CN109800512A - Rotating cylindrical shell-variable cross-section disk-pretwist blade system dynamic modeling method - Google Patents
Rotating cylindrical shell-variable cross-section disk-pretwist blade system dynamic modeling method Download PDFInfo
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Abstract
The present invention relates to a kind of rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system dynamics modelings, and this method comprises the following steps: step 1: three-dimensional system of coordinate needed for building rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system dynamics modeling；Step 2: rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system structural parameters and material parameter are measured；The kinetic energy that pre- twisted blade and pre- twisted blade are coupled with variable cross-section disk and rotating cylindrical shell is obtained according to Kinetic Energy Calculation formula；Step 4: being based on plate shell Theory of Vibration, consider centrifugation rigidifying effects of the pre- twisted blade in rotary course, obtain the potential energy for rotating pre- twisted blade；Step 5: variable cross-section disk can satisfy elastic sheet oscillation crosswise small deflection theory, and the differential equation of its transverse elasticity vibration is derived using Hamiton's principle, obtains the kinetic energy and potential energy of variable cross-section disk.Method provided by the invention has higher computational efficiency and computational accuracy.
Description
Technical field
The invention belongs to mechanical kinetics technical field more particularly to a kind of rotating cylindrical shell-variable cross-section disk-pre- twisted blades
The dynamic modeling method of system.
Background technique
Currently, existing rotating cylindrical shell-variable cross-section disk-pre- twisted blade modeling method mainly includes following two side
Method:
1, based on commercial finite element analysis software
CAD 3D model is imported into commercial finite element analysis software or directly establishes entire component in finite element software
Threedimensional model, select suitable unit and suitable material parameter, to threedimensional model carry out grid dividing, establish finite element mould
Type, setting suitably constrain and select suitable method for solving to rotating cylindrical shell-variable cross-section disk-pretwist blade system power
Characteristic is learned to be analyzed.But using existing commercial finite element analysis software to rotating cylindrical shell-variable cross-section disk-pre- twisted blade system
When system carries out dynamical property analysis, modeling process is complicated and heavy and different modeling pattern and cell type obtain
Kinetic characteristics also have larger gap, simultaneously because finite element software cannot consider rotating cylindrical shell during solution
Anamorphic effect, so large error can be generated when high-revolving.
2, based on the modeling method of Plate Theory
Currently based on I grasp knowledge, on the market it is most of all be rotating cylindrical shell and cross-section disk coupled system,
And the diameter of drum barrel is identical as the diameter of rotating cylindrical shell, plate be based on plate theory, rotating cylindrical shell be based on LOVE shell theory,
Donnell shell theory etc., whole system are based on energy method and carry out Dynamic Modeling, can not consider that disk is variable cross-section disk and rotational circle
The case where when radius of column shell is less than disk radius, the practical application structure of rotating cylindrical shell Yu disk coupled system is not met.
Currently based on plate shell Theory of Vibration, for the skill of rotating cylindrical shell-variable cross-section disk-pre- twisted blade Dynamic Modeling
Art is in space state.
Summary of the invention
(1) technical problems to be solved
For existing technical problem, the present invention provides a kind of rotating cylindrical shell-variable cross-section disk-pretwist blade system
Dynamic modeling method, can reach under the premise of guaranteeing entire coupled system mode of oscillation, consider whole system revolving
Centrifugation rigidifying, Spin softening, circumferential initial stress and coriolis force during turning influence, then using Galerkin truncation and
Hamilton variational approach obtains the differential equation of motion of the kinetic model of whole system.
