CN111563323A - Transient POD method based on minimum error of bifurcation parameter - Google Patents

Transient POD method based on minimum error of bifurcation parameter Download PDF

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CN111563323A
CN111563323A CN202010343355.6A CN202010343355A CN111563323A CN 111563323 A CN111563323 A CN 111563323A CN 202010343355 A CN202010343355 A CN 202010343355A CN 111563323 A CN111563323 A CN 111563323A
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路宽
张康宇
张昊鹏
赵仕博
靳玉林
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Northwestern Polytechnical University
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Abstract

The invention provides a transient POD method based on minimum error of bifurcation parameters, which comprises the steps of firstly, regarding a POD mode function as a function of system parameters, an initial value and sampling length, giving a reduction condition of a parameter domain by defining an average truncation error function and a total average truncation error function of the parameter domain, then utilizing the POD method to carry out parameter domain reduction on a high-dimensional nonlinear rotor-sliding bearing system, and analyzing the influence of rotating speed, the initial condition, the sampling length and mode number on the reduction. The invention can obtain the invariable order-reducing model of the high-dimensional system, so that the model keeps similar bifurcation characteristic with the original system in the parameter domain, namely the error of the bifurcation parameter of the order-reducing model and the original system is minimum, and the parameter domain order reduction of the high-dimensional complex system is realized on the parameter with the same or similar bifurcation, thereby effectively reducing the calculated amount of the high-dimensional complex system under the condition of ensuring the calculation precision.

Description

Transient POD method based on minimum error of bifurcation parameter
Technical Field
The invention relates to the field of dimension reduction of a dynamic system, in particular to a transient POD method based on minimum errors of bifurcation parameters.
Background
With the development of science and technology in China, a large-scale high-dimensional complex rotor system becomes more important, a core component rotor-bearing system of the system has a complex structure, a plurality of wheel disc stages and a severe operating environment, the system operates under complex and changeable working conditions and is a high-dimensional nonlinear system essentially, and therefore the order of the system must be reasonably reduced.
Although the traditional POD method can solve the robustness problem of the order-reduced system in a large-range parameter domain to a certain extent, the adaptive order reduction of the parameter domain is obtained by continuously updating the POD order-reduced mode or adjusting the mode number, and a low-dimensional invariant model which can approximately reflect the dynamic characteristics of a high-dimensional complex system cannot be obtained.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a transient POD method based on the minimum error of a bifurcation parameter, which can obtain a mathematical equivalent reduced order model of a high-dimensional nonlinear rotor bearing system in a parameter domain. A constant order-reduction model of a high-dimensional system is obtained by using a POD method, so that the model keeps similar bifurcation characteristics with an original system in a parameter domain, a low-dimensional equivalent model of the original system is obtained, and research and calculation are facilitated. The invention further perfects the transient POD method and provides a transient POD method based on the minimum error of the bifurcation parameter, which is called POD parameter domain order reduction method.
The invention firstly takes the POD mode function as the function of system parameter, initial value and sampling length, and provides the order-reducing condition of parameter domain by defining the average truncation error function and the total average truncation error function of parameter domain. And then, performing parameter domain order reduction on the high-dimensional nonlinear rotor-sliding bearing system by using a POD (product-oriented programming) method, and analyzing the influence of the rotating speed, initial conditions, sampling length and modal number on the order reduction.
The technical scheme adopted by the invention for solving the technical problem comprises the following steps:
one, transient POD parameter domain order reduction condition
Taking the POD reduced-order mode as a function of all state parameters of the system, and recording the set of all reduced-order systems obtained by the method as Sr(ii) a The kinetic equation of the high-dimensional complex structure system is assumed as follows:
Figure BDA0002469276420000011
in the formula, M, K and C are respectively a total mass, rigidity and damping matrix of the system; x, X,
Figure BDA0002469276420000021
Generalized displacement vector, generalized velocity vector and generalized acceleration vector with n degrees of freedom respectively,
Figure BDA0002469276420000022
the generalized force vectors of the system such as nonlinear force, external excitation and the like are adopted; order to
Figure BDA0002469276420000023
Equation (1) is expressed as a differential equation of motion of the state space:
Figure BDA0002469276420000024
wherein
Figure BDA0002469276420000025
Order to
Figure BDA0002469276420000026
Theta to ω t and transforming the balance point of the vector field to zero 0 by a translation transformation2n+1Equation (2) can be expressed as an autonomous system:
Figure BDA0002469276420000027
in the formula (3)
Figure BDA0002469276420000028
And g (0) is 0, and q (t) is a bounded solution of formula (3), according to the constant variation method:
Figure BDA0002469276420000029
where Ψ (t) is the basis solution matrix for the linear portion of equation (3), q0Q (0) is an initial value;
if POD reduced mode is
Figure BDA00024692764200000210
x0∈R2n、α∈Rl、ts∈R+Wherein x is0，α，tsRespectively as initial condition, system parameter and sampling length, and a coordinate transformation matrix composed of the first m reduced-order modes
Figure BDA00024692764200000211
The following can be obtained:
Figure BDA00024692764200000212
the primary autonomous system state space coordinates are then expressed as:
Figure BDA00024692764200000213
wherein the content of the first and second substances,
