CN109800512B - Dynamic modeling method for rotating cylindrical shell-variable cross-section disc-pre-twisted blade system - Google Patents

Dynamic modeling method for rotating cylindrical shell-variable cross-section disc-pre-twisted blade system Download PDF

Info

Publication number
CN109800512B
CN109800512B CN201910064646.9A CN201910064646A CN109800512B CN 109800512 B CN109800512 B CN 109800512B CN 201910064646 A CN201910064646 A CN 201910064646A CN 109800512 B CN109800512 B CN 109800512B
Authority
CN
China
Prior art keywords
cylindrical shell
variable cross
section
twisted blade
disc
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Active
Application number
CN201910064646.9A
Other languages
Chinese (zh)
Other versions
CN109800512A (en
Inventor
马辉
刘诗宇
崔璨
曾劲
杨桐
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Northeastern University China
Original Assignee
Northeastern University China
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Northeastern University China filed Critical Northeastern University China
Priority to CN201910064646.9A priority Critical patent/CN109800512B/en
Publication of CN109800512A publication Critical patent/CN109800512A/en
Application granted granted Critical
Publication of CN109800512B publication Critical patent/CN109800512B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Landscapes

  • Rolls And Other Rotary Bodies (AREA)
  • Turbine Rotor Nozzle Sealing (AREA)

Abstract

The invention relates to a dynamic modeling method of a rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system, which comprises the following steps: step 1: constructing a three-dimensional coordinate system required by dynamic modeling of a rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system; step 2: measuring structural parameters and material parameters of a rotary cylindrical shell-variable cross-section disc-pre-twisted blade coupling system; obtaining the pre-twisted blades and the kinetic energy of the pre-twisted blades coupled with the variable cross-section disc and the rotating cylindrical shell according to a kinetic energy calculation formula; and 4, step 4: based on a plate shell vibration theory, the influence of a centrifugal rigidization effect of the pre-twisted blade in the rotating process is considered, and the potential energy of the rotating pre-twisted blade is obtained; and 5: the variable cross-section disc can meet the small-deflection theory of transverse vibration of the elastic thin plate, and a differential equation of the transverse elastic vibration of the variable cross-section disc is deduced by using the Hamilton principle to obtain the kinetic energy and the potential energy of the variable cross-section disc. The method provided by the invention has higher calculation efficiency and calculation precision.