(2) technical solution
In order to achieve the above object, the main technical schemes that the present invention uses include:
A kind of rotating cylindrical shell-variable cross-section disk-pretwist blade system dynamic modeling method, comprising the following steps:
Step 1: three-dimensional needed for building rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system dynamics modeling to sit
Mark system；
The three-dimensional system of coordinate includes: the coordinate system of the fixed coordinate system OXYZ of entire coupled system, variable cross-section disk
oxdydzd, the moving coordinate system ox of whole system during the motionryrzr, pre- twisted blade local coordinate system oxbybzb, wheel disc with
The Geometric center coordinates system o of drum barrel intersection circle1x1y1z1, drum barrel satellite coordinate system o2x2y2z2It is sat with the curve of drum barrel mean camber
Mark system ox θ z；
Step 2: rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system structural parameters and material parameter are carried out
Measurement；
The parameter of measurement includes: pretwist length of blade L, pretwist width of blade b, pretwist vane thickness h, the installation of pre- twisted blade
Angle beta, pretwist leaf elastic modulus E, Poisson's ratio μ, pretwist leaves density ρ, variable cross-section disk internal diameter rs, first segment radius rd, second segment
Radius rD, outer radii Rd, disk first segment thickness hD1, second segment thickness hD2, third section thickness hD3, disk elastic modulus Ed、
The density p of diskd, rotating cylindrical shell length Ls, rotating cylindrical shell mean camber radius R, rotating cylindrical shell elastic modulus Es, rotation
Cylindrical shell density psWith rotating cylindrical shell Poisson's ratio υs；
The translation spring in three directions of x, y, z and the rigidity of torsional spring are respectively kc,u、kc,v、kc,wWith
Step 3: being uniform-distribution with N on setting variable cross-section diskbA identical pre- twisted blade；
After i-th of pre- twisted blade is deformed, through any point Q on pre- twisted blade i in global coordinate system OXYZ
Motion vector converted by a series of coordinates after, obtain pre- twisted blade and pre- twisted blade and become to cut according to Kinetic Energy Calculation formula
The kinetic energy of face and rotating cylindrical shell；
Step 4: being based on plate shell Theory of Vibration, consider centrifugation rigidifying effects of the pre- twisted blade in rotary course, obtain
The potential energy of pre- twisted blade is rotated out；
Step 5: variable cross-section disk can satisfy elastic sheet oscillation crosswise small deflection theory, be derived using Hamiton's principle
The differential equation of its transverse elasticity vibration, obtains the kinetic energy and potential energy of variable cross-section disk；
Step 6: it is based on Sanders shell theory, considers the coriolis force of rotating cylindrical shell, centrifugal force and circumferential initial stress
It influences, obtains the kinetic energy and potential energy of rotating cylindrical shell；
Step 7: simulation being carried out using spring and torsional spring between variable cross-section disk and drum barrel and is bolted, connection function has been obtained
Spring and torsional spring entirely rotation drum barrel-variable cross-section disk-pre- twisted blade coupled system in kinetic energy；
Step 8: according to Hamilton variation principle
Wherein, U=Ub+Us,ε+Ud+Uθ, T=Tb+Ts1+Td；
WnonFor the function that external force is done, and with δ us、δvs、δws、δu、δv、δw、δ φ and δ WdBecome as independent variable
Get rotating cylindrical shell-variable cross-section disk-pretwist blade system kinetics equation；
Step 9: using Galerkin method, be introduced into canoncial coordinates to any one on face in the rotating cylindrical shell in step 8
The displacement components u of point in a longitudinal directions, tangential direction displacement vs, radial direction displacement ws, cantilever beam radial displacement u, cross
W, bending direction corner are displaced to displacement v, swaying directionThe lateral displacement W of swaying direction corner φ and variable cross-section diskdInto
Row sliding-model control；
Obtain rotation drum barrel-variable cross-section disk-pre- twisted blade coupled system mass matrix, coriolis force matrix and rigidity square
Battle array；
Step 10: variable cross-section disk and pre- twisted blade introduce dimensionless item, to guarantee the variable cross-section disk, the pretwist leaf
Ensure when piece is coupled with rotating cylindrical shell consistent；
Nondimensionalization coefficient η=x/Li, when corresponding to pre- twisted blade, LiFor the length L of pre- twisted blade；
When corresponding rotating cylindrical shell, LiFor the length L of rotating cylindrical shells；
The nondimensionalization coefficient of variable cross-section disk is ξ=(r-rs)/(rD-rs)；
WithRespectively pre- twisted blade carries out the model function of vibration after nondimensionalization；
WithModel function of vibration respectively after variable cross-section disk nondimensionalization；
Step 11: introducing Rayleigh damping, it is micro- to obtain the movement of rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system
Divide equation:
In formula, M, G and D are respectively the mass matrix of entire coupled system, coriolis force matrix and damping matrix；
Ke、Ki、KΩ、KσAnd KcStiffness matrix caused by the structural stiffness matrix of respectively entire coupled system, acceleration,
It is centrifuged the stiffness matrix that rigidifying matrix, Spin softening matrix, rotating cylindrical shell and disk coupling spring generate；
Q and F is respectively the outer exciting force vector of pre- twisted blade canoncial coordinates vector sum；
Rayleigh damping D is to combine acquisition in proportion by mass matrix and stiffness matrix；
Formula specific as follows:
D=α M+ β K；
Wherein, α and β are acquired by following formula:
In formula, fn1、fn2Respectively the first rank of spring-variable cross-section disk-pre- twisted blade coupled system and the intrinsic frequency of second-order
Rate (Hz), ξ1And ξ2For damping ratio；
Q is the column vector of generalized coordinates composition related with the time, and expression-form is as follows:
Preferably, the kinetic energy expression in step 3 are as follows:
In formula, rQTake up an official post motion vector of the one point Q of meaning under global coordinate system for pre- twisted blade；
A is the area of section of pre- twisted blade；
IzIt is pretwist blade profile around the cross sectional moment of inertia of y-axis；
IyIt is pretwist blade profile around the cross sectional moment of inertia of z-axis；
θ is the angular displacement of wheel disc movement；
Symbol () indicates the 1 rank local derviation to the time, rDIt is the outer diameter of disk, us、vs、wsIt is rotation drum barrel mean camber whole
The displacement in three directions in body coordinate system, x, y and z be respectively pre- twisted blade local coordinate system along pretwist length of blade direction,
Thickness direction and swaying direction,It is respectively the corner of pre- twisted blade bending direction and swaying direction with φ, u, v and w are respectively
The displacement of pre- twisted blade radial, transverse direction and swaying direction in local coordinate system, WdFor the lateral displacement of disk.
