

Above formula is launched:

In formula, R _cfor the casing radius; for phasing degree; c _rub(t) be the t gap of blade-casing constantly; r _gfor blade tip orbital radius, r _g=L+R _d, the length that L is blade, R _dradius for the blade disk; Δ is the mean gap of casing when concentric with the blade track, Δ=R _c-r _g; c _minfor the minor increment between blade tip and casing inwall, if c _min0, c _minmean initial minimum clearance, if c _min<0, c _minmean initial maximum invasion depth; The angular velocity of rotation that Ω is blade; T is the time;

Solve the expression formula in the gap that can obtain blade-casing by the expansion to above formula, if consider the impact of casing pitch diameter, obtain following expression:

In formula, R _cfor the casing radius;

for phasing degree; c _rub(t) be the t gap of blade-casing constantly; r _gfor blade tip orbital radius, r _g=L+R _d, the length that L is blade, R _dradius for the blade disk; Δ is the mean gap of casing when concentric with the blade track, Δ=R _c-r _g; c _minfor the minor increment between blade tip and casing inwall, if c _min0, c _minmean initial minimum clearance, if c _min<0, c _minmean initial maximum invasion depth; n _pfor the pitch diameter number; The angular velocity of rotation that Ω is blade; T is the time;

Intrusion amount between actual rotating vane and casing inwall is:

δ=u _L(t)-c _rub(t)

In formula, u _l(t) be the t radial displacement of blade tip constantly, c _rub(t) be the t gap of blade-casing constantly, its expression formula is suc as formula shown in (14).

As δ > 0 the time, touch the generation that rubs; When δ≤0, do not touch and rub.As can be seen from the above equation, δ is a variations per hour, and gap width all will be by upper one u constantly obtained constantly for each _land c (t) _rub(t) recalculate.

The present invention will be further described to touch below in conjunction with a blade-casing example that rubs.

It is as shown in table 1 that the geometric parameter of blade and blade tip touch the simulation parameter that rubs, leaf model as shown in Figure 5, in Fig. 5, the radial displacement that u is blade, the bending displacement that v is blade, the swing displacement that w is blade, x is integration amount.

Geometric parameter and the blade tip of table 1 blade touch the simulation parameter that rubs

As seen from Figure 6, normal force and intrusion amount after the blade tip place is synthetic are nonlinear relationship, and along with the increase of intrusion amount, the amplitude of variation of the normal force after the blade tip place is synthetic reduces, and is weak characteristic.In Fig. 6 (a), under the prerequisite that is all the rigidity casing, the curve that the present invention obtains and formula (1) curve is substantially identical.Fig. 6 (b) is, at k _casing=5 * 10 ⁸during N/m, the normal force after the blade tip place under different rotating speeds is synthetic is with the change curve of intrusion amount.From Fig. 6 (b), can see, the normal force after the rotating speed increase can make the blade tip place synthetic increases.

As seen from Figure 7, normal force after the blade tip place is synthetic is nonlinearities change with rotating speed: the normal force after incipient stage blade tip place is synthetic is larger with the amplitude of rotation speed change, but the normal force after the blade tip place is synthetic is not along with rotating speed increases always, but has slowed down gradually ascendant trend and finally be tending towards a normal value.

As seen from Figure 8, the normal force after the blade tip place is synthetic is nonlinearities change with the stiffness term coefficient, and through fast rise after the stage, the normal force after the blade tip place is synthetic, along with the speed of stiffness term index variation slows down gradually, finally is tending towards k _casingδ.The increase of stiffness term coefficient is larger, illustrates that the resistant to bending ability of blade is stronger, and finally, when blade no longer bends, the normal force after the blade tip place is synthetic also no longer increases along with the increase of stiffness term coefficient.This also can explain the reason of the normal force of blade tip place after synthetic with rotation speed change: the rising of rotating speed has increased the bending stiffness of blade, has also just increased relative rigidity.Because blade and contacting of casing are non-smooth friction processes, so the formula that the present invention derives has considered to touch the impact of the normal force after synthetic on the blade tip place of rubbing characteristics in the process of rubbing.

The present invention has considered the impact of the normal force of friction force after synthetic on the blade tip place, and as seen from Figure 9, along with the increase of friction factor, the normal force after the blade tip place is synthetic has and is non-linear trend of successively decreasing.At friction factor hour, the speed that the normal force after the blade tip place is synthetic reduces is very fast, and the speed reduced afterwards slows down gradually.Fig. 9 has also illustrated: the radial force after the existence of friction force is synthetic to the blade tip place has also played certain inhibiting effect.

