CN113919104B - Method and system for acquiring nonlinear dynamic response of rotary drum - Google Patents

Method and system for acquiring nonlinear dynamic response of rotary drum Download PDF

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CN113919104B
CN113919104B CN202111513866.9A CN202111513866A CN113919104B CN 113919104 B CN113919104 B CN 113919104B CN 202111513866 A CN202111513866 A CN 202111513866A CN 113919104 B CN113919104 B CN 113919104B
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cylindrical shell
equation set
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rotary drum
shell model
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CN113919104A (en
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秦朝烨
刘云飞
褚福磊
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Tsinghua University
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    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G06F17/12—Simultaneous equations, e.g. systems of linear equations
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G06F17/13—Differential equations
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    • 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 provides a method and a system for acquiring nonlinear dynamic response of a rotary drum, wherein the method comprises the following steps: preprocessing a target rotary drum barrel to obtain parameter information of the target rotary drum barrel; respectively acquiring strain energy, kinetic energy and external force acting corresponding to the target rotary drum based on the parameter information; on the basis of strain energy, kinetic energy and external force acting, a nonlinear forced vibration partial differential control equation set corresponding to the target rotary drum is established; carrying out weighted integral processing on the control equation set, and converting the control equation set into an ordinary differential equation set with multiple coupling degrees of freedom; and carrying out numerical solution on the ordinary differential equation set to obtain a nonlinear amplitude-frequency curve and a force amplitude response curve of the target rotary drum. The invention can effectively solve the problem of nonlinear vibration response when the rotary drum of the engine is excited by multiple harmonics, and has the characteristics of high precision, strong universality and the like.

Description

Method and system for acquiring nonlinear dynamic response of rotary drum
Technical Field
The invention relates to the technical field of mechanical dynamics, in particular to a nonlinear dynamic response obtaining method and system of a rotary drum.
Background
At present, for an engine with a complex structure, a rotary drum in a rotor system is required to have enough structural strength and rigidity to bear axial thrust, internal hydraulic pressure and strong vibration; and it is required to be as light as possible in weight to improve the thrust-weight ratio of the engine, so that the thin-walled cylindrical shell is widely used in various engines with its excellent structure. However, the rotating drum is affected by manufacturing and installation errors or thermal deformation in the operation process, so that unbalance exists inevitably, the rotating drum is easily excited by multiple harmonics of complex surrounding environment, beat vibration, resonance or more complex phenomena occur, the dynamic behavior of the rotating drum is more complex, if the dynamic response of the engine cannot be accurately obtained, the normal use of a rotor system can be greatly damaged, the whole engine can be stopped in serious cases, and huge economic loss and severe social influence are caused.
Therefore, the research on the nonlinear dynamic response of the rotary thin-wall cylindrical shell structure plays an important role in ensuring the structural vibration characteristic and stable operation after the rotor system is subjected to complex multi-harmonic excitation.
At present, the dynamic research scheme of the rotary thin-wall cylindrical shell under multi-harmonic excitation is less, and most of the rotary thin-wall cylindrical shell adopts a finite element method and has no universality; in addition, when the structural model is too complex or nonlinear vibration characteristics occur, the finite element method cannot accurately acquire amplitude deformation, and the limitation is also large. In addition, some traditional analytic or numerical methods simplify the research process of dynamic response, only consider the linear vibration generated by the system, and are incapable of accurately predicting the system prediction error of large-amplitude vibration, especially the complex phenomena such as nonlinear beat vibration and resonance. For example, such errors are unacceptable for precision structures such as aircraft engine rotor systems.
Therefore, the research and establishment of the dynamic response analysis method of the rotary thin-wall cylindrical shell, which has universality and high precision and can be used in a complex multi-harmonic excitation environment, have very important significance.
Disclosure of Invention
In view of the above problems, an object of the present invention is to provide a method and a system for obtaining nonlinear dynamic response of a rotating drum, so as to solve the problems of low precision, no universality and the like of the existing method for studying vibration characteristics of a rotating thin-walled cylindrical shell excited by multiple harmonics.
The invention provides a nonlinear dynamic response obtaining method of a rotary drum, which comprises the following steps: preprocessing a target rotary drum barrel to obtain parameter information of the target rotary drum barrel; respectively acquiring strain energy, kinetic energy and external force acting corresponding to the target rotary drum based on the parameter information; on the basis of strain energy, kinetic energy and external force acting, a nonlinear forced vibration partial differential control equation set corresponding to the target rotary drum is established; carrying out weighted integral processing on the control equation set, and converting the control equation set into an ordinary differential equation set with multiple coupling degrees of freedom; and carrying out numerical solution on the ordinary differential equation set to obtain a nonlinear amplitude-frequency curve and a force amplitude response curve of the target rotary drum.
