- Email: [email protected]
Model updating using correlation analysis of strain frequency response function
Mechanical Systems and Signal Processing 70-71 (2016) 284–299
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Model updating using correlation analysis of strain frequency response function Ning Guo n, Zhichun Yang, You Jia, Le Wang Institute of Structural Dynamics and Control, School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China
a r t i c l e in f o
abstract
Article history: Received 2 February 2015 Received in revised form 17 September 2015 Accepted 27 September 2015 Available online 23 October 2015
A method is proposed to modify the structural parameters of a dynamic finite element (FE) model by using the correlation analysis for strain frequency response function (SFRF). Sensitivity analysis of correlation coefficients is used to establish the linear algebraic equations for model updating. In order to improve the accuracy of updated model, the regularization technique is used to solve the ill-posed problem in model updating procedure. Finally, a numerical study and a model updating experiment are performed to verify the feasibility and robustness of the proposed method. The results show that the updated SFRFs and experimental SFRFs agree well, especially in resonance regions. Meanwhile, the proposed method has good robustness to noise ability and remains good feasibility even the number of measurement locations reduced significantly. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Model updating Strain frequency response function Correlation analysis Sensitivity analysis Regularization technique
1. Introduction The structural dynamic strength analysis is not often taken into account in the engineering practice of structure design, thus many structural failure problems caused by excessive dynamic stress or vibration fatigue occurred in services. Therefore, attentions should be paid to the simulative evaluation of structural dynamic strength in structural design. The accuracy of simulative evaluation is affected by the accuracy of the dynamic finite element (FE) model. However, it is difficult to establish a sufficiently accurate dynamic FE model using FE modeling techniques. In order to improve the accuracy of dynamic FE model, model updating technique has been introduced and widely adopted to modify dynamic FE model using experimental data [1,2]. In the past decades, a large number of model updating methods were proposed. Almost all the dynamic parameters, such as modal frequencies, modal shapes, frequency response functions (FRFs) and dynamic response, were used in dynamic FE model updating. Modal parameters were the earliest to be used to update dynamic FE model. Mottershead et al. [3] adopted the eigenvalue sensitivity approach to update the FE models of welded joints. Steenackers and Guillaume [4] developed an updating method based on measured modal parameters, and took the uncertainty of measurement into account in their method. However, the incompleteness of measured modal parameters and modal analysis errors may significantly affect the accuracy of the updated model [2]. Compared with the model updating method based on modal parameter, the FRF-based model updating method has some advantages [5,6]: (1) the errors during the modal parameter identification can be avoided; (2) the incompleteness of measured data makes less impacts on the FRF-based model updating method; (3) the structure information included in FRF data is more than those in modal parameters. Most of the FRF-based model updating n
Corresponding author. Tel./fax: þ86 029 88460461. E-mail addresses: [email protected] (N. Guo), [email protected] (Z. Yang).
http://dx.doi.org/10.1016/j.ymssp.2015.09.036 0888-3270/& 2015 Elsevier Ltd. All rights reserved.
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
Nomenclature
HεA
FE SFRF FRF Hx
S p e RDI ik CDI ik U V
Hε ℜð∙Þ ℑð∙Þ Be αs αa HεE
Finite element Strain frequency response function Frequency response function Displacement frequency response function matrix Strain frequency response function matrix The real part of complex strain frequency response function The imaginary part of complex strain frequency response function Strain-displacement mapping matrix Shape correlation coefficient vector Amplitude correlation coefficient vector Experimental strain frequency response function matrix
285
Analytical strain frequency response function matrix Sensitivity matrix Design parameter Residual vector Row linear dependency coefficient matrix Column linear dependency coefficient matrix Unitary matrix Unitary matrix
Superscript H
Complex conjugate transpose operator of a matrix or a vector
methods modify the uncertain design parameters by minimizing the error norms between experimental and analytical FRFs [7,8]. The dynamic strength parameters, such as strain mode shapes, strain frequency response functions and dynamic strain responses, can reflect the local feature of structure well, thus these parameters are also used in dynamic FE model updating method. Ha et al. [9] developed a method of combining strain modal shapes and a closed-loop scheme to update FE model. Ip and Vickery [10] proposed a novel random walk approach using strain frequency response function (SFRF) to update the dynamic FE model, and described an investigation of analysis methods for predicting dynamic stresses in a structure. Esfandiari et al. [11] proposed a dynamic FE model updating method using incomplete SFRF, and used the least square method to solve the quasi-linear sensitivity equation in model updating. SFRF, as a structural dynamic strength parameter, is used in dynamic FE model updating in the present work. By means of the correlation analysis between experimental and analytical SFRFs at the specific critical locations of a structure, a dynamic FE model updating method is proposed. A frequency selection strategy is also introduced to improve the computational efficiency and the stability of the proposed updating method. Based on sensitivity analysis of correlation coefficients between experimental and analytical SFRFs, the linear algebraic equations for model updating are established to modify the modeling errors. Meanwhile, the ill-pose problem in updating procedure is solved by regularization technique. Finally, the feasibility and robustness of the proposed method are demonstrated by numerical study and experimental validation respectively. The rest of this current study is organized as follows: in Section 2, according to the derived SFRF expression, the correlation coefficients of experimental and analytical SFRFs are introduced. In Section 3, the dynamic FE model updating method using SFRFs is described. In order to check the feasibility and robustness of the updated method, a numerical study is adopted in Section 4. Section 5 presents an experimental example to demonstrate the efficiency of the proposed dynamic FE model updating method. Finally, some conclusions are drawn in Section 6.
2. SFRF and its correlation coefficients 2.1. Strain frequency response function A continuous structure can be discretized into a multiple of freedom (MDOF) system using FE technique, the dynamic equation for a MDOF system is, Mx€ ðt Þ þ Cx_ ðt Þ þ Kxðt Þ ¼ f ðt Þ
ð1Þ
where M,C,K are mass, viscous damping and stiffness matrices, respectively. x (t) and f (t) are displacement and force vectors, respectively. Assuming the system is subjected to a harmonic excitation, the excitation and displacement response can be expressed as, f ðt Þ ¼ Fejωt xðt Þ ¼ Xejωt
ð2Þ
286
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
Substituting Eq. (2) into Eq. (1) yields, ω2 M þ jωC þ K X ¼ F
ð3Þ
Using modal superposition technique, Eq. (3) can be rewritten as, X¼
n X
φr φTr F ¼ Hx F K ω2 M r þjωC r r¼1 r
ð4Þ
where φr is the r-th modal shape, K r ¼ φTr Kφr , M r ¼ φTr Mφr , C r ¼ φTr Cφr , n is the number of modes. According to the theory of finite element analysis, the strain at any point within i-th element can be expressed as, εei ¼ Be xi ; i ¼ 1; 2; ⋯; q
ð5Þ
where Be is strain-displacement mapping matrix, xi is the node displacement vector of the i-th element, q is the total element number. Assuming the j-th node is shared by nk elements, namely, there are nk different values of εej;k ðk ¼ 1; 2; ⋯; nk Þ. The average nodal strain εj can be expressed by nodal averaging method [12] εj ¼
nk 1 X εe nk k ¼ 1 j;k
ð6Þ
Transforming the average nodal strain of each node in local coordinate of the element to global one, yields, nk nk P P Be xek Be Gek nk 1 X k¼1 k¼1 εj ¼ ε ¼ ¼ x ¼ Tj x nk k ¼ 1 j;k nk nk
ð7Þ
where xek is the displacement vector of k-th element, Gek is the transformation matrix between the element coordinate nk P e e B Gk
and the global coordinate, Tj ¼ i ¼ 1nk
is the mapping matrix between i-th average nodal strain vector and the global dis-
placement vector. The strain of all nodes can be represented as, ε ¼ Tx
T
where ε ¼ ε1 ε2 ⋯ εN , T ¼ T1 T2 ⋯ TN Substituting Eqs. (2) and (4) into Eq. (8) yields, εð t Þ ¼ θ
T
ð8Þ , N is the total number of nodes.
n X
n X φr φTr ψεr φTr Fejωt ¼ Fejωt 2 2 M þ jωC K ω M þ jωC K ω r r r r r r r¼1 r¼1
ð9Þ
where ψεr ¼ θφr is the r-th strain modal shape. So the SFRF matrix can be expressed as, Hε ¼
n X
n X ψεr φTr ψεr φTr ¼ ¼ ℜðHε Þ þj U ℑðHε Þ 2 2 ω2 þ 2jζ ωω K ω M þ jωC ω r r r r r r¼1 r¼1 r
ð10Þ
where ωr is the r-th modal frequency; operators ℜð∙Þ and ℑð∙Þ address the real and imaginary part of complex SFRF. Thus, the SFRF between the output at dof i and input at dof j can be expressed as. ψ εir φjr ¼ ℜ Hεij þ j Uℑ Hεij 2 ω2 þ 2jζ ωω ω r r r¼1 r n n P P ψ εir φjr ðω2r ω2 Þ 2ψ εir φjr ζ r ωr ω ε ;ℑ Hεij ¼ where ℜ Hij ¼ 2 ω2 2 þ ð2ζ ω ωÞ2 2 ω2 2 þ ð2ζ ω ωÞ2 ω ω ð Þ ð Þ r r r r r¼1 r r¼1 r Hεij ðωÞ ¼
n X
ð11Þ
2.2. Correlation coefficients of SFRF In order to evaluate the correlation level between experimental and analytical SFRFs, shape correlation coefficientαs ðω; pÞ and amplitude correlation coefficient αa ðω; pÞ are introduced for any measured frequency ω [13], i.e. Eq. (12) and Eq. (13), which are derived from the Modal Assurance Criterion (MAC) value and Modal Scale Factor (MSF) value. H ∣ HεE;i ðωÞ HεA;i ðω; pÞ∣2 ð12Þ αs;i ðω; pÞ ¼ H H HεE;i ðωÞ HεE;i ðωÞ HεA;i ðω; pÞ HεA;i ðω; pÞ
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
H 2∣ HεE;i ðωÞ HεA;i ðω; pÞ∣ αa;i ðω; pÞ ¼ H H HεE;i ðωÞ HεE;i ðωÞ þ HεA;i ðω; pÞ HεA;i ðω; pÞ
287
ð13Þ
where HεE;i ðωÞand HεA;i ðω; pÞare the i-th column of the experimental and analytical SFRF matrix, respectively, p is the design parameter, superscript H is the complex conjugate transpose operator. The correlation coefficient vectors can be expressed as, 2 3 αs;1 ðω; pÞ 6 7 ⋮ ð14Þ αs ðω; pÞ ¼ 4 5 αs;m ðω; pÞ 2 6 αa ðω; pÞ ¼ 4
αa;1 ðω; pÞ ⋮ αa;m ðω; pÞ
3 7 5
ð15Þ
where subscript m is the number of column of SFRF matrix, i.e. the number of system input. According to Eqs. (12) and (13), all the values of correlation coefficient vectors lie between zero and unity, i.e. 0 r αs;i ðω; pÞ r1,0 rαa;i ðω; pÞ r 1. The correlation coefficient αs is only sensitive to discrepancies in the overall deflection shape of the structure. αa is introduced by targeting the discrepancies in amplitude, which is more stringent than αs and only becomes unity if HεE;i ðωÞ ¼ HεA;i ðωÞ [13].
