CN113065202A - Graphene variable cross-section blade dynamics modeling method - Google Patents

Graphene variable cross-section blade dynamics modeling method Download PDF

Info

Publication number
CN113065202A
CN113065202A CN202110257491.8A CN202110257491A CN113065202A CN 113065202 A CN113065202 A CN 113065202A CN 202110257491 A CN202110257491 A CN 202110257491A CN 113065202 A CN113065202 A CN 113065202A
Authority
CN
China
Prior art keywords
graphene
variable cross
section blade
blade
section
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.)
Pending
Application number
CN202110257491.8A
Other languages
Chinese (zh)
Inventor
张伟
崔派
陈杰
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.)
Beijing University of Technology
Original Assignee
Beijing University of Technology
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 Beijing University of Technology filed Critical Beijing University of Technology
Priority to CN202110257491.8A priority Critical patent/CN113065202A/en
Publication of CN113065202A publication Critical patent/CN113065202A/en
Pending legal-status Critical Current

Links

Images

Classifications

    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces

Abstract

The invention discloses a graphene variable cross-section blade dynamics modeling method, which comprises the following steps: constructing a simplified model of the graphene variable cross-section blade; performing dynamic modeling and free vibration analysis on the graphene variable cross-section blade; a Rayleigh-Ritz method is adopted to solve a dynamic equation of free vibration of the graphene variable cross-section blade, and influence of parameters of the graphene material on vibration frequency of the variable cross-section blade is analyzed. The method overcomes the defect that the influence of the graphene composite material is not considered in the traditional modeling method of the variable cross-section blade of the aero-engine, can describe the influence of the excellent mechanical property of the graphene composite material on the vibration of the variable cross-section blade, has high practicability, and provides an accurate dynamic modeling method for solving the problem of free vibration of the variable cross-section trapezoidal plate by adopting the Ritz method.

