CN112989486A - Method and device for constructing frame-lifting model - Google Patents

Method and device for constructing frame-lifting model Download PDF

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CN112989486A
CN112989486A CN202110259926.2A CN202110259926A CN112989486A CN 112989486 A CN112989486 A CN 112989486A CN 202110259926 A CN202110259926 A CN 202110259926A CN 112989486 A CN112989486 A CN 112989486A
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equation
landing gear
landing
model
order
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CN112989486B (en
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齐鹏远
廖东明
焦宗夏
王壮壮
刘晓超
李洋
尚耀星
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Beihang University
Ningbo Institute of Innovation of Beihang University
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Ningbo Institute of Innovation of Beihang University
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Abstract

The invention discloses a method and a device for constructing a drop frame model, wherein the method comprises the following steps: establishing a first landing gear beam equation based on a partial differential equation according to a geometric nonlinear composite beam, wherein the geometric nonlinear composite beam is a strut part of a landing gear; projecting the first landing beam equation onto a modal base to convert the first landing beam equation based on the partial differential equation to a second landing beam equation based on an ordinary differential equation; and performing linear order reduction on the second landing gear beam equation to obtain a low-order landing gear model. The method can directly obtain the deformation load information of each point of the undercarriage in the model, is convenient for analyzing the vibration condition of the undercarriage, and converts the nonlinear partial differential equation into the ordinary differential equation which is easy to solve in time by projecting the equation onto the modal base, thereby simplifying the order reduction process of the model to a great extent.

Description

Method and device for constructing frame-lifting model
Technical Field
The invention relates to the field of aircraft structure simulation modeling, in particular to a method and a device for building a landing frame model.
Background
The landing gear is used as a take-off and landing device of the airplane, the working environment is severe in the take-off and landing process of the airplane, the stress condition is complex, and a complete and accurate landing gear model needs to be established in the processes of airplane brake control law design, ground dynamics analysis and the like.
The existing undercarriage modeling analysis method generally simplifies the undercarriage structure and performs rigid body stress analysis, and the processing mode does not consider the deformation condition of the undercarriage and lacks deformation information of the undercarriage during vibration mechanics analysis.
Disclosure of Invention
To address at least one of the above technical problems, the present disclosure provides a method and apparatus for constructing a model of a landing gear.
In a first aspect, the present invention provides a method of constructing a model of a landing gear, the method comprising:
establishing a first landing gear beam equation based on a partial differential equation according to a geometric nonlinear composite beam, wherein the geometric nonlinear composite beam is a strut part of a landing gear;
projecting the first landing beam equation onto a modal base to convert the first landing beam equation based on the partial differential equation to a second landing beam equation based on an ordinary differential equation;
and performing linear order reduction on the second landing gear beam equation to obtain a low-order landing gear model.
Optionally, the first landing beam equation is:
Figure BDA0002969452460000021
Figure BDA0002969452460000022
wherein the content of the first and second substances,
Figure BDA0002969452460000023
representing the derivative over time, ●' representing the derivative over the length of the beam, x1＝[vTωT]TAnd x2＝[fTmT]TIn order to be a state variable, the state variable,
Figure BDA0002969452460000024
and
Figure BDA0002969452460000025
respectively the linear velocity and the angular velocity of the beam section,
Figure BDA0002969452460000026
and
Figure BDA0002969452460000027
respectively, the resultant beam cross-sectional force and moment, M and C respectively being the mass matrix and stiffness matrix of the material, variable fERepresenting externally applied forces and moments, matrix E representing initial beam curvature, operator
Figure BDA0002969452460000028
And
Figure BDA0002969452460000029
representing a linear transformation of a variable.
Alternatively,matrix of displacements at position l along the landing gear beam structure at time t
Figure BDA00029694524600000210
And a rotation matrix
Figure BDA00029694524600000211
Figure BDA00029694524600000212
Figure BDA00029694524600000213
Wherein the content of the first and second substances,
Figure BDA00029694524600000214
the symbols are represented as cross-product operators for ternary vectors.