Figure BDA00024692764200000214
by taking the formula (6) into the formula (3), the differential equation of motion of the reduced-order system in the state space can be obtained as follows:
Figure BDA00024692764200000215
in the formula (7)
Figure BDA0002469276420000031
gr(qr)＝TT×[g(q)]2n+1，gr(0)＝TT×[g(0)]2n+1＝0，
Figure BDA0002469276420000032
The general solution of the order-reduced system obtained from the ordinary differential equation basic theory is as follows:
Figure BDA0002469276420000033
in the formula psir(t) is the linear part A of the reduced order systemrBase solution matrix of qr0＝TTq (0), the solution of the reduced order system to the original system can be obtained by the formula (6):
Figure BDA0002469276420000034
the solution in equation (9) approximates the solution of the original system, i.e.:
||q(t)-qOr(t)||→0 (10)
the formula (4) and the formula (9) are introduced into the formula (10):
Figure BDA0002469276420000035
equation (11) is sufficiently small to satisfy, for any t ∈ [0, + ∞):
||Ψ(t)-ΨOr(t)||→0，ΨOr(t)＝TΨr(t)TT(12)
Ψ (t) and Ψr(t) constant matrices A and A, respectivelyrThe constructed basis solution matrix is respectively expressed as:
Figure BDA0002469276420000036
Figure BDA0002469276420000037
the formula (14) is introduced into the formula (12):
Figure BDA0002469276420000038
when m is n, TTT＝TTT＝I2n+1Then, then
||Ψ(t)-ΨOr(t)||≡0 (16)
The conversion from the expression (10) to the expression (12) is regular transformation, the model before and after transformation is completely equivalent, and the degree of freedom of the system is not reduced completely; when m < n, TTT≠TTT＝I2m+1Equation (16) cannot equal zero;
at parameter x0∈R2n,α∈Rl,ts∈R+The response of the original system is approximately expressed as
Figure BDA0002469276420000041
And α' ∈ R for system parameterslDoes not necessarily satisfy
Figure BDA0002469276420000042
i＝1,...,N；
The sample parameters that make equation (12) sufficiently small are analyzed by defining the average truncation error function as:
Figure BDA0002469276420000043
is expressed in the original system parameter x0∈R2n，α∈Rl，ts∈R+Error of response of the sampled reduced order model and the original system parameter α' in t time, wherein<·>Represents ti(i 1.. N) an average operator of N points in time;
it is assumed that the system parameters are continuously changing in the parameter domain,
Figure BDA0002469276420000048
to reduce the modal function of the parameter α
Figure BDA0002469276420000044
The method can approximate the original system in the whole parameter domain, so that the total average truncation error function of the parameter domain is defined as follows:
Figure BDA0002469276420000045
formula (18)Representing the approximation degree of POD reduced modal function obtained by sampling the original system parameters to the original system in the whole parameter domain omega when EmWhen sufficiently small, the response of the original system in the whole parameter domain omega is reduced by the first few orders of the modal shape of the parameter α
Figure BDA0002469276420000046
A linear representation;
the POD method constructs a reduced-order mode through an original system response signal, and at the moment Em(t,x0,α,ts) Is compared with the reduced order mode number m and the sampling length ts(ii) related; when the number m of reduced-order modes is determined, the total average truncation error function Em(t,x0,α,ts) Only with tsIn connection with, composed ofm{t,α′,φ(x0,α,ts) Is equal to or more than 0, and E is obtainedm(t,x0,α,ts) There must be an infimum bound, i.e. a minimum in the parameter domain; when the system parameters, initial conditions and reduced order mode number are determined, the method makes Em(t,x0,α,ts) The minimum sample length is called the optimal sample length topt；
Em(t,x0,α,ts)≥infEm(t,x0,α,ts)＝Em(t,x0,α,topt) (19)
If E ism(t,x0,α,topt) <, and sufficiently small, POD reduced-order mode function by optimal sample length
Figure BDA0002469276420000047
The method can approach the original system in the whole parameter domain, thereby obtaining a low-dimensional invariant order-reduction model and realizing the order reduction of the parameter domain of the high-dimensional complex system;
secondly, realizing parameter domain reduction by the optimal sampling length;
two different parameters omega for a parameter domain of a high dimensional complex system (n-dimensional)1,ω2∈[ωα,ωβ]Corresponding to optimal sampling lengths of
Figure BDA0002469276420000051
When in use
Figure BDA0002469276420000052
When the time is small enough, the obtained two reduced order models are equivalent, S1，S2∈Sr，
Figure BDA00024692764200000517
Both is
Figure BDA0002469276420000053
Setting two different parameters omega of the system1，ω2Respectively, the optimal sampling length is
Figure BDA0002469276420000054
And
Figure BDA0002469276420000055
due to the satisfaction of
Figure BDA0002469276420000056
Sufficiently small that there is a sufficiently small positive number > 0, such that
Figure BDA0002469276420000057
Figure BDA0002469276420000058
Wherein
Figure BDA0002469276420000059
Are all less than
Figure BDA00024692764200000510
Due to the fact that
Figure BDA00024692764200000511
Figure BDA00024692764200000512
Thus, it is possible to provide
Figure BDA00024692764200000513
In the parameter domain [ omega ]α,ωβ]Inner is zero, so there is X (X)0T, ω) in the parameter domain
Figure BDA00024692764200000514
And
Figure BDA00024692764200000515
the following formula can be obtained:
Figure BDA00024692764200000516
wherein
Figure BDA0002469276420000061
Figure BDA0002469276420000062
Thus is provided with
Figure BDA0002469276420000063
Therefore, the reduced models obtained by the optimal sampling lengths of different sampling parameters form an equivalence relation when the total average truncation error is small enough, namely the reduced models are equivalent to each other, and all the reduced models with the small enough total average truncation error can be equivalent to each other in the same way;
thirdly, reducing the order of the high-dimensional nonlinear rotor-bearing system model;
