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Coupled vibro-acoustic model updating using frequency response functions
Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Coupled vibro-acoustic model updating using frequency response functions D.V. Nehete, S.V. Modak n, K. Gupta Department of Mechanical Engineering, Indian Institute of Technology Delhi, Hauz khas, New Delhi 110016, India
a r t i c l e i n f o
abstract
Article history: Received 29 November 2014 Received in revised form 3 July 2015 Accepted 1 September 2015
Interior noise in cavities of motorized vehicles is of increasing significance due to the lightweight design of these structures. Accurate coupled vibro-acoustic FE models of such cavities are required so as to allow a reliable design and analysis. It is, however, experienced that the vibro-acoustic predictions using these models do not often correlate acceptably well with the experimental measurements and hence require model updating. Both the structural and the acoustic parameters addressing the stiffness as well as the damping modeling inaccuracies need to be considered simultaneously in the model updating framework in order to obtain an accurate estimate of these parameters. It is also noted that the acoustic absorption properties are generally frequency dependent. This makes use of modal data based methods for updating vibro-acoustic FE models difficult. In view of this, the present paper proposes a method based on vibro-acoustic frequency response functions that allow updating of a coupled FE model by considering simultaneously the parameters associated with both the structural as well as the acoustic model of the cavity. The effectiveness of the proposed method is demonstrated through numerical studies on a 3D rectangular box cavity with a flexible plate. Updating parameters related to the material property, stiffness of joints between the plate and the rectangular cavity and the properties of absorbing surfaces of the acoustic cavity are considered. The robustness of the method under presence of noise is also studied. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Updating Vibro-acoustics Interior noise Constrained optimization Absorber
1. Introduction Accurate mathematical models of vibro-acoustic systems are needed to predict vibro-acoustic response accurately. Cavities encountered in aerospace, automotive and other transportation equipment are some of the examples of such systems where forces generated during operation excite the structure causing acoustic response inside the cavity. It is often seen that the acoustic response predicted by the coupled FE model does not correlate well with the measured acoustic response. This situation in structural dynamics for the problem of predicting vibration response is addressed through updating of the FE model (Mottershead and Friswell [1]). Model updating is now considered an acceptable methodology to improve the correlation between the FE model and experimental tests data through adjustment of model parameters that are likely to be in error. Model updating has been extensively studied and various methods have been developed. For vibro-acoustic systems also, model updating techniques can be extended with the objective of improving correlation with the vibro-acoustic test data by adjusting the parameters of the coupled FE model. n
Corresponding author. Tel.: þ 91 11 26596336; fax: þ 91 11 26582053. E-mail address: [email protected] (S.V. Modak).
http://dx.doi.org/10.1016/j.ymssp.2015.09.002 0888-3270/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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Decouvreur et al. [2] proposed a method for updating 2D acoustic models using constitutive relation error. Parameters of the acoustic absorption on the interior surfaces of the cavity are updated. The method is later extended to updating of 3D acoustic models with application to a vehicle cabin (Decouvreur et al. [3]). Schedlinski et al. [4] considered updating of damping parameters of the structural model and the acoustic absorption parameters of the acoustic model. The structural damping parameters of the model are updated based on the frequency response functions measured on the structure while the acoustic parameters are updated using the acoustic frequency response measured using a volume velocity excitation source. The procedure was applied on a car cavity. Dhandole and Modak [5] developed a method for updating of acoustic FE models using measured acoustic pressure response. They further report updating of a weakly coupled structural model of a cavity and study its effect on the accuracy of the predicted vibro-acoustic frequency response (Dhandole and Modak [6]). A method for structural FE model updating incorporating two-way vibro-acoustic coupling has been recently developed Nehete et al. [7]. This method extends inverse eigen-sensitivity approach to update undamped structural FE models of cavities by incorporating vibro-acoustic coupling. The method, however, cannot be used for updating of damped models. It is noted that in a damped case, the damping can originate from both the structural as well as the acoustic domains of the cavity. Wan et al. [8] presented a method for acoustic FE model updating by linearizing the relationship between the acoustic FRFs and the acoustic absorption parameters. Modak [9] developed a formulation for direct updating of mass and stiffness matrices of the acoustic and structural domains of a coupled vibro-acoustic model. The method, however, does not take damping of the two domains into account. It is observed that a vibro-acoustic FE model may have modeling inaccuracies in both the acoustic as well as its structural model. While some of the developed methods address updating of only acoustic models, some other either consider updating of only the structural model or do not consider updating of all possible modeling inaccuracies like related to the stiffness and the damping. A couple of other methods consider updating of undamped coupled models and hence cannot be used for updating damping matrices. The present paper proposes a method for updating of coupled vibro-acoustic FE models considering simultaneous updating of all the parameters and therefore allows updating of the mass, stiffness as well as damping matrices of both the structural and the acoustic domains. The proposed method offers a framework where the modeling inaccuracies of both the models can be addressed while incorporating the two-way coupling. A numerical study of a 3D rectangular cavity with a flexible plate is presented to validate the method.
2. Coupled vibro-acoustic FE model updating In this section, a method for updating of coupled vibro-acoustic models based on frequency response functions measured on the cavity is proposed. The updating approach is based on posing the problem as a constrained minimization problem so that it could readily be solved using a standard optimization routine. The coupled updating formulation allows taking into account acoustic loading on the structure in the updating process. The method utilizes frequency response functions rather than the modal data and hence allows identification of the frequency dependent absorption properties as well along with the structural updating parameters. Consider a coupled cavity with volume V. The boundary surface S of the cavity is composed of surface S1 that is acoustically hard, surface S2 that represents the common boundary between the acoustic domain and the structure and surface S3 that is covered with sound absorbing material. The subscripts s and a are used to indicate the structural and acoustic domains respectively. In the following, symbols in bold and upper case represent matrices, those in bold and lower case represent vectors while those in lower case are scalars. Assuming the acoustic pressure fluctuations inside the cavity to be harmonic with frequency ω rad/s, the homogeneous Helmholtz equation, which is the governing differential equation, is given by,
∇2p +
ω2 p=0 c2
(1)
The boundary condition over S1 will be,
∂p =0 ∂n
(2)
The boundary condition over S3 (surface with sound absorbing material having specific acoustic admittance A) will be,
∂p = − ρ⋅jω⋅p. A ∂n
(3)
The admittance A is generally complex and is seen to be frequency dependent. In view of this, two models for the acoustic absorption coefficient are assumed. A frequency independent model given by,
A = A1 + jA2
(4)
and a frequency dependent model given by,
Aω = Aω1 + jωAω2
(5)
Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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are considered. Here A1 and Aω1 are coefficients in the real part of frequency independent and frequency dependent admittance models respectively while A2 and Aω2 are such coefficients in the imaginary parts respectively. In FE formulation of the problem, the acoustic domain of volume V is represented by an assemblage of acoustic finite elements. The pressure distribution within an element is interpolated in terms of the nodal pressures by using shape functions Na . Variational formulation based on the differential Eq. (1) and the boundary conditions (2) and (3) and a subsequent FE discretization gives the acoustic mass, stiffness and damping matrices (Filippi [10]).
Κa =
∑ ∫ ∇NaT ∇Na . dV V
Ma =
∑∫ V
Da =
1 T Na Na ⋅dV c2
∑ ∫ ρ A⋅NaT Na⋅dS S3
(6)
(7)
(8)
Here Ma , Κ a and Da are respectively the mass, stiffness and damping matrices of the acoustic domain each of the order na × na . c and ρ are the speed of sound and density of the medium of the cavity respectively. The summation sign in (6) and (7) represents assembly over the acoustic finite elements of the model, while the summation sign in (8) represents assembly over those finite elements that have a face common with surface S3. At any point on the surface S2, the normal gradient of the acoustic pressure and the normal velocity of the structure ( vn ) must satisfy the following boundary condition,
∂p = − ρ⋅jω⋅vn ∂n
(9)
This equation gives rise to the coupling matrix between the acoustic and the structure domains and is given by,
C as =
∑ ρ NaT Ns⋅dS S2
(10)
Here Ns is the shape function matrix for the structural finite elements. The summation sign in (10) represents assembly for those pairs of acoustic and structural finite elements that have a face common with surface S2. The differential equation of equilibrium for the acoustic domain, in the absence of any acoustic excitation, is written as, .
