CN105260563B - A kind of entity of impeller pre-stressed mode and axial symmetry become dimension limited element analysis technique - Google Patents
A kind of entity of impeller pre-stressed mode and axial symmetry become dimension limited element analysis technique Download PDFInfo
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Abstract
The invention belongs to Structural Dynamics technical field, specially a kind of entity of impeller pre-stressed mode and axial symmetry become dimension limited element analysis technique.The deformation of present invention three-D displacement modeling complex configuration blade, and freedom degree polycondensation is carried out to leaf model with dynamic sub-structure methods；The deformation of revolving body wheel disc simply includes the two-dimensional axial symmetric mode of circumferentially displaced component, reduces the freedom degree of wheel disc model, and keep the displacement coordination between blade and wheel disc.It is operated by two above, the freedom degree of blade and wheel disc Coupling Dynamic Model is reduced into several magnitudes, the efficiency of blade wheel structure natural mode of vibration (including pre-stressed mode) analysis improves, and calculated result is accurate.The shape of blade wheel structure is not limited by gyration period in the present invention, is also applied for being equipped with the impeller of multiple groups, the different blade of circumferential number on wheel disc, can solve complicated blade wheel structure can not carry out the problem of model analysis with sector models.
Description
Technical field
The invention belongs to technical fields such as Structural Dynamics and FEM calculations, the three-dimensional of specially a kind of blade wheel structure is real
Body blade and two-dimensional axial symmetric wheel disc become dimensional model finite element prestressed modal analysis method.
Background technique
Blade wheel structure is widely present in rotating machinery, such as electric organ, steamer device and aero-engine.To impeller knot
Structure carry out model analysis be Impeller Design important link, to guarantee high speed rotation under impeller system stablize it is significant.
For the single group blade impeller structure with characteristic gyration period, sector models is relatively mostly used to carry out model analysis at present, but fanned
Section model is often larger, it is long to calculate the time, and can not calculate the blade wheel structure for being unsatisfactory for characteristic gyration period.When to impeller
When structure optimizes, the model analysis that need to be carried out to model repeatedly is calculated, and sector models longer analysis time becomes
The bottleneck of optimization design.
The method of the present invention proposes the 3D solid blade of blade wheel structure and two-dimensional axial symmetric wheel disc becomes the pre- of dimensional model
Stress modal analysis method, its significance lies in that the prestressed modal analysis efficiency of impeller class model gyration period is greatly improved,
Its model analysis precision is not reduced simultaneously.The method of the present invention meticulously characterizes the change of complex configuration blade with three-D displacement mode
Shape carries out freedom degree polycondensation to 3D solid leaf model with dynamic sub-structure methods on this basis；The change of revolving body wheel disc
Shape simply includes the two-dimensional axial symmetric mode of circumferentially displaced component, reduces the freedom degree of wheel disc model, and keep blade with
Displacement coordination between wheel disc.It is operated, the freedom degree of blade and wheel disc Coupling Dynamic Model can be contracted by two above
Subtract several magnitudes, improve the efficiency of blade wheel structure natural mode of vibration (including pre-stressed mode) analysis, and modal analysis result with
Complete three-dimensional blades finite element model calculated result is close.The shape of blade wheel structure is not by the limit of gyration period in this method
System, for being equipped with the impeller of multiple groups, the different blade of circumferential number on wheel disc, also usable this method is analyzed,
The problem of model analysis can not be carried out with sector models to can solve complicated blade wheel structure.
Summary of the invention
The object of the present invention is to provide a kind of 3D solid blades of blade wheel structure and two-dimensional axial symmetric wheel disc to become dimension mould
Type finite element prestressed modal analysis method.This method has universality for impeller class rotationally periodic structure, passes through foundation
Become dimension finite element model with the 3D solid blade and two-dimensional axial symmetric wheel disc of Complete three-dimensional impeller pattern equivalence, reduces mould
The number of degrees of freedom of type, to improve the model analysis efficiency of the class model.