3. Model updating procedure In the following section, a model updating method based on SFRF correlation analysis is proposed. Based on the sensitivity analysis [14] of SFRF correlation coefficients, the linear algebraic equations used in model updating are established firstly. Then regularization technique is used to solve the ill-posed problem in this model updating method. Finally, the selection strategies of measurement locations and frequency points are also developed respectively. 3.1. Sensitivity analysis and model updating methodology In order to establish the linear algebraic equations used in model updating, the sensitivity of correlation coefficients should be analyzed firstly. Starting with Eqs. (12) and (13), the sensitivity of αs ðω; pÞ and αa ðω; pÞ with respect to design parameter p can be expressed respectively as, 2 ∂αs;i ðω; pÞ ∂ℜðHA;i Þ ∂ℑðHA;i Þ 2 ¼ HH H HH H 4ℜ HH ℜ HH ℑ HH E;i HA;i E;i E ∂p ∂p ð E;i E;i Þð A;i A;i Þ ∂p H ∂ℑ HA;i H ∂ℜ HA;i þ ℑ H ℜ H H þℑ HH E;i A;i E;i E;i ∂p ∂p # H ∂ℑ H ∂ℜH ∣HE;i HA;i ∣2 A;i A;i H H þ H ℜ HA;i ℑ HA;i ∂p ∂p HA;i HA;i
ð16Þ
2 ∂αa;i ðω; pÞ ∂ℜðHA;i Þ ∂ℑðHA;i Þ ¼ ∣HH H ∣ HH 2H þ HH H 4ℜ HH ℜ HH ℑHH E;i HA;i E;i E;i ∂p ∂p ð Þ A;i E;i A;i ∂p E;i E;i A;i H H ∂ℑ HA;i H ∂ℜ HA;i þ ℑ HE;i þℑ HE;i HA;i ℜ HE;i ∂p ∂p # H ∂ℑ H 2∣HE;i HA;i ∣2 A;i H H ∂ℜ HA;i þ H ℜ HA;i ℑ HA;i ∂p ∂p HE;i HE;i þ HH A;i HA;i
ð17Þ
whereHE;i ¼ HεE;i ðωÞ, HA;i ¼ HεA;i ðω; pÞ. The sensitivity of SFRF with respect to design parameter can be obtained by the following two approaches, 1. If design parameter is only related to mass matrix and (or) stiffness matrix, the sensitivity of SFRF can be obtained by analytic method based on FE model, ∂ℜ H εij ðω; pÞ ∂ℜ H εij ðω; pÞ ∂ω2 ðpÞ ∂ℜ H εij ðω; pÞ ∂φ ðpÞ ∂ℜ H εij ðω; pÞ ∂ψ ε ðpÞ r r r þ þ ¼ ∂p ∂p ∂p ∂p ∂φr ðpÞ ∂ψ εr ðpÞ ∂ω2r ðpÞ
288
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
20 n X 6B ¼ 4@ r¼1
1 ψ εir φjr 2ψ εir φjr ω2r ω2 ω2r ω2 þ 2ζ 2r ω2 C∂ω2r ðpÞ A 2 h i2 2 ∂p ω2r ω2 þ ð2ζ r ωr ωÞ2 ω2 ω2 þ ð2ζ ωr ωÞ2 r
r
3 2 2 ∂φ ð p Þ ωr ω ∂φir ðpÞ ε jr 7 þ ψ jr þψ εir 5 2 ∂p ∂p ω2r ω2 þ ð2ζ r ωr ωÞ2 ∂ℑ H εij ðω; pÞ ∂p
¼
∂ℑ H εij ðω; pÞ ∂ω2 ðpÞ
∂ω2r ðpÞ 2 0 n X 6 B ¼ 4@ r¼1
r
∂p
ð18Þ
∂ℑ H εij ðω; pÞ ∂φ ðpÞ ∂ℑ H εij ðω; pÞ ∂ψ ε ðpÞ r r þ þ ∂p ∂p ∂φr ðpÞ ∂ψ εr ðpÞ
1 ζ r ωψ εir φjr 4ψ εir φjr ζ r ωr ω ω2r ω2 þ 2ζ 2r ω2 C∂ω2r ðpÞ h i2 A 2 2 ∂p ωr ω2r ω2 þ ð2ζ r ωr ωÞ2 ω2 ω2 þ ð2ζ r ωr ωÞ2 r
3 ∂φjr ðpÞ 7 2ζr ωr ω ∂φir ðpÞ ε ε ψ jr þ ψ ir 5 2 ∂p ∂p ω2r ω2 þ ð2ζ r ωr ωÞ2 where
∂ω2r ðpÞ ∂p
¼ φTr
∂K 2 ∂M ∂p ωr ∂p
ð19Þ
φr , the first partial derivative of eigenvector can be solved by implicit method which was pro-
posed by Wang [15], i.e., n X ∂φr ðpÞ ¼ φcr cr φr þ ci φi þ dr wr ð20Þ ∂p i ¼ 1 iar ∂ω n φT ∂K ∂ωr M ω ∂M φ ∂M ∂M r P φTi ∂K wTr ∂K r ∂p r ∂p ∂ωr ∂p ωr ∂p φr ∂p ∂p M ωr ∂p φr i ∂p ∂M φ , c ¼ , d ¼ , w ¼ φi K 1 ∂K where cr ¼ 12φTr ∂M r r T T i ωi ωr ωi ∂p r ∂p ∂p M ωr ∂p φr . w Kw ω w Mw r
r
r
r
r
i
∂M The solution of ∂K ∂p and ∂p is related to the element attributes. 2. If the design parameter is viscous damping ratio, i.e. p ¼ ζ r , based on Eq. (11), the sensitivity can be derived as, ∂ℜ H εij ðω; pÞ 8ψ ir φjr ω2r ω2 ζ r ω2r ω2 ¼ ð21Þ 2 2 ∂p ω2r ω2 þ ð2ωr ωζ r Þ2
∂ℑ H εij ðω; pÞ ∂p
2 2ψ ir φjr ωr ω ð2ωr ωζ r Þ2 ω2r ω2 ¼ 2 2 ω2r ω2 þ ð2ωr ωζ r Þ2
ð22Þ
According to the sensitivity analysis of SFRF correlation coefficient, Eqs. (16) and (17) can be expanded by truncated Taylor series expansion at any frequency ω, the high order term is neglected in the expansion equation, then we get, SðωÞΔp ¼ eðωÞ 2
∂αs ðω;pÞ ∂p1
∂αs ðω;pÞ ∂p2
⋯
∂p1
∂αa ðω;pÞ ∂p2
⋯
6 whereSðωÞ ¼ 4 ∂αa ðω;pÞ
∂αs ðω;pÞ ∂pNp
3 7
∂αa ðω;pÞ 5,eðωÞ ¼ ∂pNp
ð23Þ (
1 αs ðω; pÞ 1 αa ðω; pÞ
)
,Δp ¼ Δp .
In model updating, SFRF data at sufficient measurement frequency points should be used, Eq. (23) can be transformed into a set of over-determined linear algebraic equations, which can be denoted as, SΔp ¼ e
ð24Þ 2ðN f U mÞN p
where S A C is the sensitivity matrix, Δp A C vector, Np is the number of updating parameters.
Np 1
is design parameter changes, e A C
2ðN f U mÞ1
is the residual
3.2. The solution for ill-posed problem In order to obtain stable updating parameters during iterations, the sensitivity matrix S should be well-conditioned. Otherwise, an arbitrarily small disturbance of the residual vector can cause a considerable fluctuation of the updated results. To make sure that the sensitivity matrix is well-conditioned, the row (or column) vectors in the sensitivity matrix should be nearly independent with their counterparts [16]. Then, the linear dependency coefficient between row (or column) vectors are introduced to measure the linear dependency of row (or column) vectors in the sensitivity matrix in the present work,
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
289
i.e., RDIik ¼
∣XTi Xk ∣ 12 XTi Xi XTk Xk
ð25Þ
∣Y Ti Y k ∣ 12 Y Ti Y i YTk Yk
ð26Þ
CDIik ¼
where Xi ,i ¼ 1; 2; ⋯; 2 N f Um are row vectors in the sensitivity matrix, Y i ,i ¼ 1; 2; ⋯; Np are the column vectors. If the value of RDIik (or CDIik) is close to 1, which means that the corresponding two rows (or columns) in the sensitivity matrix have similar effects, any one of these two rows (or columns) can be discarded. With the help of Tikhonov regularization technique [17,18], the potential ill-posed problem in the current study can be further mitigated, the over-determined equations (Eq. (23)) can be replaced by the following minimization problem, J λ ¼ arg min ‖SΔp e‖22 þ λ2 ‖IðΔp Δp Þ‖22 ð27Þ whereλis regularization parameter, Δp is the initial value of design parameter changes, which is assumed to be zero in the present work, I is identity matrix. Then the above minimization problem can be transformed into a following least squares problem,
2 S e min Δp λI 0 2
ð28Þ
By simplification, Eq. (28) can be written as, ST Sþ λ2 I Δp ¼ ST e
ð29Þ
The matrix S can be decomposed using singular value decomposition technique, S ¼ UΣVT ¼
Np X
ui σ i v i
ð30Þ
i¼1
where the columns of U and Vare the left and right singular vectors of matrix S, respectively. Σ ¼ diag σ 1 ; ⋯; σ Np has nonnegative diagonal elements appearing in descending order, i.e.σ 1 Z⋯ Zσ Np Z0, where σ i is the singular values of matrix S. Substituting Eq. (30) into Eq. (29), the regularized solution ΔpLSQ can be obtained, ΔpLSQ ¼
Np X uT e f i i vi σi i¼1
ð31Þ
where f i ¼ σ 2i = σ 2i þ λ2 is the filter factor for the regularization method. The regularization parameter in the filter factors can be selected by the L-curve method [18]. Assuming ξ ¼ log ‖SΔp ε‖2 , η ¼ log ‖IΔp‖2 , the L-curve function can be denoted as follows, ρðλÞ ¼
ξ0 η″ ξ″η0
ð32Þ
0 2
½ðξ Þ þðη0 Þ2 3=2
where ξ0 ,ξ″, η0 and η″ are the first-order derivative and the second-order derivative of ξ and η with respect to regularization parameter λ, respectively. The regularization parameter corresponding to the maximum of Eq. (32) is the best regularization parameter. Substituting the best regularization parameter into Eq. (31), then the design parameter changes ΔpLSQ can be obtained. The design parameters are updated iteratively using parameter changes ΔpLSQ , i.e. pnew ¼ pold þ ΔpLSQ , then the new SFRFs and correlation coefficients are solved based on the updated dynamic model. Thus, the stopping criterion used in the iterations is defined as, 8 Nf > X > 1 αs ðωi Þ þ αa ðωi Þ > 1 > Z κ1 stop > > 1 2 < N f i ¼ 1m ð33Þ Nf > X > 1 αs ðωi Þ þ αa ðωi Þ > 1 > o κ continue > 1 > 1 : N f i ¼ 1m 2
‖
‖
‖
‖
where κ1 is the pre-specified threshold.