..
¨ =0 M a p + Da p + K a p + C as u
(11)
where p is a na × 1 vector of the nodal acoustic pressures. The structure of the cavity is modeled here using Kirchoff's thin plate bending finite elements. The equilibrium equation for the structure in the presence of acoustic pressure acting on it can be written as, .
..
M s u s + Ds u s + K su s + C sa p = fs
(12)
where Ms , K s and Ds are respectively the mass, stiffness and damping matrices of the structure each of the order ns × ns . u s and fs are the ns × 1 vectors of the structural FE model degrees of freedom and the structural excitations respectively. C sa is the coupling matrix between the structure and the acoustic domains. The structural finite element discretization gives matrices Ms and K s for the structure as (Petyt [11]),
Ks =
∑∫ S2
Ms =
h3 T B DB⋅dA 12
∑ ∫ ρ0 h NsT Ns⋅dA S2
(13)
(14)
Here h and ρ0 are the thickness and the density of the plate respectively. B and D are the strain–displacement and the material property matrices respectively. Viscous damping matrix Ds can be built based on the experimentally identified modal damping factors of the structure. In case of structural/hysteretic damping also the corresponding damping matrix can be built based on the experimentally identified modal loss factors. The coupling matrix C sa is related to the matrix Cas by the following relationship,
C sa = −
1 T Cas ρ
(15)
Let ns and na represent the number of structural and acoustic degrees of freedom of the coupled FE model respectively. Combining the equilibrium equations for the structural and the acoustic domains, (Eqs. (11) and (12)), gives the following equation of equilibrium for the coupled structural-acoustic cavity, Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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4 .
..
Mu + Ku + Du = f
(16)
Here,
⎡ Ms 0 ⎤ ⎡ K s C sa ⎤ ⎡ Ds 0 ⎤ ⎧f ⎫ ⎧ us ⎫ M=⎢ ⎥, K = ⎢ ⎥, D = ⎢ ⎥, u = ⎨ ⎬ and f = ⎨ s ⎬ ⎩ pa ⎭ ⎩0⎭ ⎣ C as M a ⎦ ⎣ 0 Ka ⎦ ⎣ 0 Da ⎦
(17)
Here, M , K and D are respectively the mass, stiffness and damping matrices each of order n × n, f is n × 1 excitation vector and u is n × 1 vector of the degrees of freedom of the coupled FE model. Here, n = ns + na and represents the degrees of freedom of the coupled FE model. For updating of the coupled FE model, an objective function is framed as sum of the squares of the percentage error in the real and the imaginary parts of the vibro-acoustic frequency response functions evaluated at a chosen number of frequency points and measurement degrees of freedom. Subscripts FE and m are used to represent the FRFs related to the FE model and the measurements, respectively, while the superscripts r and i are used to represent the real and the imaginary parts of the FRFs. The objective function is defined as,
⎡ 2 i r r ⎞2⎤ ⎛ i ⎢ ⎛⎜ αFE − αm × 100⎞⎟ + ⎜ αFE − αm × 100⎟ ⎥ ∑∑⎢ ⎟⎥ ⎜ αmr ⎝ ⎠ αmi ⎠⎦ ⎝ m=1 k=1 ⎣ mk nm
f (u) =
nf
(18)
The variables nf and nm represent the numbers of frequency points and measured frequency response functions respectively. Modeling inaccuracies may exist in the structural as well as the acoustic domains of the coupled FE model. These may be related to the modeling of joints and boundary conditions, material property values, idealization and approximation of the structural details, acoustic absorption properties of the interior surfaces of the cavity and the speed of sound. Geometric details of the two domains used for building the model are assumed to be correct. Therefore, in practice, both the structural as well as the acoustic parameters may need updating to improve correlation of the FE model with the measured FRFs. The objective function given by Eq. (18) is minimized subject to the lower and upper bounds on the vector of updating parameters u . These constraints are given by,
uLB ≤ u ≤ uUB
(19)
Any known data about the maximum degree of variation or uncertainty in the vector of updating parameters can be included through above bounds. This helps to obtain a physically meaningful revision of the updating parameters. The constrained minimization of the objective function given by Eq. (18), subjected to the bounds given by Eq. (19) is carried out through a routine for constrained optimization available in MATLAB, that is based on sequential quadratic programming (SQP). This routine solves the constrained minimization problem in an iterative way to satisfy the specified convergence criterion. A flow chart of the updating method showing the important steps in its implementation is given in Fig. 1.
3. Numerical study This section presents a numerical study to evaluate the effectiveness of the proposed method. An example of a 3D rectangular cavity of size 0.261 m 0.3 m 0.686 m backed by a flexible Aluminum plate of size 0.261 m 0.3 m and thickness 0.001 m is considered. The flexible plate is modeled with 48 four-nodded Kirchhoff's thin plate bending finite elements. The Transverse displacement and the rotations (about x and y axes) are the three degrees of freedom at each of the nodes. The acoustic domain of the cavity is modeled with 480 3D eight-nodded acoustic finite elements with acoustic pressure as a nodal degree of freedom. The cavity structural FE model is coupled to the acoustic model through the coupling matrix. The modulus of elasticity and density for the plate material are taken as 7.1eþ10 N/m2 and 2700 kg/m3 respectively while the speed of sound and the density of the medium of the cavity, are taken as 343 m/sec and 1.21 kg/ m3 respectively. Different surfaces of the cavity are shown in Fig. 2a. The FE mesh of the coupled model is shown in Fig. 2b. The surfaces of the cavity are designated as front, back, left side, right side, top and bottom faces. All surfaces, except the front face which represents the flexible plate, are covered with sound absorbing materials. Boundary condition corresponding to Eq. (3) is applied on these surfaces. These surfaces of the cavity now contribute damping to the acoustic part of the cavity. However, for the purpose of numerical study, it is assumed that the structure of the cavity is undamped. An eigen-value analysis of the coupled vibro-acoustic FE model is carried out, to get an idea of the coupled natural frequencies and the mode shapes. The coupled natural frequencies are shown in Table 1. The table also shows the rigid-wall acoustic as well as the in-vacuo structural natural frequencies. The coupled natural frequencies dominated by either the acoustic or the structural resonances are indicated with letters ‘A’ and ‘S’ respectively. The ‘measured’ FRFs of the cavity are simulated by introducing certain known discrepancies in the coupled vibroacoustic FE model. The discrepancies introduced are: (a) the modulus of elasticity of plate is reduced by 9% to simulate material property error in the model (b) boundary condition error is introduced at the four edges of the plate and (c) the real Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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Coupled Vibro-acoustic Cavity Construct Acoustic FE Model
Construct Structural FE Model
Construct Coupling Matrix
Build Coupled FE Model Updated Acoustic Model parameters
Updated Structural Model parameters
Definition of Excitation
Compute FRFs
Measured FRFs
Compute Objective Function If Not Satisfied
Check for Convergence
Minimization of Objective Function with bounds If Satisfied
Updated Coupled FE Model Fig. 1. Flow chart of the updating method.