in the high-dimensional nonlinear rotor-bearing system, each disk is assumed to be a rigid disk, two ends of a rotor are supported by sliding bearings, a left end bearing has a loosening fault, and each shaft section is also assumed to be an elastic shaft with equal rigidity design, namely a left sideThe geometric center of the shaft neck, the geometric center of each disk shaft section and the geometric center of the right shaft neck are O in sequencei(i 1.. 7), left and right journal masses are concentrated at respective geometric centers (O)1，O7) The mass of the rest shaft sections is concentrated at the mass center position O of each wheel disci' (i ═ 2.. 6) with a corresponding eccentricity ei(i ═ 2.. 6); selecting the geometric center of the left end bearing as a reference point, xi，yi(i 1.. 7) displacements in horizontal and vertical directions relative to a reference point within radial planes of the left journal centroid, each disk centroid, and the right journal centroid, respectively;
mi，ci(i＝1...7)，ki(i 1.. 6) respectively represent equivalent mass, equivalent damping and equivalent stiffness of each shaft section at the corresponding position; fx，FyOil film forces of the journal in the horizontal and vertical directions, respectively; assuming that the left end loosening occurs only in the vertical direction, y8For displacement from a reference point, the loosening stiffness ksAnd loose damping csThe piecewise linear function is:
Figure BDA0002469276420000064
the differential equation of motion of the multi-disk rotor-bearing system is established as follows:
Figure BDA0002469276420000065
wherein X is [ X ] in the formula (27)1,...,x7,y1...y7,y8]TThe expression of the system mass matrix M, the damping matrix C, the stiffness matrix K and the mass matrix M is as follows:
Figure BDA0002469276420000071
Figure BDA0002469276420000072
Figure BDA0002469276420000073
the device comprises nonlinear oil film force of a left journal and a right journal, eccentric excitation of each disk and gravity, and the expression is as follows:
Figure BDA0002469276420000074
equation (25) is dimensionless processed by the following transformation relationship:
Figure BDA0002469276420000075
Figure BDA0002469276420000076
Figure BDA0002469276420000077
Figure BDA0002469276420000078
Figure BDA0002469276420000079
Figure BDA00024692764200000710
in formula (29) fx，fyIs dimensionless nonlinear oil film force in horizontal and vertical directions respectively, mu is the dynamic viscosity coefficient of lubricating oil, omega is angular frequency, L is the effective length of the bearing, R is the effective radius of the bearing, W is half the weight of the rotor system,
Figure BDA0002469276420000081
is the Sommerfeld coefficient, c is the oil film thickness, and the system parameters are shown in Table 1;
TABLE 1 rotor-bearing System parameters
Figure BDA0002469276420000082
The 4-order Runge-Kutta numerical integration is utilized to obtain the accurate numerical response of 15 degrees of freedom of the original system, and then the displacement response signals of different parameters are extracted
Figure BDA0002469276420000083
As a sampled snapshot matrix, where NsData length); the system transient motion comprises free vibration and forced vibration and has system inherent modal information, and the sampling of each parameter comprises the transient motion of the system; the eigenvectors of the autocorrelation matrix are calculated by:
Figure BDA0002469276420000084
the characteristic vectors are arranged in descending order according to the characteristic values, and then POD reduced-order mode is obtained
Figure BDA0002469276420000085
Projecting the original system to a subspace spanned by the first m-order reduced mode, namely, the following coordinate transformation relation exists:
Figure BDA0002469276420000086
the differential equation of motion for the resulting reduced order model is obtained by substituting (25) with equation (30):
Figure BDA0002469276420000087
in the formula Mr＝PTMP，Cr＝PTCP，Kr＝PTKP，Fr＝PTF, initial conditions are as follows0＝PTx0Determining;
m of known reduced order modelsr，Cr，Kr，FrRelated to system parameters, initial conditions, sampling length, reduced order mode number and the like, and the requirement that the total average truncation error is small enoughThe reduced order models obtained by sampling the signals are equivalent to each other; once the sampling snapshot signal matrix of the original system is determined, the reduced order model is determined, so that the influence of system parameters, initial conditions, sampling length and reduced order mode number on the reduced order of the parameter domain can be analyzed, the reduced order mode of the parameter domain is determined, and the invariant reduced order model of the system on the parameter domain is obtained.
The method has the advantages that the invariant reduced order model of the high-dimensional system can be obtained, so that the model keeps similar bifurcation characteristics with the original system in a parameter domain, namely the error of the bifurcation parameter of the reduced order model and the original system is minimum, and the bifurcation occurs on the same or similar parameters. By adopting the method, a low-dimensional invariant model which can approximately reflect the dynamic characteristics of the high-dimensional system can be obtained in the parameter domain, and the order reduction of the parameter domain of the high-dimensional complex system is realized, so that the calculation amount of the high-dimensional complex system is effectively reduced under the condition of ensuring the calculation precision.
Drawings
FIG. 1 is a schematic view of a high-dimensional non-linear rotor-bearing system model.
FIG. 2 shows the original system in the rotation speed range of [300,1200 ]]Graph of the response bifurcation at rad/s, the abscissa of the graph being the rotation speed and the ordinate being y1Amplitude in the direction.
In FIG. 3, FIGS. 3(a) and 3(b) are spectrograms of oil film oscillation of the original system at rotational speeds of 915rad/s and 1100rad/s, respectively, where the abscissa is rotational speed and the ordinate is y1Amplitude in the direction.
FIG. 4(a) is a bifurcation diagram of the original system; FIGS. 4(b) - (h) are bifurcation diagrams of the 2-DOF reduced order model under the conditions of a sampling rotation speed of 350rad/s and sampling lengths of 50 pi, 80 pi, 100 pi, 130 pi, 150 pi, 200 pi and 300 pi, respectively, wherein the abscissa in the diagram is the rotation speed and the ordinate is y1Amplitude in the direction.
FIGS. 5(a) and 5(b) are axial trace graphs of the original system model and the 2-DOF reduced model at a rotation speed of 900rad/s under the conditions of a sampling rotation speed of 350rad/s and an optimal sampling length, respectively, and FIGS. 5(c) and 5(d) are frequency spectrums of the two system models, wherein the abscissa is the rotation speed and the ordinate is the y1Amplitude in the direction.