290
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
3.3. Selection of response measurement locations For various experimental purposes, the exciting and response measurement locations may be totally different [19]. So the selection of response measurement locations is very important in vibration experiment. In the present study, the purpose of dynamic model updating is using the updated dynamic FE model to evaluate the structural dynamic strength. Thus the critical points (i.e. the locations of maximum dynamic strain) of a structure are chosen as the measurement locations. And the critical points are determined by numerical simulation using the updating FE model. The procedure is described as follows: 1. Obtain the strain modes by using the initial dynamic FE model of the structure. 2. Determine the critical points of the structure by examining the maximum strain in the concerned strain modes. 3. Search for the elements and the nodes corresponding to the critical points. 3.4. The frequency selection strategy for model updating The model updating method, in the present work, is based on the correlation analysis of experimental and analytical SFRFs. It is well known that there is a large number of sampling frequency points in experimental SFRFs. If all the sampling frequency points are used in model updating procedure, it will be time consuming and some frequency points are believed
Fig. 1. The sketch diagram of SFRF updating [20].
Fig. 2. Flow chart of model updating using correlation analysis of SFRFs.
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
291
to be redundant for updating model [16]. Therefore, a small number of optimum frequency points for mode updating should be selected. Some considerations for selecting the frequency points will be pursued as follows: 1. Select frequency points where the correlation level needs to be improved, the selection criterion is given as follows [13], αs;i ðωi Þ þ αa;i ðωi Þ oκ 2 2
ð34Þ
where κ 2 is the pre-specified threshold. 2. The measurement sampling frequency points between experimental and analytical resonances should not be selected. If the selected frequency is not suitable for model updating, the difference between analytical and experimental SFRF will increase with the process of model updating, thus the stability of updating method will be affected. As shown in Fig. 1, three typical frequency points are marked to explain this phenomenon: point A is located before the experimental resonant point, point B is located between the analytical and experimental resonant points, and point C is located behind the analytical resonant point. The vertical lines A–A0 , B–B0 , and C–C0 show the differences between the experimental and analytical SFRF at these frequency points, respectively. During the model updating process, it can be inferred that the difference at point B will increase initially, thus the stability problems can be introduced into the updating procedure [20]. It should be noted that the frequency points for model updating should be selected in each iteration. 3.5. Procedure of the proposed updating method To sum up, the steps for updating the dynamic model are as follows: Step 1: Establish the initial dynamic FE model, and determine the measurement locations of the structure by numerical simulation using the FE model of the structure. Step 2: Calculate the correlation coefficients based on experimental and analytical SFRFs, and construct the linear algebraic equations for model updating by using the sensitivity analysis of correlation coefficients. Step 3: Select the frequency points for model updating. Step 4: Solve Eq. (24) by using Tikhonov regularization technique, and obtain the design parameter changes. Step 5: Modify the appropriate parameters of the FE dynamic model and repeat the Step 2– 4 until the stopping criterion is satisfied. Fig. 2 Shows the flow chart of the current model updating method using correlation analysis of SFRFs.
4. Numerical study In this section, two FE models of cantilever aluminum plate are adopted to validate the feasibility and robustness of the proposed updating method. One FE model considered here is used as the model to be updated (i.e. initial model), the other one is utilized to obtain the ‘experimental’ data (i.e. theoretical model). The corresponding modeling parameters are listed in Table 1. In the updating example, the measurement locations are selected by the selection strategy described in Section 3.3, the selected measurement locations on the plate are listed in Table 2. The FE mesh of the updating structure with the locations of measurement and excitation are shown in Fig. 3. The FE model of the structure is meshed by four-node quadrilateral element based on Kirchhoff plate bending theory [12], the mass matrix of dynamic FE model is established by coupled mass matrix used in MSC. Nastran [21]. The frequency bandwidth (from 0.1 Hz to 190 Hz), which covers the first six modal frequencies, is taken as measured frequency range, the frequency resolution is 0.1 Hz. In order to check the robustness of the proposed method, two noise levels, 5% and 10%, are respectively considered in the theoretical SFRFs. The SFRF considered the level of noise is shown in Eq. (35). H εij ðωk Þ ¼ H εij ðωk Þð1 þ ηδÞ
ð35Þ
where η is the level of noise, δis a random number between 0 and 1. Table 1 The design parameters of cantilever plate. Design parameter
Elastic modulus (GPa)
Density (kg/m3)
Length (mm) Width (mm) Thickness (mm)
Modal damping ratios ζ1
Initial model 65 Theoretical model 71
2700 2700
450 –
350 –
2.0 –
ζ2
ζ3
ζ4
ζ5
ζ6
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.005 0.005 0.01 0.015 0.008
292
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
Table 2 The locations of measurement and excitation.
Measurement location
1 2 3 4 5 6 7 8 9 10 11
Exciting location
Node number
Element number
coordinate
6 7 11 14 21 135 249 272 273 277 280 55
5, 6 6, 7 10, 11 13, 14 1, 2, 19, 20 109, 110, 127, 128 217, 218, 235, 236 239, 240 240, 241 244, 245 247, 248 34, 35, 52, 53
(125 mm, 0 mm, 0 mm) (150 mm, 0 mm, 0 mm) (250 mm, 0 mm, 0 mm) (325 mm, 0 mm, 0 mm) (25 mm, 25 mm, 0 mm) (25 mm, 175 mm, 0 mm) (25 mm, 325 mm, 0 mm) (125 mm, 350 mm, 0 mm) (150 mm, 350 mm, 0 mm) (250 mm, 350 mm, 0 mm) (325 mm, 350 mm, 0 mm) (400 mm, 50 mm, 0 mm)
Fig. 3. The FE mesh of the cantilever plate.
Fig. 4. The initial correlation coefficient curves.
4.1. The frequency points selection 1. Select the measurement sampling frequency points where the correlation level needs to be improved. The initial correlation coefficient curves between theoretical SFRFs with noise level of 5% and initial analytical SFRFs are shown in Fig. 4. Using the Eq. (34), the number of frequency points for model updating is declined from 1900 to 841, where κ 2 ¼ 0:95.
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
293
The plots in Fig. 4 also confirm that the amplitude correlation coefficient αa ðωÞ is more stringent than the shape correlation coefficient αs ðωÞ, the discrepancies in the whole frequency bandwidth are less well captured by αa ðωÞ. 2. Delete the measurement sampling frequency points between experimental and initial analytical resonant points at the same mode. The first six modal frequencies of the theoretical and initial FE model are listed in Table 3, respectively. The deleting interval is defined as [ωr 0:95,ωr _e 1:05], and according to the information in Table 3, the number of measurement sampling frequency points in the deleting interval is 727, thus, the number of frequency points for model updating is further declined from 841 to 223.
4.2. The analysis for sensitivity matrix and the selection of updating parameter According to Section 4.1, the size of initial sensitivity matrix used in the present work is 446 8, and its rank is 7. The condition number of sensitivity matrix is 8.027 1014. Singular values of the sensitivity matrix are also listed in Table 4. As listed in Table 4, the singular value of sensitivity matrix decreases to zero gradually, and the ratio between the largest and smallest nonzero singular values is very large. So the initial sensitivity matrix is an ill-conditioned matrix. It can be seen from the rank (rank(S) ¼7o8) of sensitivity matrix S, linear dependency among columns exists in the sensitivity matrix S, the linear dependent index matrix can be obtained by Eq. (26), i.e. 2 6 6 6 6 6 6 6 CDIik ¼ 6 6 6 6 6 6 6 4
1
0:999
0:094
0:097
1
0:090
0:010
1
5:96 10
5
3:70 10 4
8:84 10 4
0:224
0:006
0:276
0:227
1:98 10
1
5
2:41 10
5
7 7 7 3:15 10 7 7 7 1:83 10 5 7 7 5 7 9:98 10 7 7 7 0:005 7 4 7 8:95 10 5 1 0:051
4:12 10
5
6:01 10 5
1:07 10 4
3:66 10 4
1
0:001 1
5:50 10 4 0:416
SYS:
3
0:041
1
7
As shown in the CDIik matrix, CDI12 ¼0.999, it indicates that the first column and the second column in sensitivity matrix trend to be linear dependent, which are associated with the elastic modulus and density of the updating model respectively. Thus, one of these two columns can be excluded, the matrix dimension becomes 446 7. The condition numbers and ranks of sensitivity matrix without some particular column are listed in Table 5. From Table 5, the ill-conditioned of sensitivity matrix can be improved obviously by deleting the second column, which is associated with the density of structure. Among the aforementioned design parameters, seven design parameters, i.e. elastic modulus and the first six modal damping ratios are selected as the updating parameters in this example.
Table 3 The modal frequency of cantilever plate. Mode
1
2
3
4
5
6
Initial FE model modal frequency ωr Theoretical model modal frequency ωr _e
9.757 10.197
28.632 29.924
60.589 63.324
99.614 104.110
116.140 121.382
173.927 181.777
Table 4 Singular values of the sensitivity matrix. No. Singular value
1 3.35 10
10
2
3
4
5
6
7
8
134.94
85.32
78.84
27.93
14.00
12.81
4.17 10 5
Table 5 The condition number and rank of sensitivity matrix without some particular column.
condition number rank
Without the first column
Without the second column
2.615 109 7
1.02 105 7
294
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
Table 6 The cases of the numerical study. Case
A
B
C
Details
With 5% noise level
With 10% noise level
A þonly six measurement locations(No.1~6)
Table 7 Comparison of modal frequencies among theoretical model, initial model and updated model. Mode Modal frequency/Hz
1 2 3 4 5 6
Theoretical Initial
Errors (%) Updated (Case A)
Updated (Case B)
Updated (Case C)
Errors (Case A) (%)
Errors (Case B) (%)
Errors (Case C) (%)
10.197 29.924 63.324 104.110 121.382 181.777
4.31 4.32 4.32 4.32 4.32 4.32
10.156 29.804 63.070 103.693 120.896 181.049
10.124 29.708 62.867 103.359 120.506 180.466
0.608 0.612 0.613 0.614 0.613 0.613
0.402 0.401 0.401 0.401 0.400 0.400
0.716 0.722 0.722 0.721 0.722 0.721
9.757 28.632 60.589 99.614 116.140 173.927
10.135 29.741 62.936 103.471 120.638 180.663
Fig. 5. Comparison of SFRFs: (a) case A; (b) case B; and (c) case C.