Optionally, projecting the first landing beam equation onto a modal base to convert the first landing beam equation based on the partial differential equation to a second landing beam equation based on a constant differential equation, comprising:
based on Galerkin projection method, the state variable x in the first erection beam equation1Expressed as x from each order of structural mode base and corresponding linear velocity/angular velocity of the beam section2The second landing gear beam equation with the vibration mode as the state variable is established in a form consisting of the structural mode bases of each order and the mode amplitude of the corresponding beam section force/moment.
In a second aspect, the present invention provides an apparatus for constructing a model of a landing gear, the apparatus comprising: an equation establishing module, an equation conversion module and a model reduction module, wherein,
the equation establishing module is used for establishing a first landing gear beam equation based on a partial differential equation according to a geometric nonlinear combined beam, wherein the geometric nonlinear combined beam is a strut part of a landing gear;
the equation conversion module is used for projecting the first landing beam equation to a modal base so as to convert the first landing beam equation based on the partial differential equation into a second landing beam equation based on a normal differential equation;
and the model order reduction module is used for linearly reducing the second landing gear beam equation to obtain a low-order landing gear model.
Optionally, the first landing beam equation is:
Figure BDA0002969452460000031
Figure BDA0002969452460000032
wherein the content of the first and second substances,
Figure BDA0002969452460000033
representing the derivative over time, ●' representing the derivative over the length of the beam, x1＝[vTωT]TAnd x2＝[fTmT]TIn order to be a state variable, the state variable,
Figure BDA0002969452460000034
and
Figure BDA0002969452460000035
respectively the linear velocity and the angular velocity of the beam section,
Figure BDA0002969452460000036
and
Figure BDA0002969452460000037
respectively, the resultant beam cross-sectional force and moment, M and C respectively being the mass matrix and stiffness matrix of the material, variable fERepresenting externally applied forces and moments, matrix E representing initial beam curvature, operator
Figure BDA0002969452460000038
And
Figure BDA0002969452460000039
representing a linear transformation of a variable.
Optionally, a matrix of displacements at position i along the landing gear beam structure at time t
Figure BDA00029694524600000310
And a rotation matrix
Figure BDA00029694524600000311
Figure BDA00029694524600000312
Figure BDA00029694524600000313
Wherein the content of the first and second substances,
Figure BDA00029694524600000314
the symbols are represented as cross-product operators for ternary vectors.
Optionally, the equation conversion module is configured to convert the state variable x in the first erection beam equation by using galois projection method1Expressed as x from each order of structural mode base and corresponding linear velocity/angular velocity of the beam section2The second landing gear beam equation with the vibration mode as the state variable is established in a form consisting of the structural mode bases of each order and the mode amplitude of the corresponding beam section force/moment.
In a third aspect, the present invention provides a readable storage medium having executable instructions thereon which, when executed, cause a computer to perform the method as comprised in the first aspect.
In a fourth aspect, the present invention provides a computing device comprising: one or more processors, memory, and programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors to perform the method as comprised in the first aspect.
Compared with the prior art, the invention has at least the following beneficial effects:
according to the invention, the undercarriage model is established based on the beam model, the deformation load information of each point of the undercarriage can be directly obtained in the model, the vibration condition of the undercarriage is convenient to analyze, the equation is obtained by projecting the equation to a modal base, the nonlinear partial differential equation is converted into the ordinary differential equation which is easy to solve in time, the order reduction process of the model is simplified to a great extent, the nonlinear order reduction model of the undercarriage is obtained after the order reduction of the full-order model, and the design model prediction control is guided because the accuracy of the model is greatly improved.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below, and it is obvious that the drawings in the following description are some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a flow chart of a method for constructing a landing gear model according to an embodiment of the present invention;
FIG. 2 is a schematic view of a drop frame beam model provided in accordance with an embodiment of the present invention;
fig. 3 is a block diagram of an apparatus for constructing a landing frame model according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer and more complete, the technical solutions in the embodiments of the present invention will be described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention, and based on the embodiments of the present invention, all other embodiments obtained by a person of ordinary skill in the art without creative efforts belong to the scope of the present invention.
