CN107622160B - Multi-point excitation vibration numerical simulation method based on inverse problem solving - Google Patents
Multi-point excitation vibration numerical simulation method based on inverse problem solving Download PDFInfo
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Abstract
The invention discloses a multipoint excitation vibration numerical simulation method based on inverse problem solving. The method carries out the numerical simulation calculation of the multi-point excitation vibration test in an inverse problem solving mode, mainly solves the problem that m acceleration load spectrums are accurately calculated on the basis of known n output control spectrums in the simulation calculation process of the multi-input multi-output control finite element, can effectively increase the accuracy and the simulation precision of the simulation calculation of the multi-point excitation vibration finite element, improves the calculation efficiency, avoids the safety risk of the missile entity vibration test, and reduces the test cost.
Description
Technical Field
The invention belongs to the technical field of finite element simulation calculation of mechanical engineering, and particularly relates to a multipoint excitation vibration numerical simulation method based on inverse problem solving.
Background
The multipoint excitation vibration test system is used for exciting a test piece in a vibration test along a single axial direction or multiple axial directions at a time by adopting a plurality of vibration tables. The control method of the control system is developed on the basis of a single-shaft vibration control technology. Control theories and methods such as random vibration control, multipoint sine frequency sweep control, multipoint transient vibration and the like have been developed abroad. The basic idea of the control method is as follows: the controlled system composed of the vibration table, the test piece, the clamp, the power amplifier, the sensor and the like is generally regarded as a multi-input multi-output linear time-invariant system; the physical model of the multipoint excitation vibration test system can be simplified into a typical MIMO system, and the corresponding mathematical model can represent a transfer function matrix; the dimension of a transfer function matrix of the MIMO system is n multiplied by m according to the difference between the number m of input excitation signals and the number n of output response signals, wherein a row vector represents the influence of different excitation signals on the same control point, and a column vector represents the magnitude of response signals excited by the same excitation signal at different control points.
In the field of simulation calculation, the problem of accurately simulating the finite element calculation result of a multipoint excitation vibration test system is the most important problem, but the problems of large difference, low calculation accuracy and low efficiency still exist in the current numerical simulation method of the finite element model of the multipoint excitation vibration test.
Therefore, an efficient and accurate numerical simulation method for a multipoint excitation vibration test system is needed to improve the precision of multipoint excitation numerical simulation calculation and improve the calculation accuracy and efficiency.
Disclosure of Invention
In order to solve the technical problems, the invention provides a multipoint excitation vibration numerical simulation method based on inverse problem solving, which can ensure that the result of the dynamic simulation calculation of a multipoint excitation vibration test system is more real, has higher conformity with the real vibration test result, and can be effectively applied to the vibration numerical simulation of various products.
The invention is realized by the following technical scheme:
a multipoint excitation vibration numerical simulation method based on inverse problem solving is characterized by comprising the following steps: carrying out finite element modeling on a specific engineering object, and obtaining modal parameters of a finite element model through finite element analysis;
extracting a frequency response function between each input excitation point and each output control point by adopting a white noise excitation method;
according to the preset control spectrum of each output control point, the acceleration load spectrum of each input excitation point is worked out through the load identification matrix inversion;
loading the calculated acceleration load spectrum of each input excitation point to a corresponding excitation position;
obtaining an actual acceleration response power spectral density function of n output control points, wherein n is not less than 1 and is an integer;
comparing the value with a preset acceleration power spectrum density function to perform relative error analysis, and if the analysis result meets the control spectrum requirement, finishing the calculation; if the control spectrum requirement is not met, load spectrums at m input positions which are reversely solved are corrected, wherein m is not less than 1 and is an integer.