Description

Graphene variable cross-section blade dynamics modeling method
Technical Field
The invention relates to a graphene variable cross-section blade dynamics modeling method, and belongs to the technical field of aero-engine blade dynamics.
Background
With the rapid development of the modern aviation industry, people have more and more researches on the aero-engine, and the blade is taken as one of the key parts of the aero-engine, so that the performance, the safety and the reliability of the aero-engine are directly influenced. The graphene has excellent mechanical and thermal properties, is widely applied to the engineering fields of aerospace and the like, and researches show that the mechanical properties of the material can be effectively changed by adding a small amount of graphene serving as a reinforcement into the composite material. At present, for the problem of blade dynamics modeling, the graphene composite material and the blade structure are combined, so that the vibration of the blade structure can be greatly influenced. The patent technology provides a graphene variable cross-section blade dynamics modeling method which is used for evaluating the dynamics behavior of a blade in free vibration.
Disclosure of Invention
The invention aims to illustrate the influence of excellent mechanical properties of a graphene composite material on free vibration of a variable cross-section blade of an aeroengine, and provides a graphene variable cross-section blade dynamics modeling method.
The purpose of the invention is realized by the following technical scheme:
the graphene variable cross-section blade dynamics modeling method comprises the following steps:
step 1: constructing a simplified model of the graphene variable cross-section blade;
step 2: performing dynamic modeling and free vibration analysis on the graphene variable cross-section blade;
and step 3: a Rayleigh-Ritz method is adopted to solve a dynamic equation of free vibration of the graphene variable cross-section blade, and influence of parameters of the graphene material on vibration frequency of the variable cross-section blade is analyzed.
Compared with the prior art, the graphene variable cross-section blade dynamics modeling method has the following beneficial effects:
(1) aiming at the problem of free vibration of the blades of the aero-engine, a graphene variable cross-section blade structure is constructed; the influence of the excellent mechanical property of the graphene on the vibration is considered.
(2) The method is high in applicability, the frequency of the graphene structure in free vibration can be conveniently solved, the method is high in practicability, and the engineering application value is good.
(3) Aiming at the actual working conditions of the blades of the aero-engine, the energy of free vibration of the blades with variable cross sections is calculated, a dynamic equation is obtained, and the method is convenient for solving the problem of free vibration of other plate models.
(4) The Rayleigh-Ritz method is used for solving the established variable cross-section blade dynamic equation, the energy problem of the blade during free vibration can be fully considered, and the algorithm is more efficient.
(5) This patent can go deep into the analysis graphite alkene material parameter, frequency's influence when freely vibrating to aeroengine variable cross section blade.
Drawings
FIG. 1 is a flow chart of graphene variable cross-section blade dynamics modeling according to the present invention;
FIG. 2 is a simplified schematic representation of a graphene variable cross-section blade according to the present invention;
FIG. 3 is a graph of dimensional parameters of graphene trapezoidal plates of the present invention;
FIG. 4 is a schematic diagram of a graphene variable cross-section blade coordinate system of the present invention;
FIG. 5 is a graph of the distribution of graphene according to the present invention;
FIG. 6 is a graph comparing the effect of graphene mass fraction on frequency for different distribution modes of the present invention;
FIG. 7 is a graph comparing the effect of the number of graphene layers on frequency for different distribution modes according to the present invention;
FIG. 8 is a graph comparing the effect of the size parameters of graphene small blocks on frequency in different distribution modes according to the present invention;
Detailed Description
The invention will be further described by the following specific examples in conjunction with the drawings, which are provided for illustration only and are not intended to limit the scope of the invention.
The invention provides a graphene variable cross-section blade dynamics modeling method which mainly comprises the following steps:
step 1: constructing a simplified model of the graphene variable cross-section blade;
as shown in FIG. 2, the invention simplifies the aero-engine blade to be fixed on a high-speed rotating hub with a pre-twist angle thetaRPre-installation angle thetarThe variable cross-section cantilever trapezoidal plate model. The twist angle of the blade along the x-axis may be represented as θ: (x)＝θr+ β x, where β ═ θRAnd a are the lengths of the blades. As shown in fig. 3, the top view of the blade is a trapezoidal plate, the thickness is h, the width of the joint between the blade and the hub is b, the width of the tip is c, the variation law of the whole plate width is b (x) ═ kx + c, and k is the plate width variation coefficient. The rotation speed of the blades is omega.
In order to accurately represent the position of each point on the blade in the inertial frame, three coordinate systems are established on the model, as shown in fig. 4.
Figure BDA0002968162520000031
Is an inertial coordinate system and is fixed at the center of the hub; rotating coordinate system
Figure BDA0002968162520000032
A blade root located and rotating with the blade; coordinate system of cross section
Figure BDA0002968162520000033
Located on the blade cross-section.
Step 2: performing dynamic modeling and free vibration analysis on the graphene variable cross-section blade;
the variable cross-section blade is assumed to be uniformly divided into N layers in the thickness direction, and the volume fraction of graphene in each layer is changed in a gradient or uniform manner along with the change of the number of layers. As shown in fig. 5, the distribution mode of graphene can be divided into a uniform distribution (U-shaped distribution) and a functionally gradient distribution (X-shaped distribution and O-shaped distribution) according to the content of graphene in each layer of matrix. The U-shaped distribution indicates that the content of graphene in each layer is the same; the X-type distribution shows that the content of graphene is high at the top layer and the bottom layer and low in the middle; the distribution of the O type is opposite to that of the X type, the graphene content is high in the middle, and the top layer and the bottom layer are low. In fig. 5, the color shades represent the content of graphene.
The mass fraction expressions of the kth layer of graphene under different distribution modes of the graphene are respectively as follows:
Figure BDA0002968162520000034
wherein, k is 1,2GIs the total mass fraction of graphene.
According to the rule of mixture, equivalent density
Figure BDA0002968162520000035
And poisson's ratio
Figure BDA0002968162520000036
The calculation formula of (2) is as follows:
Figure BDA0002968162520000041
where ρ isG、ρM、νG、νMThe density and the poisson ratio of the graphene and the blade matrix are respectively; the terms "G", "M" and "c" respectively represent graphene, a blade substrate and a composite material.
The equivalent elastic modulus of the k layer of the blade can be calculated by a modified Halpin-Tsai model:
Figure BDA0002968162520000042
wherein E isG、EMThe elastic modulus of the graphene and the elastic modulus of the blade matrix are respectively set;
Figure BDA0002968162520000043
the volume fraction of the k-th graphene layer is shown as the following formula:
Figure BDA0002968162520000044
coefficient etaL、ηWThe expression of (a) is:
Figure BDA0002968162520000045
ξL、ξWas geometrical parameters of grapheneThe calculation formula is as follows:
Figure BDA0002968162520000046
wherein lG、hG、wGThe length, thickness and width of the graphene nubs are respectively.
According to the first-order shear deformation theory, the displacement field of any point on the blade on the section coordinate system can be obtained:
u(x,y,z,t)＝u0(x,y,t)+zφx(x,y,t) (7a)
v(x,y,z,t)＝v0(x,y,t)+zφy(x,y,t) (7b)
w(x,y,z,t)＝w0(x,y,t) (7c)
wherein u is0、v0、w0Respectively displacement of the middle plane along the directions of x, y and z; phi is ax、φyThe rotation angles of the middle plane around the y axis and the x axis respectively.
Coordinate system of cross section
Figure BDA0002968162520000051
And a rotating coordinate system
Figure BDA0002968162520000052
The included angle between the two is theta (x), and the conversion relationship between the two is as follows:
Figure BDA0002968162520000053
rotating coordinate system
Figure BDA0002968162520000054
And inertial frame
Figure BDA0002968162520000055
The conversion relationship is as follows:
Figure BDA0002968162520000056
wherein theta is an included angle between two coordinate systems, and the expression is
Figure BDA0002968162520000057
From equations (8) and (9), the position coordinate function of any point on the blade in the inertial coordinate system can be obtained:
Figure BDA0002968162520000058
from the Von Karman geometric non-linear relationship, the strain-displacement relationship can be obtained:
Figure BDA0002968162520000059
the constitutive equation of the k-th layer of the blade in the thickness direction is as follows:
Figure BDA00029681625200000510
wherein epsilon and gamma are respectively positive strain and shear strain; sigma and tau are respectively normal stress and shear stress; and is
Figure BDA0002968162520000061
The kinetic energy of the blade in free vibration can be obtained according to the position coordinate function, and the expression is as follows:
Figure BDA0002968162520000062
the blade potential energy expression is as follows:
U＝U1+U2 (15)
wherein, U1To deform potential energy, U2Centrifugal potential energy generated by centrifugal force when the blades rotate at high speed; according to formulae (11) and (12), U1The expression is as follows:
Figure BDA0002968162520000063
the main source of centrifugal potential energy of the blades is the spanwise centrifugal force FcxChord-wise centrifugal force FcyThe expression is as follows:
Figure BDA0002968162520000064
wherein the content of the first and second substances,
Figure BDA0002968162520000065
Figure BDA0002968162520000066
and step 3: a Rayleigh-Ritz method is adopted to solve a dynamic equation of free vibration of the graphene variable cross-section blade, and influence of parameters of the graphene material on vibration frequency of the variable cross-section blade is analyzed.
Rayleigh-Ritz method scheme:
the simplified model of the rotating blade is a cantilever trapezoidal plate, the boundary conditions of the fixed edge (x ═ 0) are all 0 displacements, and the boundary conditions of the free edge (x ═ L, y ═ b (x)/2) are all zero stresses and moments. According to the boundary conditions, the displacement and rotation angle of the system are assumed to be of the form:
u0(x,y,t)＝U(x,y)eiωt (20a)
v0(x,y,t)＝V(x,y)eiωt (20b)
w0(x,y,t)＝W(x,y)eiωt (20c)
φx(x,y,t)＝Φx(x,y)eiωt (20d)
φy(x,y,t)＝Φy(x,y)eiωt (20e)
where ω is the frequency of the plate and t is the time variable; u (x, y), V (x, y), W (x, y), Φx(x,y)，Φy(x, y) is a mode shape function satisfying the boundary condition of the cantilever plate, and the expression is as follows:
Figure BDA0002968162520000071
Figure BDA0002968162520000072
Figure BDA0002968162520000073
Figure BDA0002968162520000074
Figure BDA0002968162520000075
wherein, Umn，Vmn，Wmn，Φxmn，ΦymnIs the undetermined coefficient.
Substituting equations (20) and (21) into the kinetic energy equation (14) and the potential energy equation (15), neglecting the nonlinear terms, the maximum kinetic energy T can be obtainedmaxAnd maximum potential energy Umax. According to the Rayleigh-Ritz method, the following expression can be obtained:
Figure BDA0002968162520000076
Figure BDA0002968162520000077
Figure BDA0002968162520000078
Figure BDA0002968162520000081
Figure BDA0002968162520000082
the above formula is organized into a matrix form as follows:
(K-ω2M)P＝0 (23)
wherein K is a stiffness matrix and M is a mass matrix; p is coefficient vector, and expression is P ═ U11...Umn,V11...Vmn,W11...Wmn,Φx11...Φxmn,Φy11...Φymn}T. By solving the eigenvalue and coefficient vector of the coefficient matrix of the formula (23), the natural frequency of the cantilever trapezoidal plate when freely vibrating can be obtained.
The material parameters of the graphene are as follows: eG＝1010GPa，ρG＝1062kg/m3，νG＝0.186，lG＝2.5μm，wG＝1.5μm，hG＝1.5nm，f G1%, and N10; the material parameters of the blade base body are as follows: eM＝3GPa，ρM＝1200kg/m3，νM＝0.34。
And finally, solving the dynamic characteristics of the graphene variable cross-section blade during free vibration to obtain the vibration frequency of the graphene variable cross-section blade. In FIGS. 5-8, the ordinate RF is the relative rate of change of frequency increase, calculated as
Figure BDA0002968162520000083
ωMIs the natural frequency of the blade base. As shown in fig. 5, as the mass fraction of graphene increases, the natural frequency of the variable cross-section blade in different distribution modes increases significantly. As shown in fig. 6, as the number of layers increases, the blade frequency remains the same in the U-shaped distribution mode and increases significantly in the X-shaped distribution mode; the variation trend of the frequency in the O-shaped distribution mode is opposite to that of the X-shaped distribution mode. As shown in fig. 7, when the aspect ratio of the graphene small block is not changed, the frequency in different distribution modes tends to increase with the increase of the aspect ratio; when the length-thickness ratio of the graphene small block is not changed, the frequency under different distribution modes tends to be reduced along with the increase of the length-width ratio. By comparing the results of fig. 5 to 8, it can be found that the material parameters of graphene have great influence on the variable cross-section blade frequency, and the blade frequency has the most obvious increasing trend in the X-type distribution mode.
While the foregoing is directed to the preferred embodiment of the present invention, it is not intended that the invention be limited to the embodiment and the drawings disclosed herein. Equivalents and modifications may be made without departing from the spirit of the disclosure, which is to be considered as within the scope of the invention.