As illustrated in fig. 1, the present invention provides a method of constructing a model of a landing gear, the method comprising:
establishing a first landing gear beam equation based on a partial differential equation according to a geometric nonlinear composite beam, wherein the geometric nonlinear composite beam is a strut part of a landing gear;
projecting the first landing beam equation onto a modal base to convert the first landing beam equation based on the partial differential equation to a second landing beam equation based on an ordinary differential equation;
and performing linear order reduction on the second landing gear beam equation to obtain a low-order landing gear model.
In the embodiment, deformation information of the undercarriage can be tracked and observed, and guidance is provided for control law optimization design. The specific method comprises the following steps:
(1) establishing landing gear beam equations
As shown in FIG. 2, the landing gear strut portion is considered as a set of geometrically nonlinear composite beams whose equations of motion can be described based on Euler-Bernoulli beam theory, and based on Hamilton theory, the landing gear moving beam equations described in partial differential equations are established.
Figure BDA0002969452460000051
Figure BDA0002969452460000052
Wherein the content of the first and second substances,
Figure BDA0002969452460000053
representing the derivative over time, ●' representing the derivative over beam length, ●TRepresenting transposes of matrices or vectors, x1＝[vTωT]TAnd x2＝[fTmT]TIn order to be a state variable, the state variable,
Figure BDA0002969452460000054
and
Figure BDA0002969452460000055
respectively the linear velocity and the angular velocity of the beam section,
Figure BDA0002969452460000056
and
Figure BDA0002969452460000057
respectively the resultant beam cross-sectional force and moment,
these characteristic variables are established at a position L e [0, L ] along the beam reference axisa]In the locally deformed reference frame of (2), wherein LaIs the total length of the beam structure. M and C are the mass matrix and stiffness matrix of the material, respectively. Variable fERepresenting externally applied forces and moments, matrix E representing initial beam curvature, operator
Figure BDA0002969452460000058
And
Figure BDA0002969452460000059
representing a linear transformation of a variable. Because there are many methods for linear transformation, different results can be obtained according to different transformation algorithms, so that in practical application, operators can be used
Figure BDA00029694524600000510
And
Figure BDA00029694524600000511
different algorithms can be selected according to specific requirements.
The variable of the natural speed (displacement speed and rotation speed) of the initial state of the landing gear (t ═ 0) is integrated, and the displacement matrix at the position l along the landing gear beam structure at the time t can be obtained
Figure BDA00029694524600000512
And a rotation matrix
Figure BDA00029694524600000513
Figure BDA00029694524600000514
Figure BDA00029694524600000515
Wherein the content of the first and second substances,
Figure BDA00029694524600000516
symbolic representations as cross-product operators for ternary vectors, e.g. for vectors
Figure BDA00029694524600000517
And
Figure BDA00029694524600000518
(2) projection into modal space
The established landing gear beam equation is a nonlinear partial differential equation, and in order to simplify the model solving process, the nonlinear partial differential equation can be converted into a normal differential equation which is easy to solve in real time by projecting the model onto a group of mode bases, so that the subsequent control is facilitated.
According to the structural mode, a pair of structural mode bases is taken, and the state variable x in the model can be obtained through a Galerkin projection method1And x2The model is expressed in a form consisting of each order of structural modal bases, corresponding beam section linear velocity/angular velocity and modal amplitude of force/moment, so that the landing gear beam model and the dynamic model can be respectively projected onto the modal bases, a system full-order equation with a vibration mode as a state variable is established, and rigid body displacement and rotation information can be obtained by tracking the state variable.
(3) Order reduction of model
In order to obtain a model suitable for control design, a full-order model can be partially linearized, and an original n-order model is reduced to an m-order model (m < < n). The model obtained by reducing the order comprises modal information and deformation information, and a control law can be designed by a model prediction-based method.
It should be noted that, in order to express the formula simply and accurately, the full-text formula is written according to the unified specification, and all the operation symbols mentioned are applicable to the full text. For example, ●' represents the derivative of the beam length, ● represents all factors involved in the derivation of the beam length, e.g., ● represents x in the first landing beam equation1And x2. Other operators are the same as the above, and are not described herein again.