According to another aspect of the present invention, there is provided a non-linear dynamic response acquiring system of a rotary drum, including: the parameter information acquisition unit is used for preprocessing the target rotary drum barrel and acquiring the parameter information of the target rotary drum barrel; an energy and work obtaining unit for respectively obtaining strain energy, kinetic energy and external force acting corresponding to the target rotary drum based on the parameter information; the control equation set acquisition unit is used for doing work based on strain energy, kinetic energy and external force and establishing a nonlinear forced vibration partial differential control equation set corresponding to the target rotary drum; the system comprises an ordinary differential equation set conversion unit, a control equation set calculation unit and a control equation set calculation unit, wherein the ordinary differential equation set conversion unit is used for carrying out weighted integral processing on the control equation set and converting the control equation set into an ordinary differential equation set with multiple coupling degrees of freedom; and the response curve acquisition unit is used for carrying out numerical solution on the ordinary differential equation set to acquire a nonlinear amplitude-frequency curve and a force amplitude response curve of the target rotary drum.
By utilizing the nonlinear dynamic response obtaining method and the system of the rotary drum, the parameter information of the target rotary drum is obtained by preprocessing the target rotary drum, then the strain energy, the kinetic energy and the external force which correspond to the target rotary drum are respectively obtained to do work, a nonlinear forced vibration partial differential control equation set corresponding to the target rotary drum is established, further the weighted integral processing is carried out on the control equation set, the control equation set is converted into a coupled multi-degree-of-freedom ordinary differential equation set, the ordinary differential equation set is numerically solved, a nonlinear amplitude-frequency curve and a force amplitude response curve of the target rotary drum are obtained, the method and the system can be used for solving the complicated nonlinear dynamic response of the rotary drum, the calculation is convenient, the readability is strong, and the accuracy is high.
To the accomplishment of the foregoing and related ends, one or more aspects of the invention comprise the features hereinafter fully described. The following description and the annexed drawings set forth in detail certain illustrative aspects of the invention. These aspects are indicative, however, of but a few of the various ways in which the principles of the invention may be employed. Further, the present invention is intended to include all such aspects and their equivalents.
Drawings
Other objects and results of the present invention will become more apparent and more readily appreciated as the same becomes better understood by reference to the following description taken in conjunction with the accompanying drawings. In the drawings:
FIG. 1 is a flow chart of a method of obtaining a nonlinear dynamic response of a rotating drum according to an embodiment of the present invention;
FIG. 2 is a block diagram of an equivalent rotating thin-walled cylindrical shell model according to an embodiment of the invention;
fig. 3 is a detailed flowchart of a nonlinear dynamic response acquisition method of a rotary drum according to an embodiment of the present invention;
fig. 4 shows a block schematic diagram of a non-linear dynamic response acquisition system for a rotating drum according to an embodiment of the present invention.
The same reference numbers in all figures indicate similar or corresponding features or functions.
Detailed Description
In the following description, for purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of one or more embodiments. It may be evident, however, that such embodiment(s) may be practiced without these specific details. In other instances, well-known structures and devices are shown in block diagram form in order to facilitate describing one or more embodiments.
In the description of the present invention, it is to be understood that the terms "central," "longitudinal," "lateral," "length," "width," "thickness," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," "outer," "clockwise," "counterclockwise," "axial," "radial," "circumferential," and the like are used in the orientations and positional relationships indicated in the drawings for convenience in describing the invention and to simplify the description, and are not intended to indicate or imply that the referenced devices or elements must have a particular orientation, be constructed and operated in a particular orientation, and are therefore not to be considered limiting of the invention.
To describe in detail the present invention's method and system for obtaining nonlinear dynamic response of a rotating drum, a detailed description of embodiments of the present invention will be given below with reference to the accompanying drawings.
Fig. 1 shows a schematic flow of a non-linear dynamic response acquisition method of a rotating drum according to an embodiment of the present invention.
As shown in fig. 1, the method for acquiring nonlinear dynamic response of a rotating drum according to an embodiment of the present invention mainly includes the following steps:
s110: and preprocessing the target rotary drum barrel to obtain the parameter information of the target rotary drum barrel.
In this step, the target rotating drum can be equated to a rotating thin-walled cylindrical shell model; then, acquiring parameter information of the rotary thin-wall cylindrical shell model as parameter information of a target rotary drum barrel; wherein the parameter information comprises a geometric parameter and a physical parameter.
Specifically, FIG. 2 illustrates a schematic structure of an equivalent rotating thin-walled cylindrical shell model according to an embodiment of the present invention, and a target rotating drum, for example, an aircraft engine drum rotor structure, may be equivalent to one rotating thin-walled cylindrical shell model as illustrated in FIG. 2, and then the radius of the rotating thin-walled cylindrical shell model is measuredRIs long and longLAnd thicknesshModulus of elasticityEPoisson ratiovAnd mass densityρThe parameter information may be used as parameter information of the target rotary drum.
S120: and respectively acquiring strain energy, kinetic energy and external force acting corresponding to the target rotary drum based on the parameter information.