Preferably, the potential energy of strain expression formula of the pre- twisted blade of rotation described in step 4 are as follows:
In formula, E, G, κ, A and L respectively indicate the Young's modulus of pre- twisted blade, modulus of shearing, shearing factor, cross-sectional area,
Pretwist length of blade and the centrifugal force suffered by pre- twisted blade in rotary course.
Preferably, the kinetic energy expression of variable cross-section disk described in step 5 are as follows:
The potential energy expression formula of variable cross-section disk are as follows:
In formula, Wd1、Wd2And Wd3The respectively transverse mode function of variable cross-section disk first segment, second segment and third section；
The expression formula of Laplace operator and bending stiffness is as follows:
Preferably, kinetic energy expression caused by rotating cylindrical shell described in step 6 are as follows:
Rotating cylindrical shell considers to stretch and curved strain energy expression is as follows:
The initial circumference stress expression formula generated under the action of the centrifugal force:
In formula, Nθ=ρshsΩ2R2For circumferential initial stress；
ρsFor the density of drum barrel；
hsFor the thickness of drum barrel；
R is the middle radius surface of drum barrel；
υsFor the Poisson's ratio of drum barrel；
LsFor the length of drum barrel；
EsFor the rigidity of drum barrel.
Preferably, variable cross-section disk described in step 7 couples potential energy expression formula caused by place's spring with rotating cylindrical shell are as follows:
Preferably, rotating cylindrical shell described in step 8-variable cross-section disk-pretwist blade system kinetics equation expression
Formula is as follows:
Preferably, the specific method is as follows described in step 10:
Discrete, introducing canoncial coordinates U is carried out to pre- twisted bladei(t)、Vi(t)、Wi(t)、ψi(t) and Φi(t), pretwist is obtained
The radial vibration of blade, oscillation crosswise, swing, bending pivot angle and swing corner displacement it is as follows:
In formula,WithIt is expressed as corresponding pretwist blade radial, cross
To, swing and the model function of vibration of the i-th rank of the corner of pre- twisted blade bending direction and swaying direction, N are mode truncation number.
Preferably, after step 11, the method also includes:
It is zero that external excitation vector, which is arranged, determines rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system in different rotating speeds
Under intrinsic frequency；
Accuracy according to the intrinsic frequency of acquisition to verify differential equation of motion.
Preferably,WithExpression formula such as following formula:
In formula:
βaiL is pretwist blade radial direction character root, βfiL is that pre- twisted blade is lateral, βsiL is the feature of swaying direction
Root；
βaiThe preceding 6 rank value of L is respectively as follows: 1.571,4.712,7.854,11,14.14,17.28；βfiThe preceding 6 rank value of L
It is respectively as follows: 1.875,4.686,7.819,10.89,13.9,16.81；βsiThe preceding 6 rank value of L is respectively as follows: 1.874,4.669,
7.735,10.63,13.27,15.8；
The coupled system of spring-supported flexible disk and pre- twisted blade, the lateral position of resilient disc are analyzed using hypothesis modal method
Moving indicates are as follows:
Wi c(r, θ)=Ri(r)cos(iθ) (23)
Wi s(r, θ)=Ri(r)cos(iθ) (24)
In formula, Wi cAnd Wi sThe column vector of mode composition is assumed in two orthogonal planes for disk,WithRespectively resilient disc
About the generalized coordinates of time, RiIt (r) is the model function of vibration for the disk derived according to non-uniform beam.