A kind of 3D solid blade of blade wheel structure provided by the invention and two-dimensional axial symmetric wheel disc change dimensional model are limited
First prestressed modal analysis method, specific steps include:
(1) it establishes the blade wheel structure being made of blade and wheel disc and becomes dimension finite element model；It is first by blade three-dimensional
Solid element indicates, establishes 3D solid leaf model；Wheel disc is indicated with two-dimensional axial symmetric unit, establishes two-dimensional axial symmetric wheel
Disk model；It is real again by the way that one group of connecting node is respectively set in 3D solid leaf model and two-dimensional axial symmetric wheel disc model
Existing impeller becomes the combination of dimensional model, and respectively in the partial interior of 3D solid leaf model and two-dimensional axial symmetric wheel disc model
The linear restriction relationship of one group of displacement component is established between node and connecting node, so that each connecting node is used as static determinacy about
Shu Jiedian；
(2) stiffness matrix and mass matrix after 3D solid leaf model polycondensation are obtained using dynamic sub-structure methods,
The coupling process proposed by this method, is coupled with the stiffness matrix of two-dimensional axial symmetric wheel disc model and mass matrix；Benefit
It is carried out becoming dimensional model model analysis with the kinetic model after coupling.
In the present invention, using the Coupling Deformation calculation method of 3D solid leaf model and two-dimensional axial symmetric wheel disc model,
It finds out impeller and preloads the quiet deformation under effect in centrifugal force, while is rigid with the tangent line that stiffness matrix should be GEOMETRICALLY NONLINEAR
Matrix is spent, analysis impeller becomes the pre-stressed mode under dimensional model centrifugal force field.
Technical solution of the present invention is described in detail below.
One, become the foundation of two-dimensional axial symmetric wheel disc model in dimensional model
Two-dimensional axial symmetric wheel disc model defines in cylindrical coordinate Oxr θ, and three axis of the coordinate system are referred to as x-axis (axis
To), r axis (radial direction) and θ axis (circumferential direction), coordinate and the transform mode of standard rectangular coordinate system are
Two-dimensional axial symmetric wheel disc finite element model defines in the unilateral Oxr of θ=0, is made of two-dimensional axial symmetric unit, around
X-axis rotates a circle available with the completely the same revolving body wheel disc model of threedimensional model.
It is the deformation of n revolving body wheel disc, referred to as n pitch diameter displacement model for circumferential wave number in cylindrical coordinates.Its any position
Set axial displacement u on (x, r, θ)x, radial displacement urWith circumferentially displaced uθ(the following letter of the meridian plane of θ=0 can be defined on one group
Claim 0- meridian plane) cosine displacement functionWith sinusoidal displacement functionIt is expressed as
It then only include cosine displacement function since sinn θ perseverance is zero for zero pitch diameter displacement model.
Under cylindrical coordinate, the components of strain of revolving body wheel disc n pitch diameter displacement are
Revolving body wheel disc n pitch diameter displacement potential energy and kinetic energy be
Wherein D0For elastic matrix, ρ is density.
Wheel disc hatch region on 0- meridian plane is divided into two-dimensional axial symmetric finite element grid, shares nDA node.Herein
One group of node connecting with blade is set in finite element model, and connecting node number is nJ.So-called wheel disc tie point, refers to this group node
Circular arc (referred to as revolution arc) after x-axis revolution will be connect with a corresponding node in blade finite element model.
It might as well assume the preceding n of two-dimensional axial symmetric wheel disc modelJA node is connecting node, node cosine motion vectorWith node sinusoidal displacement vectorIt is each that there are six components, including three translation displacements components and three
Rotational displacement component.By being defined linearly about between wheel disc connecting node partial interior modal displacement component adjacent thereto
Beam relationship provides condition so that each connecting node can be used as static determinacy restraint joint for leaf dish coupling.The cosine of other nodes
Motion vectorWith sinusoidal displacement vectorWithout specifically limited, the conventional finite element modeling of reference
Method.
Using finite element interpolation, the cosine displacement function of wheel disc and sinusoidal displacement function can be separated into
Wherein, N(D)(x, r) is shape function
In this way, the displacement function of entire wheel disc can be by q(D)={ q(D,c)T,q(D,s)T}TIt is expressed as
For zero pitch diameter displacement model, then motion vector q(D)Only include q(D,c).For different types of two-dimensional axial symmetric list
Formula (8) are substituted into potential energy and kinetic energy expression (4), the element stiffness of two-dimensional axial symmetric unit n pitch diameter displacement model can be obtained by member
Matrix and element mass matrix.
Element stiffness matrix is
Wherein D0For elastic matrix, nPFor cell node number.
Element mass matrix is
THE TANGENTIAL STIFFNESS MATRICES derives as follows:
Shown in cylindrical coordinates intrinsic displacement gradient formula such as formula (14)
Formula (8) substitution formula (14) can be obtained
Wherein
It is available using formula (15), formula (16) and formula (17)
Wherein G matrix can be write as cosine component G(c)With sinusoidal component G(s)Form.