Wherein the process of acquiring strain energy corresponding to the target rotating drum based on the parameter information may further include the steps of:
s121: establishing a relation model between nonlinear strain and displacement of the rotary thin-wall cylindrical shell model based on the parameter information;
specifically, the expression of the relational model between the nonlinear strain and the displacement is:
Figure 304356DEST_PATH_IMAGE001
wherein the content of the first and second substances,
Figure 765425DEST_PATH_IMAGE002
、
Figure 195269DEST_PATH_IMAGE003
and
Figure 358266DEST_PATH_IMAGE004
representing the strain of any point on the rotating thin-wall cylindrical shell model;
Figure 898969DEST_PATH_IMAGE005
、
Figure 542440DEST_PATH_IMAGE006
and
Figure 815289DEST_PATH_IMAGE007
representing the mid-plane strain of the rotating thin-wall cylindrical shell model;
Figure 544211DEST_PATH_IMAGE008
、
Figure 809976DEST_PATH_IMAGE009
and
Figure 307953DEST_PATH_IMAGE010
respectively representing the mid-plane curvature and twist in the thin-walled cylindrical shell model of revolution.
Wherein, the expression of the middle surface strain is as follows:
Figure 751704DEST_PATH_IMAGE011
the expressions for mid-plane curvature and twist are:
Figure 967922DEST_PATH_IMAGE012
wherein the content of the first and second substances,u，v，wrespectively a middle surface edge of a rotary thin-wall cylindrical shell modelx、
Figure 34448DEST_PATH_IMAGE013
、zDisplacement in the axial direction.
S122: and acquiring a stress and strain relation model under the thermal effect based on the relation model.
Specifically, the expression of the stress-strain relationship model is:
Figure 324615DEST_PATH_IMAGE014
wherein the content of the first and second substances,
Figure 267163DEST_PATH_IMAGE015
and
Figure 892049DEST_PATH_IMAGE016
representing the coefficient of thermal expansion in the corresponding direction;
Figure 312666DEST_PATH_IMAGE002
、
Figure 722918DEST_PATH_IMAGE003
and
Figure 570789DEST_PATH_IMAGE004
represents the strain of any point on the rotating thin-wall cylindrical shell model, deltaTExpressed as a change in temperature relative to the initial state;C ij the component of the elastic stiffness matrix is represented by the following specific expression:
Figure 496019DEST_PATH_IMAGE017
，
Figure 907278DEST_PATH_IMAGE018
，
Figure 234354DEST_PATH_IMAGE019
wherein the content of the first and second substances,Eit means the modulus of elasticity of the polymer,vrepresenting the poisson ratio.
S123: acquiring strain energy of the rotary thin-wall cylindrical shell model based on a stress and strain relation model; wherein the strain energy comprises elastic strain energy and additional strain energy.
Specifically, the expression formula of the elastic strain energy is as follows:
Figure 456388DEST_PATH_IMAGE020
wherein the content of the first and second substances,srepresenting the infinitesimal area of the rotating thin-walled cylindrical shell model,hrepresenting the thickness of the rotating thin-walled cylindrical shell model,
Figure 868915DEST_PATH_IMAGE021
、
Figure 631335DEST_PATH_IMAGE022
、
Figure 999868DEST_PATH_IMAGE023
the relationship between the stress and the strain is shown,
Figure 455120DEST_PATH_IMAGE024
、
Figure 292626DEST_PATH_IMAGE025
and
Figure 858737DEST_PATH_IMAGE026
representing the strain of any point on the rotating thin-wall cylindrical shell model;
the expression formula of the additional strain energy is:
Figure 84706DEST_PATH_IMAGE027
wherein the content of the first and second substances,srepresenting the infinitesimal area of the rotating thin-walled cylindrical shell model,
Figure 710859DEST_PATH_IMAGE028
，
Figure 35662DEST_PATH_IMAGE029
which represents the angular velocity of rotation of the rotating body,Rrepresenting the radius of the rotating thin-walled cylindrical shell model,ρthe mass density is expressed in terms of,u，v，wrespectively the middle surface of the rotary thin-wall cylindrical shell model is arranged alongx、
Figure 139884DEST_PATH_IMAGE013
、zDisplacement in the axial direction.
Further, the expression of the kinetic energy corresponding to the target rotary drum is:
Figure 296059DEST_PATH_IMAGE030
wherein the content of the first and second substances,Lrepresenting the length of the rotating thin-walled cylindrical shell model,
Figure 280064DEST_PATH_IMAGE028
，ρthe mass density is expressed in terms of,
Figure 888900DEST_PATH_IMAGE029
which represents the angular velocity of rotation of the rotating body,u，v，wrespectively the middle surface of the rotary thin-wall cylindrical shell model is arranged alongx、
Figure 75DEST_PATH_IMAGE013
、zDisplacement in the axial direction;
the expression of the external force acting corresponding to the target rotary drum is as follows:
Figure 10757DEST_PATH_IMAGE031
wherein the content of the first and second substances,
Figure 165663DEST_PATH_IMAGE032
representing the radially distributed force acting on a unit area, the expression:
Figure 261795DEST_PATH_IMAGE033
wherein the content of the first and second substances,fin terms of radial force amplitude (x 0,
Figure 645503DEST_PATH_IMAGE034
) Is the location of the point of applied radial force;
Figure 776270DEST_PATH_IMAGE035
representing a dirac function;ω f is the excitation frequency;Jrepresents the total number of harmonics;
Figure 102078DEST_PATH_IMAGE036
and
Figure 685506DEST_PATH_IMAGE037
respectively representing amplitude coefficients not equal to different phases of the fundamental harmonic.