Back
Top Face
y
Left Side Face z Botto mFace
x
Face
Right Side Face
17
166 Acoustic Cavity
Structural Mesh
Acoustic Mesh
Front Face (Flexible Plate) Fig. 2. 3D cavity for the numerical study (a) cavity with flexible plate (b) FE mesh of the coupled cavity.
and the imaginary parts of the admittance coefficients of the absorbers on the five cavity surfaces are perturbed. To simulate error (b), the rotational degrees of freedom for the boundary nodes are modeled by torsional springs with stiffnesses kthx1, kthx2, ktvy1 and ktvy2 as shown in Fig. 3 and these stiffnesses are reduced by 80%, 85%, 90% and 95% respectively to simulate the boundary condition error. Discrepancies between the simulated and the structural FE model of the cavity are shown in Table 2 that gives the details of the initial value and the fractional error in the parameters of the structural FE model of the cavity. These errors are implemented in the simulated model by following equation,
pm = (1 + u) pFE
(20)
Here pm is the parameter value in the simulated model, pFE is the parameter value in the FE model and u is the fractional correction factor. The discrepancies in the real and the imaginary parts of the admittance coefficients of the simulated and the acoustic FE model of the cavity are detailed in Table 3. These are used for both the frequency dependent and the independent cases. A comparison of an FRF with frequency independent and frequency dependent absorbing surfaces is shown in Fig. 4. It is seen Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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Table 1 Comparison of coupled, rigid-wall acoustic and in-vacuo structural natural frequencies. Mode number
Rigid wall acoustic natural freq. (Hz)
In-vacuo structural Coupled vibronatural freq. (Hz) acoustic natural freq. (Hz)
Mode number
Rigid wall acoustic natural freq. (Hz)
In-vacuo structural Coupled vibronatural freq. (Hz) acoustic natural freq.(Hz)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.0 – – – 251.0 – – – – 508.2 – 575.3 – 627.7 – – 664.6 710.4 – 767.6 777.9 – – 836.6
– 114.2 210.7 251.4 – 330.4 367.1 464.3 467.2 – 528.4 – 580.5 – 643.6 661.8 – – 746.9 – – 798.3 814.3 –
25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
– 879.05 – – 914.19 967.00 1015.41 1023.20 – – – 1066.31 – – 1172.89 1173.89 – 1199.46 – 1211.63 – 1256.48 1278.28 –
848.2 – 892.3 912.9 – – – – 1039.6 1041.3 1058.5 – 1098.0 1160.9 – – 1177.7 – 1205.2 – 1244.5 – – 1301.3
0.00 114.5 (S) 208.1 (S) 248.7 (S) 254.6 (A) 327.9 (S) 365.4 (S) 460.5 (S) 464.5 (S) 512.9 (A) 524.7 (S) 569.0 (A) 582.9 (S) 631.8 (A) 641.2 (S) 658.9 (S) 664.7 (A) 708.8 (A) 746.6 (S) 769.3 (A) 779.4 (A) 794.2 (S) 812.3 (S) 838.6 (A)
846.0 (S) 880.0 (A) 890.0 (S) 910.6 (S) 915.8 (A) 968.8 (A) 1016.0 (A) 1025.1 (A) 1037.1 (S) 1039.8 (S) 1053.3 (S) 1070.7 (A) 1095.5 (S) 1157.5 (S) 1171.7 (A) 1173.6 (A) 1176.5 (S) 1201.1 (A) 1204.4 (S) 1214.2 (A) 1241.9 (S) 1258.2 (A) 1279.5 (A) 1299.3 (S)
K thx2 K tvy2
K tvy1
Y x
K thx1
Fig. 3. Torsional spring stiffness updating parameters for the rotational degrees of freedom.
Table 2 Discrepancies between the simulated and the structural FE model of the cavity. Sr. no.
Structural updating parameter
Initial value
Value in the simulated model
Fractional correction factor
1 2 3 4 5
Modulus of elasticity (N/m2) Kthx1(N m/rad) Ktvy1 (N m/rad) Kthx2 (N m/rad) Ktvy2 (N m/rad)
7.1e10 1.1e4 0.9e4 1.1e4 0.9e4
6.5e10 2.2e3 1.3e3 1.1e3 0.4e3
0.09 0.80 0.85 0.90 0.95
that as the frequency increases the difference in the influence of the two models of the absorbing surfaces on the vibroacoustic response also increases. Acoustic domain controlled resonant peaks are seen not only to have more damping but also shift due to the complex admittances. Updating is carried out using a constrained optimization algorithm implemented in MATLAB minimizing an objective function as described in Section 2. A target value of the objective function equal to 1e 05 is used as the convergence criterion. The modulus of elasticity of the plate, stiffness of springs at the plate boundaries and the acoustic properties of all Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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Table 3 Discrepancies in the admittance coefficients for the five absorbing surfaces between the simulated and the acoustic FE model for the cavity. Sr. No. Absorbing surface on the cavity
Real part of admittance coefficient in FE model
Real part of admittance coefficient in simulated model
Fractional correction factor for real part of admittance coefficient
Imaginary part of admittance coefficient in FE model
Imaginary part of admittance coefficient in simulated model
Fractional correction factor for imaginary part of admittance coefficient
1 2 3 4 5
1e 9 1e 8 1e 9 1e 8 1e 9
10.0e 9 8.5e 8 9.0e 9 6.5e 8 7.0e 9
9.2 7.5 8.0 5.5 6.0
1e 12 1e 11 1e 12 1e 11 1e 12
1.5e 12 1.2e 11 1.6e 12 1.4e 11 1.7e 12
0.5 0.2 0.6 0.4 0.7
Back Side Right Side Left Side Top Side Bottom Side
Frequency independent(Blue--)& Frequency dependent(Red-)
FRF(pa/N)magnitude in db(re:1pa/N)
60 40 20 0 -20 -40 -60
0
200
400 600 800 Frequency in Hz
1000
1200
Fig. 4. Comparison of FRFs with frequency independent (blue–) and frequency dependent (red-) absorbing surfaces. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 4 Updating parameters and their correct value. Parameter no.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Updating Parameter Correct value
E
Kthx1
Ktvy1
Kthx2
Ktvy2
R1
R2
R3
R4
R5
I1
I2
I3
I4
I5
0.09
0.80
0.85
0.90
0.95
9.2
7.5
8.0
5.5
6.0
0.5
0.2
0.6
0.4
0.7
the absorbing surfaces of the cavity are updated. The modulus of elasticity of the plate (E), stiffness of springs at the plate boundaries (represented by kthx1, ktvy1, kthx2 and ktvy2) and the real parts (represented by R1, R2, R3, R4, R5) and imaginary parts (represented by I1, I2, I3, I4, I5) of the admittance coefficients of the absorbers on the five cavity surfaces of the cavity are updated. This makes a total of fifteen updating parameters as shown in Table 4. Frequency range from 0 to 1300 Hz is treated as the ‘measured’ frequency range and fourteen frequency points are used for updating in all the runs. These frequencies are: 248, 266, 497, 528, 708, 724, 847, 855, 1058, 1092, 1192, 1207, 1246 and 1297 Hz. The structural-acoustic FRFs are assumed to be ‘measured’ at twenty one nodes (72, 118, 139, 177, 207, 225, 269, 292, 345, 349, 410, 458, 461, 486, 489, 524, 548, 578, 597, 601 and 620) of the acoustic cavity with excitation of transverse degree of freedom of the structure at nodes 9, 20, 31, 41 and 52 making a total of 105 simulated FRFs. Fig. 5 shows identified values of the updating parameters with the frequency independent and dependent absorbing surfaces respectively. The updating parameter values are seen to be exact when compared with the values in the last column of Table 2 (giving correct values of the structural parameters) and with the values in the columns 5 and 8 of Table 3 (giving correct values of the acoustic parameters). Fig. 6 shows variation of objective function value over the iterations for the case of frequency dependent absorbing surfaces indicating a continuous reduction of the objective function value as the iterations progress. The experimental data generally contains some measurement noise. To investigate the effectiveness of the algorithm in the presence of noise, uniformly distributed random noise of 1%, 3% and 5% is added to the ‘measured’ coupled FRFs. For 3% noise, Fig. 7 shows plots of the identified correction factors and % error in them for the case of cavity with the frequency dependent absorbing surfaces. It is seen that some of the parameters are not identified exactly. Table 5 gives a comparison of Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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10
Fractional Correction Factors
Frequency Dependent Value Frequency Independent Value
8 6 4 2 0 -2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Parameter number. Fig. 5. Identified correction factors for updating with frequency independent and frequency dependent absorbing surfaces.