Description

Dynamic modeling method for rotating cylindrical shell-variable cross-section disc-pre-twisted blade system
Technical Field
The invention belongs to the technical field of mechanical dynamics, and particularly relates to a dynamic modeling method of a rotating cylindrical shell-variable cross-section disc-pre-twisted blade system.
Background
At present, the existing modeling method of the rotating cylindrical shell, the variable cross-section disc and the pre-twisted blade mainly comprises the following two methods:
1. commercial finite element analysis software
And importing the CAD three-dimensional model into commercial finite element analysis software or directly establishing a three-dimensional model of the whole assembly in the finite element software, selecting proper units and proper material parameters, carrying out grid division on the three-dimensional model, establishing the finite element model, setting proper constraint and selecting a proper solving method to analyze the dynamic characteristics of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade system. However, when the dynamic characteristics of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade system are analyzed by using the existing commercial finite element analysis software, the modeling process is complex and heavy, the dynamic characteristics obtained by different modeling modes and unit types have large differences, and meanwhile, the finite element software cannot consider the deformation effect of the rotating cylindrical shell in the solving process, so that a large error is generated in the high-rotating speed process.
2. Modeling method based on plate shell theory
At present, based on knowledge mastered by people, most of the coupling systems of the rotary cylindrical shell and the disc with the equal cross section are on the market, the diameter of the drum barrel is the same as that of the rotary cylindrical shell, the plate is based on a thin plate theory, the rotary cylindrical shell is based on an LOVE shell theory, a Donnell shell theory and the like, the whole system is subjected to dynamic modeling based on an energy method, the condition that the disc is the disc with the variable cross section and the radius of the rotary cylindrical shell is smaller than that of the disc cannot be considered, and the coupling systems do not conform to the practical application structure of the coupling systems of the rotary cylindrical shell and.
At present, based on a plate shell vibration theory, a dynamic modeling technology of a rotating cylindrical shell, a variable cross-section disc and a pre-twisted blade is in a blank state.
Disclosure of Invention
Technical problem to be solved
Aiming at the existing technical problems, the invention provides a dynamic modeling method of a rotating cylindrical shell-variable cross-section disc-pre-twisted blade system, which can obtain a motion differential equation of a dynamic model of the whole system by considering the influences of centrifugal rigidization, rotary softening, hoop initial stress and Coriolis force of the whole system in the rotating process and adopting a method of Galerkin truncation and Hamilton variation on the premise of ensuring the vibration mode of the whole coupling system.
(II) technical scheme
In order to achieve the purpose, the invention adopts the main technical scheme that:
a dynamic modeling method for a rotating cylindrical shell-variable cross-section disc-pre-twisted blade system comprises the following steps:
step 1: constructing a three-dimensional coordinate system required by dynamic modeling of a rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system;
the three-dimensional coordinate system includes: fixed coordinate system OXYZ of whole coupling system and coordinate system ox of variable cross-section diskdydzdAnd a moving coordinate system ox of the whole system in the motion processryrzrLocal coordinate system ox of pre-twisted bladebybzbGeometric center coordinate system o of intersection circle of wheel disc and drum barrel1x1y1z1Coordinate system o of drum2x2y2z2And a curvilinear coordinate system ox θ z of a curved surface in the drum;
step 2: measuring structural parameters and material parameters of a rotary cylindrical shell-variable cross-section disc-pre-twisted blade coupling system;
the parameters measured include: length L of pre-twisted blade, width b of pre-twisted blade, thickness h of pre-twisted blade, installation angle beta of pre-twisted blade, elastic modulus E of pre-twisted blade, Poisson ratio mu, density rho of pre-twisted blade and inner diameter r of variable cross-section discsFirst radius rdSecond radius rDOuter ring radius RdFirst section thickness h of the discD1A second section thickness hD2A third section thickness hD3Elastic modulus E of the diskdDisc density pdLength L of the rotating cylindrical shellsRadius of curved surface R in rotating cylindrical shell and elastic modulus E of rotating cylindrical shellsDensity rho of rotary cylindrical shellsAnd a Poisson ratio upsilon of the rotating cylindrical shells；
The rigidity of the translational spring and the torsional spring in the x direction, the y direction and the z direction is respectively kc,u、kc,v、kc,wAnd
Figure BDA0001955267720000021
and step 3: n is uniformly distributed on the set variable cross-section diskbThe same pre-twisted blades;
when the ith pre-twisted blade is deformed, converting a series of coordinates through a displacement vector of any point Q on the pre-twisted blade i in an overall coordinate system OXYZ, and obtaining kinetic energy of the pre-twisted blade, the variable cross-section disc and the rotating cylindrical shell according to a kinetic energy calculation formula;
and 4, step 4: based on a plate shell vibration theory, the influence of a centrifugal rigidization effect of the pre-twisted blade in the rotating process is considered, and the potential energy of the rotating pre-twisted blade is obtained;
and 5: the variable cross-section disc can meet the small-deflection theory of transverse vibration of the elastic sheet, and a differential equation of the transverse elastic vibration of the variable cross-section disc is deduced by using the Hamilton principle to obtain the kinetic energy and the potential energy of the variable cross-section disc;
step 6: based on the Sanders shell theory, the influence of Coriolis force, centrifugal force and annular initial stress of the rotating cylindrical shell is considered, and the kinetic energy and potential energy of the rotating cylindrical shell are obtained;
and 7: the variable cross-section disc and the drum are connected by adopting a spring and a torsional spring to simulate a bolt, and kinetic energy of the spring and the torsional spring which play a role in connection in the whole rotary drum-variable cross-section disc-pre-twisted blade coupling system is obtained;
and 8: according to Hamilton variation principle
Figure BDA0001955267720000031
Wherein U is U ═ Ub+Us,+Ud+Uθ，T＝Tb+Ts1+Td；
WnonActing as an external force and by us、vs、ws、u、v、w、
Figure BDA0001955267720000033
Phi and WdPerforming variational division as an independent variable to obtain a dynamic equation of a rotating cylindrical shell-variable cross-section disc-pre-twisted blade system;
and step 9: introducing regular coordinates to the displacement u of any point on the middle surface of the rotating cylindrical shell in the step 8 along the longitudinal direction by adopting a Galerkin methodsDisplacement v in tangential directionsDisplacement w in radial directionsRadial displacement u, transverse displacement v, swing direction displacement w and bending direction corner of cantilever beam