(3) beneficial effect
The beneficial effects of the present invention are: a kind of rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled systemes provided by the invention
The dynamic modeling method of system saves cost required for rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system is tested
Expense；Different rotations can be obtained after need to only modifying the structure size and material parameter of cylindrical shell, variable cross-section disk and blade in the present invention
Turn cylindrical shell-variable cross-section disk-pre- twisted blade coupled system kinetic model, it is easy to operate；The present invention considers real blade
In established angle and torsional angle influence, the variable cross-section geometric configuration of flexible disk is closer to the wheel disc in true aero-engine；This
Invention consider rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system be centrifuged in rotary course rigidifying, Spin softening,
The influence of circumferential initial stress and coriolis force, kinetic characteristics can more react rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupling
The real work state of collaboration system；With rotating cylindrical shell-variable cross-section disk-pretwist is analyzed by traditional commercial finite element software
The kinetic characteristics of blade are compared, and the present invention has higher computational efficiency and computational accuracy.In addition, the method is compared to tradition
Commercial finite element software freedom degree it is less.Compared to present existing rotating cylindrical shell-disk coupled system, what the present invention established
Model has following advantage:
(1) as long as, rotating cylindrical shell-variable cross-section disk-pretwist blade system for establishing of the present invention by variable cross-section disk degeneration at
Cross-section disk can simulate rotating cylindrical shell-cross-section disk-blade system, i.e. rotating cylindrical shell-variable cross-section disk-blade coupling
System includes existing rotating cylindrical shell-cross-section disk-blade system；
(2), rotating cylindrical shell-variable cross-section disk-pretwist blade system can be by changing between variable cross-section disk and cylindrical shell
Connecting spring rigidity, to simulate the rotating cylindrical shell under various boundary-variable cross-section disk-pre- twisted blade coupled system；
(3), in cylindrical shell-disk-blade coupled system in aero-engine, disk is mostly that the change of both sides thickness intermediate thin is cut
Face disk, the diameter of drum barrel are also usually less than the diameter of disk, meet derivation rotating cylindrical shell-variable cross-section disk-pretwist leaf herein
The modeling of piece coupled system, therefore the model of this paper is more in line with reality, rotating cylindrical shell-variable cross-section disk-pretwist before solving
Blade system differs larger problem with model in production application, is that rotating cylindrical shell-variable cross-section is designed in practical application
Disk-pretwist blade system provides theoretical support, and can be compared by the dynamic frequency of entire coupled system, avoids system
Interval of resonance reduces vibration, reasonably selects working speed, keeps system work more stable.
Simultaneously, moreover it is possible to the exciter response analysis for carrying out whole system, thus lifting system performance and stability.
Detailed description of the invention
Fig. 1 is a kind of side of rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system dynamics modeling method of the present invention
Method flow chart；
Fig. 2 is that a kind of rotating cylindrical shell-variable cross-section disk of the present invention-pre- twisted blade coupled system dynamics modeling method is implemented
Cylindrical shell-variable cross-section disk-pre- twisted blade structural schematic diagram in example；
Fig. 3 is that a kind of rotating cylindrical shell-variable cross-section disk of the present invention-pre- twisted blade coupled system dynamics modeling method is implemented
Example in rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system traveling wave frequency with revolving speed change curve；
Fig. 4 is that a kind of rotating cylindrical shell-variable cross-section disk of the present invention-pre- twisted blade coupled system dynamics modeling method is implemented
Rotating cylindrical shell-variable cross-section disk coupled system traveling wave frequency obtains result with the change curve of revolving speed and using FInite Element in example
Comparison diagram.
Specific embodiment
In order to preferably explain the present invention, in order to understand, with reference to the accompanying drawing, by specific embodiment, to this hair
It is bright to be described in detail.
As illustrated in fig. 1 and 2, the embodiment of the present invention provides a kind of rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system
Dynamic modeling method, comprising the following steps:
Step 1: three-dimensional needed for building rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system dynamics modeling to sit
Mark system；
The three-dimensional system of coordinate includes: the coordinate system of the fixed coordinate system OXYZ of entire coupled system, variable cross-section disk
oxdydzd, the moving coordinate system ox of whole system during the motionryrzr, pre- twisted blade local coordinate system oxbybzb, wheel disc with
The Geometric center coordinates system o of drum barrel intersection circle1x1y1z1, drum barrel satellite coordinate system o2x2y2z2It is sat with the curve of drum barrel mean camber
Mark system ox θ z；
Step 2: rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system structural parameters and material parameter are obtained, this
Assume that coupled system is isotropic linear elastic materials in embodiment, constitutive relation meets Hooke law, for variable cross-section disk
Thinking the preceding straight line perpendicular to middle face of deformation after deformation is still straight line, and vertical with the holding of middle face, without in the direction of face
Deformation, cylindrical shell is the cylindrical shell of thin-walled, rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system related coefficient
It is as shown in table 1:
1 rotating cylindrical shell of table-variable cross-section disk-pre- twisted blade coupled system parameter
Consider initial position, the translation of variable cross-section disk and the cross of disk of established angle of the blade on variable cross-section disk, spring
To vibration, determine that blade is deformed motion vector of the rear any point Q under global coordinate system:
In formula, β (x) is to turn round the angle that any point is rotated around x-axis on shape blade, β (x)=β0+ γ (x), β0For blade
Established angle, θ are the angular displacement of disk and drum barrel in rotary course, and φ is the angle that circumferentially rotates of drum barrel, and γ (x) is the torsion of blade
Angle.