FIG. 6(a) and FIG. 6(b) are the trace diagrams of the axes of the original system model and the 2-DOF reduced model at the rotation speed of 950rad/s respectively under the conditions of the sampling rotation speed of 350rad/s and the optimal sampling length, and FIG. 6(c) and FIG. 6(d) are the spectrograms of the two system models, wherein the abscissa is the rotation speed and the ordinate is y1Amplitude in the direction.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
One, transient POD parameter domain order reduction condition
Since the POD reduced-order mode is related to system parameters, initial conditions, sampling length, etc., the POD reduced-order mode is regarded as a function of all state parameters of the system, and the set of all the reduced-order systems obtained thereby is denoted as Sr(ii) a The kinetic equation of the high-dimensional complex structure system is assumed as follows:
Figure BDA0002469276420000101
in the formula, M, K and C are respectively a total mass, rigidity and damping matrix of the system; x, X,
Figure BDA0002469276420000102
Generalized displacement vector, generalized velocity vector and generalized acceleration vector with n degrees of freedom respectively,
Figure BDA0002469276420000103
the generalized force vectors of the system such as nonlinear force, external excitation and the like are adopted; order to
Figure BDA0002469276420000104
Equation (1) is expressed as a differential equation of motion of the state space:
Figure BDA0002469276420000105
wherein
Figure BDA0002469276420000106
Order to
Figure BDA0002469276420000107
Theta to ω t and transforming the balance point of the vector field to zero 0 by a translation transformation2n+1Equation (2) can be expressed as an autonomous system:
Figure BDA0002469276420000108
in the formula (3)
Figure BDA0002469276420000109
And g (0) is 0, and q (t) is a bounded solution of formula (3), according to the constant variation method:
Figure BDA00024692764200001010
where Ψ (t) is the basis solution matrix for the linear portion of equation (3), q0Q (0) is an initial value;
if POD reduced mode is
Figure BDA00024692764200001011
x0∈R2n、α∈Rl、ts∈R+Wherein x is0，α，tsRespectively as initial condition, system parameter and sampling length, and a coordinate transformation matrix composed of the first m reduced-order modes
Figure BDA00024692764200001012
The following can be obtained:
Figure BDA00024692764200001013
the primary autonomous system state space coordinates are then expressed as:
Figure BDA00024692764200001014
wherein the content of the first and second substances,
Figure BDA0002469276420000111
by taking the formula (6) into the formula (3), the differential equation of motion of the reduced-order system in the state space can be obtained as follows:
Figure BDA0002469276420000112
in the formula (7)
Figure BDA0002469276420000113
gr(qr)＝TT×[g(q)]2n+1，gr(0)＝TT×[g(0)]2n+1＝0，
Figure BDA0002469276420000114
The general solution of the order-reduced system obtained from the ordinary differential equation basic theory is as follows:
Figure BDA0002469276420000115
in the formula psir(t) is the linear part A of the reduced order systemrBase solution matrix of qr0＝TTq (0), the solution of the reduced order system to the original system can be obtained by the formula (6):
Figure BDA0002469276420000116
the solution in equation (9) approximates the solution of the original system, i.e.:
||q(t)-qOr(t)||→0 (10)
the formula (4) and the formula (9) are introduced into the formula (10):
Figure BDA0002469276420000117
equation (11) is sufficiently small to satisfy, for any t ∈ [0, + ∞):
||Ψ(t)-ΨOr(t)||→0，ΨOr(t)＝TΨr(t)TT(12)
Ψ (t) and Ψr(t)Constant matrices A and A, respectivelyrThe constructed basis solution matrix is respectively expressed as:
Figure BDA0002469276420000118
Figure BDA0002469276420000119
the formula (14) is introduced into the formula (12):
Figure BDA00024692764200001110
when m is n, TTT＝TTT＝I2n+1Then, then
||Ψ(t)-ΨOr(t)||≡0 (16)
The conversion from the expression (10) to the expression (12) is regular transformation, the model before and after transformation is completely equivalent, and the degree of freedom of the system is not reduced completely; when m < n, TTT≠TTT＝I2m+1Equation (16) may not equal zero, but whether P (x) is present within the error range0,α,ts) Or T (x)0,α,ts) So that equation (12) is sufficiently small. On one hand, the transient response signal contains the inherent modal information of the original system, so that P (x) sampled from the transient signal0,α,ts) Formula (12) can be made smaller; p (x) on the other hand0,α,ts) As a function of system parameters, initial conditions, and sample length;
according to the POD method, the parameter x0∈R2n,α∈Rl,ts∈R+The response of the original system can be approximated as
Figure BDA0002469276420000121
And α' ∈ R for system parameterslDoes not necessarily satisfy
Figure BDA0002469276420000122
The reason why i 1.. ang.n is not satisfied is the reduced order systemThe solution of the system differential equation has no direct approximation relation with the solution of the original physical system;
the sample parameters that make equation (12) sufficiently small are analyzed by defining the average truncation error function as:
Figure BDA0002469276420000123
is expressed in the original system parameter x0∈R2n，α∈Rl，ts∈R+Error of response of the sampled reduced order model and the original system parameter α' in t time, wherein<·>Represents ti(i 1.. said., N) averaging operator of N points in time whenmSufficiently small, it means that the response of the original system at parameter α' can be linearly represented by the first few (m) POD orders of magnitude of parameter α.