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
295
4.3. Results and discussion Using the initial sensitivity matrix determined in Sections 4.1 and 4.2, the model updating is then performed and the seven parameters are updated using the proposed model updating method. In the updating example, the stopping criterion κ1 is given as 0.99.To check the robustness of the proposed method, some case studies are considered in this example, as shown in Table 6. A comparison of the first six modal frequencies of the theoretical model, initial model and updated FE model is listed in Table 7. From Table 7, compared with the modal frequencies of the initial dynamic FE model, the modal frequencies of the updated dynamic FE models in the above cases are all closer to the theoretical ones, and the maximum error is only 0.722%. Based on this encouraging result, it is confirm that the noise and the less measurement locations are both less effect on updating the modal frequency of the FE model. In different case studies, the SFRFs obtained based on the models (including the theoretical model, initial FE model and the updated FE model) at the same measurement location (No.1) are shown in Fig. 5. From Fig. 5, compared with the initial SFRFs, the updated SFRFs obtained in all case studies show theoretical a perfect match to theoretical ones in the whole sampling frequency bandwidth. Fig. 6 shows the updated SFRFs correlation level between the updated SFRFs and the theoretical ones. It is shown that the updated correlation levels obtained in case studies are almost unity across the full spectrum, which further validates that (1) the perfect updated results are obtained by the proposed method; (2) the updating equations have insensitivity against noise; (3) the proposed updating method remains good feasibility even the number of measurement locations reduced to six (the original number of measurement locations is eleven). When the stopping criterion (33) is satisfied, a new solution of the updated dynamic FE model is computed. Table 8 shows the comparison of every updating parameter before and after the updating.
5. Experimental validation In order to further verify the feasibility and effectiveness of the proposed method, a steel cantilever stepped-beam is used in the experimental validation, as shown in Fig. 7. The schematic diagram of experiment setup is shown in Fig. 8. The
Fig. 6. The updated correlation level of SFRF.
Table 8 The comparison of updating parameters before and after the updating. Updating parameter
Initial model Theoretical model Updated model (Case A) Updated model (Case B) Updated model (Case C)
Elastic modulus/MPa
65000 71000 70132 70432 69979
Modal damping ratios ζ1
ζ2
ζ3
ζ4
ζ5
ζ6
0.01 0.01 0.0108 0.0103 0.0104
0.01 0.005 0.0051 0.0054 0.0050
0.01 0.005 0.0050 0.0050 0.0050
0.01 0.01 0.0107 0.0103 0.0111
0.01 0.015 0.0139 0.0133 0.0140
0.01 0.008 0.0081 0.0075 0.0088
296
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
Fig. 7. The experimental model and test devices.
Fig. 8. Schematic diagram of the experiment setup.
Fig. 9. The finite element model of experimental structure. Table 9 The comparison of design parameters before and after the updating. Design parameter
Initial Updated
Thickness (mm)
8.1 –
Elastic modulus (MPa)
210,000 198,500
Density (kg/m3)
7800 –
Modal damping ratio ζ1
ζ2
ζ3
ζ4
ζ5
0.01 0.015
0.01 0.0041
0.01 0.0035
0.01 0.0066
0.01 0.0060
FE model of the experimental structure is shown in Fig. 9. In the experiment, an electromagnetic exciter is employed to exert sine sweep exciting load near the root of the cantilever stepped-beam, and dynamic strain responses are measured by strain gages sensor affixed on the selected measurement locations. The sampling frequency and the spectral line used in this experiment are 1024 Hz and 4096, respectively. Besides, the bandwidth of sine sweep loading is 1–450 Hz, a single sweep time is 120 s. The quarter bridge is used to measure dynamic strain, and a measured signal is amplified by dynamic strain amplifier. According to the modeling information, seven design parameters, i.e. the elastic modulus, density, and the first five modal damping ratios of the beam, are selected as the initial updating parameters in the present work. Modeled on the updating procedure in Sections 4.1 and 4.2, the size of sensitivity matrix finally becomes 2252 6. Then, six design parameters, i.e. elastic modulus and the first five modal damping ratios are selected as the updating parameters in this updating experiment. Using the sensitivity matrix, the model updating is then performed and the six parameters are updated using the proposed model updating method, and the comparison of every design parameter before and after the updating is shown in
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
297
Fig. 10. Comparison of SFRFs: (a)H ε189;92 ; (b)Hε589;92 .
Table 9. The SFRFs obtained based on the models (including the experimental model, initial FE model and the updated FE model) at two particular measurement locations (i.e. node 189 and 589 in the FE model) are shown in Fig. 10. From Fig. 10, compared with the initial SFRFs, the updated SFRFs show a closer match to experimental ones in the whole sampling frequency bandwidth, especially in the resonance regions. Meanwhile, the updated SFRFs at the other measurement locations can be improved in the same way as H ε189;92 and Hε589;92 . The same conclusion can also be extracted from Fig. 11, which shows the initial SFRFs correlation (between the initial SFRFs and the experimental ones) and the updated SFRFs correlation (between the updated SFRFs and the experimental ones). Furthermore, compared with the initial and updated correlation coefficient curves in Fig. 11, the shape correlation coefficient curve αs remains almost the same before and after the updating, the correlation αa significantly improves across the full sampling bandwidth. Thus, in this particular circumstance, the correlation αs has the less effect for the model updating, the information for model updating is solely provided by the correlation αa .
6. Conclusions In this study, a novel dynamic model updating method based on the correlation analysis of experimental and analytical SFRFs is proposed, and verified by the model updating of a numerical study and an experimental example. The primary contributions of this work can be summarized as follows: 1. In order to ensure the updated model can be used for simulative evaluation of structural dynamic strength, SFRF, as a parameter which can reflect the structural dynamic strength features, is adopted in the proposed dynamic FE model
298
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
Fig. 11. The initial and updated lever of SFRF correlation: (a) shape correlation coefficient; and (b) magnitude correlation coefficient.
updating method. The linear algebraic equations for model updating are established based on the sensitivity analysis of correlation coefficients between experimental and analytical SFRFs. 2. A selection strategy for response measurement locations is developed by numerical simulation using the FE model of the structure. 3. In order to improve the computational efficiency and the stability of model updating method, a selection strategy of frequency points is also developed. 4. A numerical study and an experimental example are performed to verify the feasibility and robustness of the proposed updating method. From the updated results, the proposed updating method has good robustness to noise ability and remains good feasibility even the number of measurement locations reduced to six (the original number of measurement locations is 11). The updated SFRFs and the experimental SFRFs agree well, especially in resonance regions.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant no. 11402205), the Fundamental Research Funds for the Central Universities of China (Grant no. 310201401JCQ01015), the Innovation Funds of CALT for Unversities of China (Grant no. CALT201508) and the 111 Project of China (Grant no.B07050). Guo gratefully thank Professor Dong Wang for their helpful suggestions.
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
299
References [1] J.E. Mottershead, M. Link, M.I. Friswell, The sensitivity method in finite element model updating: a tutorial, Mech. Syst. Signal Process. 25 (7) (2011) 2275–2296. [2] R.M. Lin, J. Zhu, Finite element model updating using vibration test data under base excitation, J. Sound Vib. 303 (3) (2007) 596–613. [3] J.E. Mottershead, M.I. Friswell, G.H.T. Ng, et al., Geometric parameters for finite element model updating of joints and constraints, Mech. Syst. Signal Process. 10 (2) (1996) 171–182. [4] G. Steenackers, P.F. Guillaume, element model updating taking into account the uncertainty on the modal parameters estimates, J. Sound Vib. 296 (4) (2006) 919–934. [5] V. Arora, S.P. Singh, T.K. Kundra, Damped FE model updating using complex updating parameters: its use for dynamic design, J. Sound Vib. 324 (1) (2009) 350–364. [6] V. Arora, Comparative study of finite element model updating methods, J. Vib. Control 17 (23) (2011) 2023–2039. [7] A. Esfandiari, F. Bakhtiari-Nejad, M. Sanayei, et al., Structural finite element model updating using transfer function data, Comput. Struct. 88 (1) (2010) 54–64. [8] A. Esfandiari, F. Bakhtiari-Nejad, A. Rahai, et al., Structural model updating using frequency response function and quasi-linear sensitivity equation, J. Sound Vib. 326 (3) (2009) 557–573. [9] J. Ha, Y. Park, Y. Park, Model updating with closed-loop strain mode shapes, J. Guid. Control Dyn. 30 (4) (2007) 1206–1209. [10] C.U. Ip, C.A. Vickery, Dynamic stress at critical locations of a structure as a criterion for mathematical model modification[C]//In Shock and Vibration Information Center The Shock and Vibration Bulletin Part. 4: Structural Dynamics and Modal Test and Analysi, (SEE N 88-17062 09-39), 1987, pp. 29– 46. [11] A. Esfandiari, M. Sanayei, F. Bakhtiari-Nejad, et al., Finite element model updating using frequency response function of incomplete strain data, AIAA J. 48 (7) (2010) 1420–1433. [12] R.D. Cook, Concepts and Applications of Finite Element Analysis, John Wiley & Sons, Hoboken, 2007. [13] H. Grafe, Model Updating of Large Structural Dynamics Models Using Measured Response Functions, University of London, London, 1999. [14] P.G. Bakir, E. Reynders, G. De Roeck, Sensitivity-based finite element model updating using constrained optimization with a trust region algorithm, J. Sound Vib. 305 (1) (2007) 211–225. [15] B.P. Wang, Improved approximate methods for computing eigenvector derivatives in structural dynamics, AIAA J. 29 (6) (1991) 1018–1020. [16] K.S. Kwon, R.M. Lin, Frequency selection method for FRF-based model updating, J. Sound Vib. 278 (1) (2004) 285–306. [17] A.N. Tikhonov, V.Y. Arsenin, Solutions of Ill-posed Problems, Wiley, New York, 1977. [18] P.C. Hansen, Regularization tools version 4.1 (for Matlab version 7.3), Numer. Algorithms 46 (2007) 189–194. [19] A.N. Thite, D.J. Thompson, Selection of response measurement locations to improve inverse force determination, Appl. Acoust. 67 (8) (2006) 797–818. [20] S.Z. Rad, Methods of Updating Numerical Models in Structural Dynamics, University of London, London, 1997. [21] K. Blakely, MSC/NASTRAN Basic Dynamic Analysis User's Guide, Version 68, MacNeal-Schwendler, Los Angeles, 1993.
Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Model updating using correlation analysis of strain frequency response function Ning Guo n, Zhichun Yang, You Jia, Le Wang Institute of Structural Dynamics and Control, School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China
a r t i c l e in f o
abstract
Article history: Received 2 February 2015 Received in revised form 17 September 2015 Accepted 27 September 2015 Available online 23 October 2015
A method is proposed to modify the structural parameters of a dynamic finite element (FE) model by using the correlation analysis for strain frequency response function (SFRF). Sensitivity analysis of correlation coefficients is used to establish the linear algebraic equations for model updating. In order to improve the accuracy of updated model, the regularization technique is used to solve the ill-posed problem in model updating procedure. Finally, a numerical study and a model updating experiment are performed to verify the feasibility and robustness of the proposed method. The results show that the updated SFRFs and experimental SFRFs agree well, especially in resonance regions. Meanwhile, the proposed method has good robustness to noise ability and remains good feasibility even the number of measurement locations reduced significantly. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Model updating Strain frequency response function Correlation analysis Sensitivity analysis Regularization technique
1. Introduction The structural dynamic strength analysis is not often taken into account in the engineering practice of structure design, thus many structural failure problems caused by excessive dynamic stress or vibration fatigue occurred in services. Therefore, attentions should be paid to the simulative evaluation of structural dynamic strength in structural design. The accuracy of simulative evaluation is affected by the accuracy of the dynamic finite element (FE) model. However, it is difficult to establish a sufficiently accurate dynamic FE model using FE modeling techniques. In order to improve the accuracy of dynamic FE model, model updating technique has been introduced and widely adopted to modify dynamic FE model using experimental data [1,2]. In the past decades, a large number of model updating methods were proposed. Almost all the dynamic parameters, such as modal frequencies, modal shapes, frequency response functions (FRFs) and dynamic response, were used in dynamic FE model updating. Modal parameters were the earliest to be used to update dynamic FE model. Mottershead et al. [3] adopted the eigenvalue sensitivity approach to update the FE models of welded joints. Steenackers and Guillaume [4] developed an updating method based on measured modal parameters, and took the uncertainty of measurement into account in their method. However, the incompleteness of measured modal parameters and modal analysis errors may significantly affect the accuracy of the updated model [2]. Compared with the model updating method based on modal parameter, the FRF-based model updating method has some advantages [5,6]: (1) the errors during the modal parameter identification can be avoided; (2) the incompleteness of measured data makes less impacts on the FRF-based model updating method; (3) the structure information included in FRF data is more than those in modal parameters. Most of the FRF-based model updating n
Corresponding author. Tel./fax: þ86 029 88460461. E-mail addresses: [email protected] (N. Guo), [email protected] (Z. Yang).
http://dx.doi.org/10.1016/j.ymssp.2015.09.036 0888-3270/& 2015 Elsevier Ltd. All rights reserved.
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
Nomenclature
HεA
FE SFRF FRF Hx
S p e RDI ik CDI ik U V
Hε ℜð∙Þ ℑð∙Þ Be αs αa HεE
Finite element Strain frequency response function Frequency response function Displacement frequency response function matrix Strain frequency response function matrix The real part of complex strain frequency response function The imaginary part of complex strain frequency response function Strain-displacement mapping matrix Shape correlation coefficient vector Amplitude correlation coefficient vector Experimental strain frequency response function matrix
285
Analytical strain frequency response function matrix Sensitivity matrix Design parameter Residual vector Row linear dependency coefficient matrix Column linear dependency coefficient matrix Unitary matrix Unitary matrix
Superscript H
Complex conjugate transpose operator of a matrix or a vector
methods modify the uncertain design parameters by minimizing the error norms between experimental and analytical FRFs [7,8]. The dynamic strength parameters, such as strain mode shapes, strain frequency response functions and dynamic strain responses, can reflect the local feature of structure well, thus these parameters are also used in dynamic FE model updating method. Ha et al. [9] developed a method of combining strain modal shapes and a closed-loop scheme to update FE model. Ip and Vickery [10] proposed a novel random walk approach using strain frequency response function (SFRF) to update the dynamic FE model, and described an investigation of analysis methods for predicting dynamic stresses in a structure. Esfandiari et al. [11] proposed a dynamic FE model updating method using incomplete SFRF, and used the least square method to solve the quasi-linear sensitivity equation in model updating. SFRF, as a structural dynamic strength parameter, is used in dynamic FE model updating in the present work. By means of the correlation analysis between experimental and analytical SFRFs at the specific critical locations of a structure, a dynamic FE model updating method is proposed. A frequency selection strategy is also introduced to improve the computational efficiency and the stability of the proposed updating method. Based on sensitivity analysis of correlation coefficients between experimental and analytical SFRFs, the linear algebraic equations for model updating are established to modify the modeling errors. Meanwhile, the ill-pose problem in updating procedure is solved by regularization technique. Finally, the feasibility and robustness of the proposed method are demonstrated by numerical study and experimental validation respectively. The rest of this current study is organized as follows: in Section 2, according to the derived SFRF expression, the correlation coefficients of experimental and analytical SFRFs are introduced. In Section 3, the dynamic FE model updating method using SFRFs is described. In order to check the feasibility and robustness of the updated method, a numerical study is adopted in Section 4. Section 5 presents an experimental example to demonstrate the efficiency of the proposed dynamic FE model updating method. Finally, some conclusions are drawn in Section 6.
2. SFRF and its correlation coefficients 2.1. Strain frequency response function A continuous structure can be discretized into a multiple of freedom (MDOF) system using FE technique, the dynamic equation for a MDOF system is, Mx€ ðt Þ þ Cx_ ðt Þ þ Kxðt Þ ¼ f ðt Þ
ð1Þ
where M,C,K are mass, viscous damping and stiffness matrices, respectively. x (t) and f (t) are displacement and force vectors, respectively. Assuming the system is subjected to a harmonic excitation, the excitation and displacement response can be expressed as, f ðt Þ ¼ Fejωt xðt Þ ¼ Xejωt
ð2Þ
286
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
Substituting Eq. (2) into Eq. (1) yields, ω2 M þ jωC þ K X ¼ F
ð3Þ
Using modal superposition technique, Eq. (3) can be rewritten as, X¼
n X
φr φTr F ¼ Hx F K ω2 M r þjωC r r¼1 r
ð4Þ
where φr is the r-th modal shape, K r ¼ φTr Kφr , M r ¼ φTr Mφr , C r ¼ φTr Cφr , n is the number of modes. According to the theory of finite element analysis, the strain at any point within i-th element can be expressed as, εei ¼ Be xi ; i ¼ 1; 2; ⋯; q
ð5Þ
where Be is strain-displacement mapping matrix, xi is the node displacement vector of the i-th element, q is the total element number. Assuming the j-th node is shared by nk elements, namely, there are nk different values of εej;k ðk ¼ 1; 2; ⋯; nk Þ. The average nodal strain εj can be expressed by nodal averaging method [12] εj ¼
nk 1 X εe nk k ¼ 1 j;k
ð6Þ
Transforming the average nodal strain of each node in local coordinate of the element to global one, yields, nk nk P P Be xek Be Gek nk 1 X k¼1 k¼1 εj ¼ ε ¼ ¼ x ¼ Tj x nk k ¼ 1 j;k nk nk
ð7Þ
where xek is the displacement vector of k-th element, Gek is the transformation matrix between the element coordinate nk P e e B Gk
and the global coordinate, Tj ¼ i ¼ 1nk
is the mapping matrix between i-th average nodal strain vector and the global dis-
placement vector. The strain of all nodes can be represented as, ε ¼ Tx
T
where ε ¼ ε1 ε2 ⋯ εN , T ¼ T1 T2 ⋯ TN Substituting Eqs. (2) and (4) into Eq. (8) yields, εð t Þ ¼ θ
T
ð8Þ , N is the total number of nodes.
n X
n X φr φTr ψεr φTr Fejωt ¼ Fejωt 2 2 M þ jωC K ω M þ jωC K ω r r r r r r r¼1 r¼1
ð9Þ
where ψεr ¼ θφr is the r-th strain modal shape. So the SFRF matrix can be expressed as, Hε ¼
n X
n X ψεr φTr ψεr φTr ¼ ¼ ℜðHε Þ þj U ℑðHε Þ 2 2 ω2 þ 2jζ ωω K ω M þ jωC ω r r r r r r¼1 r¼1 r
ð10Þ
where ωr is the r-th modal frequency; operators ℜð∙Þ and ℑð∙Þ address the real and imaginary part of complex SFRF. Thus, the SFRF between the output at dof i and input at dof j can be expressed as. ψ εir φjr ¼ ℜ Hεij þ j Uℑ Hεij 2 ω2 þ 2jζ ωω ω r r r¼1 r n n P P ψ εir φjr ðω2r ω2 Þ 2ψ εir φjr ζ r ωr ω ε ;ℑ Hεij ¼ where ℜ Hij ¼ 2 ω2 2 þ ð2ζ ω ωÞ2 2 ω2 2 þ ð2ζ ω ωÞ2 ω ω ð Þ ð Þ r r r r r¼1 r r¼1 r Hεij ðωÞ ¼
n X
ð11Þ
2.2. Correlation coefficients of SFRF In order to evaluate the correlation level between experimental and analytical SFRFs, shape correlation coefficientαs ðω; pÞ and amplitude correlation coefficient αa ðω; pÞ are introduced for any measured frequency ω [13], i.e. Eq. (12) and Eq. (13), which are derived from the Modal Assurance Criterion (MAC) value and Modal Scale Factor (MSF) value. H ∣ HεE;i ðωÞ HεA;i ðω; pÞ∣2 ð12Þ αs;i ðω; pÞ ¼ H H HεE;i ðωÞ HεE;i ðωÞ HεA;i ðω; pÞ HεA;i ðω; pÞ
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
H 2∣ HεE;i ðωÞ HεA;i ðω; pÞ∣ αa;i ðω; pÞ ¼ H H HεE;i ðωÞ HεE;i ðωÞ þ HεA;i ðω; pÞ HεA;i ðω; pÞ
287
ð13Þ
where HεE;i ðωÞand HεA;i ðω; pÞare the i-th column of the experimental and analytical SFRF matrix, respectively, p is the design parameter, superscript H is the complex conjugate transpose operator. The correlation coefficient vectors can be expressed as, 2 3 αs;1 ðω; pÞ 6 7 ⋮ ð14Þ αs ðω; pÞ ¼ 4 5 αs;m ðω; pÞ 2 6 αa ðω; pÞ ¼ 4
αa;1 ðω; pÞ ⋮ αa;m ðω; pÞ
3 7 5
ð15Þ
where subscript m is the number of column of SFRF matrix, i.e. the number of system input. According to Eqs. (12) and (13), all the values of correlation coefficient vectors lie between zero and unity, i.e. 0 r αs;i ðω; pÞ r1,0 rαa;i ðω; pÞ r 1. The correlation coefficient αs is only sensitive to discrepancies in the overall deflection shape of the structure. αa is introduced by targeting the discrepancies in amplitude, which is more stringent than αs and only becomes unity if HεE;i ðωÞ ¼ HεA;i ðωÞ [13].