Define matrix MσFor
Wherein
Formula (8) substitution formula (22) can be obtained into each components of strain, then substitute into formula (21) each components of stress can be obtained.Then unit is pre-
Stress tangent stiffness matrix is
KTe=π ∫ ∫ (G(c)TMσG(c)+G(s)TMσG(s))rdrdx (23)
By the stiffness matrix and mass matrix that assemble available two-dimensional axial symmetric model.Due to describing revolving body wheel disc
The modal displacement vector q of deformation(D,c)And q(D,s)It is defined on meridian plane Oxr, vector dimension is much smaller than complete three-dimensional wheel disc
Model.
Two, become the foundation of three dimendional blade model in dimensional model
Assuming that only one group of blade in complete impeller pattern, blade circumferential direction number are N, by it around the shaft counterclockwise successively from
0 to N-1 numbers, and wherein starting point No. 0 blade is arbitrarily designated.
No. 0 3D solid blade is modeled in rectangular coordinate system Oxyz, coordinate origin and x-axis and is required two-dimentional in 2
Axial symmetry wheel disc model is identical, and y-axis is identical as the r axis in the plane of θ in 2=0 is required.It the position of No. 0 blade should be with two-dimentional axis
The revolving body wheel disc that symmetrical wheel disc model rotates mutually is coordinated.
Cosine displacement function is defined on No. 0 bladeWith sinusoidal displacement function
So for n pitch diameter displacement model, the displacement function on kth blade (k=0 ..., N-1) is
Wherein α=2 π/N is the differential seat angle of two neighboring blade.For zero pitch diameter displacement model, since sinnk α perseverance is
Zero, then it only include cosine displacement function.
No. 0 blade is divided into three-dimensional finite element mesh, shares nBA node.In its bottom, one group and two-dimentional axis pair are set
The node for claiming wheel disc connection should ensure that connecting node number n in connecting node number and two-dimensional axial symmetric wheel discJIt is identical, and two
The x-axis coordinate of person's corresponding points is identical.The wheel disc that so-called blade tie point, exactly this group node are all fallen within is correspondingly connected with a little
It turns round on arc.
It might as well assume the preceding n of No. 0 blade three-dimensional finite element modelJA node is connecting node, the displacement of node cosine
VectorWith node sinusoidal displacement vectorIt is each there are six component, including three translation displacements components and
Three rotational displacement components.Pass through the definition wires between blade connecting node partial interior modal displacement component adjacent thereto
Property the constraint relationship provide condition for leaf dish coupling so that each connecting node can be used as static determinacy restraint joint.Other nodes
Cosine motion vectorWith sinusoidal displacement vectorWithout specifically limited, the conventional finite element of reference
Modeling method.
Using finite element interpolation, No. 0 blade cosine displacement function and sinusoidal displacement function can be separated into
Wherein, N(B)(x, y, z) is shape function
N pitch diameter is displaced, the corresponding connection node in connecting node and wheel disc on No. 0 blade meets bottom offset such as and assists
Tune relationship
Wherein βiFor the differential seat angle being correspondingly connected in i-th of tie point in No. 0 blade and wheel disc a little in cylindrical coordinate.
The displacement function of kth blade (k=0 ..., N-1) can be by q as a result,(B)={ q(B,c)T q(B,s)T}TIt is expressed as
For zero pitch diameter displacement model, motion vector q(B)Only include q(B,c)。
Total potential energy under the n pitch diameter displacement model of N blade is
Kb0=∫ ∫ ∫ BTD0Bdxdydz (31)
Wherein B is the strain-transposed matrix indicated by shape function, D0For elastic matrix.Here with arriving
Total kinetic energy under the n pitch diameter displacement model of N blade is
Mb0=ρ ∫ ∫ ∫ N(B)TN(B)dxdydz (34)
Wherein ρ is density.
Can further calculate the n pitch diameter displacement model global stiffness matrix of N blade is with mass matrix
It is not difficult to find that Kb0And Mb0The stiffness matrix that is calculated for single 3 D entity blade conventional finite element method and
Mass matrix, referred to as single blade stiffness matrix and single blade mass matrix.If calculated prestressing force mode, Kb0It should be geometrical non-linearity
The tangent stiffness matrix of problem.
For the blade wheel structure of multiple groups blade, then each group blade No. 0 blade individually models, and can obtain the rigidity of each group blade
Matrix and mass matrix.
