CN106294975A - A kind of girder structure free vibration analysis method based on reduced-order model - Google Patents

Abstract

The invention belongs to Aero-Space complexity beam type structural elements and calculate field, it is provided that a kind of girder structure free vibration analysis method based on reduced-order model, comprise the following steps: 1) utilize finite element software to obtain Mass matrix and the Stiffness Matrix of fine structure；2) polynomial interpolating function structure is used to reduce base vector, it is achieved former complicated girder structure depression of order；3) solve the kinetics equation of reduced-order model, obtain overall frequency and the part low order frequency of structure.The method reduces base vector by polynomial interpolating function structure, with the form of positional displacement interpolation, the node in labyrinth FEM (finite element) model is condensed to cross-section centroid, it is achieved original structure depression of order, and the computational accuracy of reduced-order model can be controlled by polynomial order.The invention have the benefit that the experience not relying on user, reduce the difference selected due to the master and slave degree of freedom harmful effect to computational accuracy；It is not required to substantial amounts of matrix manipulation, improves the computational efficiency of model reduction；The reduced-order model obtained is applied widely.

Description

A kind of girder structure free vibration analysis method based on reduced-order model

Technical field

The invention belongs to Aero-Space girder structure component and calculate field, relate to a kind of girder structure based on reduced-order model Free vibration analysis method.

Background technology

Along with computational science and the development of technology, computer disposal speed and storage capacity improve constantly, but simultaneously need to The engineering calculated also is improving constantly with scale and the complexity of problem in science.For the static problems of complex engineering, permissible Set up be on a grand scale, grid is the closeest, at large describe the FEM (finite element) model of CONSTRUCTED SPECIFICATION and it is carried out structural analysis, but Due to the amount of calculation of dynamic structural analysis than static analysis big several amount rank, as a consequence it is hardly possible to use such model to tie Structure dynamic analysis.It addition, design the initial stage in structure, designer is more concerned with the deformation characteristic of whole component, too much consideration structure Details, disturbs the understanding to overall performance sometimes.Therefore, in the initial design stage, find one suitably containing less freedom The reduced-order model of degree substitutes refined model and completes the design of structure and optimization is very important.

In recent years, dynamic model depression of order is theoretical and application has been achieved for huge progress, according to different thinkings, and can be by Conventional Structural Dynamic Model order reducing method is divided three classes: agent model method, physics order reducing method, structure equivalent method.Generation Reason model refers to replace original labyrinth with the mathematic(al) representation of a little amount of calculation, but its result of calculation and high accuracy mould The result of calculation of type is close.Commonly used agent model approximation method has response surface model, Kriging model, RBF Model, artificial nerve network model etc..But utilize agent model to lose the physical significance of original model, be not easy to original knot The parameter adjustment of structure, during it addition, design variable quantity is more, the sample point of needs is many, and the structure of agent model need to be through repeatedly Amendment, computational efficiency is relatively low.Physics reduced-order model utilizes mathematics or the mechanical characteristics of original structure FEM (finite element) model, selects a combination Suitable reduces base vector, by large-scale Structural Dynamics depression of order.Conventional physics order reducing method has: Guyan static concentration Method, improvement polycondensation systems approach (IRS), modal synthesis method etc..Major part physics order reducing method is relevant with structure, works as original structure When some parameters (boundary condition, material property) of model change, new reduced-order model need to be re-established, and these depression of order moulds The Successful utilization of type relies on the experience of user mostly.Structure equivalent method is the master utilizing simple structure equivalence labyrinth Want feature, the mechanical model being simplified.Common equivalent structure method has: asymptotic homogenization, RVE method etc., should Method is for specific structure, it is desirable to simple structure can be by revising the mechanical response of some parameter approximation reflection labyrinth.