4
Objective function value
3.5 3 2.5 2 1.5 1 0.5 0
0
20
40 60 80 Iteration number
100
120
Fig. 6. Plot of variation of objective function value over the iterations.
%Error in Identified Correction Factors
Fractional Correction Factors
10 8 6 4 2 0 -2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Parameter number
100
50
0
-50
-100
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Parameter number
Fig. 7. Updating with frequency dependent absorbing surfaces with 3% noise: (a) Identified correction factors and (b) % error in the identified factors.
the identified values of the correction factors and % error in them for updating with 1%, 3% and 5% noise levels. It also gives maximum and average % error for each case. The results show that the errors are increasing in proportion to the level of the noise. However, the average % errors are seen to be far less than the maximum % error (shown in bold), indicating that the % Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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Table 5 Comparison of the identified values of the correction factors and % error in them for updating with 1%, 3% and 5% noise levels. Updating parameter no.
Correct value of updating parameter
0.09 0.80 0.85 0.90 0.95 9.2 7.5 8.0 5.5 6.0 0.5 0.2 0.6 0.4 0.7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Maximum % error Average % error
Case of 1% noise
Case of 3% noise
Identified value
% Error
Identified value
% Error
Identified value
% Error
0.08 0.87 0.81 0.85 0.93 9.12 7.48 7.80 5.28 5.94 0.45 0.17 0.57 0.36 0.65 – –
0.11 9.98 4.00 4.55 1.15 0.86 0.26 2.50 4.00 1.00 9.00 15.00 3.66 7.75 5.85 15 4.65
0.08 0.92 0.98 0.97 0.97 9.35 7.46 8.28 5.42 7.74 0.32 0.20 0.41 0.40 0.88 – –
8.33 15.12 15.41 7.77 2.31 1.63 0.53 3.50 1.45 29.00 36.00 3.00 31.33 1.50 25.71 36 12.17
0.02 0.82 0.78 0.87 0.92 7.89 6.46 7.28 4.42 6.74 0.04 0.10 0.51 0.50 0.91 – –
77.77 2.62 8.11 3.33 3.15 14.23 13.86 9.00 19.63 12.33 90.80 47.00 14.66 26.50 30.00 90 24.87
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Fig. 8. Comparison of overlays of the simulated and the FE model coupled vibro-acoustic FRFs for frequency dependent absorbing surfaces for the case of noisy data: (a) before updating (b) after updating with 1% noise (c) after updating with 3% noise and (d) after updating with 5% noise.
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Fig. 9. Response and excitation nodes in acoustic and structural domains.
error in only some of the updating parameters is on a higher side. The maximum % error in the three cases is seen to be in the imaginary part of the admittance of the absorbing surfaces. It seems that since vibro-acoustic FRFs are less sensitive to the imaginary parts as compared to the real parts, the corresponding parameters have larger error due to noise. Fig. 8 shows a comparison of the overlays of the vibro-acoustic FRF ( α166 ,17) for updating with the three noise levels. Vibro-acoustic FRF α166 ,17 represents response at node 166 in the acoustic cavity and excitation at node 17 on the structure in the transverse direction. It is seen that the updated FE model FRF matches very well with the ‘measured’ FRF and thus the FRF correlation has significantly improved after updating even though the parameters could not be identified exactly as a result of measurement noise. Fig. 9 shows response and the excitation nodes in the acoustic and structural domains where a further assessment of the FRFs after updating is done. Vibro-acoustic FRFs at six other nodes in the acoustic cavity after updating for the case of 3% simulated noise are shown in Fig. 10. Similarly, Fig. 11 shows FRFs after updating with 3% noise in the transverse direction at two nodes on the structure. FRFs in Figs. 10 and 11 are with respect to excitation of a certain node on the structure in the transverse direction. From these figures it is seen that the updating has resulted into a global improvement in the FRF correlation throughout the cavity both in the acoustic and the structural domain.
4. Conclusion A method for updating of coupled vibro-acoustic FE models is proposed in this paper. The method performs updating by minimization of an objective function defined as the error between the measured and the coupled vibro-acoustic FE model frequency response functions subjected to bounds on the updating parameters. The framework of the method allows parameters related to the stiffness, mass or damping distribution of the structural as well as the acoustic domains to be updated taking into account the two-way coupling between these two domains. The objective function is based upon frequency response functions and hence allows even frequency dependent acoustic absorption characteristics to be identified along with other parameters. Results of a numerical study on a rectangular cavity backed by a flexible plate are presented to simultaneously update the stiffness and the material property parameters of the structural domain and the acoustic parameters of the acoustic domain. Effect of simulated noise in the data is studied. Based on the results of the numerical study the proposed method is found to work satisfactorily to estimate the updating parameters and yield an updated vibro-acoustic model whose vibro-acoustic frequency response correlates well with the simulated response. A further study is planned to validate the proposed updating method using experimental data on a vibro-acoustic cavity. Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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Fig. 10. Plot of vibro-acoustic FRFs after updating with 3% noise at six other nodes in the acoustic cavity.
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References [1] J.E. Mottershead, M.I. Friswell, Model updating in structural dynamics: a survey, J. Sound Vib. 167 (1993) 347–375. [2] V. Decouvreur, Ph Bouillard, A. Deraemaeker, P. Ladeveze, Updating 2D acoustic models with the constitutive relation error method, J. Sound Vib. 278 (2004) 773–787. [3] V. Decouvreur, P. Ladeveze, Ph Bouillard, Updating 3D acoustic model with the constitutive relation error method: A two-stage approach for absorbing material characterization, J. Sound Vib. 310 (2008) 985–999. [4] C. Schedlinski, F. Wagner, K. Bohnert, M. Kuesel, D. Clasen, C. Stein, C. Glandier, M. Kaufmann, E. Tijs, Computational model updating of structural damping and acoustic absorption for coupled fluid-structure-analyses of passenger cars, in: Proceedings of International Conference on Noise and Vibration Engineering. ISMA, Leuven, Belgium, 2008. [5] S.D. Dhandole, S.V. Modak, A constrained optimization based method for acoustic finite element model updating of cavities using pressure response, Appl. Math. Model 36 (1) (2011) 399–413. [6] S. Dhandole, S.V. Modak, On improving weakly coupled cavity models for vibro-acoustic predictions and design, Appl. Acoust. 71 (2010) 876–884. [7] D.V. Nehete., S.V. Modak, K. Gupta, Structural FE model updating of cavity systems incorporating vibro-acoustic coupling, Mech. Syst. Sig. Process. 50–51 (2015) 362–379. [8] Z.M. Wan, T. Wang, Q.B. Huang, J. Wang, Acoustic finite element model updating using acoustic frequency response function, Finite Elem. Anal. Des. 87 (2014) 1–9. [9] S.V. Modak, Direct matrix updating of vibro-acoustic finite element models using modal test data, AIAA J. 52 (7) (2014) 1386–1392. [10] P. Filippi, Theoretical Acoustics and Numerical Techniques CISM Courses and Lectures, 227, Springer-Verlag, Wien, New York, 1983, 135–216. [11] M. Petyt, Introductions to Finite Element Vibration Analysis, Cambridge University Press, United Kingdom, 2003, 229–238.