Three, become dimensional model Coupling Deformation in centrifugal force field to calculate
For the Coupling Deformation problem preloaded, this method refers in particular to the problem that preloads under impeller is acted on by centrifugal force, right
Should be in the zero pitch diameter displacement model for becoming dimensional model, therefore No. 0 blade displacement vector only includes q(B,c)。
The blade wheel structure for considering single group blade, using the tie point of No. 0 leaf three-dimensional model as the interface point of static(al) polycondensation,
Static(al) polycondensation [1] is carried out using static condensation method.Assuming that blade is acted on by centrifugal force, the equilibrium equation of leaf model is
WhereinFor blade interior modal displacement；For the displacement of blade interface node；WithFor
The vector that centrifugal force suffered by node generates, is known quantity.Then internal node displacement can be gone out by blade interface node offset table, i.e.,
(38) are substituted into (37) second equations, the nodal force of interface point after static(al) polycondensation under the action of the centrifugal force can be obtained
VectorAnd Stiffness Matrix
By interface node motion vectorTwo-dimentional axis pair is transformed into from the rectangular coordinate system for defining 3D solid leaf model
Claim wheel disc model cylindrical coordinate in, i.e.,
Wherein TiReferring to formula (28), obtains transformed polycondensation stiffness matrix and force vector is
For two-dimensional axial symmetric wheel disc, equally using tie point as the interface node of polycondensation, disk can be obtained and acted in centrifugal force
Under equilibrium equation be
WhereinFor the displacement of disk internal node；For the displacement of disk interface node；WithFor node
The vector that suffered centrifugal force generates, is known quantity.ByTable goes out
Similarly obtain the force vector P of interface point after polycondensationdWith stiffness matrix Kd
Then the rigidity of composite structure and force vector are
Wherein N is the circumferential number of blade.
Using the stiffness matrix and force vector after assembling, the displacement of placing interface node can be solved
It is displaced with blade interface node
Formula (47) and formula (46) are further substituted into the inside of formula (38) and formula (43) available blade and wheel disc respectively
Modal displacement vector.
For 3D solid leaf model, strain and stress can be obtained by general finite Meta algorithm by modal displacement vector
Distribution；For two-dimensional axial symmetric wheel disc model, formula (22) can be substituted by modal displacement vector and formula (21) obtains strain and stress
Distribution.
The case where for multiple blades, then the rigidity of composite structure and force vector need to be superimposed more, calculating process class
Seemingly.
Four, become the Dynamics Coupling of blade and wheel disc in dimensional model
For n pitch diameter displacement model (n > 0), the Free Vibration Equations of single 3 D entity blade cosine motion vector are
The form for being write as matrix in block form is
WhereinFor blade interior modal displacement；For the displacement of blade interface node.
Using the Craig-Bampton method [2] of Dynamic Substructure, by single 3 D entity leaf model cosine be displaced to
Measure q(B,c)With part cosine immobile interface cylindrical coordinates ξ(c)And whole tie point cosine motion vectorsIt is expressed as
In formula (50), ΦbReferred to as immobile interface master mode meets
I.e.It is corresponding about M for vibration equation (52)biiNormalized preceding nKRank modal vector, ωbiFor corresponding circle frequency
Rate.
In formula (50), ΨbReferred to as Constrained mode meets
Formula (50) are substituted into formula (49), and premultiplication simultaneouslyStiffness matrix after polycondensation can be obtained is
Mass matrix after polycondensation is
Wherein ξ corresponds to modal coordinate, qJCorresponding each connecting node component.
By taking stiffness matrix converts as an example, by connecting node motion vectorIt transforms in Oxr θ cylindrical coordinate
Then blade reduced model single blade stiffness matrix is transformed to
Polycondensation transformation similarly is carried out to sinusoidal displacement vector, displacement relation can be obtained
Its immobile interface master mode and Constrained mode are identical as cosine motion vector polycondensation calculated result, and coordinate conversion side
Formula is identical, thus gained polycondensation matrix withIt is identical, then n pitch diameter displacement model (n > 0) global stiffness matrix polycondensation of N blade
It is transformed to
By formula (28), the interfacial displacement vector of No. 0 blade reduced model can be displaced by wheel disc tie point in cylindrical coordinate
Vector is expressed as
Note
Then have
Then the global stiffness matrix after all blade polycondensations is further transformed to
Wherein
For mass matrix, the overall quality matrix after all blade polycondensations can be obtained with identical method, i.e.,
Wherein
Finally by the stiffness matrix of transformed stiffness matrix and mass matrix general assembly to two-dimensional axial symmetric wheel disc model with
In the corresponding position of mass matrix, the global stiffness matrix and mass matrix of model are obtained, two-dimensional axial symmetric can be carried out
Wheel disc and 3D solid blade become the n pitch diameter model analysis of dimensional model.Assuming that the coupling Free Vibration Equations after general assembly are
The corresponding generalized eigenvalue problem of calculating formula (70), it is available become dimensional model n pitch diameter intrinsic frequency and its
Corresponding mode.Assuming that certain obtained rank Mode Shape vector is
Six components respectively correspond blade modal coordinate, the cosine point of connecting node displacement and the displacement of disk internal node on disk
Amount corresponds to the component of the rank mode with sinusoidal component.