Being noticeably greater than horizontal large-scale girder structure for this kind of longitudinal size of rocket, it is new that Chinese scholars proposes some Physics order reducing method: locally rigid body dynamic model order reducing method, the method is based on fine finite element, by structure It is divided into some synchronicity regions and uses rigid body mode to approximate the real displacement in each region, obtaining one group and reduce basal orientation Amount, sets up the reduced-order model that precision is higher on this basis；Model order reducing method based on beam plane cross-section assumption, the method will knot Finite element node on each cross section of structure agglomerates to the centre of form in this cross section by displacement transition matrix, thus quickly establish for Model reduction reduce base vector, and for the structure of big opening, obtain representing the warpage that section loss rates deforms by numerical method Base vector, compensate for the deficiency of plane cross-section assumption；But the reduced-order model that above two method obtains is only capable of obtaining the entirety of structure Frequency.High-order beam model order reducing method, it is many that the offset table of structure finite element node is shown as beam element modal displacement by the method Item formula function, and utilize the principle of virtual work to derive Stiffness Matrix and the Mass matrix of reduced-order model, but the application of the method needs a large amount of Matrix calculus, and existing finite element software can not be utilized, limit the range of application of the method.

Therefore, FEM (finite element) model based on existing girder structure, set up an amount of calculation low, it is possible to quick obtaining beam type is tied The entirety of structure and the reduced-order model of part low order frequency, be still that primary study direction.

Summary of the invention

Structure is lost low after present invention mainly solves existing model order reducing method complicated, computationally intensive, the model reduction of operation The problem of order frequency, proposes a kind of girder structure free vibration analysis method based on reduced-order model, and the method is simple to operate, fills Divide the FEM (finite element) model information utilizing former girder structure, reduce base vector in conjunction with polynomial interpolating function structure, it is achieved beam type is tied The depression of order of structure.Reduced-order model is obtained in that overall frequency and the part low order frequency of girder structure, and computational accuracy can be by multinomial Formula exponent number is controlled.

In order to achieve the above object, the technical scheme is that

A kind of girder structure free vibration analysis method based on reduced-order model, specifically includes following steps:

The first step, utilizes finite element software to process girder structure, obtains Mass matrix M and the Stiffness Matrix of girder structure K。

Second step, uses polynomial interpolating function structure to reduce base vector, girder structure is carried out model reduction.

2.1) polynomial interpolating function is utilized, with the form of positional displacement interpolation by the arbitrary finite element on the cross section of girder structure The displacement of node is condensed at the Ω centre of form of cross section；

Girder structure employing is axially the rectangular coordinate system of x-axis forward, and on the Ω of cross section, arbitrary finite element node j passes through this section Column u that at centre of form i of face, generalized displacement represents_jFor:

u_j=R_jq_i (1)

Wherein, u_jRepresent the motion vector of jth finite element node, q on the Ω of cross section_iRepresent the broad sense at Ω centre of form i of cross section Motion vector, R_jRepresent the displacement transition matrix of j point；

u_jExpanded form be:

$\left\{\begin{array}{c}{u}_{jx}\\ u_{jy}\\ u_{jz}\end{array}\right\}=\left[\begin{array}{ccc}{F}_{\mathrm{\τ}j}& 0& 0\\ 0& F_{\mathrm{\τ}j}& 0\\ 0& 0& F_{\mathrm{\τ}j}\end{array}\right]\left\{\begin{array}{c}{q}_{ix\mathrm{\τ}}\\ q_{iy\mathrm{\τ}}\\ q_{iz\mathrm{\τ}}\end{array}\right\},\mathrm{\τ}=1,2,...W---\left(2\right)$

Wherein, { u_jx u_jy u_jz}^TFor the finite element node j three displacement components under rectangular coordinate system Oxyz, { q_ixτ q_iyτ q_izτ}^TFor cross-section centroid i generalized displacement component under rectangular coordinate system Oxyz, F_τjExpress for polynomial interpolating function Formula, repeats subscript τ and represents summation, and W is multinomial item number.