S130: and establishing a nonlinear forced vibration partial differential control equation set corresponding to the target rotary drum based on the strain energy, the kinetic energy and the external force acting.
Based on the strain energy, the kinetic energy and the external force to do work, the process of establishing a nonlinear forced vibration partial differential control equation set corresponding to the target rotary drum comprises the following steps:
constructing a variation expression based on a Hamilton variation principle, the strain energy, the kinetic energy and the external force acting:
Figure 872905DEST_PATH_IMAGE038
wherein the content of the first and second substances,tthe time is represented by the time of day,
Figure 858179DEST_PATH_IMAGE039
the kinetic energy is represented by a representation of the kinetic energy,
Figure 167937DEST_PATH_IMAGE040
which is indicative of the elastic strain energy of the material,
Figure 422683DEST_PATH_IMAGE041
which represents the additional strain energy, is,
Figure 476089DEST_PATH_IMAGE042
the external force is shown to do work,
Figure 253552DEST_PATH_IMAGE035
representing a dirac function;
using Hamilton principle variation, namely:
Figure 734212DEST_PATH_IMAGE038
then, based on the variational expressions, the information aboutu，v，wVariation of direction to obtain the damping coefficient containing structurec d The system of nonlinear forced vibration partial differential control equations:
Figure 213604DEST_PATH_IMAGE043
（1）
Figure 70702DEST_PATH_IMAGE044
（2）
Figure 702671DEST_PATH_IMAGE045
（3）
wherein the content of the first and second substances,Rrepresenting the radius of the rotating thin-walled cylindrical shell model,hrepresenting the thickness of the rotating thin-walled cylindrical shell model,
Figure 541183DEST_PATH_IMAGE029
which represents the angular velocity of rotation of the rotating body,
Figure 586500DEST_PATH_IMAGE046
，
Figure 919392DEST_PATH_IMAGE047
，
Figure 655136DEST_PATH_IMAGE048
the moment of inertia is represented as a function of,
Figure 743177DEST_PATH_IMAGE049
which is indicative of an external force,x、
Figure 947894DEST_PATH_IMAGE013
andzare all indicative of the coordinate axes,u，v，wrespectively the middle surface of the rotary thin-wall cylindrical shell model is arranged alongx、
Figure 881215DEST_PATH_IMAGE013
、zDisplacement in the axial direction.
S140: and performing weighted integral processing on the control equation set, and converting the control equation set into an ordinary differential equation set with multiple coupling degrees of freedom.
This step may include: establishing a displacement function meeting boundary conditions; and then, converting the control equation set into a multi-degree-of-freedom coupled ordinary differential equation set by utilizing a multi-mode Galerkin method and a displacement function.
Specifically, by establishing that the boundary condition(s) is satisfied
Figure 550093DEST_PATH_IMAGE050
，
Figure 998917DEST_PATH_IMAGE051
，
Figure 425350DEST_PATH_IMAGE052
，
Figure 427941DEST_PATH_IMAGE053
) The displacement function of (2) simplifies the nonlinear system into finite dimensions, and the expression is as follows:
Figure 138277DEST_PATH_IMAGE054
wherein the content of the first and second substances,u m,n (t),v m,n (t)and w m,n (t) Representing a displacement amplitude component;M 1，N 1representing a modal truncation coefficient;min the order of the axial half-wave number,nis the circumferential wave number; subscriptcIndicating driving mode, subscriptsRepresenting the accompanying modality. In nonlinear vibration, these two modes interact through one-to-one internal resonance.
Then, a multi-mode Galerkin method is utilized to convert the control equation set into an ordinary differential equation set coupled with multiple degrees of freedom, namely, the original equation set is weighted by proper functions in sequence and is integrated on the middle surface of the rotary thin-wall cylindrical shell model, wherein the weighting functions are as follows:
Figure 302542DEST_PATH_IMAGE055
then, the system of ordinary differential equations is expressed in matrix form as:
Figure 668801DEST_PATH_IMAGE056
wherein the content of the first and second substances,
Figure 209504DEST_PATH_IMAGE057
m, C and K are denoted as stiffness matrix, damping matrix and mass matrix, respectively; g represents a gyro matrix generated by rotation;
Figure 587396DEST_PATH_IMAGE058
represent other non-linear portions; f represents the vector of the external force.
S150: and carrying out numerical solution on the ordinary differential equation set to obtain a nonlinear amplitude-frequency curve and a force amplitude response curve of the target rotary drum.