3. Model updating procedure In the following section, a model updating method based on SFRF correlation analysis is proposed. Based on the sensitivity analysis [14] of SFRF correlation coefficients, the linear algebraic equations used in model updating are established firstly. Then regularization technique is used to solve the ill-posed problem in this model updating method. Finally, the selection strategies of measurement locations and frequency points are also developed respectively. 3.1. Sensitivity analysis and model updating methodology In order to establish the linear algebraic equations used in model updating, the sensitivity of correlation coefficients should be analyzed firstly. Starting with Eqs. (12) and (13), the sensitivity of αs ðω; pÞ and αa ðω; pÞ with respect to design parameter p can be expressed respectively as, 2 ∂αs;i ðω; pÞ ∂ℜðHA;i Þ ∂ℑðHA;i Þ 2 ¼ HH H HH H 4ℜ HH ℜ HH ℑ HH E;i HA;i E;i E ∂p ∂p ð E;i E;i Þð A;i A;i Þ ∂p H ∂ℑ HA;i H ∂ℜ HA;i þ ℑ H ℜ H H þℑ HH E;i A;i E;i E;i ∂p ∂p # H ∂ℑ H ∂ℜH ∣HE;i HA;i ∣2 A;i A;i H H þ H ℜ HA;i ℑ HA;i ∂p ∂p HA;i HA;i
ð16Þ
2 ∂αa;i ðω; pÞ ∂ℜðHA;i Þ ∂ℑðHA;i Þ ¼ ∣HH H ∣ HH 2H þ HH H 4ℜ HH ℜ HH ℑHH E;i HA;i E;i E;i ∂p ∂p ð Þ A;i E;i A;i ∂p E;i E;i A;i H H ∂ℑ HA;i H ∂ℜ HA;i þ ℑ HE;i þℑ HE;i HA;i ℜ HE;i ∂p ∂p # H ∂ℑ H 2∣HE;i HA;i ∣2 A;i H H ∂ℜ HA;i þ H ℜ HA;i ℑ HA;i ∂p ∂p HE;i HE;i þ HH A;i HA;i
ð17Þ
whereHE;i ¼ HεE;i ðωÞ, HA;i ¼ HεA;i ðω; pÞ. The sensitivity of SFRF with respect to design parameter can be obtained by the following two approaches, 1. If design parameter is only related to mass matrix and (or) stiffness matrix, the sensitivity of SFRF can be obtained by analytic method based on FE model, ∂ℜ H εij ðω; pÞ ∂ℜ H εij ðω; pÞ ∂ω2 ðpÞ ∂ℜ H εij ðω; pÞ ∂φ ðpÞ ∂ℜ H εij ðω; pÞ ∂ψ ε ðpÞ r r r þ þ ¼ ∂p ∂p ∂p ∂p ∂φr ðpÞ ∂ψ εr ðpÞ ∂ω2r ðpÞ
288
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
20 n X 6B ¼ 4@ r¼1
1 ψ εir φjr 2ψ εir φjr ω2r ω2 ω2r ω2 þ 2ζ 2r ω2 C∂ω2r ðpÞ A 2 h i2 2 ∂p ω2r ω2 þ ð2ζ r ωr ωÞ2 ω2 ω2 þ ð2ζ ωr ωÞ2 r
r
3 2 2 ∂φ ð p Þ ωr ω ∂φir ðpÞ ε jr 7 þ ψ jr þψ εir 5 2 ∂p ∂p ω2r ω2 þ ð2ζ r ωr ωÞ2 ∂ℑ H εij ðω; pÞ ∂p
¼
∂ℑ H εij ðω; pÞ ∂ω2 ðpÞ
∂ω2r ðpÞ 2 0 n X 6 B ¼ 4@ r¼1
r
∂p
ð18Þ
∂ℑ H εij ðω; pÞ ∂φ ðpÞ ∂ℑ H εij ðω; pÞ ∂ψ ε ðpÞ r r þ þ ∂p ∂p ∂φr ðpÞ ∂ψ εr ðpÞ
1 ζ r ωψ εir φjr 4ψ εir φjr ζ r ωr ω ω2r ω2 þ 2ζ 2r ω2 C∂ω2r ðpÞ h i2 A 2 2 ∂p ωr ω2r ω2 þ ð2ζ r ωr ωÞ2 ω2 ω2 þ ð2ζ r ωr ωÞ2 r
3 ∂φjr ðpÞ 7 2ζr ωr ω ∂φir ðpÞ ε ε ψ jr þ ψ ir 5 2 ∂p ∂p ω2r ω2 þ ð2ζ r ωr ωÞ2 where
∂ω2r ðpÞ ∂p
¼ φTr
∂K 2 ∂M ∂p ωr ∂p
ð19Þ
φr , the first partial derivative of eigenvector can be solved by implicit method which was pro-
posed by Wang [15], i.e., n X ∂φr ðpÞ ¼ φcr cr φr þ ci φi þ dr wr ð20Þ ∂p i ¼ 1 iar ∂ω n φT ∂K ∂ωr M ω ∂M φ ∂M ∂M r P φTi ∂K wTr ∂K r ∂p r ∂p ∂ωr ∂p ωr ∂p φr ∂p ∂p M ωr ∂p φr i ∂p ∂M φ , c ¼ , d ¼ , w ¼ φi K 1 ∂K where cr ¼ 12φTr ∂M r r T T i ωi ωr ωi ∂p r ∂p ∂p M ωr ∂p φr . w Kw ω w Mw r
r
r
r
r
i
∂M The solution of ∂K ∂p and ∂p is related to the element attributes. 2. If the design parameter is viscous damping ratio, i.e. p ¼ ζ r , based on Eq. (11), the sensitivity can be derived as, ∂ℜ H εij ðω; pÞ 8ψ ir φjr ω2r ω2 ζ r ω2r ω2 ¼ ð21Þ 2 2 ∂p ω2r ω2 þ ð2ωr ωζ r Þ2
∂ℑ H εij ðω; pÞ ∂p
2 2ψ ir φjr ωr ω ð2ωr ωζ r Þ2 ω2r ω2 ¼ 2 2 ω2r ω2 þ ð2ωr ωζ r Þ2
ð22Þ
According to the sensitivity analysis of SFRF correlation coefficient, Eqs. (16) and (17) can be expanded by truncated Taylor series expansion at any frequency ω, the high order term is neglected in the expansion equation, then we get, SðωÞΔp ¼ eðωÞ 2
∂αs ðω;pÞ ∂p1
∂αs ðω;pÞ ∂p2
⋯
∂p1
∂αa ðω;pÞ ∂p2
⋯
6 whereSðωÞ ¼ 4 ∂αa ðω;pÞ
∂αs ðω;pÞ ∂pNp
3 7
∂αa ðω;pÞ 5,eðωÞ ¼ ∂pNp
ð23Þ (
1 αs ðω; pÞ 1 αa ðω; pÞ
)
,Δp ¼ Δp .
In model updating, SFRF data at sufficient measurement frequency points should be used, Eq. (23) can be transformed into a set of over-determined linear algebraic equations, which can be denoted as, SΔp ¼ e
ð24Þ 2ðN f U mÞN p
where S A C is the sensitivity matrix, Δp A C vector, Np is the number of updating parameters.
Np 1
is design parameter changes, e A C
2ðN f U mÞ1
is the residual
3.2. The solution for ill-posed problem In order to obtain stable updating parameters during iterations, the sensitivity matrix S should be well-conditioned. Otherwise, an arbitrarily small disturbance of the residual vector can cause a considerable fluctuation of the updated results. To make sure that the sensitivity matrix is well-conditioned, the row (or column) vectors in the sensitivity matrix should be nearly independent with their counterparts [16]. Then, the linear dependency coefficient between row (or column) vectors are introduced to measure the linear dependency of row (or column) vectors in the sensitivity matrix in the present work,
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
289
i.e., RDIik ¼
∣XTi Xk ∣ 12 XTi Xi XTk Xk
ð25Þ
∣Y Ti Y k ∣ 12 Y Ti Y i YTk Yk
ð26Þ
CDIik ¼
where Xi ,i ¼ 1; 2; ⋯; 2 N f Um are row vectors in the sensitivity matrix, Y i ,i ¼ 1; 2; ⋯; Np are the column vectors. If the value of RDIik (or CDIik) is close to 1, which means that the corresponding two rows (or columns) in the sensitivity matrix have similar effects, any one of these two rows (or columns) can be discarded. With the help of Tikhonov regularization technique [17,18], the potential ill-posed problem in the current study can be further mitigated, the over-determined equations (Eq. (23)) can be replaced by the following minimization problem, J λ ¼ arg min ‖SΔp e‖22 þ λ2 ‖IðΔp Δp Þ‖22 ð27Þ whereλis regularization parameter, Δp is the initial value of design parameter changes, which is assumed to be zero in the present work, I is identity matrix. Then the above minimization problem can be transformed into a following least squares problem,
2 S e min Δp λI 0 2
ð28Þ
By simplification, Eq. (28) can be written as, ST Sþ λ2 I Δp ¼ ST e
ð29Þ
The matrix S can be decomposed using singular value decomposition technique, S ¼ UΣVT ¼
Np X
ui σ i v i
ð30Þ
i¼1
where the columns of U and Vare the left and right singular vectors of matrix S, respectively. Σ ¼ diag σ 1 ; ⋯; σ Np has nonnegative diagonal elements appearing in descending order, i.e.σ 1 Z⋯ Zσ Np Z0, where σ i is the singular values of matrix S. Substituting Eq. (30) into Eq. (29), the regularized solution ΔpLSQ can be obtained, ΔpLSQ ¼
Np X uT e f i i vi σi i¼1
ð31Þ
where f i ¼ σ 2i = σ 2i þ λ2 is the filter factor for the regularization method. The regularization parameter in the filter factors can be selected by the L-curve method [18]. Assuming ξ ¼ log ‖SΔp ε‖2 , η ¼ log ‖IΔp‖2 , the L-curve function can be denoted as follows, ρðλÞ ¼
ξ0 η″ ξ″η0
ð32Þ
0 2
½ðξ Þ þðη0 Þ2 3=2
where ξ0 ,ξ″, η0 and η″ are the first-order derivative and the second-order derivative of ξ and η with respect to regularization parameter λ, respectively. The regularization parameter corresponding to the maximum of Eq. (32) is the best regularization parameter. Substituting the best regularization parameter into Eq. (31), then the design parameter changes ΔpLSQ can be obtained. The design parameters are updated iteratively using parameter changes ΔpLSQ , i.e. pnew ¼ pold þ ΔpLSQ , then the new SFRFs and correlation coefficients are solved based on the updated dynamic model. Thus, the stopping criterion used in the iterations is defined as, 8 Nf > X > 1 αs ðωi Þ þ αa ðωi Þ > 1 > Z κ1 stop > > 1 2 < N f i ¼ 1m ð33Þ Nf > X > 1 αs ðωi Þ þ αa ðωi Þ > 1 > o κ continue > 1 > 1 : N f i ¼ 1m 2
‖
‖
‖
‖
where κ1 is the pre-specified threshold.