2.2) on the Ω of cross section, all of finite element node all agglomerates at centre of form i of this cross section Ω, obtains such as formula (3) institute The displacement shown fills changes relational expression:

U_i=(u₁..., u_s)^T=T_iq_i=(R₁..., R_s)^Tq_i (3)

Wherein, U_iThe motion vector of s finite element node in expression cross section Ω, T_iRepresent the displacement conversion square of cross section Ω Battle array.

2.3) it is p cross section by whole girder structure in axial direction subdivision, then by the displacement transformational relation on p cross section Formula is assembled into reduces base vector T, it is achieved the depression of order of former girder structure.

Specifically, the base vector T that reduces that the displacement conversion relational expression on p cross section is assembled into is:

Wherein, U represent total finite element modal displacement vector, be 3n × 1 column vector (n be girder structure from By the number of degrees), Q is the finite element modal displacement vector of model after depression of order, is the column vector of 3pW × 1, U_pIn representing pth cross section The motion vector of node, T_pRepresenting the displacement transition matrix of pth cross section interior nodes, size expands to 3n × 3W, q_pRepresent pth Generalized displacement at individual cross-section centroid, size is 3W × 1.

2.4) not considering damping effect, the finite element governing equation of former girder structure analysis is expressed as:

$M\stackrel{\·\·}{U}+KU=F---\left(5\right)$

Wherein, M, K are the square formation of 3n × 3n, are Mass matrix and the Stiffness Matrix of original structure respectively, U,It is 3n dimensional vector, Being displacement structure and acceleration responsive vector respectively, 3n dimensional vector F is to act on load structurally.

2.5) in the finite element governing equation shown in formula (5), displacement transfer equation (4) is introduced, and same on equation both sides Shi Zuocheng T^T, obtain the kinetics equation after depression of order:

$M_R\stackrel{\·\·}{Q}+K_{R}Q=F_R---\left(6\right)$

Wherein, M_R=T^TMT、K_R=T^TKT is Mass matrix and the Stiffness Matrix of reduced-order model respectively, M_R、K_RFor 3pW × 3pW's Square formation, Q,Being 3pW dimensional vector, be the displacement of reduced-order model and acceleration responsive vector respectively, due to W, < < s, so depression of order The degree of freedom of model significantly declines, F_R=T^TF is the load vectors of 3pW dimension reduced-order model.

3rd step, utilizes the Mass matrix M of girder structure reduced-order model_RWith Stiffness Matrix K_R, formula (7) after calculating depression of order The frequency of girder structure and the vibration shape.

Specifically, the Structural Dynamics generalized eigenvalue equation corresponding with formula (6) is solved:

Wherein, λ,It is respectively the feature value vector of reduced-order model and the corresponding vibration shape,Tie for former beam type The vibration shape of structure.

A kind of girder structure dynamic model order reducing method that the present invention provides, for existing model order reducing method amount of calculation Greatly, operation is complicated, lose the shortcomings such as original structure low order frequency information, obtains the quality of girder structure first with finite element software Battle array and Stiffness Matrix.Then, reduce base vector by polynomial interpolating function structure, with the form of positional displacement interpolation, girder structure is had Node in limit meta-model is condensed to cross-section centroid, it is achieved original structure depression of order.Finally, solve the kinetics equation of reduced-order model, Obtain overall frequency and the part low order frequency of girder structure.

The present invention does not relies on the experience of user, decreases the difference selected due to master and slave degree of freedom to computational accuracy Harmful effect；Need not substantial amounts of matrix manipulation, improve the computational efficiency of model reduction；The reduced-order model obtained is suitable for model Enclose wide, only need to apply different edge-restraint conditions on the reduced-order model under freedom-free state, so that it may quickly calculate multiple The frequency values of the original structure of different boundary constraints.The present invention can improve the computational efficiency of girder structure, reduction initially sets The calculating cost of meter, is extremely expected to become complicated girder structure in the aerospace fields such as China's carrier rocket, Missile Design One of dynamic model order reducing method.