Wherein, due to the systemThe system is non-autonomous bivalent and cannot be directly solved, so that virtual variables can be introduced
Figure 125825DEST_PATH_IMAGE059
As the speed, a first-order non-autonomous system is obtained as follows:
Figure 589167DEST_PATH_IMAGE060
it can be seen that the above system has been reduced to first order, but is still not autonomous, and therefore can adopt the Hopf paradigm:
Figure 120511DEST_PATH_IMAGE061
without loss of generality, the first harmonic excitation and the second harmonic excitation are chosen here for elucidation, namely:
Figure 618489DEST_PATH_IMAGE062
let (1)
Figure 124556DEST_PATH_IMAGE063
,
Figure 278457DEST_PATH_IMAGE064
) = (1，0)，
Figure 368421DEST_PATH_IMAGE065
= cos(ωt)、
Figure 720905DEST_PATH_IMAGE064
(t) = sin(ωt) (ii) a Then, let cos (2) in the code of the MATLAB program definition modelωt)=
Figure 397874DEST_PATH_IMAGE066
-
Figure 773492DEST_PATH_IMAGE067
And
Figure 459688DEST_PATH_IMAGE068
thus, it is possible to obtain:
Figure 853629DEST_PATH_IMAGE069
neglecting the above system of ordinary differential equationsuAndvthe inertia term and the rotation term of the direction can be obtained only in relation to the radial directionwThe ordinary differential equation of (2); then, on the basis of MATLAB software, a numerical algorithm is written by adopting a quasi-arc length continuation method to solve the nonlinear coupling ordinary differential equation set, the result is compared with the experimental result, and if the verification is successful, an amplitude-frequency response curve and a force-amplitude response curve can be obtained. If the verification fails, the displacement function is reconstructed, and the coupled multi-degree-of-freedom ordinary differential equation set is solved again. The above operation is then carried out.
As a specific example, fig. 3 shows a detailed flow of a nonlinear dynamic response acquisition method of a rotary drum according to an embodiment of the present invention.
As shown in fig. 3, before processing a target rotary drum or a rotary drum, the target rotary drum or the rotary drum is equivalent to a rotary thin-wall cylindrical shell, and then strain energy, kinetic energy and external force acting under coriolis force, centrifugal force and thermal effect are obtained based on multi-harmonic excitation; then constructing a nonlinear forced vibration partial differential equation set on the basis of the Hamilton principle on the basis of acting of strain energy, kinetic energy and external force; further constructing an ordinary differential equation set coupling multiple degrees of freedom based on a multi-mode Galerkin method; further based on a quasi-arc length continuation method, obtaining a dynamic characteristic calculation result of the rotary thin-wall cylindrical shell, verifying the dynamic characteristic calculation result, and outputting a nonlinear amplitude-frequency and force-amplitude response curve corresponding to the rotary drum if the verification is passed; otherwise, solving the ordinary differential equation set of the coupling multiple degrees of freedom again.
The nonlinear dynamic response obtaining method of the rotary drum barrel can be used for solving the complex nonlinear dynamic response of the rotary drum barrel, such as the complex problems of nonlinear beat vibration, nonlinear multiple internal resonance and the like which are often found by engineering, and has the following advantages:
1. by adopting the modified Donnell nonlinear shell theory, the defect that the traditional Donnell nonlinear shell theory is inaccurate in wavelet number can be overcome;
2. the nonlinear dynamic response analysis of different rotary drum barrels under multi-harmonic excitation can be met only by controlling the geometric physical parameters of the rotary cylindrical shell, external excitation and other parameters, and modeling and any modification on a calculation program are not needed;
3. and the multi-mode Galerkin method and the quasi-arc length continuation method are adopted to carry out frequency solution on the nonlinear forced vibration of the rotating thin-wall cylindrical shell, so that the calculation is convenient and the readability is strong.
Corresponding to the nonlinear dynamic response acquisition method of the rotary drum, the invention also provides a nonlinear dynamic response acquisition system of the rotary drum.
In particular, fig. 4 shows a schematic logic of a non-linear dynamic response acquisition system for a rotating drum according to an embodiment of the present invention.
As shown in fig. 4, the non-linear dynamic response acquisition system 100 for a rotating drum of the present invention comprises:
a parameter information obtaining unit 101, configured to perform preprocessing on a target rotary drum to obtain parameter information of the target rotary drum;
an energy and work obtaining unit 102 configured to obtain strain energy, kinetic energy, and external force work corresponding to the target rotary drum, respectively, based on the parameter information;
a control equation set obtaining unit 103, configured to apply work based on strain energy, kinetic energy, and external force, and establish a nonlinear forced vibration partial differential control equation set corresponding to the target rotary drum;
an ordinary differential equation set conversion unit 104, configured to perform weighted integration processing on the control equation set, and convert the control equation set into an ordinary differential equation set with multiple degrees of freedom in coupling;
and the response curve acquiring unit 105 is configured to perform numerical solution on the system of ordinary differential equations to acquire a nonlinear amplitude-frequency curve and a force amplitude response curve of the target rotary drum.
