An improved OPAX method based on moving multi-band model

Mechanical Systems and Signal Processing 122 (2019) 321–341

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

An improved OPAX method based on moving multi-band model Zengwei Wang a, Ping Zhu a,⇑, Yang Shen b, Yuanyi Huang b a b

State Key Laboratory of Mechanical System and Vibration, Shanghai Jiao Tong University, Shanghai 200240, PR China SAIC-GM-Wuling Automobile Co., Ltd, Liuzhou, Guangxi 545000, PR China

a r t i c l e

i n f o

Article history: Received 26 February 2018 Received in revised form 11 October 2018 Accepted 16 December 2018 Available online 28 December 2018 Keywords: Transfer path analysis Force identification Mount stiffness Estimation uncertainty Automotive

a b s t r a c t In this paper, a transfer path analysis (TPA) method – OPAX is further improved. The new method uses multiple estimates of the dynamic mount stiffness in the multi-band OPAX method to increase the estimation accuracy. A moving multi-band model is used to establish the parametric load model. Two statistical metrics are introduced to evaluate the estimation uncertainty, and a strategy for iteratively calculating operational forces whilst quantitatively assessing estimation accuracy is proposed by combining the two metrics. A numerical case is used to illustrate the proposed method. The results show that this method not only produces better estimation than the multi-band model method, but also evaluates estimation accuracy quantitatively by itself. Then the proposed method is investigated and demonstrated by a vehicle example and a rule of thumb is formulated. Lastly, a test campaign is carried out on a full vehicle to validate the proposed method. Ó 2018 Elsevier Ltd. All rights reserved.

1. Introduction The structure of modern mechanical system is becoming increasingly complex, and has a wide variety of subsystems and moving parts. The vibrations propagate from the sources through the system following certain transfer paths [1], and are radiated to the air. Severe vibrations will cause large dynamic load and noise for machinery and equipment, resulting in fatigue damage, environmental pollution and negative impact on human health, therefore vibration and noise problem has become one of the major engineering problems to be addressed. In the open literature [2–8], a number of methods have been developed in order to identify the vibration sources, define transfer paths and determine the contribution of each source to the problem. One of these methods, the Transfer Path Analysis (TPA), has been a valuable engineering tool for analyzing vibration and noise in complex structural systems as long as noise and vibrations of products have been of interest [9]. In the past three decades, TPA methods have been under continuous development and their family members can be mainly classified as the classical TPA (the matrix-inverse and mount-stiffness methods) [10–12], Operational TPA (OPA) [13–15], Operational path analysis with exogenous inputs (OPAX) [16–18], Component-based TPA [19–21] and Global Transfer Direct Transfer (GTDT) method [22–24]. These methods all have their specific advantages and disadvantages. The classical TPA has been proved to be an efficient technique to study the vibro-acoustic behavior of complicated structures, and nowadays becomes standard practice in an NVH field, since it is a widely implemented and well-known method. The classical TPA is based on a source-path-receiver model and divides the mechanical system into two parts: an active part containing load sources and a passive part containing the receiver points. The transfer paths are represented by the corresponding frequency response functions (FRF) between the interface forces and the response of the receiver point. Two basic steps are ⇑ Corresponding author at: Room 607, Building A, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China. E-mail address: [email protected] (P. Zhu). https://doi.org/10.1016/j.ymssp.2018.12.030 0888-3270/Ó 2018 Elsevier Ltd. All rights reserved.