For two-dimensional axial symmetric wheel disc, can useSubstitution formula (8), obtains on disk
The modal displacement vector of each node, to recover the mode of entire disk.
For kth blade, utilizeIt is corresponding in rectangular coordinate system to recover it
Motion vectorI.e.
WhereinFor formula (57) homography, T is formula (61) homography.Formula (72) are substituted into formula (50) and formula (58),
Acquired results substitute into formula (29), then the node modal displacement vector of kth blade can be expressed as
Global stiffness and matter for zero pitch diameter displacement model, after change dimensional model polycondensation can be obtained with similar method
Moment matrix carries out zero pitch diameter model analysis.
The beneficial effects of the present invention are:
1, the 3D solid blade of blade wheel structure may be implemented in its method and two-dimensional axial symmetric wheel disc becomes the pre- of dimensional model
Stress model analysis improves the model analysis computational efficiency of impeller class rotationally periodic structure under the premise of guaranteeing precision.
2, its method can specify the pitch diameter number of model analysis, targeted for particular problem.Such as turn in engine
In fast model analysis, only with shaft coupled vibrations can occur for a pitch diameter mode of impeller, then can be directly right using this method
One pitch diameter mode is analyzed, and avoids calculating other extra mode.
3, the shape of blade wheel structure is not limited by gyration period in this method.Currently, being mostly used for impeller class formation
Sector models carry out model analysis, so that analyzable blade wheel structure is limited to characteristic gyration period.For being equipped on wheel disc
The impeller of multiple groups, the different blade of circumferential number, can not often mark off identical sector strucre.The blade and wheel of this method
Disc portion can Independent modeling, the shape of blade wheel structure do not limited by gyration period, thus solve complicated impeller pattern without
Method carries out the problem of model analysis with sector models.
Detailed description of the invention
Fig. 1 is the Complete three-dimensional finite element model of blade wheel structure.
Fig. 2 is the finite element model of two-dimensional axial symmetric wheel disc.
Fig. 3 is the multi-point constraint addition situation schematic diagram at two-dimensional axial symmetric wheel disc connecting node.
Fig. 4 is the finite element model of 3D solid blade.
Fig. 5 is the multi-point constraint addition situation schematic diagram at 3D solid blade bottom connecting node.
Fig. 6 is that 3D solid blade and two-dimensional axial symmetric wheel disc become dimensional model schematic diagram.
Fig. 7 is the corresponding sector finite element of blade wheel structure.
Specific embodiment
The present invention is further elaborated in conjunction with attached drawing and example.
Technology of the invention realizes that the 3D solid blade of blade wheel structure and two-dimensional axial symmetric wheel disc become the mould of dimensional model
State analysis, under the premise of guaranteeing precision, improves the model analysis computational efficiency of impeller class rotationally periodic structure.Specific implementation
Process includes the following steps:
(1) two-dimensional axial symmetric wheel disc model and 3D solid leaf model are established, matching is correspondingly connected with a little.
Three-dimensional blades model used in example is as shown in Figure 1, it shares 17 blades.Impeller material parameter are as follows: Young mould
Measure 1.18e+11Pa；Poisson's ratio 0.3；Density 4400kg/m3。
Two-dimensional axial symmetric wheel disc model and 3D solid leaf are established in modeling method proposed according to the present invention and requirement respectively
Piece model, and the certain multi-point constraint constraint of addition in each comfortable model.Wherein two-dimensional axial symmetric wheel disc model as shown in Fig. 2,
It is as shown in Figure 3 that its multi-point constraint adds situation；3D solid leaf model is as shown in figure 4, its multi-point constraint adds situation such as Fig. 5
It is shown.
3D solid leaf model and two-dimensional axial symmetric wheel disc model are placed in same view, equivalent change dimension can be obtained
The schematic diagram of model, as shown in Figure 6.As can be seen from Figure 6, the position of 3D solid blade and two-dimensional axial symmetric wheel disc model phase
Coordinate.