Accompanying drawing explanation

Fig. 1 (a) is elongated smooth barrel structure schematic diagram.

Fig. 1 (b) is the elongated barrel structure schematic diagram containing grid reinforcement.

Fig. 2 is girder structure dynamic model depression of order schematic diagram.

Fig. 3 is the flowchart of girder structure dynamic model order reducing method.

Fig. 4 is cylindrical shell structure and schematic cross-section.

Detailed description of the invention

Technical scheme and the technique effect reached for making to present invention solves the technical problem that, using are clearer, below The present invention is described in further detail in conjunction with the embodiments.

The flowchart of the girder structure dynamic model order reducing method that Fig. 3 provides for the embodiment of the present invention.Fig. 4 is cylinder The schematic diagram of shell structure, radius R=1000mm, length L=15000mm, skin thickness t=2.0mm, elasticity modulus of materials E= 73GPa, Poisson's ratio υ=0.3, density p=2.7E-6kg/mm³.Two fixed ends, girder structure FEM (finite element) model uses entity mould Type, axially divides p=40 unit, and hoop divides 82 unit, and nodes is n=6560.

The first step, utilizes finite element software to process girder structure, obtains Mass matrix M and the Stiffness Matrix of girder structure K.The present invention is directed to girder structure and carry out model reduction.Specifically, girder structure can have different version.Fig. 1 a-b For part girder structure schematic diagram, Fig. 1 a is elongated smooth barrel structure schematic diagram, and Fig. 1 b is that the elongated barrel structure containing grid reinforcement shows Being intended to, with reference to Fig. 1 a-b, the girder structure dynamic model order reducing method that the present invention provides can be carried out for various girder structures Model reduction.Fig. 2 is girder structure dynamic model depression of order schematic diagram.

Second step, uses polynomial interpolating function structure to reduce base vector, the dynamic model of girder structure is carried out depression of order.

2.1) polynomial interpolating function is utilized, with the form of positional displacement interpolation by the arbitrary finite element on the cross section of girder structure The displacement of node is condensed at cross-section centroid；

Girder structure uses rectangular coordinate system shown in Fig. 2, and girder structure is axially x-axis forward, arbitrary limited on cross section Column u that unit node j is represented by generalized displacement at this cross-section centroid i_jFor:

u_j=R_jq_i (1)

Wherein, u_jRepresent the motion vector of jth finite element node, q on the Ω of cross section_iRepresent the broad sense at Ω centre of form i of cross section Motion vector, R_jRepresent the displacement transition matrix of j point.

This example uses N=4 rank multinomial (W=15), and in the Ω of cross section, arbitrary finite element node j is by this cross-section centroid i Column u that place's generalized displacement represents_jFor:

u_jx=q_ix1+yq_ix2+zq_ix3+y²q_ix4+…+z⁴q_ix15=F_τjq_ixτ

u_jy=q_iy1+yq_iy2+zq_iy3+y²q_iy4+…+z⁴q_iy15=F_τjq_iyτ

u_jz=q_iz1+yq_iz2+zq_iz3+y²q_iz4+…+z⁴q_iz15=F_τjq_izτ

The displacement transforming relationship formula of arbitrary finite element node j in the Ω of cross section:

$R_j=\left[\begin{array}{ccc}{F}_{\mathrm{\&tau;}j}& 0& 0\\ 0& F_{\mathrm{\&tau;}j}& 0\\ 0& 0& F_{\mathrm{\&tau;}j}\end{array}\right]={\left[\begin{array}{ccc}1{\mathrm{yzy}}^2...z^4& 0& 0\\ 0& 1{\mathrm{yzy}}^2...z^4& 0\\ 0& 0& 1{\mathrm{yzy}}^2...z^4\end{array}\right]}_{3\&times;45}$

Table 1 provides F_τjExpanded form, in table, y, z are the position coordinates of finite element node j.