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Coupled vibro-acoustic model updating using frequency response functions D.V. Nehete, S.V. Modak n, K. Gupta Department of Mechanical Engineering, Indian Institute of Technology Delhi, Hauz khas, New Delhi 110016, India
a r t i c l e i n f o
abstract
Article history: Received 29 November 2014 Received in revised form 3 July 2015 Accepted 1 September 2015
Interior noise in cavities of motorized vehicles is of increasing significance due to the lightweight design of these structures. Accurate coupled vibro-acoustic FE models of such cavities are required so as to allow a reliable design and analysis. It is, however, experienced that the vibro-acoustic predictions using these models do not often correlate acceptably well with the experimental measurements and hence require model updating. Both the structural and the acoustic parameters addressing the stiffness as well as the damping modeling inaccuracies need to be considered simultaneously in the model updating framework in order to obtain an accurate estimate of these parameters. It is also noted that the acoustic absorption properties are generally frequency dependent. This makes use of modal data based methods for updating vibro-acoustic FE models difficult. In view of this, the present paper proposes a method based on vibro-acoustic frequency response functions that allow updating of a coupled FE model by considering simultaneously the parameters associated with both the structural as well as the acoustic model of the cavity. The effectiveness of the proposed method is demonstrated through numerical studies on a 3D rectangular box cavity with a flexible plate. Updating parameters related to the material property, stiffness of joints between the plate and the rectangular cavity and the properties of absorbing surfaces of the acoustic cavity are considered. The robustness of the method under presence of noise is also studied. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Updating Vibro-acoustics Interior noise Constrained optimization Absorber
1. Introduction Accurate mathematical models of vibro-acoustic systems are needed to predict vibro-acoustic response accurately. Cavities encountered in aerospace, automotive and other transportation equipment are some of the examples of such systems where forces generated during operation excite the structure causing acoustic response inside the cavity. It is often seen that the acoustic response predicted by the coupled FE model does not correlate well with the measured acoustic response. This situation in structural dynamics for the problem of predicting vibration response is addressed through updating of the FE model (Mottershead and Friswell [1]). Model updating is now considered an acceptable methodology to improve the correlation between the FE model and experimental tests data through adjustment of model parameters that are likely to be in error. Model updating has been extensively studied and various methods have been developed. For vibro-acoustic systems also, model updating techniques can be extended with the objective of improving correlation with the vibro-acoustic test data by adjusting the parameters of the coupled FE model. n
Corresponding author. Tel.: þ 91 11 26596336; fax: þ 91 11 26582053. E-mail address: [email protected] (S.V. Modak).
http://dx.doi.org/10.1016/j.ymssp.2015.09.002 0888-3270/& 2015 Elsevier Ltd. All rights reserved.
Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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Decouvreur et al. [2] proposed a method for updating 2D acoustic models using constitutive relation error. Parameters of the acoustic absorption on the interior surfaces of the cavity are updated. The method is later extended to updating of 3D acoustic models with application to a vehicle cabin (Decouvreur et al. [3]). Schedlinski et al. [4] considered updating of damping parameters of the structural model and the acoustic absorption parameters of the acoustic model. The structural damping parameters of the model are updated based on the frequency response functions measured on the structure while the acoustic parameters are updated using the acoustic frequency response measured using a volume velocity excitation source. The procedure was applied on a car cavity. Dhandole and Modak [5] developed a method for updating of acoustic FE models using measured acoustic pressure response. They further report updating of a weakly coupled structural model of a cavity and study its effect on the accuracy of the predicted vibro-acoustic frequency response (Dhandole and Modak [6]). A method for structural FE model updating incorporating two-way vibro-acoustic coupling has been recently developed Nehete et al. [7]. This method extends inverse eigen-sensitivity approach to update undamped structural FE models of cavities by incorporating vibro-acoustic coupling. The method, however, cannot be used for updating of damped models. It is noted that in a damped case, the damping can originate from both the structural as well as the acoustic domains of the cavity. Wan et al. [8] presented a method for acoustic FE model updating by linearizing the relationship between the acoustic FRFs and the acoustic absorption parameters. Modak [9] developed a formulation for direct updating of mass and stiffness matrices of the acoustic and structural domains of a coupled vibro-acoustic model. The method, however, does not take damping of the two domains into account. It is observed that a vibro-acoustic FE model may have modeling inaccuracies in both the acoustic as well as its structural model. While some of the developed methods address updating of only acoustic models, some other either consider updating of only the structural model or do not consider updating of all possible modeling inaccuracies like related to the stiffness and the damping. A couple of other methods consider updating of undamped coupled models and hence cannot be used for updating damping matrices. The present paper proposes a method for updating of coupled vibro-acoustic FE models considering simultaneous updating of all the parameters and therefore allows updating of the mass, stiffness as well as damping matrices of both the structural and the acoustic domains. The proposed method offers a framework where the modeling inaccuracies of both the models can be addressed while incorporating the two-way coupling. A numerical study of a 3D rectangular cavity with a flexible plate is presented to validate the method.
2. Coupled vibro-acoustic FE model updating In this section, a method for updating of coupled vibro-acoustic models based on frequency response functions measured on the cavity is proposed. The updating approach is based on posing the problem as a constrained minimization problem so that it could readily be solved using a standard optimization routine. The coupled updating formulation allows taking into account acoustic loading on the structure in the updating process. The method utilizes frequency response functions rather than the modal data and hence allows identification of the frequency dependent absorption properties as well along with the structural updating parameters. Consider a coupled cavity with volume V. The boundary surface S of the cavity is composed of surface S1 that is acoustically hard, surface S2 that represents the common boundary between the acoustic domain and the structure and surface S3 that is covered with sound absorbing material. The subscripts s and a are used to indicate the structural and acoustic domains respectively. In the following, symbols in bold and upper case represent matrices, those in bold and lower case represent vectors while those in lower case are scalars. Assuming the acoustic pressure fluctuations inside the cavity to be harmonic with frequency ω rad/s, the homogeneous Helmholtz equation, which is the governing differential equation, is given by,
∇2p +
ω2 p=0 c2
(1)
The boundary condition over S1 will be,
∂p =0 ∂n
(2)
The boundary condition over S3 (surface with sound absorbing material having specific acoustic admittance A) will be,
∂p = − ρ⋅jω⋅p. A ∂n
(3)
The admittance A is generally complex and is seen to be frequency dependent. In view of this, two models for the acoustic absorption coefficient are assumed. A frequency independent model given by,
A = A1 + jA2
(4)
and a frequency dependent model given by,
Aω = Aω1 + jωAω2
(5)
Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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are considered. Here A1 and Aω1 are coefficients in the real part of frequency independent and frequency dependent admittance models respectively while A2 and Aω2 are such coefficients in the imaginary parts respectively. In FE formulation of the problem, the acoustic domain of volume V is represented by an assemblage of acoustic finite elements. The pressure distribution within an element is interpolated in terms of the nodal pressures by using shape functions Na . Variational formulation based on the differential Eq. (1) and the boundary conditions (2) and (3) and a subsequent FE discretization gives the acoustic mass, stiffness and damping matrices (Filippi [10]).
Κa =
∑ ∫ ∇NaT ∇Na . dV V
Ma =
∑∫ V
Da =
1 T Na Na ⋅dV c2
∑ ∫ ρ A⋅NaT Na⋅dS S3
(6)
(7)
(8)
Here Ma , Κ a and Da are respectively the mass, stiffness and damping matrices of the acoustic domain each of the order na × na . c and ρ are the speed of sound and density of the medium of the cavity respectively. The summation sign in (6) and (7) represents assembly over the acoustic finite elements of the model, while the summation sign in (8) represents assembly over those finite elements that have a face common with surface S3. At any point on the surface S2, the normal gradient of the acoustic pressure and the normal velocity of the structure ( vn ) must satisfy the following boundary condition,
∂p = − ρ⋅jω⋅vn ∂n
(9)
This equation gives rise to the coupling matrix between the acoustic and the structure domains and is given by,
C as =
∑ ρ NaT Ns⋅dS S2
(10)
Here Ns is the shape function matrix for the structural finite elements. The summation sign in (10) represents assembly for those pairs of acoustic and structural finite elements that have a face common with surface S2. The differential equation of equilibrium for the acoustic domain, in the absence of any acoustic excitation, is written as, .
..