It is assumed that the system parameters are continuously changing in the parameter domain,
Figure BDA0002469276420000126
to reduce the modal function of the parameter α
Figure BDA0002469276420000124
The method can approximate the original system in the whole parameter domain, so that the total average truncation error function of the parameter domain is defined as follows:
Figure BDA0002469276420000125
equation (18) represents the approximation degree of POD reduced modal function obtained by sampling the original system parameters to the original system in the whole parameter domain omega, when EmSufficiently small, meaning that the response of the original system over the entire parameter domain Ω is reduced by the first few orders of the modal shape of the parameter α
Figure BDA0002469276420000131
A linear representation;
the POD method constructs a reduced-order mode through an original system response signal, and at the moment Em(t,x0,α,ts) Is compared with the reduced order mode number m and the sampling length ts(ii) related; when the number m of reduced-order modes is determined, the total average truncation error function Em(t,x0,α,ts) Only with tsIn connection with, composed ofm{t,α′,φ(x0,α,ts) Is equal to or more than 0, and E is obtainedm(t,x0,α,ts) There must be an infimum bound, i.e. a minimum in the parameter domain; when the system parameters, initial conditions and reduced order mode number are determined, the method makes Em(t,x0,α,ts) The minimum sample length is called the optimal sample length topt；
Em(t,x0,α,ts)≥infEm(t,x0,α,ts)＝Em(t,x0,α,topt) (19)
If E ism(t,x0,α,topt) <, and sufficiently small, meaning POD reduced order mode functions obtained by optimal sample length
Figure BDA0002469276420000132
The method can approach the original system in the whole parameter domain, thereby obtaining a low-dimensional invariant order-reduction model and realizing the order reduction of the parameter domain of the high-dimensional complex system;
secondly, realizing parameter domain reduction by the optimal sampling length;
because the response of different parameters of the original system is different, the POD reduced-order modes are also different, and therefore, the optimal sampling length is also different. Two different parameters omega for a parameter domain of a high dimensional complex system (n-dimensional)1,ω2∈[ωα,ωβ]Corresponding to optimal sampling lengths of
Figure BDA0002469276420000133
When in use
Figure BDA0002469276420000134
When the time is small enough, the obtained two reduced order models are equivalent, S1，S2∈Sr，
Figure BDA00024692764200001311
Both is
Figure BDA0002469276420000135
Setting two different parameters omega of the system1，ω2Respectively, the optimal sampling length is
Figure BDA0002469276420000136
And
Figure BDA0002469276420000137
due to the satisfaction of
Figure BDA0002469276420000138
Sufficiently small that there is a sufficiently small positive number > 0, such that
Figure BDA0002469276420000139
Figure BDA00024692764200001310
Wherein
Figure BDA0002469276420000141
Are all less than
Figure BDA0002469276420000142
Due to the fact that
Figure BDA0002469276420000143
Figure BDA0002469276420000144
Thus, it is possible to provide
Figure BDA0002469276420000145
In the parameter domain [ omega ]α,ωβ]Inner almost everywhere is zero, so there is X (X)0T, ω) is almost equal to in the parameter domain
Figure BDA0002469276420000146
And
Figure BDA0002469276420000147
the following formula can be obtained:
Figure BDA0002469276420000148
wherein
Figure BDA0002469276420000149
Figure BDA00024692764200001410
Thus is provided with
Figure BDA00024692764200001411
Therefore, the reduced order models obtained by the optimal sampling lengths of different sampling parameters form an equivalent relation when the total average truncation error is small enough, namely the reduced order models are equivalent to each other. The same can be satisfied that all reduced order models with sufficiently small total average truncation error are mutually equivalent.
Thirdly, reducing the order of the high-dimensional nonlinear rotor-bearing system model;
fig. 1 is a schematic representation of a high-dimensional nonlinear rotor-bearing system model, the rotor consisting of 5 impeller disks, assuming each disk is a rigid disk, the rotor is supported at both ends by sliding bearings, and there is a loose failure in the left end bearing. Similarly, assuming that each shaft section is an elastic shaft with equal rigidity, the geometric center of the left journal, the geometric center of each disk shaft section and the geometric center of the right journal are sequentially Oi(i 1.. 7), left and right journal masses are concentrated at respective geometric centers (O)1，O7) The mass of the rest shaft sections is concentrated at the mass center position O of each wheel disci' (i ═ 2.. 6) with a corresponding eccentricity ei(i ═ 2.. 6); selecting the geometric center of the left end bearing as a reference point, xi，yi(i 1.. 7) displacements in horizontal and vertical directions relative to a reference point within radial planes of the left journal centroid, each disk centroid, and the right journal centroid, respectively; m isi，ci(i＝1...7)，ki(i 1.. 6) respectively represent equivalent mass, equivalent damping and equivalent stiffness of each shaft section at the corresponding position; fx，FyOil film forces of the journal in the horizontal and vertical directions, respectively; assuming that the left end loosening occurs only in the vertical direction, y8For displacement from a reference point, the loosening stiffness ksAnd loose damping csThe piecewise linear function is:
Figure BDA0002469276420000151
the differential equation of motion of the multi-disk rotor-bearing system is established as follows:
Figure BDA0002469276420000152
wherein X is [ X ] in the formula (27)1,…,x7,y1...y7,y8]TThe expression of the system mass matrix M, the damping matrix C, the stiffness matrix K and the mass matrix M is as follows:
Figure BDA0002469276420000153
Figure BDA0002469276420000154
Figure BDA0002469276420000161
the device comprises nonlinear oil film force of a left journal and a right journal, eccentric excitation of each disk and gravity, and the expression is as follows:
Figure BDA0002469276420000162
equation (25) is dimensionless processed by the following transformation relationship:
Figure BDA0002469276420000163
Figure BDA0002469276420000164
Figure BDA0002469276420000165
Figure BDA0002469276420000166
Figure BDA0002469276420000167
Figure BDA0002469276420000168
in formula (29) fx，fyIs dimensionless nonlinear oil film force in horizontal and vertical directions respectively, mu is the dynamic viscosity coefficient of lubricating oil, omega is angular frequency, L is the effective length of the bearing, R is the effective radius of the bearing, W is half the weight of the rotor system,
Figure BDA0002469276420000169
is the Sommerfeld coefficient, c is the oil film thickness, and the system parameters are shown in Table 1;
TABLE 1 rotor-bearing System parameters
Figure BDA00024692764200001610
The 4-order Runge-Kutta numerical integration is utilized to obtain the accurate numerical response of 15 degrees of freedom of the original system, and then the displacement response signals of different parameters are extracted
Figure BDA0002469276420000171
As a sampled snapshot matrix (N)sData length). Since the transient motion of the system includes free vibration and forced vibration, and has the information of the intrinsic mode of the system, the sampling of each parameter herein includes the transient motion of the system. The eigenvectors of the autocorrelation matrix are calculated by:
Figure BDA0002469276420000172
the characteristic vectors are arranged in descending order according to the characteristic values, and then POD reduced-order mode is obtained
Figure BDA0002469276420000173
Projecting the original system to a subspace spanned by the first m-order reduced mode, namely, the following coordinate transformation relation exists:
Figure BDA0002469276420000174
substituting (25) equation (30) into the differential equation of motion for the reduced order model:
Figure BDA0002469276420000175
in the formula Mr＝PTMP，Cr＝PTCP，Kr＝PTKP，Fr＝PTF, initial conditions can be selected from u0＝PTx0And (4) determining.