By utilizing the nonlinear dynamic response obtaining method and the nonlinear dynamic response obtaining system of the rotary drum, after the target rotary drum is equivalent to a rotary thin-wall cylindrical shell structure, the nonlinear control equation set of the structure under the multi-harmonic excitation is obtained by utilizing the improved Donnell nonlinear thin-shell theory and the Hamilton variation principle, then further converting the nonlinear control equation set into a coupled multi-degree-of-freedom ordinary differential equation set by a multi-mode Galerkin method, and based on MATLAB software, the ordinary differential equation set is numerically solved by adopting a quasi-arc length continuation method to obtain nonlinear dynamic response of the rotary thin-wall cylindrical shell under the multi-harmonic excitation, the problem of nonlinear vibration response when the rotary drum of the aero-engine is subjected to multi-harmonic excitation can be effectively solved, the method has the advantages of being high in precision, strong in universality and the like, and has great significance in the aspects of safety protection of the aero-engine, structural resonance avoidance and the like.
The non-linear dynamic response acquisition method and system of a rotating drum according to the present invention are described above by way of example with reference to the accompanying drawings. However, it should be understood by those skilled in the art that various modifications can be made to the non-linear dynamic response obtaining method and system for a rotating drum proposed by the present invention without departing from the scope of the present invention. Therefore, the scope of the present invention should be determined by the contents of the appended claims.

Claims (6)

1. A method of obtaining a nonlinear dynamic response of a rotating drum, comprising:
preprocessing a target rotary drum, equivalently using the target rotary drum as a rotary thin-wall cylindrical shell model, and acquiring parameter information of the rotary thin-wall cylindrical shell model as the parameter information of the target rotary drum;
respectively acquiring strain energy, kinetic energy and external force acting corresponding to the target rotary drum based on the parameter information; wherein the process of acquiring strain energy corresponding to the target rotary drum based on the parameter information comprises:
establishing a relation model between nonlinear strain and displacement of the rotary thin-wall cylindrical shell model based on the parameter information;
obtaining a stress and strain relation model under the thermal effect based on the relation model;
acquiring strain energy of the rotary thin-wall cylindrical shell model based on the stress and strain relation model; wherein the strain energy comprises elastic strain energy and additional strain energy;
the expression of the relation model between the nonlinear strain and the displacement is as follows:
Figure FDA0003494553170000011
wherein epsilonxx、εθθAnd gammaxθRepresenting the strain of any point on the rotating thin-wall cylindrical shell model;
Figure FDA0003494553170000012
and
Figure FDA0003494553170000013
representing the mid-plane strain of the rotating thin-walled cylindrical shell model; k is a radical ofx、kθAnd kxθRespectively representing the surface curvature and torsion in the rotating thin-wall cylindrical shell model;
the expression of the mid-plane strain is as follows:
Figure FDA0003494553170000014
the expressions of the mid-plane curvature and twist are:
Figure FDA0003494553170000021
wherein u, v and w are displacements of a plane in the rotary thin-wall cylindrical shell model along the directions of the x axis, the theta axis and the z axis respectively;
based on the strain energy, the kinetic energy and the external force acting, establishing a nonlinear forced vibration partial differential control equation set corresponding to the target rotary drum;
wherein, based on the variation expression, the damping coefficient c containing the structure is obtaineddThe system of nonlinear forced vibration partial differential control equations:
Figure FDA0003494553170000022
Figure FDA0003494553170000023
Figure FDA0003494553170000024
wherein R represents a radius of the rotating thin-walled cylindrical shell model, h represents a thickness of the rotating thin-walled cylindrical shell model, Ω represents a rotation angular velocity,
Figure FDA0003494553170000025
Figure FDA0003494553170000031
I1representing the moment of inertia, F (t) representing an external force, x, theta and z representing coordinate axes, and u, v and w representing the displacements of the middle plane of the rotating thin-wall cylindrical shell model along the directions of the x, theta and z axes;
carrying out weighted integral processing on the control equation set, and converting the control equation set into an ordinary differential equation set with multiple coupling degrees of freedom; wherein the content of the first and second substances,
first, by establishing that the boundary condition M is satisfiedx＝0，NxA displacement function of 0, and simplifying the nonlinear system into a finite dimension, wherein the displacement function is represented by:
Figure FDA0003494553170000032
Figure FDA0003494553170000033
Figure FDA0003494553170000034