¨ =0 M a p + Da p + K a p + C as u
(11)
where p is a na × 1 vector of the nodal acoustic pressures. The structure of the cavity is modeled here using Kirchoff's thin plate bending finite elements. The equilibrium equation for the structure in the presence of acoustic pressure acting on it can be written as, .
..
M s u s + Ds u s + K su s + C sa p = fs
(12)
where Ms , K s and Ds are respectively the mass, stiffness and damping matrices of the structure each of the order ns × ns . u s and fs are the ns × 1 vectors of the structural FE model degrees of freedom and the structural excitations respectively. C sa is the coupling matrix between the structure and the acoustic domains. The structural finite element discretization gives matrices Ms and K s for the structure as (Petyt [11]),
Ks =
∑∫ S2
Ms =
h3 T B DB⋅dA 12
∑ ∫ ρ0 h NsT Ns⋅dA S2
(13)
(14)
Here h and ρ0 are the thickness and the density of the plate respectively. B and D are the strain–displacement and the material property matrices respectively. Viscous damping matrix Ds can be built based on the experimentally identified modal damping factors of the structure. In case of structural/hysteretic damping also the corresponding damping matrix can be built based on the experimentally identified modal loss factors. The coupling matrix C sa is related to the matrix Cas by the following relationship,
C sa = −
1 T Cas ρ
(15)
Let ns and na represent the number of structural and acoustic degrees of freedom of the coupled FE model respectively. Combining the equilibrium equations for the structural and the acoustic domains, (Eqs. (11) and (12)), gives the following equation of equilibrium for the coupled structural-acoustic cavity, Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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4 .
..
Mu + Ku + Du = f
(16)
Here,
⎡ Ms 0 ⎤ ⎡ K s C sa ⎤ ⎡ Ds 0 ⎤ ⎧f ⎫ ⎧ us ⎫ M=⎢ ⎥, K = ⎢ ⎥, D = ⎢ ⎥, u = ⎨ ⎬ and f = ⎨ s ⎬ ⎩ pa ⎭ ⎩0⎭ ⎣ C as M a ⎦ ⎣ 0 Ka ⎦ ⎣ 0 Da ⎦
(17)
Here, M , K and D are respectively the mass, stiffness and damping matrices each of order n × n, f is n × 1 excitation vector and u is n × 1 vector of the degrees of freedom of the coupled FE model. Here, n = ns + na and represents the degrees of freedom of the coupled FE model. For updating of the coupled FE model, an objective function is framed as sum of the squares of the percentage error in the real and the imaginary parts of the vibro-acoustic frequency response functions evaluated at a chosen number of frequency points and measurement degrees of freedom. Subscripts FE and m are used to represent the FRFs related to the FE model and the measurements, respectively, while the superscripts r and i are used to represent the real and the imaginary parts of the FRFs. The objective function is defined as,
⎡ 2 i r r ⎞2⎤ ⎛ i ⎢ ⎛⎜ αFE − αm × 100⎞⎟ + ⎜ αFE − αm × 100⎟ ⎥ ∑∑⎢ ⎟⎥ ⎜ αmr ⎝ ⎠ αmi ⎠⎦ ⎝ m=1 k=1 ⎣ mk nm
f (u) =
nf
(18)
The variables nf and nm represent the numbers of frequency points and measured frequency response functions respectively. Modeling inaccuracies may exist in the structural as well as the acoustic domains of the coupled FE model. These may be related to the modeling of joints and boundary conditions, material property values, idealization and approximation of the structural details, acoustic absorption properties of the interior surfaces of the cavity and the speed of sound. Geometric details of the two domains used for building the model are assumed to be correct. Therefore, in practice, both the structural as well as the acoustic parameters may need updating to improve correlation of the FE model with the measured FRFs. The objective function given by Eq. (18) is minimized subject to the lower and upper bounds on the vector of updating parameters u . These constraints are given by,
uLB ≤ u ≤ uUB
(19)
Any known data about the maximum degree of variation or uncertainty in the vector of updating parameters can be included through above bounds. This helps to obtain a physically meaningful revision of the updating parameters. The constrained minimization of the objective function given by Eq. (18), subjected to the bounds given by Eq. (19) is carried out through a routine for constrained optimization available in MATLAB, that is based on sequential quadratic programming (SQP). This routine solves the constrained minimization problem in an iterative way to satisfy the specified convergence criterion. A flow chart of the updating method showing the important steps in its implementation is given in Fig. 1.
3. Numerical study This section presents a numerical study to evaluate the effectiveness of the proposed method. An example of a 3D rectangular cavity of size 0.261 m 0.3 m 0.686 m backed by a flexible Aluminum plate of size 0.261 m 0.3 m and thickness 0.001 m is considered. The flexible plate is modeled with 48 four-nodded Kirchhoff's thin plate bending finite elements. The Transverse displacement and the rotations (about x and y axes) are the three degrees of freedom at each of the nodes. The acoustic domain of the cavity is modeled with 480 3D eight-nodded acoustic finite elements with acoustic pressure as a nodal degree of freedom. The cavity structural FE model is coupled to the acoustic model through the coupling matrix. The modulus of elasticity and density for the plate material are taken as 7.1eþ10 N/m2 and 2700 kg/m3 respectively while the speed of sound and the density of the medium of the cavity, are taken as 343 m/sec and 1.21 kg/ m3 respectively. Different surfaces of the cavity are shown in Fig. 2a. The FE mesh of the coupled model is shown in Fig. 2b. The surfaces of the cavity are designated as front, back, left side, right side, top and bottom faces. All surfaces, except the front face which represents the flexible plate, are covered with sound absorbing materials. Boundary condition corresponding to Eq. (3) is applied on these surfaces. These surfaces of the cavity now contribute damping to the acoustic part of the cavity. However, for the purpose of numerical study, it is assumed that the structure of the cavity is undamped. An eigen-value analysis of the coupled vibro-acoustic FE model is carried out, to get an idea of the coupled natural frequencies and the mode shapes. The coupled natural frequencies are shown in Table 1. The table also shows the rigid-wall acoustic as well as the in-vacuo structural natural frequencies. The coupled natural frequencies dominated by either the acoustic or the structural resonances are indicated with letters ‘A’ and ‘S’ respectively. The ‘measured’ FRFs of the cavity are simulated by introducing certain known discrepancies in the coupled vibroacoustic FE model. The discrepancies introduced are: (a) the modulus of elasticity of plate is reduced by 9% to simulate material property error in the model (b) boundary condition error is introduced at the four edges of the plate and (c) the real Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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Coupled Vibro-acoustic Cavity Construct Acoustic FE Model
Construct Structural FE Model
Construct Coupling Matrix
Build Coupled FE Model Updated Acoustic Model parameters
Updated Structural Model parameters
Definition of Excitation
Compute FRFs
Measured FRFs
Compute Objective Function If Not Satisfied
Check for Convergence
Minimization of Objective Function with bounds If Satisfied
Updated Coupled FE Model Fig. 1. Flow chart of the updating method.
