CN109299513A - A kind of Sensitivity Analysis Method of modal frequency to quality - Google Patents
A kind of Sensitivity Analysis Method of modal frequency to quality Download PDFInfo
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Abstract
The present invention provides a kind of modal frequencies to the Sensitivity Analysis Method of quality, structural texture admittance matrix simultaneously obtains preceding m rank modal frequency, quality perturbing term is added since first node of structure, acceleration admittance after acceleration admittance information substitution matrix correction formula form is perturbed, extract the frequency information of structure, obtain sensitivity of the structural modal frequency to quality, change quality perturbation point position according to node sequence to obtain to deserved sensitivity, to obtain sensitivity of the total modal frequency to quality.The method of the present invention obtains the acceleration admittance of structure by FEM calculation first, when architecture quality perturbs, finite element secondary calculating is not necessarily to using matrixing formula, it only needs initial acceleration admittance information to carry out numerical value and calculates the acceleration admittance after can be obtained perturbation, simplify computational efficiency, it is more convenient, the sensitivity quick calculation method based on acceleration admittance to quality is realized, there is engineering significance.
Description
Technical field
The present invention relates to a kind of Sensitivity Analysis Methods, and in particular to a kind of sensitivity analysis side of modal frequency to quality
Method.
Background technique
Sensitivity analysis is that a kind of evaluate causes structural response characteristic variations rate because of the change of design variable or parameter
Method.The research of structural system sensitivity is a very special field, it is current Computational Mechanics and structural engineering field
One of main direction of studying.
In practical application, the sensitivity analysis of structure occupies important in structure optimization, reliability assessment and parameter identification etc.
Effect, method can strategically be divided into discrete method and the calculus of variations.Traditional Sensitivity Analysis Method generally requires under perturbation,
FEM calculation is re-started, special structure of working as is more soft, computationally intensive when finite element quantity is more, low efficiency.Acceleration signal
Measurement is more universal, is carried out based on acceleration admittance information such as to the sensitivity analysis of mass parameter, if solving tradition spirit
Sensitivity computational efficiency is low, it has also become Practical Project problem urgently to be resolved.
Summary of the invention
Goal of the invention: in view of the above-mentioned deficiencies in the prior art, it is an object of the present invention to provide a kind of modal frequency to quality
Sensitivity Analysis Method.
Technical solution: the present invention provides a kind of modal frequencies to the Sensitivity Analysis Method of quality, comprising the following steps:
(1) finite element analysis is carried out based on Matlab software, obtains acceleration admittance function, construct acceleration admittance square
Battle array, and preceding m rank modal frequency is obtained, quality perturbing term is added since first node of structure；
(2) acceleration admittance after matrix correction formula is perturbed is substituted into based on acceleration admittance matrix in step (1)
Matrix；
(3) frequency information for extracting structure obtains sensitivity of the structural modal frequency to quality；
(4) change quality perturbation point position according to node sequence, repeat step (2) (3) and obtain to deserved sensitivity, thus
Sensitivity of the total modal frequency to quality is obtained, sensitivity curve is drawn.
Further, step (1) the following steps are included:
(11) freedom degree of structure is N, obtains the acceleration admittance matrix of structure:
Wherein, aijIt indicates under structure node j function unit pulse, the acceleration admittance function of structure node i, i=1,
2 ... N, j=1,2 ... N；
(12) in formula (1) matrix, a is according to principle from left to right, from top to bottomijAmplitude frequency diagram judges the curve
The number of local maximum is that T continues to map if T is less than m；If T is equal to m, mapping is terminated；
(13) by taking the amplitude frequency diagram screened in (12) as an example, the abscissa of the local maximum of the amplitude frequency diagram is frequency values,
Respectively pr(r=1,2 ... m), by prElement forms m dimensional vector p.
Further, step (2) the following steps are included:
(21) quality Perturbation Δ m is added at structure i-nodei(i=1,2 ... N), the acceleration of structure after perturbing at this time
Admittance matrix is A*, according to the relationship of acceleration admittance inverse of a matrix matrix and dynamic stiffness matrix, derives and obtains:
Wherein vi∈RN×1, subscript i indicate in i-th of element of column vector beL is imaginary unit, i.e. l2=-1, ω
For circular frequency；
(22) from matrix correction formula:
The relationship of the acceleration admittance of perturbation front-end geometry can be established；
(23) by formula (1), (2), (3) connection column, neutralizing can obtain quality perturbation post-acceleration admittance matrix and initial
The relationship of acceleration admittance matrix:
Further, step (3) the following steps are included:
(31) based on the acceleration admittance matrix A after perturbation*, repeat step (12), (13) extract the frequency of structure after perturbation
RateIt willElement forms m dimensional vector p*；
(32) definition obtains sensitivity function gi, indicate that the sensitivity after quality perturbation occurs at node i, function input
Amount is the frequency before and after perturbation, specific formula is as follows:
Wherein, equation left side bracket amount is input function giInput quantity, be to calculate output result on the right of equation.
The utility model has the advantages that the method for the present invention obtains the acceleration admittance of structure by FEM calculation first, work as architecture quality
When perturbing, finite element secondary calculating is not necessarily to using matrixing formula, it is only necessary to which initial acceleration admittance information carries out
Numerical value calculates the acceleration admittance after can be obtained perturbation, simplifies computational efficiency, more convenient, realizes based on acceleration admittance
To the sensitivity quick calculation method of quality, there is engineering significance.
Detailed description of the invention
Fig. 1 is 9 freedom degree spring-dampers-quality system schematic diagram in embodiment；
Fig. 2 is the acceleration admittance a of system structure11Curve；
Fig. 3 is the acceleration admittance after system structure adds Perturbation at node 1Curve；
Fig. 4 is based on acceleration admittance signal structure frequency to the sensitivity curve of quality.
Specific embodiment
Technical solution of the present invention is described in detail below, but protection scope of the present invention is not limited to the implementation
Example.
The present embodiment is verified using a simple 9 freedom degree spring-damper-quality structure, as shown in Figure 1, system
Parameter be respectively as follows: mi(i=1,2 ... 9), spring k by=4kgi(i=1,2 ... 9), damper c by=20N/mi=0.02Nm/s
(i=1,2 ... 9), and the quality Perturbation of addition is Δ m=2kg, comprising the following steps:
Step 1, finite element analysis is carried out based on Matlab software, obtains acceleration admittance function, constructed admittance matrix, obtain
Preceding 9 rank modal frequency is obtained, quality perturbing term is added since first node of structure；
1.1) freedom degree of structure is 9, and the acceleration admittance matrix for obtaining structure is calculated based on Finite Element Principle:
Wherein, aijIndicate the i=under structure node j function unit pulse, the speed frequency response function information of structure node i
1,2 ... 9, j=1,2 ... 9；
1.2) in formula (1) matrix, according to from left to right, principle from top to bottom is aijAmplitude frequency diagram judges the curve
The number of local maximum is T, if T less than 9, continues to map；If T is equal to 9, mapping is terminated；Map a first11, such as Fig. 2
It is shown, it is known that share 9 local maximums, meet and require above, therefore terminate mapping；
1.3) by taking the amplitude frequency diagram screened in (1.2) as an example, the abscissa of the local extremum of the amplitude frequency diagram is frequency values, point
It Wei not pr(r=1,2 ... 9), by prElement forms m dimensional vector p.
p1=0.059, p2=0.175, p3=0.286,
p4=0.389, p5=0.482, p6=0.562
p7=0.626, p8=0.673, p9=0.703
P=[0.0590.1750.2860.3890.4820.5620.6260.6730.703]T。
Step 2, quality perturbing term successively is added in structure different location, based on acceleration admittance matrix generation in step (1)
Enter the acceleration admittance matrix after matrixing formula is perturbed；
2.1) quality Perturbation Δ m is added everywhere in structure node 11=2, the acceleration admittance square of structure after perturbing at this time
Battle array is A*, according to the relationship of acceleration admittance inverse of a matrix matrix and dynamic stiffness matrix, derives and obtains:
λ=- 1, Δ mi> 0 (2)
L is imaginary unit,
That is l2=-1, ω is circular frequency；
2.2) from matrix correction formula:
The relationship of the acceleration admittance of perturbation front-end geometry can be established；
2.3) by formula (1), (2), (3) connection column, neutralizing can obtain quality perturbation post-acceleration admittance matrix and initial
The relationship of acceleration admittance matrix:
Due to being known terms on the right of equation, therefore the acceleration admittance matrix A after being perturbed*。
Step 3, the frequency information of structure is extracted, sensitivity of the structural modal frequency to quality is obtained；
3.1) based on the acceleration admittance after perturbation, step (1.2) are repeated, (1.3) can mapFig. 3 is seen, by Fig. 3
The frequency of structure after perturbing can be extractedIt willElement forms m dimensional vector p*；
p*=[0.059 0.172 0.277 0.371 0.46 0.544 0.615 0.667 0.703]T
3.2) definition obtains sensitivity function g1, function input quantity is the frequency of perturbation front and back, specific formula is as follows:
Wherein, equation left side bracket amount is input function giInput quantity, be to calculate output result on the right of equation.
Step 4, change quality perturbation point position according to node sequence, repeat step (1), step (2) is obtained to deserved spirit
Sensitivity draws sensitivity curve, sees Fig. 4 to obtain sensitivity of the total modal frequency to quality.
Fig. 4 the result shows that, when architecture quality changes, the maximum absolute value of sensitivity at node 2, therefore in structure
Quality is modified at node 2 influences maximum.Rapid sensitive degree analysis method of the invention breaches traditional sensitivity and needs progress more
The limitation of secondary calculating, it is only necessary to carry out a FEM calculation, be counted using the acceleration admittance signal of initial configuration
Value calculates the response of the structure after obtaining quality perturbation, more fast.
Claims (4)
1. a kind of modal frequency is to the Sensitivity Analysis Method of quality, it is characterised in that: the following steps are included:
(1) finite element analysis is carried out based on Matlab software, obtains acceleration admittance function, construct acceleration admittance matrix, and
M rank modal frequency before obtaining, adds quality perturbing term since first node of structure；
(2) the acceleration admittance matrix after matrix correction formula is perturbed is substituted into based on acceleration admittance matrix in step (1)；
(3) frequency information for extracting structure obtains sensitivity of the structural modal frequency to quality；
(4) change quality perturbation point position according to node sequence, repeat step (2) (3) and obtain to deserved sensitivity, to obtain
Sensitivity curve is drawn in sensitivity of the total modal frequency to quality.
2. a kind of modal frequency according to claim 1 is to the Sensitivity Analysis Method of quality, it is characterised in that: step
(1) the following steps are included:
(11) freedom degree of structure is N, obtains the acceleration admittance matrix of structure:
Wherein, aijIt indicates under structure node j function unit pulse, the acceleration admittance function of structure node i, i=1,2 ... N,
J=1,2 ... N；
(12) in formula (1) matrix, a is according to principle from left to right, from top to bottomijAmplitude frequency diagram judges curve part
The number of maximum is that T continues to map if T is less than m；If T is equal to m, mapping is terminated；
(13) by taking the amplitude frequency diagram screened in (12) as an example, the abscissa of the local maximum of the amplitude frequency diagram is frequency values, respectively
For pr(r=1,2 ... m), by prElement forms m dimensional vector p.
3. a kind of modal frequency according to claim 2 is to the Sensitivity Analysis Method of quality, it is characterised in that: step
(2) the following steps are included:
(21) quality Perturbation Δ m is added at structure i-nodei(i=1,2 ... N), the acceleration admittance of structure after perturbing at this time
Matrix is A*, according to the relationship of acceleration admittance inverse of a matrix matrix and dynamic stiffness matrix, derives and obtains:
Wherein vi∈RN×1, subscript i indicate in i-th of element of column vector beL is imaginary unit, i.e. l2=-1, ω is circle
Frequency；
(22) from matrix correction formula:
The relationship of the acceleration admittance of perturbation front-end geometry can be established；
(23) by formula (1), (2), (3) connection column, neutralizing can obtain quality perturbation post-acceleration admittance matrix and initial acceleration
Spend the relationship of admittance matrix:
4. a kind of modal frequency according to claim 3 is to the Sensitivity Analysis Method of quality, it is characterised in that: step
(3) the following steps are included:
(31) based on the acceleration admittance matrix A after perturbation*, repeat step (12), (13) extract the frequency of structure after perturbationIt willElement forms m dimensional vector p*；
(32) definition obtains sensitivity function gi, indicate that the sensitivity after quality perturbation occurs at node i, function input quantity is
Frequency before and after perturbing, specific formula is as follows:
Wherein, equation left side bracket amount is input function giInput quantity, be to calculate output result on the right of equation.
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| CN108984976A (en) * | 2018-08-27 | 2018-12-11 | 东南大学 | One kind being based on acceleration responsive structural sensitivity calculation method |
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| US20160252673A1 (en) * | 2014-01-23 | 2016-09-01 | Ofs Fitel, Llc | Polarization Maintaining Single-Mode Low-Loss Hollow-Core Fiber |
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