Figure BDA0001955267720000032
Swing direction rotation angle phi and lateral displacement W of variable cross-section diskdCarrying out discretization treatment;
obtaining a mass matrix, a Coriolis force matrix and a rigidity matrix of the rotary drum barrel-variable cross-section disc-pre-twisted blade coupling system;
step 10: dimensionless terms are introduced into the variable cross-section disc and the pre-twisted blades to ensure consistency when the variable cross-section disc and the pre-twisted blades are coupled with the rotating cylindrical shell;
dimensionless coefficient eta x/LiWhen corresponding to pre-twisting blades, LiThe length L of the pre-twisted blade is;
when the cylindrical shell is correspondingly rotated, LiFor the length L of the rotating cylindrical shells；
The dimensionless coefficient of the variable cross-section disk is xi ═ (r-r)s)/(rD-rs)；
Figure BDA0001955267720000041
And
Figure BDA0001955267720000042
respectively carrying out vibration mode functions after dimensionless treatment on the pre-twisted blades;
Figure BDA0001955267720000043
and
Figure BDA0001955267720000044
respectively are vibration mode functions after the non-dimensionalization of the variable cross-section disc;
step 11: introducing Rayleigh damping to obtain a motion differential equation of the rotary cylindrical shell-variable cross-section disc-pre-twisted blade coupling system:
Figure BDA0001955267720000045
in the formula, M, G and D are respectively a mass matrix, a Coriolis force matrix and a damping matrix of the whole coupling system;
Ke、Ki、KΩ、Kσand KcRespectively a structural rigidity matrix of the whole coupling system, a rigidity matrix caused by acceleration, a centrifugal rigidization matrix, a rotary softening matrix and a rigidity matrix generated by a rotary cylindrical shell and a disc coupling spring;
q and F are a pre-twisted blade regular coordinate vector and an external excitation force vector respectively;
the Rayleigh damping D is obtained by combining a mass matrix and a rigidity matrix in proportion;
specifically, the following formula:
D＝αM+βK；
wherein α and β are obtained by the following formula:
Figure BDA0001955267720000046
in the formula (f)n1、fn2First and second order natural frequencies (Hz), ξ, respectively, of a spring-variable cross-section disk-pretwisted blade coupling system1And xi2Is the damping ratio;
q is a column vector composed of generalized coordinates related to time, expressed as follows:
Figure BDA0001955267720000051
preferably, the expression of the kinetic energy in step 3 is:
Figure BDA0001955267720000052
in the formula, rQThe displacement vector of any point Q on the pre-twisted blade under the global coordinate system is obtained;
a is the cross-sectional area of the pre-twisted blade;
Izthe section inertia moment of the pre-twisted blade section around the y axis is obtained;
Iythe section inertia moment of the pre-twisted blade section around the z axis is obtained;
theta is the angular displacement of the wheel disc movement;
the symbol (-) represents a 1 st order partial derivative of time, rDIs the outer diameter of the disc, us、vs、wsIs the displacement of the curved surface in the rotary drum barrel in three directions in the overall coordinate system, x, y and z are the local coordinate system of the pre-twisted blade along the length direction, the thickness direction and the swing direction of the pre-twisted blade respectively,
Figure BDA0001955267720000061
phi is the rotation angle of the pre-twisted blade in the bending direction and the swinging direction, u, v and W are the displacement of the pre-twisted blade in the radial direction, the transverse direction and the swinging direction in the local coordinate system, WdIs the lateral displacement of the disc.
Preferably, the strain potential energy expression of the rotating pre-twisted blade in the step 4 is as follows:
Figure BDA0001955267720000062
wherein E, G, κ, a and L represent the young's modulus, shear coefficient, cross-sectional area, pre-twisted blade length and centrifugal force to which the pre-twisted blade is subjected during rotation, respectively.
Preferably, the kinetic energy expression of the variable cross-section disk in step 5 is:
Figure BDA0001955267720000063
the potential energy expression of the variable cross-section disc is as follows:
Figure BDA0001955267720000071
in the formula, Wd1、Wd2And Wd3The transverse mode shape functions of the first section, the second section and the third section of the variable cross-section disc are respectively;
the expressions for harmonic operators and flexural stiffness are as follows:
Figure BDA0001955267720000072
Figure BDA0001955267720000073
Figure BDA0001955267720000074
Figure BDA0001955267720000075
Figure BDA0001955267720000076
Figure BDA0001955267720000077
Figure BDA0001955267720000078
Figure BDA0001955267720000079
Figure BDA0001955267720000081
Figure BDA0001955267720000082
preferably, the expression of the kinetic energy generated by rotating the cylindrical shell in step 6 is as follows:
Figure BDA0001955267720000083
the strain energy expression of the rotating cylindrical shell in consideration of stretching and bending is as follows:
Figure BDA0001955267720000084
the expression of the initial hoop stress generated under the action of centrifugal force is as follows:
Figure BDA0001955267720000085
in the formula, Nθ＝ρshsΩ2R2Is the annular initial stress;
ρsis the density of the drum;
hsis the thickness of the drum;
r is the median radius of the drum;
υsis the poisson's ratio of the drum;
Lsis the length of the drum;
Esthe rigidity of the drum.
Preferably, the potential energy generated by the spring at the coupling position of the variable cross-section disc and the rotating cylindrical shell in the step 7 is expressed as follows:
Figure BDA0001955267720000091
preferably, the dynamic equation expression of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade system in step 8 is as follows:
Figure BDA0001955267720000092
Figure BDA0001955267720000101
preferably, the specific method described in step 10 is as follows:
dispersing the pre-twisted blades and introducing a regular coordinate Ui(t)、Vi(t)、Wi(t)、ψi(t) and Φi(t), obtaining the displacements of radial vibration, transverse vibration, swinging, bending swing angle and swinging rotation angle of the pre-twisted blade as follows:
Figure BDA0001955267720000102
in the formula (I), the compound is shown in the specification,
Figure BDA0001955267720000103
and
Figure BDA0001955267720000104
and respectively expressed as the mode shape functions of the ith order of the rotation angles corresponding to the radial direction, the transverse direction and the swinging of the pre-twisted blade and the bending direction and the swinging direction of the pre-twisted blade, wherein N is a mode truncation number.
Preferably, after step 11, the method further comprises:
setting an external excitation vector as zero, and determining the natural frequency of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system at different rotating speeds;
and verifying the accuracy of the motion differential equation according to the obtained natural frequency.
Preferably, the first and second electrodes are formed of a metal,
Figure BDA0001955267720000111
and
Figure BDA0001955267720000112
the expression of (a) is as follows:
Figure BDA0001955267720000113
in the formula:
Figure BDA0001955267720000114
βail is the characteristic root value, beta, of the pre-twisted blade in the radial directionfiL is the transverse direction of the pre-twisted blade, betasiL is a characteristic root value of the swing direction;
βaithe first 6 orders of L are: 1.571, 4.712, 7.854, 11, 14.14, 17.28; beta is afiThe first 6 orders of L are: 1.875, 4.686, 7.819, 10.89, 13.9, 16.81; beta is asiThe first 6 orders of L are: 1.874, 4.669, 7.735, 10.63, 13.27, 15.8;