According to the method for establishing the undercarriage model based on the beam model, provided by the invention, the deformation load information of each point of the undercarriage can be directly obtained in the model, so that the vibration condition of the undercarriage can be conveniently analyzed; the beam equation is projected to a group of modal bases to obtain a modal formula, and the nonlinear partial differential equation is converted into an ordinary differential equation which is easy to solve in time, so that the order reduction process of the model is simplified to a great extent; after the full-order model is subjected to linearization processing, a low-order nonlinear and linear description model is obtained, and the control difficulty of the original nonlinear model is reduced. The accuracy of the model is greatly improved, so that the method has guiding significance for designing model prediction control.
As shown in fig. 3, the present invention provides an apparatus for constructing a model of a landing gear, the apparatus comprising: an equation establishing module, an equation conversion module and a model reduction module, wherein,
the equation establishing module is used for regarding a strut part of the landing gear as a geometric nonlinear composite beam and establishing a first landing gear beam equation based on a partial differential equation according to the geometric nonlinear composite beam;
the equation conversion module is used for projecting the first landing beam equation to a modal base, and further converting the first landing beam equation based on the partial differential equation into a second landing beam equation based on a normal differential equation;
and the model order reduction module is used for linearly reducing the second landing gear beam equation to obtain a low-order landing gear model.
In one embodiment of the present invention, the first landing beam equation is:
Figure BDA0002969452460000071
Figure BDA0002969452460000072
wherein the content of the first and second substances,
Figure BDA0002969452460000073
representing the derivative over time, ●' representing the derivative over the length of the beam, x1＝[vTωT]TAnd x2＝[fTmT]TIn order to be a state variable, the state variable,
Figure BDA0002969452460000074
and
Figure BDA0002969452460000075
respectively the linear velocity and the angular velocity of the beam section,
Figure BDA0002969452460000076
and
Figure BDA0002969452460000077
respectively, the resultant beam cross-sectional force and moment, M and C respectively being the mass matrix and stiffness matrix of the material, variable fERepresenting externally applied forces and moments, matrix E representing initial beam curvature, operator
Figure BDA0002969452460000078
And
Figure BDA0002969452460000079
representing a linear transformation of a variable.
In one embodiment of the invention, the inherent speed variation of the initial reference shape of the landing gear is integrated to obtain a matrix of displacements at position l along the landing gear beam structure at time t
Figure BDA00029694524600000710
And a rotation matrix
Figure BDA00029694524600000711
Figure BDA00029694524600000712
Figure BDA00029694524600000713
Wherein the content of the first and second substances,
Figure BDA00029694524600000714
the symbols are represented as cross-product operators for ternary vectors.
In an embodiment of the invention, the equation conversion module is configured to convert the state variable x in the first beam landing equation by Galerkin projection method1Expressed as x from each order of structural mode base and corresponding linear velocity/angular velocity of the beam section2The second landing gear beam equation with the vibration mode as the state variable is established in a form consisting of the structural mode bases of each order and the mode amplitude of the corresponding beam section force/moment.
It should be understood that the various techniques described herein may be implemented in connection with hardware or software or, alternatively, with a combination of both. Thus, the methods and apparatus of the present invention, or certain aspects or portions thereof, may take the form of program code (i.e., instructions) embodied in tangible media, such as floppy diskettes, CD-ROMs, hard drives, or any other machine-readable storage medium, wherein, when the program is loaded into and executed by a machine, such as a computer, the machine becomes an apparatus for practicing the invention.
In the case of program code execution on programmable computers, the computing device will generally include a processor, a storage medium readable by the processor (including volatile and non-volatile memory and/or storage elements), at least one input device, and at least one output device. Wherein the memory is configured to store program code; the processor is configured to perform the various methods of the present invention according to instructions in the program code stored in the memory.
By way of example, and not limitation, computer readable media may comprise computer storage media and communication media. Computer-readable media includes both computer storage media and communication media. Computer storage media store information such as computer readable instructions, data structures, program modules or other data. Communication media typically embodies computer readable instructions, data structures, program modules or other data in a modulated data signal such as a carrier wave or other transport mechanism and includes any information delivery media. Combinations of any of the above are also included within the scope of computer readable media.