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typically required to build a classical TPA model: the identification of interface loads during in-operation tests and measurement of FRFs between response in points of interest and points where these forces act. The main drawbacks of the classical TPA are related to measurement issues, namely the problem of having to remove the active part when conducting FRF measurements and the difficulties associated with the measurement or indirect determination of operational forces [25]. Therefore the classical TPA still remains a time consuming and complex procedure, which make it still an active research area. In order to overcome the limitations of the classical TPA, a variety of methods are developed on the basis of the classical TPA theoretical framework to be free of some classical TPA measurement difficulties or increase accuracy. A method for automatic identification of subsystems is proposed recently, and the subdivision of a mechanical system or definition of transfer paths can be performed in an automatic or algorithmic way by using this method [8]. In literature [1,26–29], some methods are proposed for estimating the passive subsystem FRFs through the global system FRFs. Theoretically speaking these methods give a quite good combination of the accuracy and efficiency, since it doesn’t require disassembling the global system but gives causality analysis. It is especially suitable for the situation where all the global FRFs involved can be easily measured. In practice, in order to measure the global system FRFs, forces must be applied at locations of interest. However in some cases, it is difficult even impossible to reach some locations due to access, and consequently the global system FRFs are not available. When it comes to this, the OPAX method is a better choice than the classical TPA. OPAX makes use of in-operation data complemented with a minimal set of extra tests with forced excitation for identifying operational forces, and no dynamic stiffness test or operational force measurement is required. Therefore OPAX provides a compromise between path accuracy and measurement time [16]. In OPAX, in order to improve the condition number of the system equations, a multi-band model of mount stiffness with prior parameters which are regarded as a constant complex mount stiffness for a given frequency band, is used for identifying operational forces. OPAX has been successfully validated by several numerical and industrial cases [30], and also against the mount stiffness method and matrix inversion techniques [31]. In this paper, the OPAX method is further developed. Multiple estimates for the dynamic mount stiffness in the multiband OPAX method are used to increase the accuracy of the proposed method. In the proposed method, a moving multiband model is used to establish the parametric load model. Each moving multi-band model contains several multi-band models, and it results in several estimations at each frequency point. Consequently the deterministic estimation problem becomes an uncertain estimation problem. Two statistical metrics are introduced to evaluate the estimation uncertainty, and a strategy for iteratively calculating operational forces whilst quantitatively assessing estimation accuracy is proposed by combining the two metrics. The next section starts with a brief introduction of the TPA theory and the multi-band model based OPAX. In Section 3 the OPAX method based on the moving multi-band model is developed and numerically verified. After the theoretical section, a vehicle example is employed to illustrate the theoretical results and experimental case study is conducted to validate the proposed method. The paper is concluded with a summary in Section 6.

2. Theories of the OPAX method 2.1. Basic TPA theory A typical model of the classical TPA is shown in Fig. 1. The basic assumption of this model is that it divides the global system into two parts: an active part and a passive part. Energy generated by the structural or the acoustic loads of the active part propagates through mount connections or the airborne paths to the receiver points of the passive part. The separation into ‘‘source” and ‘‘path” is the key to the use of TPA results for identifying dominant causes and putting forward solutions. The responses at the receiver point Y k ðxÞ are expressed as a sum of contributions due to individual path

Active part

Qj ( )

X an ( )

pj ( )

H kn ( )

F1 ( )

F2 ( )

Fn ( ) X pn ( )

H kj ( )

X a1 ( )

X a2 ( )

X p2 ( )

X p1 ( )

Hk2 ( ) H k1 ( )

Yk ( )

Passive part

Fig. 1. A typical model of the classical TPA.

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or source, and each path contribution is considered as the result of the individual load acting on the localized interface [32]. Accordingly, it can be formulated as follows:

Y k ðxÞ ¼

n X

Hki ðxÞF i ðxÞ þ

p X

i¼1

Hkj ðxÞQ j ðxÞ

ð1Þ

j¼1

where Fi (x) (i = 1,. . .,n) denotes the structural interface load, Qi (x) (i = 1,. . .,p) are the acoustic loads (typically volume velocities or accelerations), Hki (x) and Hkj (x) are the corresponding FRFs. The FRFs represent their corresponding transfer paths of noise or vibrations. As mentioned in Section 1, it typically requires two steps to build a classical TPA model: the estimation of FRF and the identification of operational forces. The FRF is estimated from experimental tests by use of a hammer or shaker for the structural loads, or a loudspeaker for acoustic loads, with the system disassembled. The decoupled FRFs are usually measured in reciprocal way using reciprocity techniques. The identification of operational forces is the main factor in the accuracy of TPA. At present, three measurement methods are in use for identifying the operational forces. The first one obtains the interface forces in a direct way by using force transducers, which is not possible in the majority of practical cases, since the load cells require enough space and well-defined support surfaces. The second one is the mount stiffness method, which calculates operational interface force by multiplying the relative displacement and the mount dynamic stiffness. The third one is the matrix inversion method. It identifies the interface forces by combining operational accelerations of nearby points with the corresponding force-acceleration FRF matrix. Because of the large number of measurements, this method costs a lot of time. 2.2. OPAX based on multi-band model OPAX uses a parametric model for identifying operational loads. This can provide a quick troubleshooting by using a simple model based on a small amount of measurement data or increase TPA accuracy by using a more complex model together with additional measurements. Two basic parametric load models are the single degree of freedom (SDOF) model and multiband model, which are shown in Fig. 2. The SDOF model is suitable for the situation where the mount or acoustic source behavior is known a priori. The operational force is expressed as



 aai ðxÞ  api ðxÞ F i ð xÞ ¼ K i ð xÞ x2

ð2Þ

K i ðxÞ ¼ mi x2 þ jci x þ ki

ð3Þ

where

mi, ci and ki represent the dynamic mass, damping coefficient and static stiffness of a given mount, respectively. aai ðxÞ and api ðxÞ are the active and passive-side accelerations at the mount connections respectively. When the mount or acoustic source behavior is not known a priori, the multi-band model is used. Different from the SDOF, it assumes a constant complex mount stiffness for a given frequency band in the multi-band model:

K i ðxÞ ¼ ki ; Q j ðxÞ ¼ hj

ð4Þ

When identifying operational forces, operational measurements and FRF measurements should have been conducted only leaving the dynamic stiffness unknown. Considering the structural path contribution (uq ðxÞ) separately, Eq. (1) can be rewritten as:

uq ðxÞ ¼

n X

Hqi ðxÞF i ðxÞ ¼

i¼1

where

n X

ki Gqi ðxÞ

ð5Þ

i¼1



aai ðxÞ  api ðxÞ Gqi ðxÞ ¼ Hqi ðxÞ x2

 ð6Þ

It can also be written as a matrix notation

2

k1

3

6k 7 6 27   7 ½ Gq1 ðxÞ Gq2 ðxÞ    Gqn ðxÞ 6 6 .. 7 ¼ uq ðxÞ 4 . 5

ð7Þ

kn For a typical automotive application case with data consisting of m orders and r (r = bw, the frequency bandwidth) frequency points in the s-th band [xs,min xs,max], the following system of equations can be formulated for the q-th indicator:

    Aq ½X  ¼ Bq

ð8Þ

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K

mount stiffness

K

mount stiffness

SDOF model

multi-band model

f

(a) Single DOF model

f

(b) Multi-band model

Fig. 2. Two mount models of OPAX.

where

2

Gq1 ðx11 Þ

Gq2 ðx11 Þ



Gqn ðx11 Þ

3

6 * G ðx Þ G ðx Þ    G ðx Þ + 7 q1 12 q2 12 qn 12 7 6 7 6 7 6 .. .. .. .. 7 6 . . . . 7 6 7 6 6 Gq1 ðx1m Þ Gq2 ðx1m Þ    Gqn ðx1m Þ 7 7 6 6 Gq1 ðx21 Þ Gq2 ðx21 Þ    Gqn ðx21 Þ 7 7 6 +7 6* 6 Gq1 ðx22 Þ Gq2 ðx22 Þ    Gqn ðx22 Þ 7 7 6 7   6 .. .. .. .. 7; Aq ¼ 6 . . . . 7 6 7 6 6 Gq1 ðx2m Þ Gq2 ðx2m Þ    Gqn ðx2m Þ 7 7 6 7 6 .. 7 6 . 7 6 6 G ðx Þ G ðx Þ    G ðx Þ 7 7 6 q1 r1 q2 r1 qn r1 7 6* 6 Gq1 ðxr2 Þ Gq2 ðxr2 Þ    Gqn ðxr2 Þ + 7 7 6 7 6 .. .. .. .. 7 6 5 4 . . . . Gq1 ðxrm Þ Gq2 ðxrm Þ    Gqn ðxrm Þ