M of the above known reduced order modelr，Cr，Kr，FrThe method is related to system parameters, initial conditions, sampling length, reduced order mode number and the like, and the mutual equivalence of reduced order models obtained by sampling signals with small enough total average truncation errors is met. Once the original system sampling snapshot signal matrix is determined, a reduced order model is determined. Therefore, the influence of system parameters, initial conditions, sampling length, reduced order modal number and the like on the reduction of the parameter domain can be analyzed, the reduced order mode of the parameter domain is determined, and the invariant reduced order model of the system in the parameter domain is obtained.
Fourthly, numerically verifying the effectiveness of the POD parameter domain reduction;
selecting the rotating speed as the parameter of the system parameter domain reduction, wherein the rotating speed range is [300,1200 ]]rad/s, and obtaining a bifurcation diagram of the original system in the rotating speed range through numerical integration, as shown in figure 2. From FIG. 2, the response of the original system at the rotation speed ωcBifurcating at 915rad/s at a rotational speed [300,915]rad/s, the system is in periodic 1 motion; at a rotational speed [915,1200]rad/s, the system shows a complex branched characteristic, the vibration is large in the rotating speed domain, and the system is mainly characterized in that a response frequency spectrum has a large peak value close to half the rotating shaft frequency. FIG. 3 is a frequency spectrum diagram corresponding to two rotation speeds in an oil film oscillation area of an original system, wherein when the rotation speed is 915rad/s, the system just generates oil film instability, and the frequency spectrum comprises a rotation shaft frequency component of 0.473 times; when the rotating speed is 1100rad/s, the amplitude of the frequency component of the rotating shaft which is 0.469 times of that of the rotating shaft is large, and the system generates a violent oil film oscillation phenomenon.
For parameter domain order reduction, the order reduction model and the original system should have basically the same bifurcation characteristics in the parameter domain, i.e. the bifurcation graphs have similar structures, similar bifurcation points and consistent bifurcation behaviors, so the influence of sampling parameters on the parameter domain order reduction is mainly analyzed by comparing the order reduction model under different sampling conditions with the bifurcation graphs of the original system.
The influence of the sampling length on the parameter domain reduction is firstly analyzed. Let the initial value of the original system state vector be xi＝yi＝0.69，y8＝0.25，
Figure BDA0002469276420000181
Obtaining the response signals of the original system in different time lengths at a fixed rotating speed to obtain a reduced order model under the corresponding sampling length, and controlling the rotating speed [300,1200 ]]And comparing the bifurcation characteristics of the original system in the rad/s range. FIG. 4 is a bifurcation diagram of the original system and 2 DOF reduced models obtained at a sampling rotation speed of 350rad/s and different sampling lengths, wherein the bifurcation points of the reduced models with sampling lengths of 50 π, 80 π, 100 π, 130 π, 150 π, 200 π, 300 π are 1200rad/s, 1100rad/s, 1005rad/s, 910rad/s, 865rad/s, 775rad/s, and 655/s, respectively, and gradually decrease with the increase of the sampling lengths until convergence. Andcomparing the bifurcation diagrams of the original system, easily finding that the bifurcation diagram of the obtained reduced order model has a similar structure with the original system only when the sampling length is 130 pi, the bifurcation point is closest in the rotating speed domain, and the error with the original system is minimum; and the difference between the reduced order model under other sampling lengths and the bifurcation graph structure of the original system is larger, and the bifurcation points are far away in the rotating speed domain. Therefore, the 2-degree-of-freedom reduced model obtained under the sampling length can reflect the bifurcation characteristics of 15 degrees of freedom of the original system in a parameter domain, and the reduced models under other sampling lengths cannot reflect the bifurcation characteristics of the original system. Therefore, the sampling length can be regarded as the optimal sampling length with the sampling rotation speed of 350rad/s and the reduced-order mode number of 2.
As can be seen from FIGS. 5 and 6, the axis locus and the spectrogram of the reduced order model are substantially similar to those of the original system, and both of them move in a period of 1 at a rotation speed of 900 rad/s; at 950rad/s, both of them are almost periodic motion, the main frequency components of the original system and the reduced order model are respectively 0.445 times and 0.453 times of the spindle frequency, about half of the spindle frequency, the spindle frequency component amplitude is weak, these characteristics are consistent with the oil film oscillation phenomenon of the original system, and as can be seen from fig. 4e, the oil film oscillation area of the reduced order model is consistent with the original system. Therefore, the 2-degree-of-freedom unchanged order reduction model obtained by the optimal sampling length of the rotating speed of 350rad/s can reflect the main dynamic characteristics of the original system in the parameter domain, and the parameter domain order reduction of the system can be realized.