It should be appreciated that in the foregoing description of exemplary embodiments of the invention, various features of the invention are sometimes grouped together in a single embodiment, figure, or description thereof for the purpose of streamlining the invention and aiding in the understanding of one or more of the various inventive aspects. However, the method of the invention should not be construed to reflect the intent: that the invention as claimed requires more features than are expressly recited in each claim. Rather, as the following claims reflect, inventive aspects lie in less than all features of a single foregoing inventive embodiment. Thus, the claims following the detailed description are hereby expressly incorporated into this detailed description, with each claim standing on its own as a separate embodiment of this invention.
Those skilled in the art will appreciate that the modules or units or components of the apparatus in the examples invented herein may be arranged in an apparatus as described in this embodiment or alternatively may be located in one or more apparatuses different from the apparatus in this example. The modules in the foregoing examples may be combined into one module or may be further divided into multiple sub-modules.
Those skilled in the art will appreciate that the modules in the device in an embodiment may be adaptively changed and disposed in one or more devices different from the embodiment. The modules or units or components of the embodiments may be combined into one module or unit or component, and furthermore they may be divided into a plurality of sub-modules or sub-units or sub-components. All of the features of the invention in this specification (including any accompanying claims, abstract and drawings), and all of the processes or elements of any method or apparatus so invented, may be combined in any combination, except combinations where at least some of such features and/or processes or elements are mutually exclusive. Each feature of the invention in this specification (including any accompanying claims, abstract and drawings) may be replaced by alternative features serving the same, equivalent or similar purpose, unless expressly stated otherwise.
Furthermore, those skilled in the art will appreciate that while some embodiments described herein include some features included in other embodiments, rather than other features, combinations of features of different embodiments are meant to be within the scope of the invention and form different embodiments. For example, in the following claims, any of the claimed embodiments may be used in any combination.
Furthermore, some of the described embodiments are described herein as a method or combination of method elements that can be performed by a processor of a computer system or by other means of performing the described functions. A processor having the necessary instructions for carrying out the method or method elements thus forms a means for carrying out the method or method elements. Further, the elements of the apparatus embodiments described herein are examples of the following apparatus: the apparatus is used to implement the functions performed by the elements for the purpose of carrying out the invention.
As used herein, unless otherwise specified the use of the ordinal adjectives "first", "second", "third", etc., to describe a common object, merely indicate that different instances of like objects are being referred to, and are not intended to imply that the objects so described must be in a given sequence, either temporally, spatially, in ranking, or in any other manner.
While the invention has been described with respect to a limited number of embodiments, those skilled in the art, having benefit of this description, will appreciate that other embodiments can be devised which do not depart from the scope of the invention as described herein. Furthermore, it should be noted that the language used in the specification has been principally selected for readability and instructional purposes, and may not have been selected to delineate or circumscribe the inventive subject matter. Accordingly, many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the appended claims. The present invention is to be considered as illustrative and not restrictive in character, with the scope of the invention being indicated by the appended claims.

Claims (10)

1. A method of constructing a landing gear model, the method comprising:
establishing a first landing gear beam equation based on a partial differential equation according to a geometric nonlinear composite beam, wherein the geometric nonlinear composite beam is a strut part of a landing gear;
projecting the first landing beam equation onto a modal base to convert the first landing beam equation based on the partial differential equation to a second landing beam equation based on an ordinary differential equation;
and performing linear order reduction on the second landing gear beam equation to obtain a low-order landing gear model.