290
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
3.3. Selection of response measurement locations For various experimental purposes, the exciting and response measurement locations may be totally different [19]. So the selection of response measurement locations is very important in vibration experiment. In the present study, the purpose of dynamic model updating is using the updated dynamic FE model to evaluate the structural dynamic strength. Thus the critical points (i.e. the locations of maximum dynamic strain) of a structure are chosen as the measurement locations. And the critical points are determined by numerical simulation using the updating FE model. The procedure is described as follows: 1. Obtain the strain modes by using the initial dynamic FE model of the structure. 2. Determine the critical points of the structure by examining the maximum strain in the concerned strain modes. 3. Search for the elements and the nodes corresponding to the critical points. 3.4. The frequency selection strategy for model updating The model updating method, in the present work, is based on the correlation analysis of experimental and analytical SFRFs. It is well known that there is a large number of sampling frequency points in experimental SFRFs. If all the sampling frequency points are used in model updating procedure, it will be time consuming and some frequency points are believed
Fig. 1. The sketch diagram of SFRF updating [20].
Fig. 2. Flow chart of model updating using correlation analysis of SFRFs.
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
291
to be redundant for updating model [16]. Therefore, a small number of optimum frequency points for mode updating should be selected. Some considerations for selecting the frequency points will be pursued as follows: 1. Select frequency points where the correlation level needs to be improved, the selection criterion is given as follows [13], αs;i ðωi Þ þ αa;i ðωi Þ oκ 2 2
ð34Þ
where κ 2 is the pre-specified threshold. 2. The measurement sampling frequency points between experimental and analytical resonances should not be selected. If the selected frequency is not suitable for model updating, the difference between analytical and experimental SFRF will increase with the process of model updating, thus the stability of updating method will be affected. As shown in Fig. 1, three typical frequency points are marked to explain this phenomenon: point A is located before the experimental resonant point, point B is located between the analytical and experimental resonant points, and point C is located behind the analytical resonant point. The vertical lines A–A0 , B–B0 , and C–C0 show the differences between the experimental and analytical SFRF at these frequency points, respectively. During the model updating process, it can be inferred that the difference at point B will increase initially, thus the stability problems can be introduced into the updating procedure [20]. It should be noted that the frequency points for model updating should be selected in each iteration. 3.5. Procedure of the proposed updating method To sum up, the steps for updating the dynamic model are as follows: Step 1: Establish the initial dynamic FE model, and determine the measurement locations of the structure by numerical simulation using the FE model of the structure. Step 2: Calculate the correlation coefficients based on experimental and analytical SFRFs, and construct the linear algebraic equations for model updating by using the sensitivity analysis of correlation coefficients. Step 3: Select the frequency points for model updating. Step 4: Solve Eq. (24) by using Tikhonov regularization technique, and obtain the design parameter changes. Step 5: Modify the appropriate parameters of the FE dynamic model and repeat the Step 2– 4 until the stopping criterion is satisfied. Fig. 2 Shows the flow chart of the current model updating method using correlation analysis of SFRFs.
4. Numerical study In this section, two FE models of cantilever aluminum plate are adopted to validate the feasibility and robustness of the proposed updating method. One FE model considered here is used as the model to be updated (i.e. initial model), the other one is utilized to obtain the ‘experimental’ data (i.e. theoretical model). The corresponding modeling parameters are listed in Table 1. In the updating example, the measurement locations are selected by the selection strategy described in Section 3.3, the selected measurement locations on the plate are listed in Table 2. The FE mesh of the updating structure with the locations of measurement and excitation are shown in Fig. 3. The FE model of the structure is meshed by four-node quadrilateral element based on Kirchhoff plate bending theory [12], the mass matrix of dynamic FE model is established by coupled mass matrix used in MSC. Nastran [21]. The frequency bandwidth (from 0.1 Hz to 190 Hz), which covers the first six modal frequencies, is taken as measured frequency range, the frequency resolution is 0.1 Hz. In order to check the robustness of the proposed method, two noise levels, 5% and 10%, are respectively considered in the theoretical SFRFs. The SFRF considered the level of noise is shown in Eq. (35). H εij ðωk Þ ¼ H εij ðωk Þð1 þ ηδÞ
ð35Þ
where η is the level of noise, δis a random number between 0 and 1. Table 1 The design parameters of cantilever plate. Design parameter
Elastic modulus (GPa)
Density (kg/m3)
Length (mm) Width (mm) Thickness (mm)
Modal damping ratios ζ1
Initial model 65 Theoretical model 71
2700 2700
450 –
350 –
2.0 –
ζ2
ζ3
ζ4
ζ5
ζ6
0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.005 0.005 0.01 0.015 0.008
292
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
Table 2 The locations of measurement and excitation.
Measurement location
1 2 3 4 5 6 7 8 9 10 11
Exciting location
Node number
Element number
coordinate
6 7 11 14 21 135 249 272 273 277 280 55
5, 6 6, 7 10, 11 13, 14 1, 2, 19, 20 109, 110, 127, 128 217, 218, 235, 236 239, 240 240, 241 244, 245 247, 248 34, 35, 52, 53
(125 mm, 0 mm, 0 mm) (150 mm, 0 mm, 0 mm) (250 mm, 0 mm, 0 mm) (325 mm, 0 mm, 0 mm) (25 mm, 25 mm, 0 mm) (25 mm, 175 mm, 0 mm) (25 mm, 325 mm, 0 mm) (125 mm, 350 mm, 0 mm) (150 mm, 350 mm, 0 mm) (250 mm, 350 mm, 0 mm) (325 mm, 350 mm, 0 mm) (400 mm, 50 mm, 0 mm)
Fig. 3. The FE mesh of the cantilever plate.
Fig. 4. The initial correlation coefficient curves.
4.1. The frequency points selection 1. Select the measurement sampling frequency points where the correlation level needs to be improved. The initial correlation coefficient curves between theoretical SFRFs with noise level of 5% and initial analytical SFRFs are shown in Fig. 4. Using the Eq. (34), the number of frequency points for model updating is declined from 1900 to 841, where κ 2 ¼ 0:95.
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
293
The plots in Fig. 4 also confirm that the amplitude correlation coefficient αa ðωÞ is more stringent than the shape correlation coefficient αs ðωÞ, the discrepancies in the whole frequency bandwidth are less well captured by αa ðωÞ. 2. Delete the measurement sampling frequency points between experimental and initial analytical resonant points at the same mode. The first six modal frequencies of the theoretical and initial FE model are listed in Table 3, respectively. The deleting interval is defined as [ωr 0:95,ωr _e 1:05], and according to the information in Table 3, the number of measurement sampling frequency points in the deleting interval is 727, thus, the number of frequency points for model updating is further declined from 841 to 223.
4.2. The analysis for sensitivity matrix and the selection of updating parameter According to Section 4.1, the size of initial sensitivity matrix used in the present work is 446 8, and its rank is 7. The condition number of sensitivity matrix is 8.027 1014. Singular values of the sensitivity matrix are also listed in Table 4. As listed in Table 4, the singular value of sensitivity matrix decreases to zero gradually, and the ratio between the largest and smallest nonzero singular values is very large. So the initial sensitivity matrix is an ill-conditioned matrix. It can be seen from the rank (rank(S) ¼7o8) of sensitivity matrix S, linear dependency among columns exists in the sensitivity matrix S, the linear dependent index matrix can be obtained by Eq. (26), i.e. 2 6 6 6 6 6 6 6 CDIik ¼ 6 6 6 6 6 6 6 4
1
0:999
0:094
0:097
1
0:090
0:010
1
5:96 10
5
3:70 10 4
8:84 10 4
0:224
0:006
0:276
0:227
1:98 10
1
5
2:41 10
5
7 7 7 3:15 10 7 7 7 1:83 10 5 7 7 5 7 9:98 10 7 7 7 0:005 7 4 7 8:95 10 5 1 0:051
4:12 10
5
6:01 10 5
1:07 10 4
3:66 10 4
1
0:001 1
5:50 10 4 0:416
SYS:
3
0:041
1
7
As shown in the CDIik matrix, CDI12 ¼0.999, it indicates that the first column and the second column in sensitivity matrix trend to be linear dependent, which are associated with the elastic modulus and density of the updating model respectively. Thus, one of these two columns can be excluded, the matrix dimension becomes 446 7. The condition numbers and ranks of sensitivity matrix without some particular column are listed in Table 5. From Table 5, the ill-conditioned of sensitivity matrix can be improved obviously by deleting the second column, which is associated with the density of structure. Among the aforementioned design parameters, seven design parameters, i.e. elastic modulus and the first six modal damping ratios are selected as the updating parameters in this example.
Table 3 The modal frequency of cantilever plate. Mode
1
2
3
4
5
6
Initial FE model modal frequency ωr Theoretical model modal frequency ωr _e
9.757 10.197
28.632 29.924
60.589 63.324
99.614 104.110
116.140 121.382
173.927 181.777
Table 4 Singular values of the sensitivity matrix. No. Singular value
1 3.35 10
10
2
3
4
5
6
7
8
134.94
85.32
78.84
27.93
14.00
12.81
4.17 10 5
Table 5 The condition number and rank of sensitivity matrix without some particular column.
condition number rank
Without the first column
Without the second column
2.615 109 7
1.02 105 7
294
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
Table 6 The cases of the numerical study. Case
A
B
C
Details
With 5% noise level
With 10% noise level
A þonly six measurement locations(No.1~6)
Table 7 Comparison of modal frequencies among theoretical model, initial model and updated model. Mode Modal frequency/Hz
1 2 3 4 5 6
Theoretical Initial
Errors (%) Updated (Case A)
Updated (Case B)
Updated (Case C)
Errors (Case A) (%)
Errors (Case B) (%)
Errors (Case C) (%)
10.197 29.924 63.324 104.110 121.382 181.777
4.31 4.32 4.32 4.32 4.32 4.32
10.156 29.804 63.070 103.693 120.896 181.049
10.124 29.708 62.867 103.359 120.506 180.466
0.608 0.612 0.613 0.614 0.613 0.613
0.402 0.401 0.401 0.401 0.400 0.400
0.716 0.722 0.722 0.721 0.722 0.721
9.757 28.632 60.589 99.614 116.140 173.927
10.135 29.741 62.936 103.471 120.638 180.663
Fig. 5. Comparison of SFRFs: (a) case A; (b) case B; and (c) case C.