Step 3: being uniform-distribution with N on setting variable cross-section diskbA identical pre- twisted blade；
After i-th of pre- twisted blade is deformed, through any point Q on pre- twisted blade i in global coordinate system OXYZ
Motion vector converted by a series of coordinates after, according to Kinetic Energy Calculation formula obtain pre- twisted blade and pre- twisted blade and disk and
The kinetic energy expression of cylindrical shell coupling are as follows:
Step 4: being based on plate shell Theory of Vibration, consider centrifugation rigidifying effects of the pre- twisted blade in rotary course, obtain
The potential energy of pre- twisted blade is rotated out；
In formula, E, G, κ, A and L respectively indicate Young's modulus, modulus of shearing, shearing factor, cross-sectional area and the leaf of blade
Leaf length.
Step 5: variable cross-section disk can satisfy elastic sheet oscillation crosswise small deflection theory, be derived using Hamiton's principle
The differential equation of its transverse elasticity vibration, obtains the kinetic energy and potential energy of variable cross-section disk；
For the spring between variable cross-section disk and cylindrical shell, kinetic energy Uc:
Step 6: it is based on Sanders shell theory, considers the coriolis force of rotating cylindrical shell, centrifugal force and circumferential initial stress
It influences, obtains the kinetic energy and potential energy of rotating cylindrical shell；
The kinetic energy of rotating variable cross-section disk:
The potential energy of rotating variable cross-section disk:
In formula, Wd1、Wd2And Wd3Respectively variable cross-section disk first and second and three sections of transverse mode function.
Step 7: simulation being carried out using spring and torsional spring between variable cross-section disk and drum barrel and is bolted, connection function has been obtained
Spring and torsional spring entirely rotation drum barrel-variable cross-section disk-pre- twisted blade coupled system in kinetic energy；
Step 8: according to Hamilton variation principleWherein U=Ub+Us,ε+Ud+Uθ, T=
Tb+Ts1+Td, WnonFor the function that external force is done, and with δ us、δvs、δws、δu、δv、δw、δ φ and δ WdIt is carried out as independent variable
Variation obtains rotating cylindrical shell-variable cross-section disk-pretwist blade system kinetics equation.
Step 9: using Galerkin method, be introduced into canoncial coordinates to any one on face in the rotating cylindrical shell in step 8
The displacement components u of point in a longitudinal directions, tangential direction displacement vsWith the displacement w of radial directions, cantilever beam radial displacement u, it is horizontal
W, bending direction corner are displaced to displacement v, swaying directionThe lateral displacement W of swaying direction corner φ and variable cross-section diskdInto
Row sliding-model control, obtain rotation drum barrel-variable cross-section disk-pre- twisted blade coupled system mass matrix, coriolis force matrix and just
Spend matrix.It is specific as follows:
Discrete, introducing canoncial coordinates U is carried out to bladei(t)、Vi(t)、Wi(t)、ψi(t) and Φi(t), the diameter of blade is obtained
Displacement to vibration, oscillation crosswise, swing, bending pivot angle and swing corner is as follows:
In formula,WithIt is expressed as corresponding blade radial, transverse direction, pendulum
The model function of vibration of i-th rank of the corner of dynamic and blade bending direction and swaying direction, N are mode truncation number.
Expression are as follows:
In formula:
βaiL is blade radial direction character root, βfiL is that blade is lateral, βsiL is the feature root of swaying direction.βaiL
Preceding 6 rank value be respectively as follows: 1.571,4.712,7.854,11,14.14,17.28；βfiThe preceding 6 rank value of L is respectively as follows:
1.875,4.686,7.819,10.89,13.9,16.81；βsiThe preceding 6 rank value of L is respectively as follows: 1.874,4.669,7.735,
10.63,13.27,15.8.
The coupled system of spring-supported flexible disk and blade is analyzed using hypothesis modal method, the lateral displacement of resilient disc can
To indicate are as follows:
Wi c(r, θ)=Ri(r)cos(iθ) (34)
Wi s(r, θ)=Ri(r)cos(iθ) (35)
In formula, Wi cAnd Wi sThe column vector of mode composition is assumed in two orthogonal planes for disk,WithIt is respectively elastic
Generalized coordinates of the disk about the time, Ri(r) be the disk derived according to non-uniform beam model function of vibration.
Step 10: dimensionless item being introduced for variable cross-section disk and pre- twisted blade, guarantees consistency when coupling with cylindrical shell
And correctness.
Nondimensionalization coefficient η=x/Li, for blade, LiFor the length L of blade；For cylindrical shell, LiFor Ls
For the length of cylindrical shell；The nondimensionalization coefficient of variable cross-section disk is ξ=(r-rs)/(rD-rs).Therefore,WithRespectively
Model function of vibration after carrying out nondimensionalization for blade；WithModel function of vibration respectively after variable cross-section disk nondimensionalization.