Table 1F_τjExpanded form

2.2) on the Ω of cross section, s (s=164) individual finite element node all agglomerates to, at centre of form i of this cross section Ω, obtain such as formula (3) displacement shown in fills changes relational expression:

T_i=(R₁..., R₁₆₄)^T _492×45

2.3) it is p=40 cross section by whole girder structure in axial direction subdivision, then by the displacement on p=40 cross section Conversion relational expression is assembled into and reduces base vector T:

2.4) not considering damping effect, the finite element governing equation of former girder structure analysis is expressed as:

$M\stackrel{\&CenterDot;\&CenterDot;}{U}+KU=F$

Wherein, M, K are the square formation of 19680 × 19680, are Mass matrix and the Stiffness Matrix of former girder structure respectively, U,It is 19680 dimensional vectors, are displacement and the acceleration responsive vector of girder structure respectively, and 19680 dimensional vector F are to act on structure On load.

2.5) in above-mentioned finite element governing equation, displacement transition matrix T is introduced, and premultiplication T simultaneously on equation both sides^T, Kinetics equation after depression of order:

$M_R\stackrel{\&CenterDot;\&CenterDot;}{Q}+K_{R}Q=F_R$

Wherein, M_R=T^TMT、K_R=T^TKT is Mass matrix and the Stiffness Matrix of reduced-order model respectively, M_R、K_RIt is 1800 × 1800 Square formation, Q,It is 1800 dimensional vectors, is displacement and acceleration responsive vector, the F of reduced-order model respectively_R=T^TF is 1800 dimensions The load vectors of reduced-order model, degree of freedom reduces about 91%, and computational efficiency increases substantially.

3rd step, utilizes the Mass matrix M of girder structure reduced-order model_RWith Stiffness Matrix K_R, calculate the frequency of girder structure and shake Type.

Solve generalized eigenvalue equation (7) and obtain frequency and the formation of girder structure.Result of calculation is as shown in table 2, table 2 In provide first three rank corner frequency, first three rank torsional frequency and the part low order frequency that girder structure is overall, wherein L represents the vibration shape Along half wave number of girder structure axis direction, M represents the wave number in circumferentially direction.With the calculating of Lanczos method in ANSYS software Result is standard, uses Guyan static condensation algorithm in ANSYS software to carry out model reduction, to make comparisons simultaneously.Relatively three meters Calculation result is it is found that the computational accuracy of two reduced-order models is all acceptable, and error, within 10%, can meet engineering The requirement calculated, but the computational accuracy of the model order reduction of present invention proposition is higher than Guyan condensing method, especially for knot The high order of frequency of structure.

The elongated smooth barrel structure result of calculation contrast that table 2 embodiment of the present invention provides

Last it is noted that various embodiments above is only in order to illustrate technical scheme, it is not intended to limit；To the greatest extent The present invention has been described in detail by pipe with reference to foregoing embodiments, it will be understood by those within the art that: it is right Technical scheme described in foregoing embodiments is modified, or the most some or all of technical characteristic is carried out equivalent replaces Change, do not make the essence of appropriate technical solution depart from the scope of various embodiments of the present invention technical scheme.

Claims (1)

1. a girder structure free vibration analysis method based on reduced-order model, it is characterised in that following steps:

The first step, utilizes finite element software to process girder structure, obtains Mass matrix M and Stiffness Matrix K of girder structure；

Second step, uses polynomial interpolating function structure to reduce base vector, girder structure is carried out model reduction；

2.1) polynomial interpolating function is utilized, with the form of positional displacement interpolation by the arbitrary finite element node on the cross section of girder structure Displacement be condensed at the Ω centre of form of cross section；

Girder structure employing is axially the rectangular coordinate system of x-axis forward, and on the Ω of cross section, arbitrary finite element node j is by this cross section shape Column u that at heart i, generalized displacement represents_jFor:

u_j=R_jq_i (1)

Wherein, u_jRepresent the motion vector of jth finite element node, q on the Ω of cross section_iRepresent the generalized displacement at Ω centre of form i of cross section Vector, R_jRepresent the displacement transition matrix of j point；

u_jExpanded form be:

$\left\{\begin{array}{c}{u}_{jx}\\ u_{jy}\\ u_{jz}\end{array}\right\}=\left[\begin{array}{ccc}{F}_{\mathrm{\&tau;}j}& 0& 0\\ 0& F_{\mathrm{\&tau;}j}& 0\\ 0& 0& F_{\mathrm{\&tau;}j}\end{array}\right]\left\{\begin{array}{c}{q}_{ix\mathrm{\&tau;}}\\ q_{iy\mathrm{\&tau;}}\\ q_{iz\mathrm{\&tau;}}\end{array}\right\},\mathrm{\&tau;}=1,2,...W---\left(2\right)$

Wherein, { u_jx u_jy u_jz}^TFor the finite element node j three displacement components under rectangular coordinate system Oxyz, { q_ixτ q_iyτ q_izτ}^TFor cross-section centroid i generalized displacement component under rectangular coordinate system Oxyz, F_τjFor polynomial interpolating function expression formula, weight Purgation again mark τ represents summation, and W is multinomial item number；

2.2) on the Ω of cross section, all of finite element node all agglomerates at centre of form i of this cross section Ω, obtains as shown in formula (3) Displacement fills changes relational expression:

U_i=(u₁..., u_s)^T=T_iq_i=(R₁..., R_s)^Tq_i (3)

Wherein, U_iThe motion vector of s finite element node in expression cross section Ω, T_iRepresent the displacement transition matrix of cross section Ω；

2.3) it is p cross section by whole girder structure in axial direction subdivision, then by the displacement conversion relational expression group on p cross section Dress up and reduce base vector T, it is achieved the depression of order of former girder structure；

The base vector T that reduces that the described displacement conversion relational expression on p cross section is assembled into is:

Wherein, U represents total finite element modal displacement vector, is the column vector of 3n × 1, and n is the degree of freedom of girder structure Number；Q is the finite element modal displacement vector of model after depression of order, is the column vector of 3pW × 1；U_pRepresent pth cross section interior nodes Motion vector；T_pRepresenting the displacement transition matrix of pth cross section interior nodes, size expands to 3n × 3W；q_pRepresent pth cross section Generalized displacement at the centre of form, size is 3W × 1；

2.4) not considering damping effect, the finite element governing equation of former girder structure analysis is expressed as:

$M\stackrel{\&CenterDot;\&CenterDot;}{U}+KU=F---\left(5\right)$

Wherein, M, K are the square formation of 3n × 3n, are Mass matrix and the Stiffness Matrix of original structure respectively；U、It is 3n dimensional vector, respectively It is displacement structure and acceleration responsive vector；3n dimensional vector F is to act on load structurally；

2.5) in the finite element governing equation shown in formula (5), displacement transfer equation (4) is introduced, and the most left on equation both sides Take advantage of T^T, obtain the kinetics equation after depression of order:

$M_R\stackrel{\&CenterDot;\&CenterDot;}{Q}+K_{R}Q=F_R---\left(6\right)$

Wherein, M_R=T^TMT、K_R=T^TKT is Mass matrix and the Stiffness Matrix of reduced-order model respectively；M_R、K_RSquare formation for 3pW × 3pW； Q、Being 3pW dimensional vector, be the displacement of reduced-order model and acceleration responsive vector respectively, due to W, < < s, so reduced-order model Degree of freedom significantly decline；F_R=T^TF is the load vectors of 3pW dimension reduced-order model；

3rd step, utilizes second step 2.5) the Mass matrix M of girder structure reduced-order model that obtains_RWith Stiffness Matrix K_R, by formula (7) Calculate frequency and the vibration shape of the girder structure after depression of order；

Solve the Structural Dynamics generalized eigenvalue equation corresponding with formula (6):

Wherein, λ,It is respectively the feature value vector of reduced-order model and the corresponding vibration shape；For shaking of former girder structure Type.