Claims (4)

1. A graphene variable cross-section blade dynamics modeling method comprises the following steps:
step 1: constructing a simplified model of the graphene variable cross-section blade;
step 2: performing dynamic modeling and free vibration analysis on the graphene variable cross-section blade;
and step 3: a Rayleigh-Ritz method is adopted to solve a dynamic equation of free vibration of the graphene variable cross-section blade, and influence of parameters of the graphene material on vibration frequency of the variable cross-section blade is analyzed.
2. The graphene variable cross-section blade dynamics modeling method of claim 1, wherein: in the step 1, the simplified model of the graphene variable cross-section blade is fixed on a high-speed rotating hub and provided with a pre-torsion angle thetaRPre-installation angle thetarThe variable cross section cantilever trapezoidal plate.
3. The graphene variable cross-section blade dynamics modeling method of claim 1, wherein: in the step 2, the graphene adopts three different distribution modes of U-shaped distribution, X-shaped distribution and O-shaped distribution, and calculates material related parameters by utilizing a Halpin-Tsai model and a mixing rule; and constructing a kinetic energy and potential energy equation when the blade freely vibrates.
4. The graphene deployable antenna thermal structure dynamics modeling method of claim 1, wherein: in the step 3, assuming that displacement and corner forms meeting the boundary conditions of the cantilever are met, solving a dynamic equation of free vibration of the graphene variable cross-section blade by adopting a Rayleigh-Ritz method.
CN202110257491.8A 2021-03-09 2021-03-09 Graphene variable cross-section blade dynamics modeling method Pending CN113065202A (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202110257491.8A CN113065202A (en) 2021-03-09 2021-03-09 Graphene variable cross-section blade dynamics modeling method