the coupling system of the spring-supported flexible disk and the pre-twisted blade is analyzed by a hypothetical mode method, and the transverse displacement of the elastic disk is expressed as:
Figure BDA0001955267720000115
Wi c(r,θ)＝Ri(r)cos(iθ) (23)
Wi s(r,θ)＝Ri(r)cos(iθ) (24)
in the formula, Wi cAnd Wi sColumn vectors of modal composition are assumed for the disc in two orthogonal planes,
Figure BDA0001955267720000116
and
Figure BDA0001955267720000117
respectively, the generalized coordinates of the elastic disk with respect to time, Ri(r) is the disk mode shape function derived from a variable cross section beam.
(III) advantageous effects
The invention has the beneficial effects that: the dynamic modeling method of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system saves the cost required by the experiment of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system; according to the invention, the dynamic models of different rotary cylindrical shell-variable cross-section disc-pre-twisted blade coupling systems can be obtained only by modifying the structural sizes and material parameters of the cylindrical shell, the variable cross-section disc and the blades, and the operation is simple and convenient; the influence of the installation angle and the torsion angle in the real blade is considered, and the variable cross-section geometric configuration of the flexible disk is closer to a wheel disk in a real aero-engine; the invention considers the influences of centrifugal rigidization, rotary softening, annular initial stress and Coriolis force of the rotary cylindrical shell-variable cross-section disc-pre-twisted blade coupling system in the rotating process, and the dynamic characteristics of the coupling system can better reflect the real working state of the rotary cylindrical shell-variable cross-section disc-pre-twisted blade coupling system; compared with the method for analyzing the dynamic characteristics of the rotating cylindrical shell, the variable cross-section disc and the pre-twisted blade by means of traditional commercial finite element software, the method has higher calculation efficiency and calculation precision. Furthermore, this method has fewer degrees of freedom compared to conventional commercial finite element software. Compared with the existing rotating cylindrical shell-disk coupling system, the model established by the invention has the following advantages:
(1) the rotating cylindrical shell-variable cross-section disc-pre-twisted blade system established by the invention can simulate the rotating cylindrical shell-equal cross-section disc-blade system as long as the variable cross-section disc is degenerated into the equal cross-section disc, namely the rotating cylindrical shell-variable cross-section disc-blade coupling system comprises the existing rotating cylindrical shell-equal cross-section disc-blade system;
(2) the rotating cylindrical shell-variable cross-section disc-pre-twisted blade system can simulate the rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system under different boundary conditions by changing the rigidity of a connecting spring between the variable cross-section disc and the cylindrical shell;
(3) in a cylindrical shell-disc-blade coupling system in an aircraft engine, discs are variable cross-section discs with thick two sides and thin middle parts, the diameter of a drum barrel is usually smaller than that of the disc, and the method accords with the derivation of modeling of a rotary cylindrical shell-variable cross-section disc-pre-twisted blade coupling system, so that the model of the text is more practical, the problem that the model of the rotary cylindrical shell-variable cross-section disc-pre-twisted blade system is larger in difference with that of practical production application is solved, theoretical support is provided for designing the rotary cylindrical shell-variable cross-section disc-pre-twisted blade system in practical application, the resonant interval of the system is avoided through dynamic frequency comparison of the whole coupling system, vibration is reduced, the working speed is reasonably selected, and the system works more stably.
Meanwhile, excitation response analysis of the whole system can be carried out, and therefore system performance and stability are improved.
Drawings
FIG. 1 is a flow chart of a method of a dynamic modeling method of a rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system according to the invention;
FIG. 2 is a schematic structural diagram of a cylindrical shell-variable cross-section disk-pre-twisted blade in an embodiment of a dynamic modeling method of a rotating cylindrical shell-variable cross-section disk-pre-twisted blade coupling system according to the invention;
FIG. 3 is a graph showing the variation of the traveling wave frequency of the rotating cylindrical shell-variable cross-section disk-pre-twisted blade coupling system with the rotating speed in the embodiment of the dynamic modeling method of the rotating cylindrical shell-variable cross-section disk-pre-twisted blade coupling system according to the present invention;
FIG. 4 is a graph comparing a variation curve of a traveling wave frequency of a rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system with a rotation speed in an embodiment of a dynamic modeling method of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system according to the present invention with a result obtained by a finite element method.
Detailed Description
For the purpose of better explaining the present invention and to facilitate understanding, the present invention will be described in detail by way of specific embodiments with reference to the accompanying drawings.
As shown in fig. 1 and 2, an embodiment of the present invention provides a dynamic modeling method for a rotating cylindrical shell-variable cross-section disk-pre-twisted blade coupling system, including the following steps:
step 1: constructing a three-dimensional coordinate system required by dynamic modeling of a rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system;
the three-dimensional coordinate system includes: fixed coordinate system OXYZ of whole coupling system and coordinate system ox of variable cross-section diskdydzdAnd a moving coordinate system ox of the whole system in the motion processryrzrLocal coordinate system ox of pre-twisted bladebybzbGeometric center coordinate system o of intersection circle of wheel disc and drum barrel1x1y1z1Coordinate system o of drum2x2y2z2And a curvilinear coordinate system ox θ z of a curved surface in the drum;
step 2: the structural parameters and the material parameters of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system are obtained, in this embodiment, the coupling system is assumed to be an isotropic linear elastic material, the constitutive relation satisfies the Hooke law, for the variable cross-section disc, a straight line perpendicular to the middle plane before deformation is still a straight line after deformation, the straight line is perpendicular to the middle plane and has no deformation in the middle plane direction, the cylindrical shells are thin-walled cylindrical shells, and the correlation coefficients of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system are shown in table 1:
TABLE 1 rotating cylindrical Shell-variable Cross-section disk-pretwisted blade coupling System parameters
Figure BDA0001955267720000141