Step 11: introducing Rayleigh damping, obtain torsional spring-variable cross-section disk-pre- twisted blade coupled system motion
Equation:
In formula, M, G and D are respectively the mass matrix of entire coupled system, coriolis force matrix and damping matrix, Ke、Ki、KΩ、
KσAnd KcStiffness matrix caused by the structural stiffness matrix of respectively entire coupled system, acceleration, centrifugation rigidifying matrix, rotation
Soften the stiffness matrix that matrix and cylindrical shell and disk coupling spring generate.Q and F is respectively the outer exciting of blade canoncial coordinates vector sum
Force vector.M,G,Ke、Ki、KΩ、KσAnd KcExpression be respectively as follows:
The expression formula of rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system mass matrix:
Each element expression in mass matrix are as follows:
The expression formula of rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system coriolis force matrix:
Each element expression in coriolis force matrix are as follows:
The expression formula of rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system structural stiffness matrix:
Each element expression in structural stiffness matrix are as follows:
The expression formula of stiffness matrix caused by rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system acceleration:
Each element expression in stiffness matrix caused by acceleration are as follows:
The expression formula of rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system centrifugation rigidifying matrix:
The each element expression being centrifuged in rigidifying matrix are as follows:
The expression formula of rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system Spin softening stiffness matrix:
Each element expression in Spin softening stiffness matrix matrix are as follows:
The expression formula of the stiffness matrix of rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system coupling spring:
Each element expression in Coupling stiffness matrix matrix are as follows:
Rayleigh damping D is the D=α M+ β K as made of mass matrix and stiffness matrix composite construction in proportion, wherein α and β
It can be acquired by following formula:
In formula, fn1、fn2Respectively the first rank of blade and second-order intrinsic frequency (Hz), ξ1And ξ2For damping ratio.
The eigenvalue λ for calculating the characteristic equation determinant of coefficient takes the absolute value of its imaginary part divided by 2 π, and carries out from small to large
Sequence obtains one group of natural frequency ωk, wherein k indicates rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system mode
Kth rank, k=1,2 ....
Effectiveness of the invention is verified by finite element model, table 2 and table 3 list the rotating cylindrical under stationary state respectively
Intrinsic frequency after shell-variable cross-section disk-pre- twisted blade coupled system intrinsic frequency and change structural parameters is obtained with finite element software
The comparison and error size of the result arrived.
Fig. 3 is illustrated using rotating cylindrical shell drawn by context of methods-variable cross-section disk-pre- twisted blade coupled system traveling wave frequency
With the change curve of revolving speed.
Fig. 4 illustrates the rotating cylindrical shell drawn by context of methods-variable cross-section disk coupled system traveling wave frequency with the change of revolving speed
Change curve and the comparison diagram of result is obtained using FInite Element.
Table 2 rotates drum barrel-variable cross-section disk-blade coupled system intrinsic frequency comparison
The different draw ratios of table 3 are to rotation drum barrel-variable cross-section disk-blade coupled system intrinsic frequency influence
Result through this embodiment can be obtained to draw a conclusion:
(1) variable cross-section disk intrinsic frequency increases with the increase of revolving speed in rotating cylindrical shell-variable cross-section disk coupled system, says
Bright is also centrifugation rigidifying effect for rotating cylindrical shell-variable cross-section disk coupling obviously in Spin softening effect；
(2) as the increase of drum barrel draw ratio, the intrinsic frequency of drum barrel itself reduce, the intrinsic frequency of variable cross-section disk has
The intrinsic frequency of reduction by a small margin, blade is almost unchanged, illustrates the structural parameters for changing drum barrel in coupled system
Blade has little effect；
(3) former order frequencies of analytic modell analytical model and ANSYS model coincide preferably, to demonstrate the power established herein
Learn the reliability and validity of model.
The technical principle of the invention is described above in combination with a specific embodiment, these descriptions are intended merely to explain of the invention
Principle shall not be construed in any way as a limitation of the scope of protection of the invention.Based on explaining herein, those skilled in the art
It can associate with other specific embodiments of the invention without creative labor, these modes fall within this hair
Within bright protection scope.