2

uq ðx11 Þ

3

6 * u ðx Þ + 7 q 12 7 6 7 6 7 6 .. 7 6 . 7 6 7 6 6 uq ðx1m Þ 7 7 6 6 uq ðx21 Þ 7 7 6 +7 6* 6 uq ðx22 Þ 7 7 6 7   6 .. 7; Bq ¼ 6 . 7 6 7 6 6 uq ðx2m Þ 7 7 6 7 6 7 6 ... 7 6 6 u ðx Þ 7 7 6 q r1 7 6* 6 uq ðxr2 Þ + 7 7 6 7 6 .. 7 6 5 4 .

3 k1 6k 7 6 27 7 ½X  ¼ 6 6 .. 7 4 . 5 2

ð9Þ

kn

uq ðxrm Þ

xij (xs,min  xij  xs,max) is the i-th (i = 1,2,. . .,r) frequency point of the band, and j (j = 1,2,. . .,m) is the label of orders. Then an overdetermined set of equations can be formulated for all the indicators. The whole system of equations in the frequency band can be written as: ½A½X  ¼ ½B

ð10Þ

where

2

A1

3

6A 7 6 27 7 ½ A ¼ 6 6 .. 7; 4 . 5 Av

2

B1

3

6B 7 6 27 7 ½B ¼ 6 6 .. 7 4 . 5

ð11Þ

Bv

v is the number of indicators. An estimator is used to solve the equation: ½X  ¼ ½Aþ ½B

ð12Þ

When the dynamic stiffness in each band is obtained, the dynamic stiffness in the complete frequency range of interest can be determined by using a linear interpolation method. It is obvious that the wider the frequency bands are, the more frequency points are included, and the whole system of equations becomes more overdetermined, giving more robust estimation of operational forces. More details can be found in [16,31]. 3. The moving multi-band OPAX method 3.1. Moving multi-band model The moving multi-band model (MMB) is proposed based on the multi-band model (MB). The term ‘‘moving” refers to moving the original multi-band model several frequency points (the frequency band interval). Therefore a moving multiband model contains several multi-band models. The establishment process of a moving multi-band model for the n-th dynamic mount stiffness is schematically shown in Fig. 3. The frequency band interval and the number of multi-band models should be determined by a rule of thumb, which will be discussed in the next section.

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MB by moving the original MB f

The original MB

Kn

MB by moving the original MB 2 f

Kn

Kn f

2 f

f

f

f

MMB

Kn

bw

f

f

the frequency bandwidth

f frequency band interval

Fig. 3. The establishment process of a moving multi-band model.

In a multi-band model, the frequency range is split into many non-overlapping bands, and a system of equations is formulated for each band accounting for all frequency points involved in the band and all indicators (see Eqs. (8) and (10)). The dynamic mount stiffness in a band can be solved by using an estimator, and then a mathematical interpolation method is applied in order to give approximate dynamic stiffness values for the frequency range of interest. Generally, a linear interpolation method is used. Each multi-band model obtains one estimated value for one frequency point. Since a moving multiband model contains several multi-band models, there will be several estimated values of dynamic mount stiffness at one frequency point. In practice, because of nonlinearity, random noise effects and impact of unmeasured load components, multiple estimates calculated by multi-band models of a moving multi-band model may be different. Suppose that there are t multi-band models in a moving multi-band model, and the estimated value of each independent multi-band model is denoted by wi (x) (1  i  t) . wi is a complex parameter in general. The estimation process using the moving multi-band model is shown in Fig. 4. It can be found that the deterministic estimation of the dynamic mount stiffness becomes accordingly an uncertain identification problem. The assumption of the multi-band model is that the dynamic mount stiffness should change slowly with the frequency. It is obvious that the moving multi-band model is not limited to the identification of dynamic mount stiffness (or operational forces), and it can also be used for identifying other objects which don’t fluctuate violently with frequency (or other variables). 3.2. Estimation uncertainty metric and accuracy metric Two statistical metrics are introduced to deal with this uncertain identification problem: the coefficient of variation (CV) and the relative absolute error (RAE), the former quantifying the estimation uncertainty of the dynamic mount stiffness and the latter evaluating the estimation accuracy of the dynamic mount stiffness [33]. The formulation of CV is shown below:

SDn ðxÞ 1  ¼   CV n ðxÞ ¼       wn ðxÞ wn ðxÞ

SDn ðxÞ ¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 uP      u t  wn;i ðxÞ  wn ðxÞ t i¼1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 uP      u t  wn;i ðxÞ  wn ðxÞ t i¼1

t

t

ð13Þ

ð14Þ

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Kn

bw

f

f

Kn

Kn

1

the frequency bandwidth

f frequency band interval

Kn

2

f

t

f

f

Fig. 4. The estimation process of the moving multi-band model.



where wn;i ðxÞ is the estimated value obtained from the i-th independent multi-band model, and wn ðxÞ is the average value of these estimated values for the n-th path. SDn (x) is the standard deviation (SD) for the n-th path. CV is used to quantify the degree of dispersion of the estimate value wn;i ðxÞ. The calculation process of the CV metric is shown in Fig. 5. The formulation of RAE is

RAEn ðxÞ ¼

    ^  jK n ðxÞj  K n ðxÞ

ð15Þ

jK n ðxÞj

^ n (x) is the finally determined value by the movwhere Kn (x) is true or measured dynamic stiffness value of the n-th path, K ing multi-band model. In order to evaluate the uncertainty and accuracy of estimate values in the frequency range of interest, another two metrics are also introduced: the overall coefficient of variation (OCV) and the relative root mean square error (RRMSE) [34]. Equations for these two metrics are given below, respectively.

Kn

Kn

1

f

Kn

1

Kn

2

t

f

t

2

f

CVn

f Fig. 5. The calculation process of the CV metric.

f

Z. Wang et al. / Mechanical Systems and Signal Processing 122 (2019) 321–341

OCV n ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sP fe 2 x¼fs CV n ðxÞ

RRMSEn ¼

ð16Þ

fr

1 

Kn

327

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2ffi  uP ^  u fe jK n ðxÞj  K n ðxÞ t x¼fs

fr

ð17Þ



where K n is the mean value of Kn (x) in the frequency range [fe, fs] (fr = fe-fs). Essentially, these two metrics are used to assess the uncertainty and accuracy of the mount dynamic stiffness for a certain path overall, and small values of both of them are preferred. 3.3. Proposed parametric load identification strategy Based on the assumption that multiple estimates for the dynamic mount stiffness can increase the estimation accuracy, a strategy combining the moving multi-band model with the CV metric is proposed for the identification of the dynamic mount stiffness. The key point of this strategy is to calculate the CV as the quality indicator playing the role of the RAE metric. This potentially requires that there should be a good consistency between the uncertainty metric and the error metric. It is noted that RRMSE or RAE would not be available to an operator in a real OPAX measurement because it requires knowledge of the true dynamic stiffness which is unknown. When the CV value is small, the mean value of estimate values at the corresponding frequency point is selected as the accurate value, and then it is taken into the original equations as priori parameters to calculate the dynamic mount stiffness for other transfer paths at the same frequency points. A flowchart of the identification process of the dynamic mount stiffness is shown Fig. 6.

Fig. 6. Flowchart of the dynamic stiffness identification strategy.

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K1

mount stiffness

Kn

multi-band model

a f

f

K1

Kn

b CV1

c

f the coefficient of variation

f

CVn

CVd f

Kˆ 1

f

Kˆ n

final estimate value

d f

f

Fig. 7. Illustration of the dynamic stiffness identification process.