Considering the installation angle of the blade on the variable cross-section disc, the initial position of the spring, the translation of the variable cross-section disc and the transverse vibration of the disc, determining the displacement vector of any point Q of the deformed blade under a global coordinate system:
Figure BDA0001955267720000142
wherein β (x) is an angle of rotation of any point on the twisted blade about the x-axis, and β (x) ═ β0+γ(x)，β0Is the mounting angle of the blade, theta is the angular displacement of the disc and drum during rotation, phi is the circumferential rotation angle of the drum, and gamma (x) is the twist angle of the blade.
And step 3: n is uniformly distributed on the set variable cross-section diskbThe same pre-twisted blades;
after the ith pre-twisted blade is deformed, after a series of coordinate conversions are carried out on the displacement vector of any point Q on the pre-twisted blade i in the overall coordinate system OXYZ, the kinetic energy expressions of the pre-twisted blade and the coupling of the pre-twisted blade, the disc and the cylindrical shell are obtained according to a kinetic energy calculation formula as follows:
Figure BDA0001955267720000151
and 4, step 4: based on a plate shell vibration theory, the influence of a centrifugal rigidization effect of the pre-twisted blade in the rotating process is considered, and the potential energy of the rotating pre-twisted blade is obtained;
Figure BDA0001955267720000152
in the formula, E, G, κ, a and L represent the young's modulus, shear coefficient, cross-sectional area and blade length of the blade, respectively.
And 5: the variable cross-section disc can meet the small-deflection theory of transverse vibration of the elastic sheet, and a differential equation of the transverse elastic vibration of the variable cross-section disc is deduced by using the Hamilton principle to obtain the kinetic energy and the potential energy of the variable cross-section disc;
for a spring between a variable cross-section disc and a cylindrical shell, the kinetic energy is Uc：
Figure BDA0001955267720000161
Step 6: based on the Sanders shell theory, the influence of Coriolis force, centrifugal force and annular initial stress of the rotating cylindrical shell is considered, and the kinetic energy and potential energy of the rotating cylindrical shell are obtained;
kinetic energy of rotating variable cross-section disc:
Figure BDA0001955267720000162
potential energy of the rotating variable cross-section disc:
Figure BDA0001955267720000163
in the formula, Wd1、Wd2And Wd3The transverse mode shape functions of the first section, the second section and the third section of the variable cross-section disk are respectively.
And 7: the variable cross-section disc and the drum are connected by adopting a spring and a torsional spring to simulate a bolt, and kinetic energy of the spring and the torsional spring which play a role in connection in the whole rotary drum-variable cross-section disc-pre-twisted blade coupling system is obtained;
and 8: according to Hamilton variation principle
Figure BDA0001955267720000164
Wherein U is U ═ Ub+Us,+Ud+Uθ，T＝Tb+Ts1+Td，WnonActing as an external force and by us、vs、ws、u、v、w、
Figure BDA0001955267720000165
Phi and WdAnd performing variation as an independent variable to obtain a dynamic equation of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade system.
Figure BDA0001955267720000171
Figure BDA0001955267720000181
And step 9: introducing regular coordinates to the displacement u of any point on the middle surface of the rotating cylindrical shell in the step 8 along the longitudinal direction by adopting a Galerkin methodsDisplacement v in tangential directionsAnd a displacement w in the radial directionsRadial displacement u, transverse displacement v, swing direction displacement w and bending direction corner of cantilever beam
Figure BDA0001955267720000182
Swing direction rotation angle phi and variable cross-section discTransverse displacement W ofdAnd carrying out discretization treatment to obtain a mass matrix, a Coriolis force matrix and a rigidity matrix of the rotary drum barrel-variable cross-section disc-pre-twisted blade coupling system. The method comprises the following specific steps:
dispersing the blades and introducing a regular coordinate Ui(t)、Vi(t)、Wi(t)、ψi(t) and Φi(t), obtaining the displacements of the radial vibration, the transverse vibration, the swing, the bending swing angle and the swing rotation angle of the blade as follows:
Figure BDA0001955267720000183
in the formula (I), the compound is shown in the specification,
Figure BDA0001955267720000184
and
Figure BDA0001955267720000185
and respectively expressed as the mode shape function of the ith order of the rotation angles corresponding to the radial direction, the transverse direction and the swing direction of the blade and the bending direction and the swing direction of the blade, and N is the mode truncation number.
The specific expression is as follows:
Figure BDA0001955267720000191
in the formula:
Figure BDA0001955267720000192
βail is the characteristic root value of the radial direction of the blade, betafiL is transverse to the blade, betasiAnd L is a characteristic root value of the swinging direction. Beta is aaiThe first 6 orders of L are: 1.571, 4.712, 7.854, 11, 14.14, 17.28; beta is afiThe first 6 orders of L are: 1.875, 4.686, 7.819, 10.89, 13.9, 16.81; beta is asiThe first 6 orders of L are: 1.874,4.669,7.735, 10.63, 13.27, 15.8.
Using a hypothetical mode method to analyze the coupled system of the spring-supported flexible disk and the blade, the lateral displacement of the flexible disk can be expressed as:
Figure BDA0001955267720000193
Wi c(r,θ)＝Ri(r)cos(iθ) (34)
Wi s(r,θ)＝Ri(r)cos(iθ) (35)
in the formula, Wi cAnd Wi sColumn vectors of modal composition are assumed for the disc in two orthogonal planes,
Figure BDA0001955267720000194
and
Figure BDA0001955267720000195
respectively, the generalized coordinates of the elastic disk with respect to time, Ri(r) is the mode shape function of the disc derived from a variable cross section beam.
Step 10: dimensionless terms are introduced to the variable cross-section disc and the pre-twisted blades, so that the consistency and the correctness when the variable cross-section disc and the pre-twisted blades are coupled with a cylindrical shell are ensured.
Dimensionless coefficient eta x/LiFor the blade, LiIs the length L of the blade; for a cylindrical shell, LiIs LsIs the length of the cylindrical shell; the dimensionless coefficient of the variable cross-section disk is xi ═ (r-r)s)/(rD-rs). Therefore, the temperature of the molten metal is controlled,
Figure BDA0001955267720000196
and
Figure BDA0001955267720000197
respectively carrying out vibration mode functions after dimensionless operation on the blades;
Figure BDA0001955267720000198
and
Figure BDA0001955267720000199
respectively, the vibration mode functions of the variable cross-section disc after dimensionless.
Step 11: introducing Rayleigh damping to obtain a motion differential equation of the rotary spring-variable cross-section disc-pre-twisted blade coupling system:
Figure BDA0001955267720000201
where M, G and D are the mass matrix, Coriolis force matrix and damping matrix, respectively, of the entire coupling system, Ke、Ki、KΩ、KσAnd KcThe matrix is respectively a structural rigidity matrix of the whole coupling system, a rigidity matrix caused by acceleration, a centrifugal rigidization matrix, a rotation softening matrix and a rigidity matrix generated by a cylindrical shell and disc coupling spring. And q and F are a blade regular coordinate vector and an external excitation force vector respectively. M, G, Ke、Ki、KΩ、KσAnd KcThe specific expressions of (a) are respectively:
an expression of a mass matrix of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system is as follows:
Figure BDA0001955267720000202
the expression of each element in the quality matrix is as follows:
Figure BDA0001955267720000203
Figure BDA0001955267720000204
Figure BDA0001955267720000205
Figure BDA0001955267720000211
Figure BDA0001955267720000212
Figure BDA0001955267720000213
Figure BDA0001955267720000214
Figure BDA0001955267720000215
Figure BDA0001955267720000216
Figure BDA0001955267720000217
Figure BDA0001955267720000218
Figure BDA0001955267720000219
Figure BDA00019552677200002110
Figure BDA00019552677200002111
Figure BDA00019552677200002112
Figure BDA00019552677200002113
Figure BDA00019552677200002114
Figure BDA00019552677200002115
Figure BDA00019552677200002116
Figure BDA0001955267720000221
Figure BDA0001955267720000222
Figure BDA0001955267720000223
Figure BDA0001955267720000224
Figure BDA0001955267720000225
Figure BDA0001955267720000226
the expression of the Coriolis force matrix of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system is as follows:
Figure BDA0001955267720000227
the expressions of the elements in the coriolis force matrix are:
Figure BDA0001955267720000228
Figure BDA0001955267720000229
Figure BDA00019552677200002210
Figure BDA00019552677200002211
Figure BDA00019552677200002212
Figure BDA0001955267720000231
Figure BDA0001955267720000232
Figure BDA0001955267720000233
Figure BDA0001955267720000234
Figure BDA0001955267720000235
Figure BDA0001955267720000236
an expression of a structural rigidity matrix of the rotary cylindrical shell-variable cross-section disc-pre-twisted blade coupling system is as follows:
Figure BDA0001955267720000237
the expressions of the elements in the structural rigidity matrix are as follows:
Figure BDA0001955267720000241
Figure BDA0001955267720000242
Figure BDA0001955267720000243
Figure BDA0001955267720000244
Figure BDA0001955267720000245
Figure BDA0001955267720000246
Figure BDA0001955267720000247
Figure BDA0001955267720000248
Figure BDA0001955267720000249
Figure BDA00019552677200002410
Figure BDA00019552677200002411
Figure BDA00019552677200002412
Figure BDA00019552677200002413
Figure BDA00019552677200002414
Figure BDA00019552677200002415
Figure BDA00019552677200002416
Figure BDA0001955267720000251
Figure BDA0001955267720000252
Figure BDA0001955267720000253
an expression of a rigidity matrix caused by acceleration of a rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system is as follows:
Figure BDA0001955267720000261
the expressions of the elements in the stiffness matrix caused by the acceleration are as follows:
Figure BDA0001955267720000262
Figure BDA0001955267720000263
an expression of a centrifugal rigidization matrix of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system is as follows:
Figure BDA0001955267720000264
the expressions of the elements in the centrifugal rigidization matrix are as follows:
Figure BDA0001955267720000271
Figure BDA0001955267720000272
Figure BDA0001955267720000273
Figure BDA0001955267720000274
Figure BDA0001955267720000275
Figure BDA0001955267720000276
Figure BDA0001955267720000277
Figure BDA0001955267720000278
Figure BDA0001955267720000279
Figure BDA00019552677200002710
Figure BDA00019552677200002711
Figure BDA00019552677200002712
Figure BDA00019552677200002713
Figure BDA00019552677200002714
Figure BDA00019552677200002715
Figure BDA00019552677200002716
Figure BDA00019552677200002717
Figure BDA00019552677200002718
Figure BDA00019552677200002719
an expression of a rotational softening rigidity matrix of the rotational cylindrical shell-variable cross-section disc-pre-twisted blade coupling system:
Figure BDA0001955267720000281
the expressions of the elements in the matrix of the rotational softening stiffness are as follows:
Figure BDA0001955267720000282
Figure BDA0001955267720000283
Figure BDA0001955267720000284
Figure BDA0001955267720000285
an expression of a stiffness matrix of a coupling spring of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system is as follows:
Figure BDA0001955267720000291
the expressions of the elements in the coupling stiffness matrix are as follows:
Figure BDA0001955267720000292
Figure BDA0001955267720000293
Figure BDA0001955267720000294
Figure BDA0001955267720000295
Figure BDA0001955267720000296
Figure BDA0001955267720000297
Figure BDA0001955267720000298
Figure BDA0001955267720000299
the Rayleigh damping D is constructed by combining a mass matrix and a rigidity matrix in proportion, wherein D is alpha M + beta K, and alpha and beta can be obtained by the following formula:
Figure BDA00019552677200002910
in the formula (f)n1、fn2First and second order natural frequencies (Hz), ξ, respectively of the blade1And xi2Is the damping ratio.
Calculating the characteristic value lambda of the characteristic equation coefficient determinant, dividing the absolute value of the imaginary part by 2 pi, and sequencing from small to large to obtain a group of natural frequencies omegakWhere k denotes the kth order of the mode of the rotating cylindrical shell-variable cross-section disk-pretwisted blade coupling system, and k is 1,2, ….
The effectiveness of the invention is verified by a finite element model, and tables 2 and 3 respectively list the comparison of the natural frequency of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system in a static state and the natural frequency after the structural parameters are changed with the result obtained by finite element software, and the error magnitude.
FIG. 3 shows the traveling wave frequency versus rotational speed for a rotating cylindrical shell-variable cross-section disk-pre-twisted blade coupling system plotted using the methods herein.
FIG. 4 is a graph showing the variation of the traveling wave frequency of the rotating cylindrical shell-variable cross-section disk coupling system with the rotation speed according to the method of the present invention compared with the result obtained by the finite element method.
TABLE 2 natural frequency comparison of rotating drum-variable cross-section disk-blade coupling system
Figure BDA0001955267720000301
TABLE 3 Effect of different aspect ratios on the natural frequency of the rotating drum-variable-section disk-blade coupling System
Figure BDA0001955267720000302
Figure BDA0001955267720000311
The following conclusions can be drawn from the results of this example:
(1) the natural frequency of the variable cross-section disc in the rotating cylindrical shell-variable cross-section disc coupling system is increased along with the increase of the rotating speed, which shows that the centrifugal rigidization effect is obvious in the rotating softening effect for the rotating cylindrical shell-variable cross-section disc coupling;
(2) along with the increase of the length-diameter ratio of the drum, the natural frequency of the drum is reduced, the natural frequency of the variable cross-section disc is reduced by a small amplitude, and the natural frequency of the blade is almost not changed, which shows that the change of the structural parameters of the drum has almost no influence on the blade in the coupling system;
(3) the analytical model is well matched with the first few orders of frequency of the ANSYS model, so that the reliability and the effectiveness of the dynamic model established by the method are verified.
The technical principles of the present invention have been described above in connection with specific embodiments, which are intended to explain the principles of the present invention and should not be construed as limiting the scope of the present invention in any way. Based on the explanations herein, those skilled in the art will be able to conceive of other embodiments of the present invention without inventive efforts, which shall fall within the scope of the present invention.