Claims (10)
1. a kind of rotating cylindrical shell-variable cross-section disk-pretwist blade system dynamic modeling method, which is characterized in that including with
Lower step:
Step 1: three-dimensional system of coordinate needed for building rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system dynamics modeling；
The three-dimensional system of coordinate includes: the coordinate system ox of the fixed coordinate system OXYZ of entire coupled system, variable cross-section diskdydzd, it is whole
The moving coordinate system ox of a system during the motionryrzr, pre- twisted blade local coordinate system oxbybzb, wheel disc intersects with drum barrel
Round Geometric center coordinates system o1x1y1z1, drum barrel satellite coordinate system o2x2y2z2With the curvilinear coordinate system ox θ z of drum barrel mean camber；
Step 2: rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system structural parameters and material parameter are measured；
The parameter of measurement include: pretwist length of blade L, pretwist width of blade b, pretwist vane thickness h, pretwist blade angle β,
Pretwist leaf elastic modulus E, Poisson's ratio μ, pretwist leaves density ρ, variable cross-section disk internal diameter rs, first segment radius rd, second segment radius
rD, outer radii Rd, disk first segment thickness hD1, second segment thickness hD2, third section thickness hD3, disk elastic modulus Ed, disk
Density pd, rotating cylindrical shell length Ls, rotating cylindrical shell mean camber radius R, rotating cylindrical shell elastic modulus Es, rotating cylindrical
Shell density psWith rotating cylindrical shell Poisson's ratio υs；
The translation spring in three directions of x, y, z and the rigidity of torsional spring are respectively kc,u、kc,v、kc,wWith
Step 3: being uniform-distribution with N on setting variable cross-section diskbA identical pre- twisted blade；
After i-th of pre- twisted blade is deformed, pass through position of any point Q in global coordinate system OXYZ on pre- twisted blade i
After the amount of shifting to is converted by a series of coordinates, pre- twisted blade and pre- twisted blade and variable cross-section disk are obtained according to Kinetic Energy Calculation formula
With the kinetic energy of rotating cylindrical shell coupling；
Step 4: being based on plate shell Theory of Vibration, consider centrifugation rigidifying effects of the pre- twisted blade in rotary course, obtain rotation
Turn the potential energy of pre- twisted blade；
Step 5: variable cross-section disk can satisfy elastic sheet oscillation crosswise small deflection theory, derive its cross using Hamiton's principle
To the differential equation of elastic vibration, the kinetic energy and potential energy of variable cross-section disk are obtained；
Step 6: being based on Sanders shell theory, consider the coriolis force of rotating cylindrical shell, the shadow of centrifugal force and circumferential initial stress
It rings, obtains the kinetic energy and potential energy of rotating cylindrical shell；
Step 7: simulation being carried out using spring and torsional spring between variable cross-section disk and drum barrel and is bolted, the bullet of connection function has been obtained
The kinetic energy of spring and torsional spring in entirely rotation drum barrel-variable cross-section disk-pre- twisted blade coupled system；
Step 8: according to Hamilton variation principle
Wherein, U=Ub+Us,ε+Ud+Uθ, T=Tb+Ts1+Td；
WnonFor the function that external force is done, and with δ us、δvs、δws、δu、δv、δw、δ φ and δ WdVariation is carried out as independent variable to obtain
To rotating cylindrical shell-variable cross-section disk-pretwist blade system kinetics equation；
Step 9: using Galerkin method, be introduced into canoncial coordinates to any point edge on face in the rotating cylindrical shell in step 8
The displacement components u of longitudinal directions, tangential direction displacement vs, radial direction displacement ws, the radial displacement u of cantilever beam, lateral position
Move v, swaying direction is displaced w, bending direction cornerThe lateral displacement W of swaying direction corner φ and variable cross-section diskdCarry out from
Dispersion processing；
Obtain rotation drum barrel-variable cross-section disk-pre- twisted blade coupled system mass matrix, coriolis force matrix and stiffness matrix；
Step 10: variable cross-section disk and pre- twisted blade introduce dimensionless item, to guarantee the variable cross-section disk, the pre- twisted blade with
Rotating cylindrical shell ensures consistent when coupling；
Nondimensionalization coefficient η=x/Li, when corresponding to pre- twisted blade, LiFor the length L of pre- twisted blade；
When corresponding rotating cylindrical shell, LiFor the length L of rotating cylindrical shells；
The nondimensionalization coefficient of variable cross-section disk is ξ=(r-rs)/(rD-rs)；
WithRespectively pre- twisted blade carries out the model function of vibration after nondimensionalization；
WithModel function of vibration respectively after variable cross-section disk nondimensionalization；
Step 11: introducing Rayleigh damping, obtain rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system motion side
Journey:
In formula, M, G and D are respectively the mass matrix of entire coupled system, coriolis force matrix and damping matrix；
Ke、Ki、KΩ、KσAnd KcStiffness matrix caused by the structural stiffness matrix of respectively entire coupled system, acceleration, centrifugation are rigid
Change the stiffness matrix that matrix, Spin softening matrix, rotating cylindrical shell and disk coupling spring generate；
Q and F is respectively the outer exciting force vector of pre- twisted blade canoncial coordinates vector sum；
Rayleigh damping D is to combine acquisition in proportion by mass matrix and stiffness matrix；
Formula specific as follows:
D=α M+ β K；
Wherein, α and β are acquired by following formula:
In formula, fn1、fn2Respectively the first rank of spring-variable cross-section disk-pre- twisted blade coupled system and second-order intrinsic frequency
(Hz), ξ1And ξ2For damping ratio；
Q is the column vector of generalized coordinates composition related with the time, and expression-form is as follows:
2. modeling method according to claim 1, which is characterized in that the kinetic energy expression in step 3 are as follows:
In formula, rQTake up an official post motion vector of the one point Q of meaning under global coordinate system for pre- twisted blade；
A is the area of section of pre- twisted blade；
IzIt is pretwist blade profile around the cross sectional moment of inertia of y-axis；
IyIt is pretwist blade profile around the cross sectional moment of inertia of z-axis；
θ is the angular displacement of wheel disc movement；
Symbol () indicates the 1 rank local derviation to the time, rDIt is the outer diameter of disk, us、vs、wsIt is rotation drum barrel mean camber in whole coordinate
The displacement in three directions in system, x, y and z are respectively pre- twisted blade local coordinate system along pretwist length of blade direction, thickness side
To and swaying direction,It is respectively the corner of pre- twisted blade bending direction and swaying direction with φ, u, v and w are respectively pretwist leaf
The displacement of piece radial, transverse direction and swaying direction in local coordinate system, WdFor the lateral displacement of disk.