The corresponding connection node information of 3D solid leaf model and two-dimensional axial symmetric wheel disc model is as shown in table 1.From table 1
In it can be seen that, two model connecting node numbers are identical, and the x-axis coordinate of corresponding connection node is essentially identical, and error is very much
Two millimeters within.Therefore the modeling that the 3D solid blade of this example and two-dimensional axial symmetric wheel disc model meet the method for the present invention is wanted
It asks.
1 corresponding connection node information of table
(2) leaf dish coupled modes proposed by the present invention are utilized, calculates and becomes dimensional model mode.
This example is calculated so that impeller becomes a pitch diameter mode of dimensional model as an example.The reference object of calculated result is quotient
With software Nastran to the FEM modal analysis and modal of same impeller sector models.
Sector models used in Nastran are as shown in fig. 7, be 1st/17th of complete impeller pattern.
The fixed pitch diameter FEM modal analysis and modal of impeller center is as shown in table 2.In general, change dimension proposed by the present invention
The error for the sector models result that model FEM modal analysis and modal and Nastran are calculated is within 1%.When becoming the calculating of dimensional model
Between be much smaller than the three-dimensional sectors Nastran model, shorten to Nastran calculate the time 1/10th within.
The example shows that the 3D solid blade for the blade wheel structure that the method for the present invention proposes and two-dimensional axial symmetric wheel disc become dimension
It spends model and its modal analysis method is effective, calculate three-dimensional sectors model result in result and existing business software Nastran
It is close, the time is calculated much smaller than the time used in Nastran, is capable of the analysis efficiency of biggish raising impeller class formation without losing points
Analyse precision.
2 one pitch diameter FEM modal analysis and modal of table
Bibliography
[1]Wilson,Edward L.The static condensation algorithm[J].International
Journal for Numerical Methods in Engineering,1974,8(1):198-203.
[2]M.C.C.Bampton,R.R.Craig Jr.Coupling of substructures for dynamic
analysis.AIAA J[J].AIAA Journal,1968,6:1313-1319。
Claims (3)
1. a kind of entity of impeller pre-stressed mode and axial symmetry become dimension limited element analysis technique, which is characterized in that specific steps
It is as follows:
(1) it establishes the blade wheel structure being made of blade and wheel disc and becomes dimension finite element model；It is first by blade 3D solid
Unit indicates, establishes 3D solid leaf model；Wheel disc is indicated with two-dimensional axial symmetric unit, establishes two-dimensional axial symmetric wheel disc mould
Type；Again by the way that one group of connecting node is respectively set in 3D solid leaf model and two-dimensional axial symmetric wheel disc model, and respectively
One group of position is established between 3D solid leaf model and the partial interior node and connecting node of two-dimensional axial symmetric wheel disc model
The linear restriction relationship of component is moved, so that each connecting node is used as static determinacy restraint joint, realizes that impeller becomes dimensional model
Combination；
(2) stiffness matrix and mass matrix after 3D solid leaf model polycondensation are obtained using dynamic sub-structure methods, with two
The stiffness matrix and mass matrix for tieing up axial symmetry wheel disc model are coupled, and carry out change dimension using the kinetic model after coupling
Model model analysis；Wherein:
In step (1), the modeling method of two-dimensional axial symmetric wheel disc model is as follows:
Two-dimensional axial symmetric wheel disc model defines in cylindrical coordinate Oxr θ, and three axis of the coordinate system are respectively by axial x-axis, diameter
To r axis and circumferential θ axis composition；Two-dimensional axial symmetric wheel disc finite element model defines in the section Oxr of θ=0, by two-dimentional axis
Symmetrical cell is constituted, and is rotated around x axis and is obtained within one week the revolving body wheel disc model completely the same with threedimensional model；
It is the deformation of n revolving body wheel disc, referred to as n pitch diameter displacement model for circumferential wave number；In cylindrical coordinates, any position (x, r,
Axial displacement u on θ)x, radial displacement urWith circumferentially displaced uθThe meridian plane of θ=0, the i.e. cosine of 0- meridian plane are defined on one group
Displacement functionWith sinusoidal displacement functionIt is expressed as:
It then only include cosine displacement function since sinn θ perseverance is zero for zero pitch diameter displacement model；
The components of strain under cylindrical coordinate indicate are as follows:
The potential energy and kinetic energy of the n pitch diameter displacement of revolving body wheel disc indicate are as follows:
Wherein D is elastic matrix, and ρ is density；
Wheel disc hatch region on 0- meridian plane is divided into two-dimensional axial symmetric finite element grid, shares nDA node；In two-dimentional axis pair