2. A method of constructing a landing gear model according to claim 1, wherein the first landing gear beam equation is:
Figure FDA0002969452450000011
Figure FDA0002969452450000012
wherein the content of the first and second substances,
Figure FDA0002969452450000013
representing the derivative over time, ●' representing the derivative over the length of the beam, x1＝[vTωT]TAnd x2＝[fTmT]TIn order to be a state variable, the state variable,
Figure FDA0002969452450000014
and
Figure FDA0002969452450000015
respectively the linear velocity and the angular velocity of the beam section,
Figure FDA0002969452450000016
and
Figure FDA0002969452450000017
respectively, the resultant beam cross-sectional force and moment, M and C respectively being the mass matrix and stiffness matrix of the material, variable fERepresenting externally applied forces and moments, matrix E representing initial beam curvature, operator
Figure FDA0002969452450000018
And
Figure FDA0002969452450000019
representing a linear transformation of a variable.
3. A method of constructing a landing gear model according to claim 2, wherein the matrix of displacements at position i along the landing gear beam structure at time t
Figure FDA00029694524500000110
And a rotation matrix
Figure FDA00029694524500000111
Figure FDA00029694524500000112
Figure FDA00029694524500000113
Wherein the content of the first and second substances,
Figure FDA00029694524500000114
the symbols are represented as cross-product operators for ternary vectors.
4. A method of constructing a landing gear model according to any of claims 2 to 3, wherein projecting the first landing beam equation onto a modal basis to convert the first landing beam equation based on the partial differential equation to a second landing beam equation based on a normal differential equation comprises:
based on the Galerkin projection method, the state variable x in the first erection beam equation1Expressed as x from each order of structural mode base and corresponding linear velocity/angular velocity of the beam section2The second landing gear beam equation with the vibration mode as the state variable is established in a form consisting of the structural mode bases of each order and the mode amplitude of the corresponding beam section force/moment.
5. An apparatus for constructing a model of a landing gear, the apparatus comprising: an equation establishing module, an equation conversion module and a model reduction module, wherein,
the equation establishing module is used for establishing a first landing gear beam equation based on a partial differential equation according to a geometric nonlinear combined beam, wherein the geometric nonlinear combined beam is a strut part of a landing gear;
the equation conversion module is used for projecting the first landing beam equation to a modal base so as to convert the first landing beam equation based on the partial differential equation into a second landing beam equation based on a normal differential equation;
and the model order reduction module is used for linearly reducing the second landing gear beam equation to obtain a low-order landing gear model.
6. An apparatus for constructing a landing gear model according to claim 5, wherein the first landing gear beam equation is:
Figure FDA0002969452450000021
Figure FDA0002969452450000022
wherein the content of the first and second substances,
Figure FDA0002969452450000023
representing the derivative over time, ●' representing the derivative over the length of the beam, x1＝[vTωT]TAnd x2＝[fTmT]TIn order to be a state variable, the state variable,
Figure FDA0002969452450000024
and
Figure FDA0002969452450000025
respectively the linear velocity and the angular velocity of the beam section,
Figure FDA0002969452450000026
and
Figure FDA0002969452450000027
respectively, the resultant beam cross-sectional force and moment, M and C respectively being the mass matrix and stiffness matrix of the material, variable fERepresenting externally applied forces and moments, matrix E representing initial beam curvature, operator
Figure FDA0002969452450000028
And
Figure FDA0002969452450000029
representing a linear transformation of a variable.
7. An arrangement for constructing a landing gear model according to claim 6, wherein the matrix of displacements at position i along the landing gear beam structure at time t is a matrix
Figure FDA0002969452450000031
And a rotation matrix
Figure FDA0002969452450000032
Figure FDA0002969452450000033
Figure FDA0002969452450000034
Wherein the content of the first and second substances,
Figure FDA0002969452450000035
the symbols are represented as cross-product operators for ternary vectors.
8. The apparatus for constructing a landing frame model according to any one of claims 6-7, wherein the equation transformation module is configured to transform the state variable x in the first landing beam equation based on Galerkin projection method1Expressed as x from each order of structural mode base and corresponding linear velocity/angular velocity of the beam section2The second vibration mode is established as a state variable by a mode composed of a mode amplitude of each order structure mode base and corresponding beam section force/momentLanding gear beam equations.
9. A readable storage medium having executable instructions thereon that, when executed, cause a computer to perform the method as included in any one of claims 1-4.
10. A computing device, comprising: one or more processors, memory, and programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors to perform the method as recited in any of claims 1-4.
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