Claims (1)

1. A transient POD method based on minimum error of bifurcation parameters is characterized by comprising the following steps:
one, transient POD parameter domain order reduction condition
Taking the POD reduced-order mode as a function of all state parameters of the system, and recording the set of all reduced-order systems obtained by the method as Sr(ii) a The kinetic equation of the high-dimensional complex structure system is assumed as follows:
Figure FDA0002469276410000011
wherein M, K and C are respectively a total mass, rigidity and damping matrix of the system；X、
Figure FDA0002469276410000012
Generalized displacement vector, generalized velocity vector and generalized acceleration vector with n degrees of freedom respectively,
Figure FDA0002469276410000013
the generalized force vectors of the system such as nonlinear force, external excitation and the like are adopted; order to
Figure FDA0002469276410000014
Equation (1) is expressed as a differential equation of motion of the state space:
Figure FDA0002469276410000015
wherein
Figure FDA0002469276410000016
Order to
Figure FDA0002469276410000017
Theta to ω t and transforming the balance point of the vector field to zero 0 by a translation transformation2n+1Equation (2) can be expressed as an autonomous system:
Figure FDA0002469276410000018
in the formula (3)
Figure FDA0002469276410000019
g∈C1And g (0) is 0, q (t) is a bounded solution of formula (3), according to the constant variation method:
Figure FDA00024692764100000110
where Ψ (t) is the basis solution matrix for the linear portion of equation (3), q0Q (0) is an initial value;
if POD reduced mode is
Figure FDA00024692764100000111
x0∈R2n、α∈Rl、ts∈R+Wherein x is0，α，tsRespectively as initial condition, system parameter and sampling length, and a coordinate transformation matrix composed of the first m reduced-order modes
Figure FDA00024692764100000112
The following can be obtained:
Figure FDA00024692764100000113
the primary autonomous system state space coordinates are then expressed as:
Figure FDA0002469276410000021
wherein the content of the first and second substances,
Figure FDA0002469276410000022
by taking the formula (6) into the formula (3), the differential equation of motion of the reduced-order system in the state space can be obtained as follows:
Figure FDA0002469276410000023
in the formula (7)
Figure FDA0002469276410000024
gr(qr)＝TT×[g(q)]2n+1，gr(0)＝TT×[g(0)]2n+1＝0，
Figure FDA0002469276410000025
The general solution of the order-reduced system obtained from the ordinary differential equation basic theory is as follows:
Figure FDA0002469276410000026
in the formula psir(t) is the linear part A of the reduced order systemrBase solution matrix of qr0＝TTq (0), the solution of the reduced order system to the original system can be obtained by the formula (6):
Figure FDA0002469276410000027
the solution in equation (9) approximates the solution of the original system, i.e.:
||q(t)-qOr(t)||→0 (10)
the formula (4) and the formula (9) are introduced into the formula (10):
Figure FDA0002469276410000028
equation (11) is sufficiently small to satisfy, for any t ∈ [0, + ∞):
||Ψ(t)-ΨOr(t)||→0，ΨOr(t)＝TΨr(t)TT(12)
Ψ (t) and Ψr(t) constant matrices A and A, respectivelyrThe constructed basis solution matrix is respectively expressed as:
Figure FDA0002469276410000029
Figure FDA00024692764100000210
the formula (14) is introduced into the formula (12):
Figure FDA0002469276410000031
when m is n, TTT＝TTT＝I2n+1Then, then
||Ψ(t)-ΨOr(t)||≡0 (16)
The conversion from the expression (10) to the expression (12) is regular transformation, the model before and after transformation is completely equivalent, and the degree of freedom of the system is not reduced completely; when m < n, TTT≠TTT＝I2m+1Equation (16) cannot equal zero;
at parameter x0∈R2n,α∈Rl,ts∈R+The response of the original system is approximately expressed as
Figure FDA0002469276410000032
And α' ∈ R for system parameterslDoes not necessarily satisfy
Figure FDA0002469276410000033
The sample parameters that make equation (12) sufficiently small are analyzed by defining the average truncation error function as:
Figure FDA0002469276410000034
is expressed in the original system parameter x0∈R2n，α∈Rl，ts∈R+Error of response of the sampled reduced order model and the original system parameter α' in t time, wherein<·>Represents ti(i 1.. N) an average operator of N points in time;
it is assumed that the system parameters are continuously changing in the parameter domain,
Figure FDA0002469276410000035
to reduce the modal function of the parameter α
Figure FDA0002469276410000036
The method can approximate the original system in the whole parameter domain, so that the total average truncation error function of the parameter domain is defined as follows:
Figure FDA0002469276410000037
equation (18) represents the approximation degree of POD reduced modal function obtained by sampling the original system parameters to the original system in the whole parameter domain omega, when EmWhen sufficiently small, the response of the original system in the whole parameter domain omega is reduced by the first few orders of the modal shape of the parameter α
Figure FDA0002469276410000038
A linear representation;
the POD method constructs a reduced-order mode through an original system response signal, and at the moment Em(t,x0,α,ts) Is compared with the reduced order mode number m and the sampling length ts(ii) related; when the number m of reduced-order modes is determined, the total average truncation error function Em(t,x0,α,ts) Only with tsIn connection with, composed ofm{t,α′,φ(x0,α,ts) Is equal to or more than 0, and E is obtainedm(t,x0,α,ts) There must be an infimum bound, i.e. a minimum in the parameter domain; when the system parameters, initial conditions and reduced order mode number are determined, the method makes Em(t,x0,α,ts) The minimum sample length is called the optimal sample length topt；
Em(t,x0,α,ts)≥infEm(t,x0,α,ts)＝Em(t,x0,α,topt) (19)
If E ism(t,x0,α,topt) <, and sufficiently small, POD reduced-order mode function by optimal sample length
Figure FDA0002469276410000041
The method can approach the original system in the whole parameter domain, thereby obtaining a low-dimensional invariant order-reduction model and realizing the order reduction of the parameter domain of the high-dimensional complex system;
secondly, realizing parameter domain reduction by the optimal sampling length;
two different parameters omega for a parameter domain of a high dimensional complex system (n-dimensional)1,ω2∈[ωα,ωβ]Corresponding to optimal sampling lengths of
Figure FDA0002469276410000042
When in use
Figure FDA0002469276410000043
When the time is small enough, the obtained two reduced order models are equivalent, S1，S2∈Sr，
Figure FDA00024692764100000415
Both is
Figure FDA0002469276410000044
Setting two different parameters omega of the system1，ω2Respectively, the optimal sampling length is
Figure FDA0002469276410000045
And
Figure FDA0002469276410000046
due to the satisfaction of
Figure FDA0002469276410000047
Sufficiently small that there is a sufficiently small positive number > 0, such that
Figure FDA0002469276410000048
Figure FDA0002469276410000049
Wherein
Figure FDA00024692764100000410
Are all less than
Figure FDA00024692764100000411
Due to the fact that
Figure FDA00024692764100000412
Figure FDA00024692764100000413
Thus, it is possible to provide
Figure FDA00024692764100000414
In the parameter domain [ omega ]α,ωβ]Inner is zero, so there is X (X)0T, ω) in the parameter domain
Figure FDA0002469276410000051
And
Figure FDA0002469276410000052
the following formula can be obtained:
Figure FDA0002469276410000053
wherein
Figure FDA0002469276410000054
Figure FDA0002469276410000055
Thus is provided with
Figure FDA0002469276410000056
Therefore, the reduced models obtained by the optimal sampling lengths of different sampling parameters form an equivalence relation when the total average truncation error is small enough, namely the reduced models are equivalent to each other, and all the reduced models with the small enough total average truncation error can be equivalent to each other in the same way;
thirdly, reducing the order of the high-dimensional nonlinear rotor-bearing system model;
in the high-dimensional nonlinear rotor-bearing system, each disk is assumed to be a rigid disk, two ends of a rotor are supported by sliding bearings, a left-end bearing has a loosening fault, each shaft section is also assumed to be an elastic shaft with equal rigidity, and the geometric center of a left shaft neck, the geometric center of each disk shaft section and the geometric center of a right shaft neck are sequentially Oi(i-1 … 7) with the left and right journal masses centered at respective geometric centers (O)1，O7) The mass of the rest shaft sections is concentrated at the mass center position O of each wheel disci' (i ═ 2.. 6) with a corresponding eccentricity ei(i-2 … 6); selecting the geometric center of the left end bearing as a reference point, xi，yi(i 1.. 7) displacements in horizontal and vertical directions relative to a reference point within radial planes of the left journal centroid, each disk centroid, and the right journal centroid, respectively;
mi，ci(i＝1...7)，ki(i 1.. 6) respectively represent equivalent mass, equivalent damping and equivalent stiffness of each shaft section at the corresponding position; fx，FyOil film forces of the journal in the horizontal and vertical directions, respectively; assuming that the left end loosening occurs only in the vertical direction, y8For displacement from a reference point, the loosening stiffness ksAnd loose damping csThe piecewise linear function is:
Figure FDA0002469276410000061
the differential equation of motion of the multi-disk rotor-bearing system is established as follows:
Figure FDA0002469276410000062
wherein X is [ X ] in the formula (27)1,...,x7,y1...y7,y8]TThe expression of the system mass matrix M, the damping matrix C, the stiffness matrix K and the mass matrix M is as follows:
Figure FDA0002469276410000063
Figure FDA0002469276410000064
Figure FDA0002469276410000065
the device comprises nonlinear oil film force of a left journal and a right journal, eccentric excitation of each disk and gravity, and the expression is as follows:
Figure FDA0002469276410000066
equation (25) is dimensionless processed by the following transformation relationship:
Figure FDA0002469276410000071
in formula (29) fx，fyIs dimensionless nonlinear oil film force in horizontal and vertical directions respectively, mu is the dynamic viscosity coefficient of lubricating oil, omega is angular frequency, L is the effective length of the bearing, R is the effective radius of the bearing, W is half the weight of the rotor system,
Figure FDA0002469276410000072
is the Sommerfeld coefficient, c is the oil film thickness, and the system parameters are shown in Table 1;
TABLE 1 rotor-bearing System parameters
Figure FDA0002469276410000073
The 4-order Runge-Kutta numerical integration is utilized to obtain the accurate numerical response of 15 degrees of freedom of the original system, and then the displacement response signals of different parameters are extracted
Figure FDA0002469276410000074
As a sampled snapshot matrix, where NsData length); the system transient motion comprises free vibration and forced vibration and has system inherent modal information, and the sampling of each parameter comprises the transient motion of the system; the eigenvectors of the autocorrelation matrix are calculated by:
Figure FDA0002469276410000081
the characteristic vectors are arranged in descending order according to the characteristic values, and then POD reduced-order mode is obtained
Figure FDA0002469276410000082
Projecting the original system to a subspace spanned by the first m-order reduced mode, namely, the following coordinate transformation relation exists:
Figure FDA0002469276410000083
the differential equation of motion for the resulting reduced order model is obtained by substituting (25) with equation (30):
Figure FDA0002469276410000084
in the formula Mr＝PTMP，Cr＝PTCP，Kr＝PTKP，Fr＝PTF, initial conditions are as follows0＝PTx0Determining;
m of known reduced order modelsr，Cr，Kr，FrThe method is related to system parameters, initial conditions, sampling length, reduced order mode number and the like, and the mutual equivalence of reduced order models obtained by sampling signals with small enough total average truncation errors is met; once the sampling snapshot signal matrix of the original system is determined, the reduced order model is determined, so that the influence of system parameters, initial conditions, sampling length and reduced order mode number on the reduced order of the parameter domain can be analyzed, the reduced order mode of the parameter domain is determined, and the invariant reduced order model of the system on the parameter domain is obtained.
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Publication number Priority date Publication date Assignee Title
CN113254863A (en) * 2021-05-07 2021-08-13 西北工业大学 Self-adaptive orthogonal decomposition method for order reduction of multi-disk rotor system model

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* Cited by examiner, † Cited by third party
Title
KUAN LU: "A modified nonlinear POD method for order reduction based on transient time series", 《NONLINEAR DYNAMICS》 *
靳玉林: "基于POD方法的模型降阶研究及其在复杂转子-轴承系统中的应用", 《中国博士学位论文全文数据库电子期刊 工程科技II辑》 *

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN113254863A (en) * 2021-05-07 2021-08-13 西北工业大学 Self-adaptive orthogonal decomposition method for order reduction of multi-disk rotor system model
CN113254863B (en) * 2021-05-07 2023-11-07 西北工业大学 Adaptive orthogonal decomposition method for model reduction of multi-disk rotor system

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