wherein u, v and w are displacements of a plane in the rotating thin-wall cylindrical shell model along the directions of the x axis, the theta axis and the z axis respectively, and um,n(t)，vm,n(t) and wm,n(t) respectively represent displacement amplitude components in the corresponding axial directions; m1、N1Representing a modal truncation coefficient; m is the axial half wave number, and n is the circumferential wave number; subscript c denotes the drive mode, subscript s denotes the companion mode; l represents the length of the rotating thin-wall cylindrical shell model, and t represents time;
then, by using a multi-mode Galerkin method and the displacement function, the control equation set is converted into an ordinary differential equation set coupled by multiple degrees of freedom, wherein the matrix form of the ordinary differential equation set is expressed as:
Figure FDA0003494553170000035
wherein, l ═ { u, v, w }TM, C and K denote a stiffness matrix, a damping matrix and a mass matrix, respectively; g represents a gyro matrix generated by rotation, Γ represents other nonlinear parts, and F represents a vector of an external force;
performing numerical solution on the ordinary differential equation set by adopting a quasi-arc length continuation method to obtain a nonlinear amplitude-frequency curve and a force amplitude response curve of the target rotary drum; wherein the content of the first and second substances,
converting the control equation set into a multi-degree-of-freedom coupled ordinary differential equation set by utilizing a multi-mode Galerkin method and the displacement function, wherein the multi-degree-of-freedom coupled ordinary differential equation set comprises the following steps: weighting the control equation system by using proper weighting functions in turn, wherein the weighting functions are as follows:
Figure FDA0003494553170000041
Figure FDA0003494553170000042
Figure FDA0003494553170000043
then, the ordinary differential equation set is solved numerically, and a nonlinear amplitude-frequency curve and a force-amplitude response curve of the target rotary drum are obtained, wherein the steps of:
introducing virtual variables
Figure FDA0003494553170000044
As the speed, a first-order non-autonomous system is obtained as follows:
Figure FDA0003494553170000045
using the Hopf paradigm:
Figure FDA0003494553170000046
wherein f (t) { f cos (ω t) + f cos (2 ω t), 0, 0}TLet (η)0，γ0)＝(1，0)，η(t)＝cos(ωt)、γ0(t) ═ sin (ω t); then, in the code of the MATLAB program definition model, cos (2 ω t) ═ η2-γ2And sin (2 ω t) ═ η/2, so the system of ordinary differential equations after deformation can be found as:
Figure FDA0003494553170000051
and acquiring a nonlinear amplitude-frequency curve and a force amplitude response curve of the target rotary drum based on the deformed ordinary differential equation set.
2. The method of claim 1, wherein the parameter information includes geometric parameters and physical parameters.
3. The method of obtaining nonlinear dynamic response of a rotating drum according to claim 1, wherein the elastic strain energy is expressed by the formula:
Figure FDA0003494553170000052
wherein s represents the infinitesimal area of the rotating thin-walled cylindrical shell model, h represents the thickness of the rotating thin-walled cylindrical shell model, and σ representsxx、σθθ、σxθDenotes the stress-strain relationship,. epsilonxx、εθθAnd gammaxθRepresenting the strain of any point on the rotating thin-wall cylindrical shell model;
the expression formula of the additional strain energy is:
Figure FDA0003494553170000053
wherein s represents the infinitesimal area of the rotating thin-wall cylindrical shell model,
Figure FDA0003494553170000054
Ω represents a rotation angular velocity, R represents a radius of the rotating thin-walled cylindrical shell model, ρ represents a mass density, and u, v, and w are displacements of a plane in the rotating thin-walled cylindrical shell model in directions along x, θ, and z axes, respectively.
4. The method of obtaining a nonlinear dynamic response of a rotating drum of claim 1, wherein the expression of the kinetic energy corresponding to the target rotating drum is:
Figure FDA0003494553170000061
wherein L represents the length of the rotating thin-walled cylindrical shell model,
Figure FDA0003494553170000062
rho represents mass density, omega represents rotation angular velocity, and u, v and w are displacements of a plane in the rotating thin-wall cylindrical shell model along the directions of x, theta and z axes respectively;
the expression of the external force acting corresponding to the target rotary drum is as follows:
Figure FDA0003494553170000063
wherein q isFRepresenting the radially distributed force acting on a unit area, the expression:
Figure FDA0003494553170000064
wherein f is the radial force amplitude, (x)0，θ0) Is the location of the point of applied radial force; δ represents a dirac function; omegafIs the excitation frequency; j represents the total number of harmonics;
Figure FDA0003494553170000065
and
Figure FDA0003494553170000066
respectively representing different phases not equal to the fundamental harmonicThe amplitude factor.
5. The method of obtaining a nonlinear dynamic response of a rotating drum according to claim 1, wherein the process of establishing a nonlinear forced vibration partial differential control equation system corresponding to the target rotating drum based on the strain energy, the kinetic energy, and the external force work comprises:
constructing the variation expression based on Hamilton variation principle, the strain energy, the kinetic energy and the external force acting:
Figure FDA0003494553170000067
wherein t represents time, ΠkRepresenting said kinetic energy, Πs1Indicating said elastic strain energy, Πs2Indicates extra strain energy,. pifRepresents the external force acting, and delta represents the dirac function.
