CN111814274A - Polynomial dimension decomposition multi-degree-of-freedom rotor power system statistical moment analysis method - Google Patents

Polynomial dimension decomposition multi-degree-of-freedom rotor power system statistical moment analysis method Download PDF

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CN111814274A
CN111814274A CN202010683642.1A CN202010683642A CN111814274A CN 111814274 A CN111814274 A CN 111814274A CN 202010683642 A CN202010683642 A CN 202010683642A CN 111814274 A CN111814274 A CN 111814274A
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路宽
张昊鹏
张康宇
赵仕博
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Northwestern Polytechnical University
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Abstract

The invention provides a statistical moment analysis method of a multi-degree-of-freedom rotor power system based on polynomial dimension decomposition. The method is suitable for the rotor system model with a large uncertainty value, and the calculation cost is low; PDD and a harmonic balance method are combined to calculate rotor response with random variables, and compared with the traditional MCS method, the method has the advantages that the calculation cost is obviously reduced; the invention is the first statistical moment analysis method for researching the problem of uncertain rotor dynamics by adopting a PDD method, and has higher accuracy. The result obtained by the technical scheme of the invention is well matched with the result obtained by adopting the MCS method, and the invention is proved to have higher accuracy.

Description

Polynomial dimension decomposition multi-degree-of-freedom rotor power system statistical moment analysis method
Technical Field
The invention relates to the field of dynamics and control, in particular to a statistical moment analysis method of a multi-degree-of-freedom rotor power system based on Polynomial Dimension Decomposition (PDD).
Background
Rotor systems are one of the most important components in mechanical systems such as aircraft engines, gas turbines, large generators, and the like. Typically, the rotor system comprises a plurality of rotating discs, is very complex and has a certain uncertainty. Parameters such as system stiffness, damping, etc. are generally not precise, but fluctuate within a certain range.
The study of uncertainty problems plays a crucial role in obtaining the dynamics of structures. Monte Carlo Simulations (MCS) have been used in the past to analyze uncertainty quantification in random systems. The MCS method uses the distribution of input parameters to generate a large number of true responses, but its computation is quite extensive. The Probability Density Function (PDF) is approximated from the true response, but is very expensive to compute. In addition, many other order reduction methods also take into account the uncertainty of such problems, such as perturbation methods, the noelman method, the polynomial chaotic expansion method, and the Polynomial Dimension Decomposition (PDD) method.
The dimension decomposition is finite, layered and convergent expansion of a multidimensional output function with an increasing dimension of an input variable. The PDD method is proposed by raman, which is a method based on a random process and a representation of variables in an orthogonal basis set of random variables. The PDD method has been widely used in various research fields, such as design sensitivity analysis of moment, design sensitivity analysis of reliability, stochastic analysis, and design under uncertainty.
In previous work, the PDD method was used to study engineering mechanics models, probabilistic fracture mechanics models, etc. The PDD method can better solve the engineering problem with uncertainty, but when the PDD method is applied to different engineering problems, the PDD method needs to be corrected properly. In the field of rotor dynamics, no relevant research for analyzing a rotor system with uncertainty by applying a PDD method has been found. Therefore, it is necessary to develop a fast and accurate method suitable for a rotor model by modifying the existing PDD method. Based on the background, the invention provides a multi-degree-of-freedom rotor power system statistical moment analysis method based on a polynomial dimension decomposition method.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a polynomial dimension decomposition method for analyzing the statistical moment of a multi-degree-of-freedom rotor power system. The invention provides an uncertainty dynamical system response analysis method for a complex rotor system, namely a statistical moment analysis method based on a PDD method. The method comprises the steps of firstly providing approximate dimension decomposition of uncertain dynamical system response, then calculating corresponding parameters by using a dimension reduction integration method, and finally calculating to obtain the front second moment of the uncertain dynamical system response, thereby achieving the purpose of accurately and quickly analyzing the uncertain rotor system.
The technical scheme adopted by the invention for solving the technical problem comprises the following specific steps:
step one, deducing a PDD method for a dynamics system containing uncertainty and verifying the accuracy of the PDD method;
the dynamic system is described by a mass matrix M of n x n, a damping matrix C and a stiffness matrix K, the external excitation forces acting on the system are described by F (t), and y (t) is a vector of degrees of freedom;
assuming that the stiffness matrix is uncertain, it is expressed as:
Figure BDA0002586702020000021
wherein ξKStandard deviation for stiffness K; covKCoefficient of variance for stiffness K; adding the parameters of the marking line as approximate parameters; underlined parameters are parameters with uncertainty;
the dynamic system is expressed by the following formula:
Figure BDA0002586702020000022
let the simple harmonic of the external excitation be F (t) ═ F0eiωtThe steady state response of the system is then y (t) YeiωtWherein
Figure BDA0002586702020000023
And Y is a solution of the following formula:
(-ω2M+iωC+K)Y＝F0(3)
k and Y are arbitrary stiffness and vectors, described by the moments of the system, and the amplitude of equation (3) is Y1+iY2L, wherein Y1And Y2Are the real and imaginary parts of Y;
an S variable approximation dimension decomposition of the response y (x) is represented as:
Figure BDA0002586702020000024
equation (4) is considered to be a finite hierarchical extension of the output function, where the output function is represented by an input variable of increasing dimensionality, where y0Is a constant value, yi(xi) Is a univariate component function representing the input variable xiContribution of the individual effects to y (x). In the same way as above, the first and second,
Figure BDA0002586702020000025
is a function of the binary component of the signal,
Figure BDA0002586702020000026
is a function of the ternary component of the signal,
Figure BDA0002586702020000027
is an S-component function. When S → N, then the response converges to a determined function y (x);
the approximate expression of the first three variable component functions is:
Figure BDA0002586702020000028
Figure BDA0002586702020000031
Figure BDA0002586702020000032
wherein alpha isij,
Figure BDA0002586702020000033
Are the corresponding coefficients, the degrees of freedom of y (t) are arbitrary, y (t) is a solution of equation (2);
the S variable of a PDD method in response to y (X) is approximately expressed as:
Figure BDA0002586702020000034
when m → ∞, formula (8) approaches y (x), in the mean square sense, S ═ N; then the dimension reduction method is used to calculate the coefficient y0And
Figure BDA0002586702020000035
the corresponding coefficients in the univariate method are represented by equations (9) to (11):
Figure BDA0002586702020000036
Figure BDA0002586702020000037
Figure BDA0002586702020000038
the univariate result approximates the result of the MCS method;
performing numerical calculation on the model by combining PDD and HB methods; applying the PDD method to a model containing uncertainty, all physical parameters are uncertain, and the corresponding uncertainty form is similar to equation (1); the formula is written as:
Figure BDA0002586702020000039
response toy(t) and excitationF(t) is expressed as an n-th order finite Fourier series, and when n approaches infinity, the response is approximated as an exact solution;
Figure BDA00025867020200000310
Figure BDA00025867020200000311
whereinA 0，F 0AndA i，B i(i ═ 1.. times, n) are the corresponding simple harmonic parameters, so:
Figure BDA0002586702020000041
Figure BDA0002586702020000042
substituting equations (13) to (16) into equation (12) yields:
PY＝Q(17)
wherein,
Figure BDA0002586702020000049
Figure BDA0002586702020000043
matrix arrayPIn order of
Figure BDA0002586702020000044
M is a degree of freedom of the optical system,
Figure BDA0002586702020000045
is the HB order; the mass, damping, stiffness and external excitation matrices may all be indeterminate, written in the form of equation (1); the expression of the amplitude-frequency response Y is obtained from equation (17), and the corresponding coefficients of the bivariate PDD method are represented by equations (20) to (23);
Figure BDA0002586702020000046
Figure BDA0002586702020000047
Figure BDA0002586702020000048
Figure BDA0002586702020000051
the first and second moment equations of the PDD method are listed in equations (24) and (25):
Figure BDA0002586702020000052
Figure BDA0002586702020000053
establishing a three-degree-of-freedom spring model by using Newton's second law, comparing amplitude-frequency response obtained by a PDD method with a reference result obtained by an MCS (modulation and coding scheme), calculating to obtain an average value and a Standard Deviation (SD), and assuming that the rigidity k is uncertain andkis the mean value of the stiffness;
k＝k(1+covkξk) (26)
stiffness matrix in equation (1)
Figure BDA0002586702020000054
Expressed by equation (27):
Figure BDA0002586702020000055
the disturbance appears along with the increase of the polynomial order, but the vibration value is reduced, and when the polynomial order tends to be infinite, the disturbance disappears;
step two, calculating a simple rotor system model by PDD and HB methods;
the kinetic equation is similar to equation (12), mass, damping, stiffness, external excitation matrix written as:
Figure BDA0002586702020000056
Figure BDA0002586702020000057
in the above equation, the detailed parameters are expressed as follows:
m1,m2,m3the equivalent mass of the rotor supported at the left and the right and the equivalent rotor mass in the magnetic disk are respectively; c. C1,c2,c3Damping coefficients of the rotor in the left bearing, the disc and the right bearing respectively;
o1,o4the geometric centers of the left support and the right support are respectively;
o2,o3the geometric center and the gravity center of the magnetic disk are respectively;
k is the stiffness of the elastic shaft;
omega is the rotation speed;
writing the condition that the Fourier series is 1 as an HB-1 method, wherein the specific expression is as follows:
Figure BDA0002586702020000061
F(t)＝F 0+F 1cosωt+F 2sinωt (29)
therefore, the temperature of the molten metal is controlled,
Figure BDA0002586702020000062
Figure BDA0002586702020000063
wherein,
F 0＝(0 -m1g 0 -m2g 0 -m3g)T
F 1＝(0 0 m2eω20 0 0)T
F 2＝(0 0 0 m2eω20 0)T
substituting equations (28), (29) into equation (12) yields:
KA 0＝F 0(32)
KA 1+CB 1ω-MA 1ω2＝F 1(33)
KB 1-CA 1ω-MB 1ω2＝F 2(34)
ignoring the static part, we get equation (35):
Figure BDA0002586702020000064
wherein,
A 1＝(A11A12A13A14A15A16)T,B 1＝(B11B12B13B14B15B16)Tsolving the formula (35) to obtain a rotor system amplitude expression, as shown in the formula (36):
Figure BDA0002586702020000065
step three, a PDD method of the rotor system containing nonlinearity is obtained through popularization;
the HB-1 method was used to calculate the solution, and is specifically expressed as follows:
x(t)＝A 0+A 1cosωt+B 1sinωt (37)
therefore, the temperature of the molten metal is controlled,
Figure BDA0002586702020000071
Figure BDA0002586702020000072
Figure BDA0002586702020000073
wherein,
Figure BDA0002586702020000074
Figure BDA0002586702020000075
Figure BDA0002586702020000076
substituting equations (37) - (43) into equation (1) yields the following equation:
Figure BDA0002586702020000077
Figure BDA0002586702020000078
Figure BDA0002586702020000079
the specific parameters are as follows:
F 0＝(0 -m1g 0 -m2g 0 -m3g)T
F 1＝(0 0 m2eω20 0 0)T
F 2＝(0 0 0 m2eω20 0)T
equations (47), (48) are obtained by calculating the above parameters, as follows:
Figure BDA00025867020200000710
Figure BDA00025867020200000711
wherein,A 1＝[a1a2a3a4a5a6]T,B 1＝[b1b2b3b4b5b6]T
solving the nonlinear algebraic equation to obtain an amplitude expression as follows:
Figure BDA00025867020200000712
the first two moments of the amplitude-frequency response are obtained by the PDD method and the MCS method, compared with the linear rotor model, the influence of the third nonlinear model on the dynamic characteristics is larger, and the result obtained by the bivariate PDD method is consistent with the result obtained by the MCS method.
The invention has the beneficial effects that:
(1) the method is suitable for the rotor system model with a large uncertainty value, and the calculation cost is low; combining the PDD and Harmonic Balancing (HB) methods to calculate a rotor response with random variables, the present invention provides a significant reduction in computational cost compared to conventional MCS methods.
(2) The invention is the first statistical moment analysis method for researching the problem of uncertain rotor dynamics by adopting a PDD method, and has higher accuracy. The result obtained by the technical scheme of the invention is well matched with the result obtained by adopting the MCS method, and the invention is proved to have higher accuracy.
Drawings
FIG. 1 is a three-degree-of-freedom spring model diagram taking into account stiffness uncertainty.
Fig. 2 shows an amplitude-frequency characteristic curve (solid line) and an accurate amplitude-frequency characteristic curve (dotted line) of the three-degree-of-freedom spring model calculated by the MCS method, in which the abscissa is frequency and the ordinate is the mean value and standard deviation of the MCS, respectively, fig. 2(a) is a first-order statistical moment amplitude-frequency curve, and fig. 2(b) is a second-order statistical moment amplitude-frequency curve.
FIG. 3 is a comparison of amplitude-frequency response results calculated based on the PDD method with MCS; amplitude-frequency characteristics of PDD (dashed line) and MCS (solid line); the abscissa of the graph is the frequency and the ordinate is the mean and standard deviation curves of the PDD for a polynomial order of 2, respectively. Fig. 3(a) is a first-order statistical magnitude-frequency curve, and fig. 3(b) is a second-order statistical magnitude-frequency curve.
FIG. 4 is a comparison of amplitude-frequency response results calculated based on the PDD method with MCS; amplitude-frequency characteristics of PDD (dashed line) and MCS (solid line); the abscissa of the graph is the frequency and the ordinate is the mean and standard deviation curves of the PDD for a polynomial order of 9, respectively. Fig. 4(a) is a first-order statistical magnitude-frequency curve, and fig. 4(b) is a second-order statistical magnitude-frequency curve.
FIG. 5 is a schematic diagram of a six-DOF rotor system model.
Fig. 6 is an amplitude-frequency characteristic curve of four indeterminate quantities, a univariate PDD (solid line), a bivariate PDD (dotted line) and an MCS (point), in which the abscissa is frequency and the ordinate is the mean and standard deviation of the second-order PDD, respectively, fig. 6(a) is a first-order statistical moment amplitude-frequency curve, and fig. 6(b) is a second-order statistical moment amplitude-frequency curve.
Fig. 7 is a graph showing amplitude-frequency characteristics of four indeterminate quantities, a single-variable PDD (solid line), a double-variable PDD (dotted line) and an MCS (point), in which the abscissa represents frequency and the ordinate represents the mean and standard deviation of the nine-step PDD, respectively. Fig. 7(a) is a first-order statistical magnitude-frequency curve, and fig. 7(b) is a second-order statistical magnitude-frequency curve.
Fig. 8 is an amplitude-frequency characteristic curve of nine indeterminate quantities, a univariate PDD (solid line), a bivariate PDD (dotted line) and an MCS (point), in which the abscissa represents frequency and the ordinate represents the mean and standard deviation of the second-order PDD, respectively. Fig. 8(a) is a first-order statistical magnitude-frequency curve, and fig. 8(b) is a second-order statistical magnitude-frequency curve.
Fig. 9 is an amplitude-frequency characteristic curve of nine indeterminate quantities, a single-variable PDD (solid line), a double-variable PDD (dotted line), and an MCS (point), in which the abscissa represents frequency and the ordinate represents the mean and standard deviation of the nine-step PDD, respectively. Fig. 9(a) is a first-order statistical magnitude-frequency curve, and fig. 9(b) is a second-order statistical magnitude-frequency curve.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
The invention specifically comprises the following contents:
first, a PDD method for a rotor system with uncertainty is derived. Establishing a dynamic equation containing uncertainty; obtaining an S variable approximate dimension decomposition of the response y (x) by using a PDD method and an HB method
Figure BDA0002586702020000093
Carrying out approximate Fourier polynomial expansion on the S variable component function in the decomposition, and calculating corresponding parameters by using a dimension reduction integral method; calculating by using a calculation formula of the first two-order statistical moment
Figure BDA0002586702020000091
The first second moment of (a), the approximate response of the system.
The method shown in the first step is then used for the calculation of a model of rotor dynamics with uncertainty. Establishing a model, assuming that a rotor system rigidity, damping and mass matrix is uncertain, and representing each parameter matrix by using a standard deviation and a variance coefficient of the rotor system; according to the excitation form, obtaining a response form, and calculating the front second moment of the stiffness matrix, the damping matrix and the mass matrix; the derived PDD method is used with the model and its accuracy is verified against the MCS method.
And finally, based on the second step, the PDD method of the rotor system containing nonlinearity is obtained by popularization, and the purpose of rapidly and accurately analyzing the multi-degree-of-freedom complex rotor dynamics model is achieved.
Firstly, deriving a PDD method for a dynamical system containing uncertainty and verifying the accuracy of the PDD method
The dynamic system is described by a mass matrix M of n x n, a damping matrix C and a stiffness matrix K, the external excitation forces acting on the system are described by F (t), and y (t) is a vector of degrees of freedom;
assuming that the stiffness matrix is uncertain, it is expressed as:
Figure BDA0002586702020000092
wherein ξKStandard deviation for stiffness K; covKCoefficient of variance for stiffness K; adding the parameters of the marking line as approximate parameters; the underlined parameters are parameters with uncertainty.
The dynamic system is expressed by the following formula:
Figure BDA0002586702020000101
let the simple harmonic of the external excitation be F (t) ═ F0eiωtThe steady state response of the system is then y (t) YeiωtWherein
Figure BDA0002586702020000102
And Y is a solution of the following formula:
(-ω2M+iωC+K)Y＝F0(3)
k and Y are arbitrary stiffness and vectors, described by the moments of the system, and the amplitude of equation (3) is Y1+iY2L, wherein Y1And Y2Are the real and imaginary parts of Y.
The PDD method is an uncertain order reduction quantification method and is widely applied to a random system. An S variable approximation dimension decomposition of the response y (x) is represented as:
Figure BDA0002586702020000103
it is treated as a finite hierarchical extension of the output function, where the output function is represented by an input variable of increasing dimension, where y0Is a constant value, yi(xi) Is a univariate component function representing the input variable xiContribution of the individual effects to y (x). In the same way as above, the first and second,
Figure BDA0002586702020000104
is a function of the binary component of the signal,
Figure BDA0002586702020000105
is a function of the ternary component of the signal,
Figure BDA0002586702020000106
is an S-component function. When S → N, the response converges to a determined function y (x).
The approximate expression of the first three variable component functions is:
Figure BDA0002586702020000107
Figure BDA0002586702020000108
Figure BDA0002586702020000109
wherein alpha isij,
Figure BDA00025867020200001010
Are the corresponding coefficients, the degrees of freedom of y (t) are arbitrary, and y (t) is a solution of the kinetic equation (2).
The S variable of a PDD method in response to y (X) is approximately expressed as:
Figure BDA0002586702020000111
when m → ∞, formula (8) approaches y (x), in the mean square sense, S ═ N; then the dimension reduction method is used to calculate the coefficient y0And
Figure BDA0002586702020000112
the corresponding coefficients in the univariate method are represented by equations (9) to (11):
Figure BDA0002586702020000113
Figure BDA0002586702020000114
Figure BDA0002586702020000115
the univariate result may well approximate the result of the MCS method.
And (4) carrying out numerical calculation on the model by combining PDD and HB methods. Applying the PDD method to a model containing an uncertainty, all physical parameters may be uncertain and the corresponding uncertainty form is similar to equation (1); the formula is written as:
Figure BDA0002586702020000116
response toy(t) and excitationF(t) is expressed as an n-th order finite Fourier series, and as n approaches infinity, the response is approximated as an exact solution.
Figure BDA0002586702020000117
Figure BDA0002586702020000118
WhereinA 0，F 0AndA i，B i(i ═ 1.. times, n) are the corresponding simple harmonic parameters, so:
Figure BDA0002586702020000119
Figure BDA00025867020200001110
substituting equations (13) to (16) into equation (12) yields:
PY＝Q(17)
wherein,
Figure BDA00025867020200001210
Figure BDA0002586702020000121
matrix arrayPIn order of
Figure BDA0002586702020000122
M is a degree of freedom of the optical system,
Figure BDA0002586702020000123
is the HB order. The mass, damping, stiffness and external excitation matrix may all be indeterminate, which is written in the form of equation (1). The expression of the amplitude-frequency response Y is obtained from equation (17). The corresponding coefficients of the bivariate PDD method are represented by equations (20-23);
Figure BDA0002586702020000124
Figure BDA0002586702020000125
Figure BDA0002586702020000126
Figure BDA0002586702020000127
the first moment and second moment formulas of the PDD method are listed in equations (24) and (25).
Figure BDA0002586702020000128
Figure BDA0002586702020000129
By using a simple dynamic modelThe accuracy of the method is verified. A three-degree-of-freedom spring model was created using newton's second law, as shown in fig. 1, and its stiffness was considered uncertain. Comparing the amplitude-frequency response obtained by the PDD method with the reference result obtained by the MCS, calculating the average value and Standard Deviation (SD) of the calculation example shown in figure 1, assuming that the rigidity k is uncertain, andkis the average of the stiffness.
k＝k(1+covkξk) (26)
Stiffness matrix in equation (1)
Figure BDA0002586702020000131
Expressed by equation (27):
Figure BDA0002586702020000132
TABLE 1 spring System parameters
Figure BDA0002586702020000133
The parameters of the system are listed in Table 1, and the MCS result is to use 10000 random variables xikAnd (4) obtaining a sample. Mean and SD (first and second moments) are plotted in fig. 2, and deterministic responses are also plotted in fig. 2 (2). The polynomial orders are 2 and 9, the amplitude-frequency response is calculated based on the PDD method, and the results are shown in fig. 1 and 2. As shown in fig. 3 and 4, they closely matched the MCS results except for perturbations around the resonant frequency. Perturbations occur as the polynomial order increases, but the vibration value decreases. As the polynomial order approaches infinity, the perturbation will disappear.
In most cases, the response of the powertrain system is not smooth, and it is clear that disturbances will occur when we use a polynomial approximation response function.
Secondly, calculating a simple rotor system model by PDD and HB methods
A more common six-degree-of-freedom rotor system model (as shown in fig. 5) is taken as an example to illustrate how to calculate a simple rotor system model by using PDD and HB methods, and its general kinetic equation is similar to equation (12), and its mass, damping, stiffness, external excitation matrix are written as:
Figure BDA0002586702020000134
Figure BDA0002586702020000141
in the above equation, the detailed parameters are expressed as follows:
m1,m2,m3: the equivalent mass of the rotor supported at the left and the right and the equivalent rotor mass in the magnetic disk are respectively;
c1,c2,c3: damping coefficients of the rotor in the left bearing, the disc and the right bearing respectively;
o1,o4: the geometric centers of the left support and the right support are respectively;
o2,o3: the geometric center and the gravity center of the magnetic disk are respectively;
k: the stiffness of the elastic shaft;
ω: a rotational speed;
the case where the Fourier series is 1 is written as the HB-1 method. The specific expression is as follows:
Figure BDA0002586702020000142
F(t)＝F 0+F 1cosωt+F 2sinωt (29)
therefore, the temperature of the molten metal is controlled,
Figure BDA0002586702020000143
Figure BDA0002586702020000144
wherein,
F 0＝(0 -m1g 0 -m2g 0 -m3g)T
F 1＝(0 0 m2eω20 0 0)T
F 2＝(0 0 0 m2eω20 0)T
substituting equations (28) (29) into equation (12) yields:
KA 0＝F 0(32)
KA 1+CB 1ω-MA 1ω2＝F 1(33)
KB 1-CA 1ω-MB 1ω2＝F 2(34)
ignoring the static part, we get equation (35):
Figure BDA0002586702020000151
wherein,
A 1＝(A11A12A13A14A15A16)T,B 1＝(B11B12B13B14B15B16)T
solving the formula (35) to obtain the rotor system amplitude expression, such as the formula (36)
Figure BDA0002586702020000152
(1) Four uncertainty values
The mass matrix M, the rigidity matrix K, the damping matrix C and the eccentricity e respectively comprise a random variable, so that four random variables exist in the six-degree-of-freedom rotor system, and the uncertainty of the four random variables is similar to the formula (27).
The PDD method will be used for the case of calculating polynomial orders of 2 and 9, respectively, the results are shown in fig. 6 and 7, the first two moments (Mean and standard deviation SD) match the MCS results completely, and the bivariate PDD is significantly better than the univariate PDD. The PDD result seems to approach the MCS result better as the polynomial order increases. The results of the rotor system with four random variables further verify the accuracy of the PDD method.
(2) Nine uncertainty
The nine uncertainty parameters of the rotor system model were all considered uncertain and all cov values would be set to 10%, as shown in table 2. While the PDD method will be generalized to a more difficult case. Table 3 shows the cov values for the indeterminate amounts.
Table 2: rotor system parameters
Figure BDA0002586702020000153
Table 3: cov value of engineering parameter
Figure BDA0002586702020000154
Compared with fig. 6, the nine uncertain dynamics are more complex than the four-random-variable case, the relative error of the univariate PDD is larger than the bivariate, and SD is more obvious than Mean.
Fig. 9 discusses the first two moments (Mean and SD) of the amplitude-frequency response. As the polynomial order increases, the result approximates the reference solution obtained by the MCS method. The results of the rotor model with 9 random variables once verified the efficiency of the PDD method.
The PDD method is applicable to the following two rotor models: one is a rotor model with four random variables; the other is a model with nine random variables. Obviously, as the number of random variables increases, univariate PDD does not work well, while bivariate PDD may better approximate MCS results. The PDD method is applied to the rotor model for the first time, and the accuracy of the PDD method is verified through the result.
Third, a PDD method for obtaining a rotor system containing nonlinearity by popularization
Based on the above linear rotor model, in fig. 4, in order to meet the requirements of practical engineering, three times of nonlinear stiffness support will be considered in the system model. We consider that four non-linear stiffnesses are uncertain and that the other random variables increase to thirteen. The HB-1 method will be used to calculate the solution, as expressed in detail below:
x(t)＝A 0+A 1cosωt+B 1sinωt (37)
therefore, the temperature of the molten metal is controlled,
Figure BDA0002586702020000161
Figure BDA0002586702020000162
Figure BDA0002586702020000163
wherein,
Figure BDA0002586702020000164
Figure BDA0002586702020000165
Figure BDA0002586702020000166
substituting equations (37) - (43) into equation (1) yields the following equation:
Figure BDA0002586702020000167
Figure BDA0002586702020000168
Figure BDA0002586702020000169
the specific parameters are as follows:
F 0＝(0 -m1g 0 -m2g 0 -m3g)T
F 1＝(0 0 m2eω20 0 0)T
F 2＝(0 0 0 m2eω20 0)T
equations (47), (48) are obtained by calculating the above parameters, as follows:
Figure BDA0002586702020000171
Figure BDA0002586702020000172
wherein,A 1＝[a1a2a3a4a5a6]T,B 1＝[b1b2b3b4b5b6]T
solving the nonlinear algebraic equation to obtain an amplitude expression as follows:
Figure BDA0002586702020000173
considering first the case of cov value of 3%, the amount of calculation increases greatly due to the large nonlinear effect, so only the case of polynomial order of 2 is considered. The first two moments of the amplitude-frequency response can be obtained by the PDD method and the MCS method. The cubic non-linear model has a greater influence on the dynamics than the linear rotor model, where many disturbances are generated. The results obtained by the bivariate PDD method are very similar to those obtained by the MCS method and much better than those of the univariate method.
The cov value for the nonlinear stiffness was further increased to 0.1. The amplitude-frequency response of the first two statistical moments is obtained by the PDD method and the MCS method. When cov is increased to 0.1, the amplitude-frequency characteristic of the first order statistical moment is more fluctuating and the dynamics are more complex. The results of the univariate PDD method cannot be approximated to those obtained by the MCS method, with relatively large errors, whereas the bivariate PDD method is much better. The comparison of the two methods verifies the accuracy of the PDD method in the application of the rotor system.
The PDD method is applied to a nonlinear rotor system for the first time, and the accuracy of the PDD method is verified by comparing with the MCS method. Meanwhile, the PDD method is suitable for a rotor system model with a large uncertain value, and the robustness of the rotor system model is verified. Compared with the MCS method, the PDD method can greatly reduce the calculated amount, and the results of the univariate and bivariate methods are well matched with the results of the MCS method.

Claims (1)

1. A polynomial dimension decomposed multi-degree-of-freedom rotor power system statistical moment analysis method is characterized by comprising the following steps:
step one, deducing a PDD method for a dynamics system containing uncertainty and verifying the accuracy of the PDD method;
the dynamic system is described by a mass matrix M of n x n, a damping matrix C and a stiffness matrix K, the external excitation forces acting on the system are described by F (t), and y (t) is a vector of degrees of freedom;
assuming that the stiffness matrix is uncertain, it is expressed as:
Figure FDA0002586702010000011
wherein ξKStandard deviation for stiffness K; covKCoefficient of variance for stiffness K; adding the parameters of the marking line as approximate parameters; underlined parameters are parameters with uncertainty;
the dynamic system is expressed by the following formula:
Figure FDA0002586702010000012
let the simple harmonic of the external excitation be F (t) ═ F0eiωtThe steady state response of the system is then y (t) YeiωtWherein
Figure FDA0002586702010000013
And Y is a solution of the following formula:
(-ω2M+iωC+K)Y＝F0(3)
k and Y are arbitrary stiffness and vectors, described by the moments of the system, and the amplitude of equation (3) is Y1+iY2L, wherein Y1And Y2Are the real and imaginary parts of Y;
an S variable approximation dimension decomposition of the response y (x) is represented as:
Figure FDA0002586702010000014
equation (4) is considered to be a finite hierarchical extension of the output function, where the output function is represented by an input variable of increasing dimensionality, where y0Is a constant value, yi(xi) Is a univariate component function representing the input variable xiContribution of sole effect to y (x); in the same way as above, the first and second,
Figure FDA0002586702010000015
is a function of the binary component of the signal,
Figure FDA0002586702010000016
is a function of the ternary component of the signal,
Figure FDA0002586702010000017
is an S-component function; when S → N, then the response converges to a determined function y (x);
the approximate expression of the first three variable component functions is:
Figure FDA0002586702010000021
Figure FDA0002586702010000022
Figure FDA0002586702010000023
wherein alpha isij,
Figure FDA0002586702010000024
Are the corresponding coefficients, the degrees of freedom of y (t) are arbitrary, y (t) is a solution of equation (2);
the S variable of a PDD method in response to y (X) is approximately expressed as:
Figure FDA0002586702010000025
when m → ∞, formula (8) approaches y (x), in the mean square sense, S ═ N; then the dimension reduction method is used to calculate the coefficient y0And
Figure FDA0002586702010000026
the corresponding coefficients in the univariate method are represented by equations (9) to (11):
Figure FDA0002586702010000027
Figure FDA0002586702010000028
Figure FDA0002586702010000029
the univariate result approximates the result of the MCS method;
performing numerical calculation on the model by combining PDD and HB methods; applying the PDD method to a model containing uncertainty, all physical parameters are uncertain, and the corresponding uncertainty form is similar to equation (1); the formula is written as:
Figure FDA00025867020100000210
response toy(t) and excitationF(t) is expressed as an n-th order finite Fourier series, and when n approaches infinity, the response is approximated as an exact solution;
Figure FDA00025867020100000211
Figure FDA0002586702010000031
whereinA 0，F 0AndA i，B i(i ═ 1.. times, n) are the corresponding simple harmonic parameters, so:
Figure FDA0002586702010000032
Figure FDA0002586702010000033
substituting equations (13) to (16) into equation (12) yields:
PY＝Q(17)
wherein,
Figure FDA0002586702010000034
Figure FDA0002586702010000035
matrix arrayPIn order of
Figure FDA0002586702010000036
M is a degree of freedom of the optical system,
Figure FDA0002586702010000037
is the HB order; the mass, damping, stiffness and external excitation matrices may all be indeterminate, written in the form of equation (1); the expression of the amplitude-frequency response Y is obtained from equation (17), and the corresponding coefficients of the bivariate PDD method are represented by equations (20) to (23);
Figure FDA0002586702010000038
Figure FDA0002586702010000039
Figure FDA0002586702010000041
Figure FDA0002586702010000042
the first and second moment equations of the PDD method are listed in equations (24) and (25):
Figure FDA0002586702010000043
Figure FDA0002586702010000044
establishing a three-degree-of-freedom spring model by using Newton's second law, comparing amplitude-frequency response obtained by a PDD method with a reference result obtained by an MCS (modulation and coding scheme), calculating to obtain an average value and a Standard Deviation (SD), and assuming that the rigidity k is uncertain andkis the mean value of the stiffness;
k＝k(1+covkξk) (26)
stiffness matrix in equation (1)
Figure FDA0002586702010000045
Expressed by equation (27):
Figure FDA0002586702010000046
the disturbance appears along with the increase of the polynomial order, but the vibration value is reduced, and when the polynomial order tends to be infinite, the disturbance disappears;
step two, calculating a simple rotor system model by PDD and HB methods;
the kinetic equation is similar to equation (12), mass, damping, stiffness, external excitation matrix written as:
Figure FDA0002586702010000047
Figure FDA0002586702010000051
in the above equation, the detailed parameters are expressed as follows:
m1,m2,m3the equivalent mass of the rotor supported at the left and the right and the equivalent rotor mass in the magnetic disk are respectively;
c1,c2,c3damping coefficients of the rotor in the left bearing, the disc and the right bearing respectively;
o1,o4the geometric centers of the left support and the right support are respectively;
o2,o3the geometric center and the gravity center of the magnetic disk are respectively;
k is the stiffness of the elastic shaft;
omega is the rotation speed;
writing the condition that the Fourier series is 1 as an HB-1 method, wherein the specific expression is as follows:
Figure FDA0002586702010000052
F(t)＝F 0+F 1cosωt+F 2sinωt (29)
therefore, the temperature of the molten metal is controlled,
Figure FDA0002586702010000053
Figure FDA0002586702010000054
wherein,
F 0＝(0 -m1g 0 -m2g 0-m3g)T
F 1＝(0 0 m2eω20 0 0)T
F 2＝(0 0 0 m2eω20 0)T
substituting equations (28), (29) into equation (12) yields:
KA 0＝F 0(32)
KA 1+CB 1ω-MA 1ω2＝F 1(33)
KB 1-CA 1ω-MB 1ω2＝F 2(34)
ignoring the static part, we get equation (35):
Figure FDA0002586702010000061
wherein,
A 1＝(A11A12A13A14A15A16)T,B 1＝(B11B12B13B14B15B16)T
solving the formula (35) to obtain a rotor system amplitude expression, as shown in the formula (36):
Figure FDA0002586702010000062
step three, a PDD method of the rotor system containing nonlinearity is obtained through popularization;
the HB-1 method was used to calculate the solution, and is specifically expressed as follows:
x(t)＝A 0+A 1cosωt+B 1sinωt (37)
therefore, the temperature of the molten metal is controlled,
Figure FDA0002586702010000063
Figure FDA0002586702010000064
Figure FDA0002586702010000065
wherein,
Figure FDA0002586702010000066
Figure FDA0002586702010000067
Figure FDA0002586702010000068
substituting equations (37) - (43) into equation (1) yields the following equation:
Figure FDA0002586702010000069
Figure FDA00025867020100000610
Figure FDA00025867020100000611
the specific parameters are as follows:
F 0＝(0 -m1g 0 -m2g 0 -m3g)T
F 1＝(0 0 m2eω20 0 0)T
F 2＝(0 0 0 m2eω20 0)T
equations (47), (48) are obtained by calculating the above parameters, as follows:
Figure FDA0002586702010000071
Figure FDA0002586702010000072
wherein,A 1＝[a1a2a3a4a5a6]T,B 1＝[b1b2b3b4b5b6]T
solving the nonlinear algebraic equation to obtain an amplitude expression as follows:
Figure FDA0002586702010000073
the first two moments of the amplitude-frequency response are obtained by the PDD method and the MCS method, compared with the linear rotor model, the influence of the third nonlinear model on the dynamic characteristics is larger, and the result obtained by the bivariate PDD method is consistent with the result obtained by the MCS method.
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