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
295
4.3. Results and discussion Using the initial sensitivity matrix determined in Sections 4.1 and 4.2, the model updating is then performed and the seven parameters are updated using the proposed model updating method. In the updating example, the stopping criterion κ1 is given as 0.99.To check the robustness of the proposed method, some case studies are considered in this example, as shown in Table 6. A comparison of the first six modal frequencies of the theoretical model, initial model and updated FE model is listed in Table 7. From Table 7, compared with the modal frequencies of the initial dynamic FE model, the modal frequencies of the updated dynamic FE models in the above cases are all closer to the theoretical ones, and the maximum error is only 0.722%. Based on this encouraging result, it is confirm that the noise and the less measurement locations are both less effect on updating the modal frequency of the FE model. In different case studies, the SFRFs obtained based on the models (including the theoretical model, initial FE model and the updated FE model) at the same measurement location (No.1) are shown in Fig. 5. From Fig. 5, compared with the initial SFRFs, the updated SFRFs obtained in all case studies show theoretical a perfect match to theoretical ones in the whole sampling frequency bandwidth. Fig. 6 shows the updated SFRFs correlation level between the updated SFRFs and the theoretical ones. It is shown that the updated correlation levels obtained in case studies are almost unity across the full spectrum, which further validates that (1) the perfect updated results are obtained by the proposed method; (2) the updating equations have insensitivity against noise; (3) the proposed updating method remains good feasibility even the number of measurement locations reduced to six (the original number of measurement locations is eleven). When the stopping criterion (33) is satisfied, a new solution of the updated dynamic FE model is computed. Table 8 shows the comparison of every updating parameter before and after the updating.
5. Experimental validation In order to further verify the feasibility and effectiveness of the proposed method, a steel cantilever stepped-beam is used in the experimental validation, as shown in Fig. 7. The schematic diagram of experiment setup is shown in Fig. 8. The
Fig. 6. The updated correlation level of SFRF.
Table 8 The comparison of updating parameters before and after the updating. Updating parameter
Initial model Theoretical model Updated model (Case A) Updated model (Case B) Updated model (Case C)
Elastic modulus/MPa
65000 71000 70132 70432 69979
Modal damping ratios ζ1
ζ2
ζ3
ζ4
ζ5
ζ6
0.01 0.01 0.0108 0.0103 0.0104
0.01 0.005 0.0051 0.0054 0.0050
0.01 0.005 0.0050 0.0050 0.0050
0.01 0.01 0.0107 0.0103 0.0111
0.01 0.015 0.0139 0.0133 0.0140
0.01 0.008 0.0081 0.0075 0.0088
296
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
Fig. 7. The experimental model and test devices.
Fig. 8. Schematic diagram of the experiment setup.
Fig. 9. The finite element model of experimental structure. Table 9 The comparison of design parameters before and after the updating. Design parameter
Initial Updated
Thickness (mm)
8.1 –
Elastic modulus (MPa)
210,000 198,500
Density (kg/m3)
7800 –
Modal damping ratio ζ1
ζ2
ζ3
ζ4
ζ5
0.01 0.015
0.01 0.0041
0.01 0.0035
0.01 0.0066
0.01 0.0060
FE model of the experimental structure is shown in Fig. 9. In the experiment, an electromagnetic exciter is employed to exert sine sweep exciting load near the root of the cantilever stepped-beam, and dynamic strain responses are measured by strain gages sensor affixed on the selected measurement locations. The sampling frequency and the spectral line used in this experiment are 1024 Hz and 4096, respectively. Besides, the bandwidth of sine sweep loading is 1–450 Hz, a single sweep time is 120 s. The quarter bridge is used to measure dynamic strain, and a measured signal is amplified by dynamic strain amplifier. According to the modeling information, seven design parameters, i.e. the elastic modulus, density, and the first five modal damping ratios of the beam, are selected as the initial updating parameters in the present work. Modeled on the updating procedure in Sections 4.1 and 4.2, the size of sensitivity matrix finally becomes 2252 6. Then, six design parameters, i.e. elastic modulus and the first five modal damping ratios are selected as the updating parameters in this updating experiment. Using the sensitivity matrix, the model updating is then performed and the six parameters are updated using the proposed model updating method, and the comparison of every design parameter before and after the updating is shown in
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
297
Fig. 10. Comparison of SFRFs: (a)H ε189;92 ; (b)Hε589;92 .
Table 9. The SFRFs obtained based on the models (including the experimental model, initial FE model and the updated FE model) at two particular measurement locations (i.e. node 189 and 589 in the FE model) are shown in Fig. 10. From Fig. 10, compared with the initial SFRFs, the updated SFRFs show a closer match to experimental ones in the whole sampling frequency bandwidth, especially in the resonance regions. Meanwhile, the updated SFRFs at the other measurement locations can be improved in the same way as H ε189;92 and Hε589;92 . The same conclusion can also be extracted from Fig. 11, which shows the initial SFRFs correlation (between the initial SFRFs and the experimental ones) and the updated SFRFs correlation (between the updated SFRFs and the experimental ones). Furthermore, compared with the initial and updated correlation coefficient curves in Fig. 11, the shape correlation coefficient curve αs remains almost the same before and after the updating, the correlation αa significantly improves across the full sampling bandwidth. Thus, in this particular circumstance, the correlation αs has the less effect for the model updating, the information for model updating is solely provided by the correlation αa .
6. Conclusions In this study, a novel dynamic model updating method based on the correlation analysis of experimental and analytical SFRFs is proposed, and verified by the model updating of a numerical study and an experimental example. The primary contributions of this work can be summarized as follows: 1. In order to ensure the updated model can be used for simulative evaluation of structural dynamic strength, SFRF, as a parameter which can reflect the structural dynamic strength features, is adopted in the proposed dynamic FE model
298
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
Fig. 11. The initial and updated lever of SFRF correlation: (a) shape correlation coefficient; and (b) magnitude correlation coefficient.
updating method. The linear algebraic equations for model updating are established based on the sensitivity analysis of correlation coefficients between experimental and analytical SFRFs. 2. A selection strategy for response measurement locations is developed by numerical simulation using the FE model of the structure. 3. In order to improve the computational efficiency and the stability of model updating method, a selection strategy of frequency points is also developed. 4. A numerical study and an experimental example are performed to verify the feasibility and robustness of the proposed updating method. From the updated results, the proposed updating method has good robustness to noise ability and remains good feasibility even the number of measurement locations reduced to six (the original number of measurement locations is 11). The updated SFRFs and the experimental SFRFs agree well, especially in resonance regions.
Acknowledgments This work was supported by the National Natural Science Foundation of China (Grant no. 11402205), the Fundamental Research Funds for the Central Universities of China (Grant no. 310201401JCQ01015), the Innovation Funds of CALT for Unversities of China (Grant no. CALT201508) and the 111 Project of China (Grant no.B07050). Guo gratefully thank Professor Dong Wang for their helpful suggestions.
N. Guo et al. / Mechanical Systems and Signal Processing 70-71 (2016) 284–299
299
References [1] J.E. Mottershead, M. Link, M.I. Friswell, The sensitivity method in finite element model updating: a tutorial, Mech. Syst. Signal Process. 25 (7) (2011) 2275–2296. [2] R.M. Lin, J. Zhu, Finite element model updating using vibration test data under base excitation, J. Sound Vib. 303 (3) (2007) 596–613. [3] J.E. Mottershead, M.I. Friswell, G.H.T. Ng, et al., Geometric parameters for finite element model updating of joints and constraints, Mech. Syst. Signal Process. 10 (2) (1996) 171–182. [4] G. Steenackers, P.F. Guillaume, element model updating taking into account the uncertainty on the modal parameters estimates, J. Sound Vib. 296 (4) (2006) 919–934. [5] V. Arora, S.P. Singh, T.K. Kundra, Damped FE model updating using complex updating parameters: its use for dynamic design, J. Sound Vib. 324 (1) (2009) 350–364. [6] V. Arora, Comparative study of finite element model updating methods, J. Vib. Control 17 (23) (2011) 2023–2039. [7] A. Esfandiari, F. Bakhtiari-Nejad, M. Sanayei, et al., Structural finite element model updating using transfer function data, Comput. Struct. 88 (1) (2010) 54–64. [8] A. Esfandiari, F. Bakhtiari-Nejad, A. Rahai, et al., Structural model updating using frequency response function and quasi-linear sensitivity equation, J. Sound Vib. 326 (3) (2009) 557–573. [9] J. Ha, Y. Park, Y. Park, Model updating with closed-loop strain mode shapes, J. Guid. Control Dyn. 30 (4) (2007) 1206–1209. [10] C.U. Ip, C.A. Vickery, Dynamic stress at critical locations of a structure as a criterion for mathematical model modification[C]//In Shock and Vibration Information Center The Shock and Vibration Bulletin Part. 4: Structural Dynamics and Modal Test and Analysi, (SEE N 88-17062 09-39), 1987, pp. 29– 46. [11] A. Esfandiari, M. Sanayei, F. Bakhtiari-Nejad, et al., Finite element model updating using frequency response function of incomplete strain data, AIAA J. 48 (7) (2010) 1420–1433. [12] R.D. Cook, Concepts and Applications of Finite Element Analysis, John Wiley & Sons, Hoboken, 2007. [13] H. Grafe, Model Updating of Large Structural Dynamics Models Using Measured Response Functions, University of London, London, 1999. [14] P.G. Bakir, E. Reynders, G. De Roeck, Sensitivity-based finite element model updating using constrained optimization with a trust region algorithm, J. Sound Vib. 305 (1) (2007) 211–225. [15] B.P. Wang, Improved approximate methods for computing eigenvector derivatives in structural dynamics, AIAA J. 29 (6) (1991) 1018–1020. [16] K.S. Kwon, R.M. Lin, Frequency selection method for FRF-based model updating, J. Sound Vib. 278 (1) (2004) 285–306. [17] A.N. Tikhonov, V.Y. Arsenin, Solutions of Ill-posed Problems, Wiley, New York, 1977. [18] P.C. Hansen, Regularization tools version 4.1 (for Matlab version 7.3), Numer. Algorithms 46 (2007) 189–194. [19] A.N. Thite, D.J. Thompson, Selection of response measurement locations to improve inverse force determination, Appl. Acoust. 67 (8) (2006) 797–818. [20] S.Z. Rad, Methods of Updating Numerical Models in Structural Dynamics, University of London, London, 1997. [21] K. Blakely, MSC/NASTRAN Basic Dynamic Analysis User's Guide, Version 68, MacNeal-Schwendler, Los Angeles, 1993.