Back
Top Face
y
Left Side Face z Botto mFace
x
Face
Right Side Face
17
166 Acoustic Cavity
Structural Mesh
Acoustic Mesh
Front Face (Flexible Plate) Fig. 2. 3D cavity for the numerical study (a) cavity with flexible plate (b) FE mesh of the coupled cavity.
and the imaginary parts of the admittance coefficients of the absorbers on the five cavity surfaces are perturbed. To simulate error (b), the rotational degrees of freedom for the boundary nodes are modeled by torsional springs with stiffnesses kthx1, kthx2, ktvy1 and ktvy2 as shown in Fig. 3 and these stiffnesses are reduced by 80%, 85%, 90% and 95% respectively to simulate the boundary condition error. Discrepancies between the simulated and the structural FE model of the cavity are shown in Table 2 that gives the details of the initial value and the fractional error in the parameters of the structural FE model of the cavity. These errors are implemented in the simulated model by following equation,
pm = (1 + u) pFE
(20)
Here pm is the parameter value in the simulated model, pFE is the parameter value in the FE model and u is the fractional correction factor. The discrepancies in the real and the imaginary parts of the admittance coefficients of the simulated and the acoustic FE model of the cavity are detailed in Table 3. These are used for both the frequency dependent and the independent cases. A comparison of an FRF with frequency independent and frequency dependent absorbing surfaces is shown in Fig. 4. It is seen Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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Table 1 Comparison of coupled, rigid-wall acoustic and in-vacuo structural natural frequencies. Mode number
Rigid wall acoustic natural freq. (Hz)
In-vacuo structural Coupled vibronatural freq. (Hz) acoustic natural freq. (Hz)
Mode number
Rigid wall acoustic natural freq. (Hz)
In-vacuo structural Coupled vibronatural freq. (Hz) acoustic natural freq.(Hz)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0.0 – – – 251.0 – – – – 508.2 – 575.3 – 627.7 – – 664.6 710.4 – 767.6 777.9 – – 836.6
– 114.2 210.7 251.4 – 330.4 367.1 464.3 467.2 – 528.4 – 580.5 – 643.6 661.8 – – 746.9 – – 798.3 814.3 –
25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48
– 879.05 – – 914.19 967.00 1015.41 1023.20 – – – 1066.31 – – 1172.89 1173.89 – 1199.46 – 1211.63 – 1256.48 1278.28 –
848.2 – 892.3 912.9 – – – – 1039.6 1041.3 1058.5 – 1098.0 1160.9 – – 1177.7 – 1205.2 – 1244.5 – – 1301.3
0.00 114.5 (S) 208.1 (S) 248.7 (S) 254.6 (A) 327.9 (S) 365.4 (S) 460.5 (S) 464.5 (S) 512.9 (A) 524.7 (S) 569.0 (A) 582.9 (S) 631.8 (A) 641.2 (S) 658.9 (S) 664.7 (A) 708.8 (A) 746.6 (S) 769.3 (A) 779.4 (A) 794.2 (S) 812.3 (S) 838.6 (A)
846.0 (S) 880.0 (A) 890.0 (S) 910.6 (S) 915.8 (A) 968.8 (A) 1016.0 (A) 1025.1 (A) 1037.1 (S) 1039.8 (S) 1053.3 (S) 1070.7 (A) 1095.5 (S) 1157.5 (S) 1171.7 (A) 1173.6 (A) 1176.5 (S) 1201.1 (A) 1204.4 (S) 1214.2 (A) 1241.9 (S) 1258.2 (A) 1279.5 (A) 1299.3 (S)
K thx2 K tvy2
K tvy1
Y x
K thx1
Fig. 3. Torsional spring stiffness updating parameters for the rotational degrees of freedom.
Table 2 Discrepancies between the simulated and the structural FE model of the cavity. Sr. no.
Structural updating parameter
Initial value
Value in the simulated model
Fractional correction factor
1 2 3 4 5
Modulus of elasticity (N/m2) Kthx1(N m/rad) Ktvy1 (N m/rad) Kthx2 (N m/rad) Ktvy2 (N m/rad)
7.1e10 1.1e4 0.9e4 1.1e4 0.9e4
6.5e10 2.2e3 1.3e3 1.1e3 0.4e3
0.09 0.80 0.85 0.90 0.95
that as the frequency increases the difference in the influence of the two models of the absorbing surfaces on the vibroacoustic response also increases. Acoustic domain controlled resonant peaks are seen not only to have more damping but also shift due to the complex admittances. Updating is carried out using a constrained optimization algorithm implemented in MATLAB minimizing an objective function as described in Section 2. A target value of the objective function equal to 1e 05 is used as the convergence criterion. The modulus of elasticity of the plate, stiffness of springs at the plate boundaries and the acoustic properties of all Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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Table 3 Discrepancies in the admittance coefficients for the five absorbing surfaces between the simulated and the acoustic FE model for the cavity. Sr. No. Absorbing surface on the cavity
Real part of admittance coefficient in FE model
Real part of admittance coefficient in simulated model
Fractional correction factor for real part of admittance coefficient
Imaginary part of admittance coefficient in FE model
Imaginary part of admittance coefficient in simulated model
Fractional correction factor for imaginary part of admittance coefficient
1 2 3 4 5
1e 9 1e 8 1e 9 1e 8 1e 9
10.0e 9 8.5e 8 9.0e 9 6.5e 8 7.0e 9
9.2 7.5 8.0 5.5 6.0
1e 12 1e 11 1e 12 1e 11 1e 12
1.5e 12 1.2e 11 1.6e 12 1.4e 11 1.7e 12
0.5 0.2 0.6 0.4 0.7
Back Side Right Side Left Side Top Side Bottom Side
Frequency independent(Blue--)& Frequency dependent(Red-)
FRF(pa/N)magnitude in db(re:1pa/N)
60 40 20 0 -20 -40 -60
0
200
400 600 800 Frequency in Hz
1000
1200
Fig. 4. Comparison of FRFs with frequency independent (blue–) and frequency dependent (red-) absorbing surfaces. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Table 4 Updating parameters and their correct value. Parameter no.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Updating Parameter Correct value
E
Kthx1
Ktvy1
Kthx2
Ktvy2
R1
R2
R3
R4
R5
I1
I2
I3
I4
I5
0.09
0.80
0.85
0.90
0.95
9.2
7.5
8.0
5.5
6.0
0.5
0.2
0.6
0.4
0.7
the absorbing surfaces of the cavity are updated. The modulus of elasticity of the plate (E), stiffness of springs at the plate boundaries (represented by kthx1, ktvy1, kthx2 and ktvy2) and the real parts (represented by R1, R2, R3, R4, R5) and imaginary parts (represented by I1, I2, I3, I4, I5) of the admittance coefficients of the absorbers on the five cavity surfaces of the cavity are updated. This makes a total of fifteen updating parameters as shown in Table 4. Frequency range from 0 to 1300 Hz is treated as the ‘measured’ frequency range and fourteen frequency points are used for updating in all the runs. These frequencies are: 248, 266, 497, 528, 708, 724, 847, 855, 1058, 1092, 1192, 1207, 1246 and 1297 Hz. The structural-acoustic FRFs are assumed to be ‘measured’ at twenty one nodes (72, 118, 139, 177, 207, 225, 269, 292, 345, 349, 410, 458, 461, 486, 489, 524, 548, 578, 597, 601 and 620) of the acoustic cavity with excitation of transverse degree of freedom of the structure at nodes 9, 20, 31, 41 and 52 making a total of 105 simulated FRFs. Fig. 5 shows identified values of the updating parameters with the frequency independent and dependent absorbing surfaces respectively. The updating parameter values are seen to be exact when compared with the values in the last column of Table 2 (giving correct values of the structural parameters) and with the values in the columns 5 and 8 of Table 3 (giving correct values of the acoustic parameters). Fig. 6 shows variation of objective function value over the iterations for the case of frequency dependent absorbing surfaces indicating a continuous reduction of the objective function value as the iterations progress. The experimental data generally contains some measurement noise. To investigate the effectiveness of the algorithm in the presence of noise, uniformly distributed random noise of 1%, 3% and 5% is added to the ‘measured’ coupled FRFs. For 3% noise, Fig. 7 shows plots of the identified correction factors and % error in them for the case of cavity with the frequency dependent absorbing surfaces. It is seen that some of the parameters are not identified exactly. Table 5 gives a comparison of Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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10
Fractional Correction Factors
Frequency Dependent Value Frequency Independent Value
8 6 4 2 0 -2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Parameter number. Fig. 5. Identified correction factors for updating with frequency independent and frequency dependent absorbing surfaces.
4
Objective function value
3.5 3 2.5 2 1.5 1 0.5 0
0
20
40 60 80 Iteration number
100
120
Fig. 6. Plot of variation of objective function value over the iterations.