Claims (10)

1. A dynamic modeling method for a rotating cylindrical shell-variable cross-section disc-pre-twisted blade system is characterized by comprising the following steps of:
step 1: constructing a three-dimensional coordinate system required by dynamic modeling of a rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system;
the three-dimensional coordinate system includes: fixed coordinate system OXYZ of whole coupling system and coordinate system ox of variable cross-section diskdydzdAnd a moving coordinate system ox of the whole system in the motion processryrzrLocal coordinate system ox of pre-twisted bladebybzbGeometric center coordinate system o of intersection circle of wheel disc and drum barrel1x1y1z1Coordinate system o of drum2x2y2z2And a curvilinear coordinate system ox θ z of a curved surface in the drum;
step 2: measuring structural parameters and material parameters of a rotary cylindrical shell-variable cross-section disc-pre-twisted blade coupling system;
the parameters measured include: length L of pre-twisted blade, width b of pre-twisted blade, thickness h of pre-twisted blade, installation angle beta of pre-twisted blade, elastic modulus E of pre-twisted blade, Poisson ratio mu, density rho of pre-twisted blade and inner diameter r of variable cross-section discsFirst radius rdSecond radius rDOuter ring radius RdFirst section thickness h of the discD1A second section thickness hD2A third section thickness hD3Elastic modulus E of the diskdDisc density pdLength L of the rotating cylindrical shellsRadius of curved surface R in rotating cylindrical shell and elastic modulus E of rotating cylindrical shellsDensity rho of rotary cylindrical shellsAnd a Poisson ratio upsilon of the rotating cylindrical shells；
The rigidity of the translational spring and the torsional spring in the x direction, the y direction and the z direction is respectively kc,u、kc,v、kc,wAnd
Figure FDA0002623191360000011
and step 3: n is uniformly distributed on the set variable cross-section diskbThe same pre-twisted blades;
after the ith pre-twisted blade is deformed, converting a series of coordinates through a displacement vector of any point Q on the pre-twisted blade i in the overall coordinate system OXYZ, and obtaining the kinetic energy of the pre-twisted blade and the coupling of the pre-twisted blade, the variable cross-section disc and the rotating cylindrical shell according to a kinetic energy calculation formula;
and 4, step 4: based on a plate shell vibration theory, the influence of a centrifugal rigidization effect of the pre-twisted blade in the rotating process is considered, and the potential energy of the rotating pre-twisted blade is obtained;
and 5: the variable cross-section disc can meet the small-deflection theory of transverse vibration of the elastic sheet, and a differential equation of the transverse elastic vibration of the variable cross-section disc is deduced by using the Hamilton principle to obtain the kinetic energy and the potential energy of the variable cross-section disc;
step 6: based on the Sanders shell theory, the influence of Coriolis force, centrifugal force and annular initial stress of the rotating cylindrical shell is considered, and the kinetic energy and potential energy of the rotating cylindrical shell are obtained;
and 7: the variable cross-section disc and the drum are connected by adopting a spring and a torsional spring to simulate a bolt, and kinetic energy of the spring and the torsional spring which play a role in connection in the whole rotary drum-variable cross-section disc-pre-twisted blade coupling system is obtained;
and 8: according to Hamilton variation principle
Figure FDA0002623191360000021
Wherein U is U ═ Ub+Us,+Ud+Uθ，T＝Tb+Ts1+Td；
WnonActing as an external force and by us、vs、ws、u、v、w、
Figure FDA0002623191360000022
Phi and WdPerforming variational division as an independent variable to obtain a dynamic equation of a rotating cylindrical shell-variable cross-section disc-pre-twisted blade system;
and step 9: introducing regular coordinates to the displacement u of any point on the middle surface of the rotating cylindrical shell in the step 8 along the longitudinal direction by adopting a Galerkin methodsDisplacement v in tangential directionsDisplacement w in radial directionsRadial displacement u, transverse displacement v, swing direction displacement w and bending direction corner of cantilever beam
Figure FDA0002623191360000023
Swing direction rotation angle phi and lateral displacement W of variable cross-section diskdGo on to separateDispersing treatment;
obtaining a mass matrix, a Coriolis force matrix and a rigidity matrix of the rotary drum barrel-variable cross-section disc-pre-twisted blade coupling system;
step 10: dimensionless terms are introduced into the variable cross-section disc and the pre-twisted blades to ensure consistency when the variable cross-section disc and the pre-twisted blades are coupled with the rotating cylindrical shell;
dimensionless coefficient eta x/LiWhen corresponding to pre-twisting blades, LiThe length L of the pre-twisted blade is;
when the cylindrical shell is correspondingly rotated, LiFor the length L of the rotating cylindrical shells；
The dimensionless coefficient of the variable cross-section disk is xi ═ (r-r)s)/(rD-rs)；
Step 11: introducing Rayleigh damping to obtain a motion differential equation of the rotary cylindrical shell-variable cross-section disc-pre-twisted blade coupling system:
Figure FDA0002623191360000024
in the formula, M, G and D are respectively a mass matrix, a Coriolis force matrix and a damping matrix of the whole coupling system;
Ke、Ki、KΩ、Kσand KcRespectively a structural rigidity matrix of the whole coupling system, a rigidity matrix caused by acceleration, a centrifugal rigidization matrix, a rotary softening matrix and a rigidity matrix generated by a rotary cylindrical shell and a disc coupling spring;
q and F are a pre-twisted blade regular coordinate vector and an external excitation force vector respectively;
the damping matrix D is obtained by combining the mass matrix and the rigidity matrix in proportion;
specifically, the following formula:
D＝αM+βK；
wherein α and β are obtained by the following formula:
Figure FDA0002623191360000031
in the formula (f)n1、fn2First and second order natural frequencies (Hz), ξ, respectively, of a spring-variable cross-section disk-pretwisted blade coupling system1And xi2Is the damping ratio;
q is a column vector composed of generalized coordinates related to time, expressed as follows:
Figure FDA0002623191360000032
2. the modeling method of claim 1, wherein the kinetic energy expression in step 3 is:
Figure FDA0002623191360000041
in the formula, rQThe displacement vector of any point Q on the pre-twisted blade under the global coordinate system is obtained;
a is the cross-sectional area of the pre-twisted blade;
Izthe section inertia moment of the pre-twisted blade section around the y axis is obtained;
Iythe section inertia moment of the pre-twisted blade section around the z axis is obtained;
theta is the angular displacement of the wheel disc movement;
the symbol (-) represents a 1 st order partial derivative of time, rDIs the second radius of the disc, us、vs、wsIs the displacement of the curved surface in the rotating cylindrical shell in three directions in the global coordinate system, x, y and z are the local coordinate system of the pre-twisted blade along the length direction, the thickness direction and the swing direction of the pre-twisted blade respectively,
Figure FDA0002623191360000042
phi is the rotation angle of the cantilever beam in the bending direction and the swinging direction, and u, v and w are the displacement of the cantilever beam in the radial direction, the transverse direction and the swinging direction in the local coordinate system，WdIs the lateral displacement of the disc.
3. The modeling method according to claim 1, wherein the strain potential energy expression of the rotating pretwisted blade in the step 4 is as follows:
Figure FDA0002623191360000051
in the formula, E, G, κ, A and L represent the Young's modulus, shear coefficient, cross-sectional area and length of the pre-twisted blade, respectively.
4. The modeling method of claim 1, wherein the kinetic energy expression of the variable cross-section disk in step 5 is:
Figure FDA0002623191360000052
the potential energy expression of the variable cross-section disc is as follows:
Figure FDA0002623191360000053
in the formula, Wd1、Wd2And Wd3The transverse mode shape functions of the first section, the second section and the third section of the variable cross-section disc are respectively; the expressions for harmonic operators and flexural stiffness are as follows:
Figure FDA0002623191360000061
Figure FDA0002623191360000062
Figure FDA0002623191360000063
Figure FDA0002623191360000064
Figure FDA0002623191360000065
Figure FDA0002623191360000066
Figure FDA0002623191360000067
Figure FDA0002623191360000068
Figure FDA0002623191360000069
Figure FDA00026231913600000610
5. the modeling method of claim 1, wherein the kinetic energy generated by rotating the cylindrical shell in step 6 is expressed as:
Figure FDA00026231913600000611
the strain energy expression of the rotating cylindrical shell in consideration of stretching and bending is as follows:
Figure FDA0002623191360000071
the expression of the initial hoop stress generated under the action of centrifugal force is as follows:
Figure FDA0002623191360000072
in the formula, Nθ＝ρshsΩ2R2Is the annular initial stress;
ρsthe density of the drum of the rotary cylindrical shell; h issIs the thickness of the drum of the rotary cylindrical shell;
r is the radius of the middle curved surface of the drum barrel of the rotary cylindrical shell;
υsis the Poisson's ratio of the rotating cylindrical shell drum;
Lsis the length of the drum of the rotating cylindrical shell;
Esthe rigidity of the drum of the rotating cylindrical shell.
6. The modeling method of claim 1, wherein the potential energy generated by the spring at the coupling position of the variable cross-section disc and the rotating cylindrical shell in the step 7 is expressed as:
Figure FDA0002623191360000073
7. the modeling method according to claim 1, wherein the dynamic equation of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade system in step 8 is expressed as follows:
Figure FDA0002623191360000081
Figure FDA0002623191360000091
8. the modeling method of claim 1, wherein the specific method of step 10 is as follows:
dispersing the pre-twisted blades and introducing a regular coordinate Ui(t)、Vi(t)、Wi(t)、ψi(t) and Φi(t), obtaining the displacements of radial vibration, transverse vibration, swinging, bending swing angle and swinging rotation angle of the pre-twisted blade as follows:
Figure FDA0002623191360000092
in the formula (I), the compound is shown in the specification,
Figure FDA0002623191360000093
and
Figure FDA0002623191360000094
and respectively expressed as the mode shape functions of the ith order of the rotation angles corresponding to the radial direction, the transverse direction and the swinging of the pre-twisted blade and the bending direction and the swinging direction of the pre-twisted blade, wherein N is a mode truncation number.
9. The modeling method of claim 1, wherein after step 11, the method further comprises:
setting an external excitation vector as zero, and determining the natural frequency of the rotating cylindrical shell-variable cross-section disc-pre-twisted blade coupling system at different rotating speeds;
and verifying the accuracy of the motion differential equation according to the obtained natural frequency.
10. The modeling method of claim 8,
Figure FDA0002623191360000101
Figure FDA0002623191360000102
and
Figure FDA0002623191360000103
the expression of (a) is as follows:
Figure FDA0002623191360000104
in the formula:
Figure FDA0002623191360000105
βail is the characteristic root value, beta, of the pre-twisted blade in the radial directionfiL is the transverse direction of the pre-twisted blade, betasiL is a characteristic root value of the swing direction;
βaithe first 6 orders of L are: 1.571, 4.712, 7.854, 11, 14.14, 17.28; beta is afiThe first 6 orders of L are: 1.875, 4.686, 7.819, 10.89, 13.9, 16.81; beta is asiThe first 6 orders of L are: 1.874, 4.669, 7.735, 10.63, 13.27, 15.8;
the coupling system of the spring-supported flexible disk and the pre-twisted blade is analyzed by a hypothetical mode method, and the transverse displacement of the elastic disk is expressed as:
Figure FDA0002623191360000106
Wi c(r,θ)＝Ri(r)cos(iθ) (23)
Wi s(r,θ)＝Ri(r)cos(iθ) (24)
in the formula, Wi cAnd Wi sIs a plateA column vector of modal contributions is assumed in two orthogonal planes,
Figure FDA0002623191360000107
and
Figure FDA0002623191360000108
respectively, the generalized coordinates of the elastic disk with respect to time, Ri(r) is the disk mode shape function derived from a variable cross section beam.
CN201910064646.9A 2019-01-23 2019-01-23 Dynamic modeling method for rotating cylindrical shell-variable cross-section disc-pre-twisted blade system Active CN109800512B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN201910064646.9A CN109800512B (en) 2019-01-23 2019-01-23 Dynamic modeling method for rotating cylindrical shell-variable cross-section disc-pre-twisted blade system