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202110257491.8A CN113065202A (en) 2021-03-09 2021-03-09 Graphene variable cross-section blade dynamics modeling method

Publications (1)

Publication Number Publication Date
CN113065202A true CN113065202A (en) 2021-07-02

Family

ID=76560027

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202110257491.8A Pending CN113065202A (en) 2021-03-09 2021-03-09 Graphene variable cross-section blade dynamics modeling method

Country Status (1)

Country Link
CN (1) CN113065202A (en)

Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN108108578A (en) * 2018-01-30 2018-06-01 南京理工大学 The numerical algorithm of FG-GRC bucking of plate load factors based on gridless routing
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

Patent Citations (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN108108578A (en) * 2018-01-30 2018-06-01 南京理工大学 The numerical algorithm of FG-GRC bucking of plate load factors based on gridless routing
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

Non-Patent Citations (3)

* Cited by examiner, † Cited by third party
Title
JIE CHEN 等: "Free Vibrations of Functionally Graded Graphene-Reinforced Composite Blades with Varying Cross-Sections", 《INTERNATIONAL JOURNAL OF STRUCTURAL STABILITY AND DYNAMICS》, pages 1 - 29 *
崔派;张伟;陈杰: "功能梯度石墨烯变截面叶片的非线性动力学研究", 北京力学会第二十七届学术年会 *
张伟;袁双;郭翔鹰;: "变截面旋转叶片的非线性动力学研究", 动力学与控制学报, no. 04 *

Similar Documents

Publication Publication Date Title
Niu et al. Free vibration of rotating pretwisted functionally graded composite cylindrical panel reinforced with graphene platelets
Bahaadini et al. Aeroelastic analysis of functionally graded rotating blades reinforced with graphene nanoplatelets in supersonic flow
Yao et al. Analysis on nonlinear oscillations and resonant responses of a compressor blade
Ramesh et al. Free vibration analysis of pre-twisted rotating FGM beams
Akbaş Geometrically nonlinear static analysis of edge cracked Timoshenko beams composed of functionally graded material
Li et al. Application of piezoelectric fiber composite actuator to aircraft wing for aerodynamic performance improvement
Liu et al. Free vibration analysis of rotating pretwisted functionally graded sandwich blades
Yao et al. Nonlinear dynamics of the high-speed rotating plate
Farhadi et al. Aeroelastic behavior of cantilevered rotating rectangular plates
Xu et al. Influence of the boundary relaxation on the free vibration of rotating composite laminated Timoshenko beams
Zhang et al. Nonlinear dynamic responses of functionally graded graphene platelet reinforced composite cantilever rotating warping plate
Zhou et al. Higher-order stability analysis of a rotating BDFG tapered beam with time-varying velocity
Zhang et al. New modeling on vibrations and bifurcations of FGGP reinforced pretwisted composite rotating blade under axial aerodynamic force: Theoretical and numerical researches
Doeva et al. Exact solution for the deflection of composite beams under non-uniformly distributed loads
Librescu et al. Material grading for improved aeroelastic stability in composite wings
Ramesh et al. Free vibration analysis of rotating functionally-graded cantilever beams
CN113065202A (en) Graphene variable cross-section blade dynamics modeling method
Drazumeric et al. On bimodal flutter behavior of a flexible airfoil
Yahea et al. Free vibration of laminated composite plates in thermal environment using a simple four variable plate theory
Hao et al. Transient response of high dimensional nonlinear dynamic system for a rotating cantilever twisted plate
Gu et al. Nonlinear analysis of a rotating pre-twisted composite blade reinforced with functionally graded graphene platelets under axial and transverse excitations
Hong et al. Structural design and multi-objective evaluation of composite bladed propeller
Murugan et al. Morping helicopter rotor blade with curvilinear fiber composites
Ramesh et al. Experimental and numerical investigation of post-flutter limit cycle oscillations on a cantilevered flat plate
Liang et al. Nonlinear isogeometric analysis of axially functionally graded graphene platelet-reinforced composite curved beams

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