Claim that one group of node connecting with blade is arranged in wheel disc finite element model, connecting node number is nJ, so-called wheel disc tie point refers to this
Circular arc of the group node after x-axis revolution, i.e. revolution arc will be connect with a corresponding node in blade finite element model；
Assuming that the preceding n of two-dimensional axial symmetric wheel disc modelJA node is connecting node, node cosine motion vectorAnd section
Point sinusoidal displacement vectorIt is each that there are six components, including three translation displacements components and three angle of rotation positions
Move component；By defining linear the constraint relationship between wheel disc connecting node partial interior modal displacement component adjacent thereto,
So that each connecting node can be used as static determinacy restraint joint, condition is provided for leaf dish coupling；The cosine of other nodes be displaced to
AmountWith sinusoidal displacement vectorWithout specifically limited, the conventional finite element modeling method of reference；
Using finite element interpolation, the cosine displacement function of wheel disc and sinusoidal displacement function are separated into:
Wherein N(D)(x, r) is shape function；
In this way, the displacement function of entire wheel disc is by q(D)={ q(D,c)T,q(D,s)T}TIt indicates, for zero pitch diameter displacement model, then position
The amount of shifting to q(D)Only include q(D,c)；For different types of two-dimensional axial symmetric unit, different shape functions is substituted into potential energy and kinetic energy
Expression formula obtains the element stiffness matrix and element mass matrix of two-dimensional axial symmetric unit n pitch diameter displacement model；By assembling
To the stiffness matrix and mass matrix of two-dimensional axial symmetric wheel disc model；
In step (1), the modeling method of 3D solid leaf model is as follows:
1. blade circumferential direction number is N, by it around the shaft counterclockwise successively from 0 to N-1 for there was only the impeller pattern of one group of blade
Number, wherein starting point No. 0 blade is arbitrarily designated；
No. 0 blade is modeled in rectangular coordinate system Oxyz, coordinate origin and x-axis are identical as two-dimensional axial symmetric wheel disc model,
And y-axis is identical as the r axis in the plane of θ=0 in two-dimensional axial symmetric wheel disc model；The position of No. 0 blade and two-dimensional axial symmetric wheel disc
The revolving body wheel disc that model rotates mutually is coordinated；
Cosine displacement function is defined on No. 0 bladeWith sinusoidal displacement functionSo
For n pitch diameter displacement model, kth blade, k=0 ..., N-1, on displacement function are as follows:
Wherein α=2 π/N is the differential seat angle of two neighboring blade；For zero pitch diameter displacement model, since sinnk α perseverance is zero, then
It only include cosine displacement function；
No. 0 blade is divided into three-dimensional finite element mesh, shares nBA node；In its bottom, one group and two-dimensional axial symmetric wheel are set
The node of disk model connection, guarantees connecting node number n in connecting node number and two-dimensional axial symmetric wheel discJIt is identical, and the two
The x-axis coordinate of corresponding points is identical；So-called blade tie point is exactly that a wheel disc all falling within of this group node is correspondingly connected with returning a little
Turn on arc；
Assuming that the preceding n of No. 0 blade three-dimensional finite element modelJA node is connecting node, node cosine motion vectorWith
Node sinusoidal displacement vectorIt is each that there are six components, including three translation displacements components and three angles of rotation
Displacement component；It is closed by defining linear restriction between blade connecting node partial interior modal displacement component adjacent thereto
System provides condition so that each connecting node can be used as static determinacy restraint joint for leaf dish coupling；The cosine of other nodes is displaced
VectorWith sinusoidal displacement vectorWithout specifically limited, the conventional finite element modeling side of reference
Method；
Using finite element interpolation, No. 0 blade cosine displacement function and sinusoidal displacement function can be separated into:
Wherein: N(B)(x, y, z) is shape function；
N pitch diameter is displaced, the corresponding connection node in connecting node and wheel disc on No. 0 blade meets following displacement coordination and closes
System:
Wherein βiFor the angle being correspondingly connected in i-th of tie point in No. 0 blade and two-dimensional axial symmetric wheel disc a little in cylindrical coordinate
Difference；
The displacement function of all N blades is by q as a result,(B)={ q(B,c)T,q(B,s)T}TIt indicates, for zero pitch diameter displacement model,
Motion vector q(B)Only include q(B,c)；Then the potential energy under the n pitch diameter displacement model of N blade and kinetic energy are by q(B)Quadratic form table
Out, further calculate N blade n pitch diameter displacement model global stiffness matrix and mass matrix are as follows:
Wherein Kb0And Mb0For stiffness matrix and mass matrix that single 3 D entity blade conventional finite element method is calculated, claim
For single blade stiffness matrix and single blade mass matrix；If calculated prestressing force mode, Kb0It should be the tangent line of GEOMETRICALLY NONLINEAR
Stiffness matrix；
2. then No. 0 blade of each group blade individually models for the blade wheel structure of multiple groups blade, the rigidity square of each group blade is obtained
Battle array and mass matrix；
In step (2), the method that stiffness matrix and mass matrix in model analysis are coupled is as follows:
For n pitch diameter displacement model, n > 0, using the Craig-Bampton method of Dynamic Substructure, by single 3 D entity leaf
Piece model cosine motion vector q(B,c)With part cosine immobile interface cylindrical coordinates ξ(c)And whole tie point cosine motion vectors
It indicates, gained Kb0Polycondensation stiffness matrix are as follows:
Wherein ξ corresponds to immobile interface cylindrical coordinates, qJCorresponding each connecting node component；
By connecting node motion vectorIt transforms in Oxr θ cylindrical coordinate,
Then blade reduced model single blade stiffness matrix converts are as follows:
Wherein nKRetain mode number for blade,It is n for sizeKUnit matrix；Polycondensation similarly is carried out to sinusoidal displacement vector
Transformation, gained polycondensation matrix withIt is identical；The then n pitch diameter displacement model of N blade, 0 global stiffness matrix polycondensation of n > transformation
Are as follows:
Interfacial displacement vector in No. 0 blade reduced model can be indicated by wheel disc tie point motion vector are as follows:
It is denoted as:
Then have
Then the global stiffness matrix after all blade polycondensations further converts are as follows:
Wherein:
The transformed mass matrix of 3D solid leaf model polycondensation is obtained with identical method；
Finally by the stiffness matrix and moment of mass of transformed stiffness matrix and mass matrix general assembly to two-dimensional axial symmetric disk model
In battle array, the stiffness matrix and mass matrix of model totality are obtained；Using dynamic coupling model corresponding to the two matrixes,
The n pitch diameter displacement model for becoming dimensional model to 3D solid blade and two-dimensional axial symmetric wheel disc carries out model analysis；Pass through mode
Restore, obtains 3D solid blade and modal displacement of the two-dimensional axial symmetric wheel disc under each rank n pitch diameter frequency；
For zero pitch diameter displacement model, global stiffness and mass matrix after obtaining change dimensional model polycondensation with same method,
Carry out zero pitch diameter model analysis.
2. the entity of impeller pre-stressed mode according to claim 1 and axial symmetry become dimension limited element analysis technique, special
Sign is:
Using the Coupling Deformation calculation method of 3D solid leaf model and two-dimensional axial symmetric wheel disc model, finds out impeller and be centrifuged
Power preloads the quiet deformation under effect, while should be the tangent stiffness matrix of GEOMETRICALLY NONLINEAR with stiffness matrix, analyzes leaf
Wheel becomes the pre-stressed mode under dimensional model centrifugal force field.
3. the entity of impeller pre-stressed mode according to claim 2 and axial symmetry become dimension limited element analysis technique, special
Sign is that the calculation method of 3D solid blade and two-dimensional axial symmetric wheel disc Coupling Deformation is as follows in centrifugal force field:
For the Coupling Deformation problem preloaded, refers in particular to impeller and preload problem under by centrifugal force effect, correspond to and become dimension
Zero pitch diameter displacement model of model, therefore No. 0 blade displacement vector only includes q(B,c)；
The blade wheel structure for considering single group blade, using the tie point of No. 0 leaf three-dimensional model as the interface point of static(al) polycondensation, to one
A blade by centrifugal force carries out static(al) polycondensation with static condensation method, obtains the node force vector of interface point after polycondensationAnd rigidity
Battle arrayBy interface node motion vectorTwo-dimensional axial symmetric is transformed into from the rectangular coordinate system for defining 3D solid leaf model
In the cylindrical coordinate Oxr θ of wheel disc model modeling, i.e.,
Obtain transformed polycondensation stiffness matrix and force vector are as follows:
For two-dimensional axial symmetric wheel disc, same progress static(al) polycondensation obtains the force vector P of interface point after polycondensationdWith stiffness matrix Kd；
Then blade wheel structure becomes the interfacial displacement vector of dimensional model from following equation solution:
(Kd+NKb)qJ=Pd+NPb
Wherein N is the circumferential number of blade,The internal displacement of blade and wheel disc then uses static condensation method root
According to qJIt calculates；
The case where for multiple blades, the then rigidity and force vector for being superimposed more composite structures are calculated.
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