6. A non-linear dynamic response acquisition system for a rotating drum, comprising:
the parameter information acquisition unit is used for preprocessing a target rotary drum, equivalently using the target rotary drum as a rotary thin-wall cylindrical shell model, and acquiring parameter information of the rotary thin-wall cylindrical shell model as the parameter information of the target rotary drum;
an energy and work obtaining unit, configured to obtain strain energy, kinetic energy, and external force work corresponding to the target rotary drum, respectively, based on the parameter information; wherein the process of acquiring strain energy corresponding to the target rotary drum based on the parameter information comprises:
establishing a relation model between nonlinear strain and displacement of the rotary thin-wall cylindrical shell model based on the parameter information;
obtaining a stress and strain relation model under the thermal effect based on the relation model;
acquiring strain energy of the rotary thin-wall cylindrical shell model based on the stress and strain relation model; wherein the strain energy comprises elastic strain energy and additional strain energy;
the expression of the relation model between the nonlinear strain and the displacement is as follows:
Figure FDA0003494553170000071
wherein epsilonxx、εθθAnd gammaxθRepresenting the strain of any point on the rotating thin-wall cylindrical shell model;
Figure FDA0003494553170000072
and
Figure FDA0003494553170000073
representing the mid-plane strain of the rotating thin-walled cylindrical shell model; k is a radical ofx、kθAnd kxθRespectively representing the surface curvature and torsion in the rotating thin-wall cylindrical shell model;
the expression of the mid-plane strain is as follows:
Figure FDA0003494553170000074
the expressions of the mid-plane curvature and twist are:
Figure FDA0003494553170000081
wherein u, v and w are displacements of a plane in the rotary thin-wall cylindrical shell model along the directions of the x axis, the theta axis and the z axis respectively;
a control equation set obtaining unit, configured to apply work based on the strain energy, the kinetic energy, and the external force, and establish a nonlinear forced vibration partial differential control equation set corresponding to the target rotary drum;
wherein, the expression is based on variationFormula (I) obtaining the damping coefficient c containing the structuredThe system of nonlinear forced vibration partial differential control equations:
Figure FDA0003494553170000082
Figure FDA0003494553170000083
Figure FDA0003494553170000084
wherein R represents a radius of the rotating thin-walled cylindrical shell model, h represents a thickness of the rotating thin-walled cylindrical shell model, Ω represents a rotation angular velocity,
Figure FDA0003494553170000085
Figure FDA0003494553170000091
I1representing the moment of inertia, F (t) representing an external force, x, theta and z representing coordinate axes, and u, v and w representing the displacements of the middle plane of the rotating thin-wall cylindrical shell model along the directions of the x, theta and z axes;
the system comprises an ordinary differential equation set conversion unit, a control equation set calculation unit and a control equation set calculation unit, wherein the ordinary differential equation set conversion unit is used for carrying out weighted integral processing on the control equation set and converting the control equation set into an ordinary differential equation set with multiple coupling degrees of freedom; wherein the content of the first and second substances,
first, by establishing that the boundary condition M is satisfiedx＝0，NxA displacement function of 0, and simplifying the nonlinear system into a finite dimension, wherein the displacement function is represented by:
Figure FDA0003494553170000092
Figure FDA0003494553170000093
Figure FDA0003494553170000094
wherein u, v and w are displacements of a plane in the rotating thin-wall cylindrical shell model along the directions of the x axis, the theta axis and the z axis respectively, and um,n(t)，vm,n(t) and wm,n(t) respectively represent displacement amplitude components in the corresponding axial directions; m1、N1Representing a modal truncation coefficient; m is the axial half wave number, and n is the circumferential wave number; subscript c denotes the drive mode, subscript s denotes the companion mode; l represents the length of the rotating thin-wall cylindrical shell model, and t represents time;
then, by using a multi-mode Galerkin method and the displacement function, the control equation set is converted into an ordinary differential equation set coupled by multiple degrees of freedom, wherein the matrix form of the ordinary differential equation set is expressed as:
Figure FDA0003494553170000095
wherein, l ═ { u, v, w }TM, C and K denote a stiffness matrix, a damping matrix and a mass matrix, respectively; g represents a gyro matrix generated by rotation, Γ represents other nonlinear parts, and F represents a vector of an external force;
the response curve acquisition unit is used for carrying out numerical solution on the ordinary differential equation set by adopting a quasi-arc length continuation method to acquire a nonlinear amplitude-frequency curve and a force amplitude response curve of the target rotary drum; wherein the content of the first and second substances,
converting the control equation set into a multi-degree-of-freedom coupled ordinary differential equation set by utilizing a multi-mode Galerkin method and the displacement function, wherein the multi-degree-of-freedom coupled ordinary differential equation set comprises the following steps: weighting the control equation system by using proper weighting functions in turn, wherein the weighting functions are as follows:
Figure FDA0003494553170000101
Figure FDA0003494553170000102
Figure FDA0003494553170000103
then, the ordinary differential equation set is solved numerically, and a nonlinear amplitude-frequency curve and a force-amplitude response curve of the target rotary drum are obtained, wherein the steps of:
introducing virtual variables
Figure FDA0003494553170000104
As the speed, a first-order non-autonomous system is obtained as follows:
Figure FDA0003494553170000105
using the Hopf paradigm:
Figure FDA0003494553170000106
wherein the content of the first and second substances,
Figure FDA0003494553170000107
let (eta)0，γ0)＝(1，0)，η(t)＝cos(ωt)、γ0(t) ═ sin (ω t); then, in the code of the MATLAB program definition model, cos (2 ω t) ═ η2-γ2And sin (2 ω t) ═ η/2, so the system of ordinary differential equations after deformation can be found as:
Figure FDA0003494553170000111
and acquiring a nonlinear amplitude-frequency curve and a force amplitude response curve of the target rotary drum based on the deformed ordinary differential equation set.
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