In the above technical solution, according to the superposition principle, each output of the linear system is formed by superposing responses corresponding to respective discrete inputs, and the system has m inputs xiWhere i is 1,2 …, m, then for each output ykWhere k is 1,2 …, n, there are m impulse response functions hki(t); and for n outputs, there is an n × m impulse response function, an n × m order impulse response function matrix [ h (t)]Comprises the following steps:
the input is represented as an m x 1 order array { x (t) } ═ xi(t) representing the output as an n × 1 order array { y (t) } ═ yk(t) }, then:
transposing the above equation to obtain:
the autocorrelation function and cross-correlation function of n outputs form the following output correlation function matrix [ Ryy(τ)]：
Wherein:
correspondingly, the self-spectrum and cross-spectrum of n outputs form an output power spectrum matrix Syy(ω)，Syy(ω) Fourier transforming equation (4) to obtain:
the auto-and cross-correlations of the n inputs form an input correlation matrix as follows:
the self-spectrum and cross-spectrum of m inputs form an input power spectrum matrix Sxx(ω)，Sxx(ω) Fourier transforming both ends of equation (6) to obtain:
when m inputs are not correlated, the self-spectrum of the multi-input multi-output system response point for m inputs and n outputs is represented as:
in the formula:
representing m input self-power spectral density functions;
a response self-power spectral density function representing the n outputs;
representing a frequency response function of the r input to the k output;
representing the conjugate of the corresponding frequency response function;
representing the square of the magnitude of the corresponding frequency response function;
according to equation (8), the multiple-input multiple-output relationship is represented in the form of a matrix:
the load identification matrix is written as:
in the formula (10), the compound represented by the formula (10),
the square of the frequency response function amplitude of the nth input to the kth output is represented and obtained by a white noise excitation method; and the positive sign means that the matrix is inverted when m is equal to n, the generalized inverse is obtained when m is not equal to n, the matrix is seen from the upper matrix, if n output control spectrums are known, m acceleration load spectrums required to be loaded are obtained in a reverse mode, and when m is equal to n, namely the number of exciting forces is the same as the number of control measuring points, the exciting force is obtainedA unique solution to the spectrum; and when m is less than n, namely the number of the exciting forces is less than the number of the control measuring points, solving the least square solution of the acceleration load spectrum.
In the above technical solution, the white noise excitation method is: when m inputs are simultaneously excited by a unit white noise acceleration power spectrum with the amplitude of 1, i.e.
The self-spectrum of the n output response points is represented as follows:
wherein
Representing the response self-power spectrum density function of the kth output when m inputs are excited by a unit white noise acceleration power spectrum with the amplitude of 1;
according to the superposition principle, white noise responses of n output control points of the linear system are formed by superposing responses generated by respectively loading corresponding m white noise power spectrum excitations.
Compared with the prior art, the invention has the following beneficial effects:
the invention adopts the idea of inverse solution to carry out numerical calculation on the finite element simulation model of the multi-input multi-output vibration test, can effectively improve the accuracy of the finite element calculation, improves the efficiency of the vibration simulation model calculation, and realizes the high consistency of the simulation calculation result and the actual test result.
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FIG. 1 is a schematic diagram of a multiple-input multiple-output system according to the present invention;
fig. 2 is a flowchart of a multipoint excitation vibration numerical simulation method based on inverse problem solution according to an embodiment of the present invention.
Detailed Description
The present invention will be described in detail with reference to specific examples. The following examples will assist those skilled in the art in further understanding the invention, but are not intended to limit the invention in any way. It should be noted that variations and modifications can be made by persons skilled in the art without departing from the spirit of the invention. All falling within the scope of the present invention.
As shown in FIG. 1, FIG. 1 shows the excitation and response relationship of a multi-input multi-output system having m inputs xi(i-1, 2 …, m), then for each output yk(k-1, 2 …, n) having m impulse response functions hki(t)(i＝1,2···,m)。
As shown in fig. 2, the multipoint excitation vibration numerical simulation method based on the inverse problem solution mainly includes the following steps: carrying out finite element modeling on a specific engineering object, and obtaining modal parameters of a finite element model through finite element analysis;
extracting a frequency response function between each input excitation point and each output control point by adopting a white noise excitation method;
according to the preset control spectrum of each output control point, the acceleration load spectrum of each input excitation point is worked out through the load identification matrix inversion;
loading the calculated acceleration load spectrum of each input excitation point to a corresponding excitation position;
obtaining an actual acceleration response power spectral density function of n output control points, wherein n is not less than 1 and is an integer;
comparing the value with a preset acceleration power spectrum density function to perform relative error analysis, and if the analysis result meets the control spectrum requirement, finishing the calculation; if the control spectrum requirement is not met, load spectrums at m input positions which are reversely solved are corrected, wherein m is not less than 1 and is an integer.
The specific implementation process of the invention is as follows:
step one, establishing a finite element model. The multi-point excitation vibration test system model is divided into three parts: moving coil, anchor clamps and aircraft. Before the finite element model is established, a geometric model needs to be established, and the complexity of the component model is reduced as follows:
1) in order to facilitate finite element analysis, characteristics of chamfers, fillets, threaded holes and the like which have small influence on overall response are deleted, and partial small round holes are filled to facilitate meshing and calculation;
2) in order to ensure that the mass of each part is consistent with the mass center of the original part, performing mass equivalence on the model according to a mass center coordinate formula in theoretical mechanics according to a given mass and mass center distribution table;
3) material parameters are equivalently processed according to actual attributes;
step two, white noise excitation method extraction
The white noise excitation method is that unit white noise acceleration power spectrum excitation with the amplitude value of 1 is applied to the excitation position of the simulation model after the vibration system is corrected, so that a frequency response function of the r input to the k output is obtained
(k＝1,2…,n，r＝1,2…,m)；
Thirdly, obtaining frequency response functions of m inputs to n outputs through the second step, reversely solving basic acceleration load spectrums which should be loaded at the m inputs by utilizing n preset output spectrums before simulation calculation;
step four, loading the basic acceleration load spectrums of the m input positions obtained in the step three to corresponding excitation input positions, and obtaining actual acceleration response power spectrum density functions of the n output control points;
and fifthly, comparing the obtained actual acceleration response power spectrum density functions of the n output control points with a preset acceleration power spectrum density function to perform relative error analysis, finishing the calculation if the analysis result meets the control spectrum requirement, and correcting the m reversely-obtained load spectrums if the control spectrum requirement is not met.
Specifically, according to the superposition principle, each output of the linear system can be formed by superposing responses corresponding to various discrete inputs, and the system has m outputsiWhere i is 1,2 …, m, then for each output ykWhere k is 1,2 …, n, mImpulse response function hki(t); and for n outputs, there is an n × m impulse response function, an n × m order impulse response function matrix [ h (t)]Comprises the following steps:
the input is represented as an m x 1 order array { x (t) } ═ xi(t) representing the output as an n × 1 order array { y (t) } ═ yk(t) }, then:
transposing the above equation to obtain:
the autocorrelation function and cross-correlation function of n outputs form the following output correlation function matrix [ Ryy(τ)]：
Wherein:
correspondingly, the self-spectrum and cross-spectrum of n outputs form an output power spectrum matrix Syy(ω)，Syy(ω) Fourier transforming equation (4) to obtain:
the auto-and cross-correlations of the n inputs form an input correlation matrix as follows:
the self-spectrum and cross-spectrum of m inputs form an input power spectrum matrix Sxx(ω)，Sxx(ω) Fourier transforming both ends of equation (6) to obtain:
when m inputs are uncorrelated with each other, the self-spectrum of the mimo system response point for m inputs and n outputs can be represented as:
in the formula:
representing m input self-power spectral density functions;
a response self-power spectral density function representing the n outputs;
representing a frequency response function of the r input to the k output;
representing the conjugate of the corresponding frequency response function;
representing the square of the magnitude of the corresponding frequency response function;
according to equation (8), the multiple-input multiple-output relationship is represented in the form of a matrix:
the load identification matrix is written as:
in the formula (10), the compound represented by the formula (10),
the square of the frequency response function amplitude of the nth input to the kth output is represented and obtained by a white noise excitation method; the positive sign means that the matrix is inverted when m is equal to n, the generalized inverse is solved when m is not equal to n, the matrix is seen from the upper matrix, if n output control spectrums are known, m acceleration load spectrums required to be loaded are solved in a reverse mode, and when m is equal to n, namely the number of exciting forces is the same as the number of control measuring points, the only solution of the exciting spectrums is obtained; and when m is less than n, namely the number of the exciting forces is less than the number of the control measuring points, solving the least square solution of the acceleration load spectrum.
The white noise excitation method comprises the following steps: when m inputs are simultaneously excited by a unit white noise acceleration power spectrum with the amplitude of 1, i.e.
The self-spectrum of the n output response points is represented as follows:
wherein
Representing the response self-power spectrum density function of the kth output when m inputs are excited by a unit white noise acceleration power spectrum with the amplitude of 1;
according to the superposition principle, white noise responses of n output control points of the linear system are formed by superposing responses generated by respectively loading corresponding m white noise power spectrum excitations.
The method can quickly obtain the frequency response function between the excitation input point and the control point of the multi-point excitation vibration simulation model, and obtain the acceleration load spectrum at the input point through inverse solution operation.
The foregoing description of specific embodiments of the present invention has been presented. It is to be understood that the present invention is not limited to the specific embodiments described above, and that various changes and modifications may be made by one skilled in the art within the scope of the appended claims without departing from the spirit of the invention.
Claims (2)
1. A multipoint excitation vibration numerical simulation method based on inverse problem solving is characterized by comprising the following steps: carrying out finite element modeling on a specific engineering object, and obtaining modal parameters of a finite element model through finite element analysis;
extracting a frequency response function between each input excitation point and each output control point by adopting a white noise excitation method;
according to the preset control spectrum of each output control point, the acceleration load spectrum of each input excitation point is worked out through the load identification matrix inversion;
loading the calculated acceleration load spectrum of each input excitation point to a corresponding excitation position;
obtaining an actual acceleration response power spectral density function of n output control points, wherein n is not less than 1 and is an integer;
comparing the actual acceleration response power spectral density functions of the n output control points with a preset acceleration power spectral density function to perform relative error analysis, and if the analysis result meets the control spectrum requirement, finishing the calculation; if the control spectrum requirement is not met, correcting the reversely solved load spectrums at the m input positions, wherein m is not less than 1 and is an integer;
according to the superposition principle, each output of the linear system is regarded as the response superposition of each independent input, and the system has m inputs xiWhere i is 1,2 …, m, then for each output ykWhere k is 1,2 …, n, there are m impulse response functions hki(t); and corresponding to n outputs, there are n × m orders of pulsesImpulse response function matrix, an impulse response function matrix of order n x m [ h (t)]Comprises the following steps:
the input is represented as an m x 1 order array { x (t) } ═ xi(t) representing the output as an n × 1 order array { y (t) } ═ yk(t) }, then:
transposing the above equation to obtain:
the autocorrelation function and cross-correlation function of n outputs form the following output correlation function matrix [ Ryy(τ)]：
Wherein:
correspondingly, the self-spectrum and cross-spectrum of n outputs form an output power spectrum matrix Syy(ω)，Syy(ω) Fourier transforming equation (4) to obtain:
the auto-and cross-correlations of the m inputs form an input correlation matrix as follows:
the self-spectrum and cross-spectrum of m inputs form an input power spectrum matrix Sxx(ω)，Sxx(ω) Fourier transforming both ends of equation (6) to obtain:
when m inputs are not correlated, the self-spectrum of the multi-input multi-output system response point for m inputs and n outputs is represented as:
in the formula:
representing m input self-power spectral density functions;
a response self-power spectral density function representing the n outputs;
representing a frequency response function of the r input to the k output;
representing the square of the magnitude of the corresponding frequency response function;
according to equation (8), the multiple-input multiple-output relationship is represented in the form of a matrix:
the load identification matrix is written as:
in the formula (10), the compound represented by the formula (10),
the square of the frequency response function amplitude of the nth input to the kth output is represented and obtained by a white noise excitation method; the positive sign means that the matrix is inverted when m is equal to n, the generalized inverse is solved when m is not equal to n, the matrix is seen from the upper matrix, if n output control spectrums are known, m acceleration load spectrums required to be loaded are solved in a reverse mode, and when m is equal to n, namely the number of exciting forces is the same as the number of control measuring points, the only solution of the exciting spectrums is obtained; and when m is less than n, namely the number of the exciting forces is less than the number of the control measuring points, solving the least square solution of the acceleration load spectrum.
2. The method for multipoint excitation vibration numerical simulation based on inverse problem solution according to claim 1, wherein the white noise excitation method is as follows: when m inputs are simultaneously excited by a unit white noise acceleration power spectrum with the amplitude of 1, i.e.
The self-spectrum of the n output response points is represented as follows:
wherein
Representing the response self-power spectrum density function of the kth output when m inputs are excited by a unit white noise acceleration power spectrum with the amplitude of 1;
according to the superposition principle, white noise responses of n output control points of the linear system are formed by superposing responses generated by respectively loading corresponding m white noise power spectrum excitations.
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| CN114577425B (en) * | 2022-03-07 | 2023-12-05 | 中国工程物理研究院总体工程研究所 | Transfer function identification method for multi-input multi-output vibration test system |
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