As seen from Figure 10, when casing rigidity hour, the intrusion amount is exactly mainly casing offset distance radially; After casing rigidity increases, casing offset distance radially reduces to some extent, shows thus: due to the increase of casing rigidity, make blade also start to occur to give birth to shape; And, when casing rigidity is larger, casing offset distance radially is almost 0, show that casing now is equivalent to be completely fixed, all be converted into the elastic potential energy of blade by blade institute work.And, under same casing rigidity, high-revolving translational movement is greater than slow-revving.

The Changing Pattern of the tangential force after the blade tip place is synthetic is identical with the Changing Pattern of normal force after above-mentioned blade tip place synthesizes, at this Ao Shu no longer.

Claims (2)

1. a rotating vane-casing touches definite method of the power of rubbing, and it is characterized in that, comprises the steps:

Step 1: physical dimension and service condition to blade and casing are measured, comprising the length L of blade, the width b of blade, the thickness h of blade, the radius R of blade disk _d, the casing radius R _c, the minor increment c between blade tip and casing inwall _minand rotating speed Ω;

Step 2: according to the law of conservation of mechanical energy, set up the functional equation that blade and casing are carved at a time:

U _e+U _c=W

In formula, U _efor the flexural deformation energy of blade, U _cfor centrifugal potential energy, W is blade tip radial force and transverse force work;

Step 3: set up the dynamic balance relation of blade and casing, obtain the radial force F at blade tip place _nexpression formula:

$F_n=\frac{5}{3}{\mathrm{L\&Gamma;k}}_{\mathrm{ca}\sin g}\frac{\frac{\mathrm{\&delta;}}{L}+\frac{5\mathrm{\&Gamma;}}{6}-\frac{\sqrt{15}}{6}\sqrt{\frac{15}{9}{\mathrm{\&Gamma;}}^2+4({\mathrm{\&mu;}}^2-\mathrm{\&Gamma;})\frac{\mathrm{\&delta;}}{L}}}{\frac{10\mathrm{\&Gamma;}}{3}-\frac{5}{3}{\mathrm{\&mu;}}^2+\frac{\mathrm{\&delta;}}{L}}$

In formula, Γ is the stiffness term coefficient,

k _casingfor casing rigidity; The elastic modulus that E is blade, the cross sectional moment of inertia that I is blade, the density that ρ is blade, the area of section that A is blade, the length that L is blade, R _dfor the radius of blade disk, the friction factor that μ is surface of contact, δ is the intrusion amount, the angular velocity of rotation that Ω is blade;

Step 4: according to the flexural deformation of blade, by blade tip radial force F _naccording to the normal direction of blade tip with tangentially decomposed:

-F _L=-F _ncos[v′(L,t)]

-F _T=-F _nsin[v′(L,t)]

In formula, F _lfor radial force F _ndecompose the power on the blade tip normal direction, F _tfor radial force F _ndecompose the power of blade tip on tangential, the bending displacement angle that v ' (L, t) is blade tip, the length that L is blade, t is the time;

Step 5: finally the stressed of blade tip place synthesized:

${\stackrel{\&OverBar;}{F}}_n=-F_L=-F_n\cos \left[v^{\&prime;}(L,t)\right]$

${\stackrel{\&OverBar;}{F}}_t=-\mathrm{\&mu;}{\stackrel{\&OverBar;}{F}}_n-F_T=-\mathrm{\&mu;}{\stackrel{\&OverBar;}{F}}_n-F_n\sin \left[v^{\&prime;}(L,t)\right]$

In formula,

normal force for the blade tip place after synthetic,

tangential force for the blade tip place after synthetic, F _lfor radial force F _ndecompose the power on the blade tip normal direction, F _tfor radial force F _ndecompose the power of blade tip on tangential, the bending displacement angle that v ' (L, t) is blade tip, the length that L is blade, t is the time, the friction factor that μ is surface of contact.

2. rotating vane-casing according to claim 1 touches definite method of the power of rubbing, and it is characterized in that the mathematical model adopted for the intrusion amount δ described in determining step three is:

δ=u _L(t)-c _rub(t)

In formula, u _l(t) be the t radial displacement of blade tip constantly, c _rub(t) be the t gap of blade-casing constantly, its expression formula is:

In formula, R _cfor the casing radius; r _gfor blade tip orbital radius, r _g=L+R _d, the length that L is blade, R _dradius for the blade disk; Δ is the mean gap of casing when concentric with the blade track, Δ=R _c-r _g; c _minfor the minor increment between blade tip and casing inwall, wherein, c _minthe initial minimum clearance of 0 expression, c _min<0 means initial maximum invasion depth;

for phasing degree; n _pfor the pitch diameter number; T is the time; The angular velocity of rotation that Ω is blade.

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