%Error in Identified Correction Factors
Fractional Correction Factors
10 8 6 4 2 0 -2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Parameter number
100
50
0
-50
-100
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Parameter number
Fig. 7. Updating with frequency dependent absorbing surfaces with 3% noise: (a) Identified correction factors and (b) % error in the identified factors.
the identified values of the correction factors and % error in them for updating with 1%, 3% and 5% noise levels. It also gives maximum and average % error for each case. The results show that the errors are increasing in proportion to the level of the noise. However, the average % errors are seen to be far less than the maximum % error (shown in bold), indicating that the % Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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Table 5 Comparison of the identified values of the correction factors and % error in them for updating with 1%, 3% and 5% noise levels. Updating parameter no.
Correct value of updating parameter
0.09 0.80 0.85 0.90 0.95 9.2 7.5 8.0 5.5 6.0 0.5 0.2 0.6 0.4 0.7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Maximum % error Average % error
Case of 1% noise
Case of 3% noise
Identified value
% Error
Identified value
% Error
Identified value
% Error
0.08 0.87 0.81 0.85 0.93 9.12 7.48 7.80 5.28 5.94 0.45 0.17 0.57 0.36 0.65 – –
0.11 9.98 4.00 4.55 1.15 0.86 0.26 2.50 4.00 1.00 9.00 15.00 3.66 7.75 5.85 15 4.65
0.08 0.92 0.98 0.97 0.97 9.35 7.46 8.28 5.42 7.74 0.32 0.20 0.41 0.40 0.88 – –
8.33 15.12 15.41 7.77 2.31 1.63 0.53 3.50 1.45 29.00 36.00 3.00 31.33 1.50 25.71 36 12.17
0.02 0.82 0.78 0.87 0.92 7.89 6.46 7.28 4.42 6.74 0.04 0.10 0.51 0.50 0.91 – –
77.77 2.62 8.11 3.33 3.15 14.23 13.86 9.00 19.63 12.33 90.80 47.00 14.66 26.50 30.00 90 24.87
FE model(Blue ---)& Simulated (Red -) FRF(pa/N)magnitude in db(re:1pa/N)
FRF(pa/N)magnitude in db(re:1pa/N)
FE model(Blue ---)& Simulated (Red -) 80 60 40 20 0 -20 -40 -60
0
200
400 600 800 1000 Frequency in Hz
1200
80 60 40 20 0 -20 -40 -60
0
60 40 20 0 -20 -40 0
200
400
600
800
Frequency in Hz
1000
1200
200
400 600 800 1000 Frequency in Hz
1200
FE model(Blue ---)& Simulated (Red -) FRF(pa/N)magnitude in db(re:1pa/N)
FRF(pa/N)magnitude in db(re:1pa/N)
FE model(Blue ---)& Simulated (Red -) 80
-60
Case of 5% noise
80 60 40 20 0 -20 -40 -60
0
200
400
600
800
1000
1200
Frequency in Hz
Fig. 8. Comparison of overlays of the simulated and the FE model coupled vibro-acoustic FRFs for frequency dependent absorbing surfaces for the case of noisy data: (a) before updating (b) after updating with 1% noise (c) after updating with 3% noise and (d) after updating with 5% noise.
Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
10
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Fig. 9. Response and excitation nodes in acoustic and structural domains.
error in only some of the updating parameters is on a higher side. The maximum % error in the three cases is seen to be in the imaginary part of the admittance of the absorbing surfaces. It seems that since vibro-acoustic FRFs are less sensitive to the imaginary parts as compared to the real parts, the corresponding parameters have larger error due to noise. Fig. 8 shows a comparison of the overlays of the vibro-acoustic FRF ( α166 ,17) for updating with the three noise levels. Vibro-acoustic FRF α166 ,17 represents response at node 166 in the acoustic cavity and excitation at node 17 on the structure in the transverse direction. It is seen that the updated FE model FRF matches very well with the ‘measured’ FRF and thus the FRF correlation has significantly improved after updating even though the parameters could not be identified exactly as a result of measurement noise. Fig. 9 shows response and the excitation nodes in the acoustic and structural domains where a further assessment of the FRFs after updating is done. Vibro-acoustic FRFs at six other nodes in the acoustic cavity after updating for the case of 3% simulated noise are shown in Fig. 10. Similarly, Fig. 11 shows FRFs after updating with 3% noise in the transverse direction at two nodes on the structure. FRFs in Figs. 10 and 11 are with respect to excitation of a certain node on the structure in the transverse direction. From these figures it is seen that the updating has resulted into a global improvement in the FRF correlation throughout the cavity both in the acoustic and the structural domain.
4. Conclusion A method for updating of coupled vibro-acoustic FE models is proposed in this paper. The method performs updating by minimization of an objective function defined as the error between the measured and the coupled vibro-acoustic FE model frequency response functions subjected to bounds on the updating parameters. The framework of the method allows parameters related to the stiffness, mass or damping distribution of the structural as well as the acoustic domains to be updated taking into account the two-way coupling between these two domains. The objective function is based upon frequency response functions and hence allows even frequency dependent acoustic absorption characteristics to be identified along with other parameters. Results of a numerical study on a rectangular cavity backed by a flexible plate are presented to simultaneously update the stiffness and the material property parameters of the structural domain and the acoustic parameters of the acoustic domain. Effect of simulated noise in the data is studied. Based on the results of the numerical study the proposed method is found to work satisfactorily to estimate the updating parameters and yield an updated vibro-acoustic model whose vibro-acoustic frequency response correlates well with the simulated response. A further study is planned to validate the proposed updating method using experimental data on a vibro-acoustic cavity. Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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60 40 20 0 -20 -40 -60
FE model(Blue ---)& Simulated (Red -) FRF(pa/N)magnitude in db(re:1pa/N)
FRF(pa/N)magnitude in db(re:1pa/N)
FE model(Blue ---)& Simulated (Red -) 80
0
200
400 600 800 1000 Frequency in Hz
1200
80 60 40 20 0 -20 -40 -60
0
200
Vibro-acoustic FRF α 272,47
60 40 20 0 -20 -40 200
400 600 800 1000 Frequency in Hz
1200
80 60 40 20 0 -20 -40
0
200
Vibro-acoustic FRF α 388,47 FRF(pa/N)magnitude in db(re:1pa/N)
FRF(pa/N)magnitude in db(re:1pa/N)
60 40 20 0 -20 -40 200
400 600 800 1000 Frequency in Hz
Vibro-acoustic FRF α111,47
1200
FE model(Blue ---)& Simulated (Red -)
FE model(Blue ---)& Simulated (Red -)
0
400 600 800 1000 Frequency in Hz
Vibro-acoustic FRF α143,47
80
-60
1200
FE model(Blue ---)& Simulated (Red -) FRF(pa/N)magnitude in db(re:1pa/N)
FRF(pa/N)magnitude in db(re:1pa/N)
FE model(Blue ---)& Simulated (Red -)
0
400 600 800 1000 Frequency in Hz
Vibro-acoustic FRF α 535,47
80
-60
11
1200
80 60 40 20 0 -20 -40
0
200
400 600 800 Frequency in Hz
1000
1200
Vibro-acoustic FRF α 432,47
Fig. 10. Plot of vibro-acoustic FRFs after updating with 3% noise at six other nodes in the acoustic cavity.
Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i
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12
-80 -100 -120 -140 -160 -180
0
200
400
600
800
1000
1200
FRF(m/sec2/N)magnitude in db(re:1m/sec2/N
FRF(m/sec2/N)magnitude in db(re:1m/sec2/N
FE model(Blue ---)& Simulated (Red -) -60
FE model(Blue ---)& Simulated (Red -) -120 -140 -160 -180 -200 -220 -240
0
200
400
600
800
Frequency in Hz
Frequency in Hz
Structural FRF α17 ,47
Structural FRF α19,19
1000
1200
Fig. 11. FRFs in the transverse direction at two nodes on the structure after updating with 3% noise.
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Please cite this article as: D.V. Nehete, et al., Coupled vibro-acoustic model updating using frequency response functions, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.09.002i