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN201910064646.9A CN109800512B (en) 2019-01-23 2019-01-23 Dynamic modeling method for rotating cylindrical shell-variable cross-section disc-pre-twisted blade system

Publications (2)

Publication Number Publication Date
CN109800512A CN109800512A (en) 2019-05-24
CN109800512B true CN109800512B (en) 2020-11-10

Family

ID=66560105

Family Applications (1)

Application Number Title Priority Date Filing Date
CN201910064646.9A Active CN109800512B (en) 2019-01-23 2019-01-23 Dynamic modeling method for rotating cylindrical shell-variable cross-section disc-pre-twisted blade system

Country Status (1)

Country Link
CN (1) CN109800512B (en)

Families Citing this family (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN110795783B (en) * 2019-09-27 2022-07-29 哈尔滨工程大学 Method for forecasting natural frequency of liquid-filled cylindrical shell
CN112270065B (en) * 2020-08-26 2023-09-08 天津大学 Dynamic stability prediction method for eccentric rotary annular periodic structure
CN113190930B (en) * 2021-05-26 2022-09-30 清华大学 Coupling dynamics modeling and analyzing method for flexible shaft-disk-shell connection rotor system
CN114235242B (en) * 2021-12-02 2024-06-18 中国航发沈阳发动机研究所 Blade two-dimensional mass moment measuring device
CN113919104B (en) * 2021-12-13 2022-04-26 清华大学 Method and system for acquiring nonlinear dynamic response of rotary drum
CN116522546B (en) * 2023-07-04 2023-09-15 太原理工大学 Container configuration reverse solving method for uniformly removing blade part through barreling finishing

Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JP2000034130A (en) * 1998-07-14 2000-02-02 Shinetsu Quartz Prod Co Ltd Production of quartz glass pipe and apparatus therefor
CN104112042A (en) * 2014-08-26 2014-10-22 同济大学 Measuring method for diamond coating film-substrate interface bonding strength based on molecular dynamics
CN104166758A (en) * 2014-08-07 2014-11-26 东北大学 Determination method for inherent frequency of rotor-blade coupled system
CN108804853A (en) * 2018-06-28 2018-11-13 东北大学 Shape convex multifunction dynamic modeling method is turned round under elastic bearing based on non-uniform beam
CN108897973A (en) * 2018-07-23 2018-11-27 东北大学 A kind of spring-variable cross-section disk-blade system dynamic modeling method

Family Cites Families (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101532906A (en) * 2009-04-27 2009-09-16 东南大学 Method for analyzing fluid dynamics and structural mechanics of wind generator blades
CN103970033B (en) * 2014-05-20 2016-08-17 河海大学常州校区 The method realizing robot solid modelling and the emulation of blade laser detection based on MATLAB

Patent Citations (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JP2000034130A (en) * 1998-07-14 2000-02-02 Shinetsu Quartz Prod Co Ltd Production of quartz glass pipe and apparatus therefor
CN104166758A (en) * 2014-08-07 2014-11-26 东北大学 Determination method for inherent frequency of rotor-blade coupled system
CN104112042A (en) * 2014-08-26 2014-10-22 同济大学 Measuring method for diamond coating film-substrate interface bonding strength based on molecular dynamics
CN108804853A (en) * 2018-06-28 2018-11-13 东北大学 Shape convex multifunction dynamic modeling method is turned round under elastic bearing based on non-uniform beam
CN108897973A (en) * 2018-07-23 2018-11-27 东北大学 A kind of spring-variable cross-section disk-blade system dynamic modeling method

Also Published As

Publication number Publication date
CN109800512A (en) 2019-05-24

Similar Documents

Publication Publication Date Title
CN109800512B (en) Dynamic modeling method for rotating cylindrical shell-variable cross-section disc-pre-twisted blade system
Gu et al. Free vibration of rotating cantilever pre-twisted panel with initial exponential function type geometric imperfection
CN108897973B (en) Dynamics modeling method of spring-variable cross-section disc-blade system
CN108804853B (en) Elastic support lower torsional shoulder blade dynamic modeling method based on variable cross-section beam
Banerjee Dynamic stiffness formulation and free vibration analysis of centrifugally stiffened Timoshenko beams
Chun et al. Vibration analysis of shaft-bladed disk system by using substructure synthesis and assumed modes method
CN104166758B (en) Determination method for inherent frequency of rotor-blade coupled system
Schmiechen Travelling wave speed coincidence
CN110457740B (en) Response characteristic analysis method of mechanical structure under basic excitation
CN110610049B (en) Method for analyzing mechanical characteristics of blade and casing system under rub-impact fault
CN107194032B (en) Torsional blade dynamic modeling method based on installation angle
CN109614707B (en) Dynamic modeling method based on stepped shaft-flexible disk coupling system
CN113190930B (en) Coupling dynamics modeling and analyzing method for flexible shaft-disk-shell connection rotor system
CN112149245B (en) Flexible shaft-disc system coupling dynamics modeling and analyzing method
CN111382503A (en) Vibration analysis method and system for flexible circular ring rotating under elastic support
CN110532732B (en) Method for determining rubbing relationship between blade and casing
Lee et al. Dynamic response of coupled shaft torsion and blade bending in rotor blade system
Zhang et al. Study on localization influences of frequency veering on vibration of mistuned bladed disk
CN113868797A (en) Dynamic design method for tuned mass damper array in blade disc vibration reduction structure
Riso Output-Based Prediction of Whirl Flutter Limit-Cycle Oscillations Considering Multiple Varying Parameters
Williams et al. Three dimensional unsteady aerodynamics and aeroelastic response of advanced turboprops
Wu et al. An analytical model of a rotating radial cantilever beam considering the coupling between bending, stretching, and torsion
CN113919104B (en) Method and system for acquiring nonlinear dynamic response of rotary drum
Jia On the bending coupled natural frequencies of a spinning, multispan Timoshenko shaft carrying elastic disks
CN111783258B (en) Inherent characteristic and pitch diameter vibration prediction method of thin rim gear system

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant