CN107908596A - A kind of mode Method of determining the optimum based on singular value decomposition - Google Patents

A kind of mode Method of determining the optimum based on singular value decomposition Download PDF

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CN107908596A
CN107908596A CN201711055573.4A CN201711055573A CN107908596A CN 107908596 A CN107908596 A CN 107908596A CN 201711055573 A CN201711055573 A CN 201711055573A CN 107908596 A CN107908596 A CN 107908596A
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费庆国
朱锐
姜东�
曹芝腑
杭晓晨
季熠
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Southeast University
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    • G—PHYSICS
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Abstract

The present invention provides a kind of mode Method of determining the optimum based on singular value decomposition, Hankel matrix Hs are constructed using the impulse response signal of measurement in modal test, singular value decomposition, i.e. H=U Σ V are carried out to matrixT, wherein ∑ is diagonal matrix, and the unusual value information in matrix ∑ calculates singular value percentage, and determine mode determines rank index RSVP, in RSVPIt is the true mode order value of structure when being up to minimum value.The present invention is theoretical based on singular value decomposition, constructs Hankle matrixes using impulse response signal, singular value decomposition is carried out to it, determines rank index R by carry out that processing proposes model to singular valueSVP, according to can effectively determine mode true order value the characteristics of determining rank index curvilinear motion, that improves mode determines rank precision, efficiently against conventional method unusual value mutation unobvious affected by noise, is not easy the problem of definite order, has engineering significance.

Description

A kind of mode Method of determining the optimum based on singular value decomposition
Technical field
The present invention relates to a kind of modal test, and in particular to a kind of mode order method.
Background technology
In Modal Parameter Identification process, how to determine that the mode order of system is most important.If the order selected is less than The real mode order of system, it is possible to omit true mode；If the order selected is higher than the real mode order of system, More false mode can then be presented in recognition result, true mode is interfered.
The unconspicuous feature of mutation being likely to occur for traditional singular value curve, and become with the increase curve of order To the phenomenon of horizontal asymptote, sometimes it is difficult to determine the true mode order problem of system.How the true of system is effectively determined Mode, it is particularly significant to later stage Modal Parameter Identification accuracy, it has also become Practical Project problem urgently to be resolved hurrily.
The content 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 based on singular value decomposition Mode Method of determining the optimum.
Technical solution：The present invention provides a kind of mode Method of determining the optimum based on singular value decomposition, comprise the following steps：
(1) the impulse response signal construction Hankel matrix Hs of measurement are utilized in modal test；
(2) singular value decomposition, i.e. H=U Σ V are carried out to Hankel matrix HsT, wherein ∑ is diagonal matrix；
(3) the unusual value information in matrix ∑ calculates singular value percentage；
(4) singular value percent value is utilized, determine mode determines rank index RSVP, in RSVPIt is structure when being up to minimum value True mode order value.
Further, exemplified by encouraging the pulse signal at point j and response point i, its Hankel matrix is step (1)
Wherein, s=m+n-2, HmnExpression row dimension is m, the Hankel matrixes that row dimension is n, hij(s Δs t) is s Impulse response function of the time Δt between point j and response point i is encouraged.
Further, step (2) comprises the following steps：
(2.1) singular value decomposition is carried out to formula (1)：
Hmn=Umn∑nnVnn T (2)
Wherein, UmnFor unitary matrice, dimension is m × n, VnnFor n × n rank unitary matrice, ∑nnDiagonal matrix, dimension for n × N, element σ on diagonall(l=1,2...n) is represented：
(2.2) meet in formula (3)
σ1＞ σ2＞ ... ＞ σl＞ σl+1＞ ... ＞ σn (4)。
Further, step (3) is according to ∑nnIn singular value σl(l=1,2 ... n), define singular value percentage Pk：
Wherein, n is ∑nnThe dimension of matrix, k are selected order, reflect selected order increase to system by percentage Contribution amount, the index can effectively react the selected order of institute proportion in the structure.
Further, step (4) by the use of singular value percentage adjacent increment ratio transformation the characteristics of as mode determine rank Index：
Wherein,Expression mode when selected order is l determines the desired value of rank, Δ PL+1, lIt is expressed as Pl+1-Pl；
Describe with mode order l changesCurve, curve tend towards stability after fluctuation, are close to 1, occurMost The mode order of small value is required true mode order.
Because proportion of the order in modal parameter selected by the reflection of singular value percentage, adjacent increment ratio can be in true mould Nearby there is huge fluctuation in type order, this is because order is noise contribution after true mode, its increment in same magnitude, Model determines rank index RSVPIt will tend towards stability, and be close to 1, and place is added since the magnitude of increment is different in true mode, and can go out An existing minimum value, will appear from lowest point in curve is mode order.
Beneficial effect：The present invention is theoretical based on singular value decomposition, Hankle matrixes is constructed using impulse response signal, to it Singular value decomposition is carried out, proposes that model determines rank index R by carrying out processing to singular valueSVP, according to determining rank index curvilinear motion The characteristics of can effectively determine the true order value of mode, that improves mode determines rank precision, efficiently against conventional method by noise The problem of influencing unusual value mutation unobvious, being not easy to determine order, there is engineering significance.
Brief description of the drawings
Fig. 1 is six degree of freedom spring-damper-quality system schematic diagram in the actual example of the present invention；
Fig. 2 is mode order index curve when taking different orders.
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.
Embodiment：A kind of mode Method of determining the optimum based on singular value decomposition, using a simple six degree of freedom spring-resistance Buddhist nun-quality system is verified, as shown in Figure 1, the parameter of system is respectively：Mass block quality is m1=m6=20kg, mi=10kg (i=2,3 ... 5), spring rate kj=8 × 106KN/m, (j=1,2 ... 7).Six ranks frequency is determined by mode superposition method Rate, damping value ci(i=1,2 ... value 7) make damping ratios be 2.00%.In mass block m4Upper application unit pulse excitation, In mi(i=1,2 ... 6) on gather vibration displacement response.Analysis the frequency f=300Hz, hits N=6000 of model.With m5Point Exemplified by output response, signal after 10% white Gaussian noise is added, generates impulse response signal.
Hankel matrix Hs are constructed using above-mentioned impulse response signal, with the pulse signal at excitation point 4 and response point 5 Exemplified by signal, its Hankel matrix is
Wherein s=m+n-2, HmnRepresenting matrix row dimension takes m=3000, and row dimension takes n=3000, Δ t=1/300 seconds, h45(m Δs t) is impulse response function of the m time Δts between point j and response point i is encouraged.
Singular value decomposition is carried out to formula (1)：
H3000,3000=U3000,3000∑3000,3000V3000,3000 T (2)
Wherein ∑nnIt is diagonal matrix, dimension is 3000 × 3000, element σ on diagonall(l=1,2...n) is represented：
Meet in formula (3)
σ1＞ σ2＞ ... ＞ σl＞ σl+1＞ ... ＞ σ3000 (4)
According to ∑nnIn singular value σl(l=1,2 ... n), define singular value percentage Pk：
K is selected order, reflects that selected order increase can be effectively anti-to systematic contributions amount, the index by percentage Answering the selected order of institute, proportion, practical structures mode order are limited in the structure, and to take into account computational efficiency, this example k obtains 40 and is Can.
On the basis of obtaining singular value percent value in formula (5), become using the ratio of singular value percentage adjacent increment The characteristics of changing determines rank index as mode：
WhereinExpression mode when selected order is l determines the desired value of rank, Δ PL+1, lIt is expressed as Pl+1-Pl.It is because strange Proportion of the order in modal parameter selected by different value percentage reflection, adjacent increment ratio can occur near true model order Huge fluctuation, this is because order is noise contribution after true mode, for its increment in same magnitude, model determines rank index (RSVP) will tend towards stability, it is close to 1, and place is added since the magnitude of increment is different in true mode, it may appear that a minimum Value, is shown in Fig. 2, and abscissa represents mode order l in figure, and ordinate is mode order indexAnalyzed from Fig. 2, curve Reach minimum after at 12, therefore mode order should take 12, it is consistent with theoretical value, demonstrate this method accuracy.

Claims (5)

  1. A kind of 1. mode Method of determining the optimum based on singular value decomposition, it is characterised in that：Comprise the following steps：
    (1) the impulse response signal construction Hankel matrix Hs of measurement are utilized in modal test；
    (2) singular value decomposition, i.e. H=U Σ V are carried out to Hankel matrix HsT, wherein ∑ is diagonal matrix；
    (3) the unusual value information in matrix ∑ calculates singular value percentage；
    (4) singular value percent value is utilized, determine mode determines rank index RSVP, in RSVPIt is the true of structure when being up to minimum value Real mode order value.
  2. 2. the mode Method of determining the optimum according to claim 1 based on singular value decomposition, it is characterised in that：Step (1) is with sharp Exemplified by encouraging the pulse signal at point j and response point i, its Hankel matrix is
    <mrow> <mtable> <mtr> <mtd> <msub> <mi>H</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>&amp;Delta;</mi> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mo>(</mo> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>&amp;CenterDot;</mo> <mi>&amp;Delta;</mi> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>&amp;Delta;</mi> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mo>&amp;CenterDot;</mo> <mi>&amp;Delta;</mi> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>n</mi> <mo>&amp;CenterDot;</mo> <mi>&amp;Delta;</mi> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> <mtd> <mo>.</mo> </mtd> <mtd> <mrow></mrow> </mtd> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mo>(</mo> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> <mo>&amp;CenterDot;</mo> <mi>&amp;Delta;</mi> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>m</mi> <mo>&amp;CenterDot;</mo> <mi>&amp;Delta;</mi> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mn>...</mn> </mtd> <mtd> <mrow> <msub> <mi>h</mi> <mrow> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>s</mi> <mo>&amp;CenterDot;</mo> <mi>&amp;Delta;</mi> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, s=m+n-2, HmnExpression row dimension is m, the Hankel matrixes that row dimension is n, hij(when s Δs t) is s Δ t The impulse response function being engraved between excitation point j and response point i.
  3. 3. the mode Method of determining the optimum according to claim 2 based on singular value decomposition, it is characterised in that：Step (2) includes Following steps：
    (2.1) singular value decomposition is carried out to formula (1)：
    Hmn=Umn∑nnVnn T (2)
    Wherein, UmnFor unitary matrice, dimension is m × n, VnnFor n × n rank unitary matrice, ∑nnIt is diagonal matrix, dimension is n × n, right Element σ on linea angulatal(l=1,2...n) is represented：
    (2.2) meet in formula (3)
    σ1＞ σ2＞ ... ＞ σl＞ σl+1＞ ... ＞ σn (4)。
  4. 4. the mode Method of determining the optimum according to claim 3 based on singular value decomposition, it is characterised in that：Step (3) basis ∑nnIn singular value σl(l=1,2 ... n), define singular value percentage Pk：
    <mrow> <msub> <mi>P</mi> <mi>k</mi> </msub> <mo>=</mo> <mfrac> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>k</mi> </munderover> <msub> <mi>&amp;sigma;</mi> <mi>l</mi> </msub> </mrow> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>l</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <msub> <mi>&amp;sigma;</mi> <mi>l</mi> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>
    Wherein, n is ∑nnThe dimension of matrix, k are selected order, reflect selected order increase to systematic contributions by percentage Amount, the index can effectively react the selected order of institute proportion in the structure.
  5. 5. the mode Method of determining the optimum according to claim 4 based on singular value decomposition, it is characterised in that：Step (4) utilizes The characteristics of ratio transformation of singular value percentage adjacent increment, determines rank index as mode：
    <mrow> <msubsup> <mi>R</mi> <mi>l</mi> <mrow> <mi>S</mi> <mi>V</mi> <mi>P</mi> </mrow> </msubsup> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;Delta;P</mi> <mrow> <mi>l</mi> <mo>+</mo> <mn>2</mn> <mo>,</mo> <mi>l</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>&amp;Delta;P</mi> <mrow> <mi>l</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mi>l</mi> </mrow> </msub> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>
    Wherein,Expression mode when selected order is l determines the desired value of rank, Δ PL+1, lIt is expressed as Pl+1-Pl；
    Describe with mode order l changesCurve, curve tend towards stability after fluctuation, are close to 1, occurMinimum value Mode order be required true mode order.
CN201711055573.4A 2017-10-31 2017-10-31 A kind of mode Method of determining the optimum based on singular value decomposition Pending CN107908596A (en)

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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN109063190A (en) * 2018-08-29 2018-12-21 百度在线网络技术（北京）有限公司 Method and apparatus for handling data sequence
CN111125626A (en) * 2019-12-12 2020-05-08 东南大学 Model order fixing method based on S-shaped function random subspace identification

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
姚志远: "大型工程结构模态识别的理论和方法研究", 《中国优秀博硕士论文全文数据库（博士）工程科技II辑》 *

Cited By (3)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN109063190A (en) * 2018-08-29 2018-12-21 百度在线网络技术（北京）有限公司 Method and apparatus for handling data sequence
CN111125626A (en) * 2019-12-12 2020-05-08 东南大学 Model order fixing method based on S-shaped function random subspace identification
CN111125626B (en) * 2019-12-12 2022-11-25 东南大学 Model order fixing method based on S-shaped function random subspace identification

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