3. modeling method according to claim 1, which is characterized in that the strain of the pre- twisted blade of rotation described in step 4
Potential energy expression formula are as follows:
In formula, E, G, κ, A and L respectively indicate the Young's modulus, modulus of shearing, shearing factor, cross-sectional area, pretwist of pre- twisted blade
Length of blade and the centrifugal force suffered by pre- twisted blade in rotary course.
4. modeling method according to claim 1, which is characterized in that the kinetic energy of variable cross-section disk described in step 5 is expressed
Formula are as follows:
The potential energy expression formula of variable cross-section disk are as follows:
In formula, Wd1、Wd2And Wd3The respectively transverse mode function of variable cross-section disk first segment, second segment and third section；
The expression formula of Laplace operator and bending stiffness is as follows:
5. modeling method according to claim 1, which is characterized in that rotation rotating cylindrical shell described in step 6 is produced
Raw kinetic energy expression are as follows:
Rotating cylindrical shell considers to stretch and curved strain energy expression is as follows:
The initial circumference stress expression formula generated under the action of the centrifugal force:
In formula, Nθ=ρshsΩ2R2For circumferential initial stress；
ρsFor the density of drum barrel；
hsFor the thickness of drum barrel；
R is the middle radius surface of drum barrel；
υsFor the Poisson's ratio of drum barrel；
LsFor the length of drum barrel；
EsFor the rigidity of drum barrel.
6. modeling method according to claim 1, which is characterized in that variable cross-section disk described in step 7 and rotating cylindrical shell
Potential energy expression formula caused by spring at coupling are as follows:
7. modeling method according to claim 1, which is characterized in that rotating cylindrical shell-variable cross-section described in step 8
Disk-pretwist blade system kinetics equation expression formula is as follows:
8. modeling method according to claim 1, which is characterized in that the specific method is as follows described in step 10:
Discrete, introducing canoncial coordinates U is carried out to pre- twisted bladei(t)、Vi(t)、Wi(t)、ψi(t) and Φi(t), pre- twisted blade is obtained
Radial vibration, oscillation crosswise, swing, bending pivot angle and swing corner displacement it is as follows:
In formula,WithIt is expressed as corresponding pretwist blade radial, transverse direction, pendulum
The model function of vibration of i-th rank of the corner of dynamic and pre- twisted blade bending direction and swaying direction, N are mode truncation number.
9. modeling method according to claim 1, which is characterized in that after step 11, the method also includes:
It is zero that external excitation vector, which is arranged, determines rotating cylindrical shell-variable cross-section disk-pre- twisted blade coupled system under different rotating speeds
Intrinsic frequency；
Accuracy according to the intrinsic frequency of acquisition to verify differential equation of motion.
10. modeling method according to claim 8, which is characterized in that With
Expression formula such as following formula:
In formula:
βaiL is pretwist blade radial direction character root, βfiL is that pre- twisted blade is lateral, βsiL is the feature root of swaying direction；
βaiThe preceding 6 rank value of L is respectively as follows: 1.571,4.712,7.854,11,14.14,17.28；βfiThe preceding 6 rank value of L is distinguished
Are as follows: 1.875,4.686,7.819,10.89,13.9,16.81；βsiThe preceding 6 rank value of L is respectively as follows: 1.874,4.669,7.735,
10.63,13.27,15.8；
The coupled system of spring-supported flexible disk and pre- twisted blade, the lateral displacement table of resilient disc are analyzed using hypothesis modal method
It is shown as:
Wi c(r, θ)=Ri(r)cos(iθ) (23)
Wi s(r, θ)=Ri(r)cos(iθ) (24)
In formula, Wi cAnd Wi sThe column vector of mode composition is assumed in two orthogonal planes for disk,WithRespectively resilient disc about
The generalized coordinates of time, RiIt (r) is the model function of vibration for the disk derived according to non-uniform beam.
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