The process starts from operational measurements and FRF measurements. In this phase, the path responses, target and indicator responses and the corresponding FRFs are firstly measured. Next, it builds several independent multi-band models with priori known parameters (e.g. some mounts have equal stiffness in two or three directions or the dynamic stiffness in one direction is known). Generally, more known and accurate parameters can reduce the number of model parameters and improve the conditioning of equations to be solved. This phase is shown in Fig. 7(a). Then multiple estimates and the mean value of them can be obtained by using the moving multi-band model. It is followed by calculating the CV value at each frequency point for all transfer paths. These two phases are shown in Fig. 7(b) and (c). Once the CV value is smaller than a cer-

k08

c08 m8

k58

c58 k68 m5

k25

m6 c25 k26

m2 k12

c68 k78

c78 m7

c26 k27 m3

c12 k13

c27 m4

c13 k14

c14

m1 k01

c01

Fig. 8. Eight degrees-of-freedom system. The active part contains mass elements m5, m6 and m8, and the passive part includes m1, m2, m3, m4 and m7.

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Table 1 Parameters of the eight degrees-of-freedom system. Mass elements m (kg)

m1 = 2.0 (3.0, 4.0), m2 = 1.0 (2.5, 1.6) , m3 = 5.5 (2.6, 1.0) , m4 = 3.0 (1.9, 5.0), m5 = 6.0 (2.0, 4.2) , m6 = 8.0 (6.8, 3.9) , m7 = 2.5 (4.8, 3.5), m8 = 3.5 (1.5, 8.1)

Stiffness elements k (kN mm1)

k01 = 0.7 (0.34, 0.67), k12 = 0.4 (0.55, 0.21), k13 = 0.5 (0.1, 0.28), k14 = 0.35 (0.13, 0.32), k25 = 0.15 (0.65, 0.4), k36 = 0.13 (0.2, 0.3), k47 = 0.6 (0.1, 0.25), k58 = 0.27 (0.6, 0.35), k68 = 0.39 (0.55, 0.4), k78 = 0.48 (0.3, 0.67), k08 = 0.29 (0.8, 0.4)

Damping elements c (kg s1)

c01 = 1.1 (12.5, 8.7), c12 = 3.6 (6.0, 11.0), c13 = 6.2 (4.0, 10.0), c14 = 4.5 (3.5, 7.9), c25 = 9.0 (5.0, 6.3), c36 = 8.3 (3.5, 4.7), c47 = 11.3 (10.0, 9.4), c58 = 14.0 (1.4, 4.6), c68 = 16.3 (1.9, 2.5), c78 = 5.2 (6.0, 3.7), c08 = 7.0 (13.0, 2.5)

tain threshold CVd, the corresponding mean value is considered to be or close to the true dynamic stiffness, and is regarded as ^ n (x). This is shown in Fig. 7(d). the accurate value. The mean of estimated values therefore is taken as the determined value K Finally, the determined values are taken as priori parameters and brought into the equations for the next loop. The loop continues until all the parameters are identified. If the number of loop is up to the preset value, the mean value in the last loop is taken as the determined value. ^ n (x) is not the original estimate value of each multi-band model but the mean It is noted that the determined value K value of them. It is chosen according to the corresponding CV metric. Therefore in the next loop a new formula is developed with little number of model parameters, and this further improves estimation accuracy. Even if the CV metric is always larger than the threshold CVd, the moving multi-band OPAX method is back to a common multi-band OPAX method.

Fig. 9. Results calculated with P1: (a) estimates of K25; (b) estimates of K26; (c) estimates of K78; (d) CV for K25; (e) CV for K26; (f) CV for K78; (g) RAE for K25; (h) RAE for K26; (i) RAE for K78.

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3.4. Numerical example A numerical example of eight degrees-of-freedom system is used to illustrate the proposed strategy and reveal the relationship between the uncertainty metric and the error metric. The numerical example is presented in Fig. 8, and parameters used for this system are described in Table 1. There are three sets of parameters in Table 1. The one outside the parentheses is the first set, and the one on the left within the parentheses is the second set, and the last is the third one. They are denoted by P1, P2 and P3, respectively. The system is divided into two parts: an active part and a passive part. The active part contains three mass elements (m5, m6 and m8) and the corresponding visco-elastic elements. The passive part contains five mass elements and the corresponding visco-elastic elements. Each connection between these two parts is regarded as a transfer path, thus there are three transfer paths. The dynamic stiffness values of three paths are denoted by K25, K26 and K78, respectively. Suppose that there are some forces acting on the active part, and FRFs of the passive part between interface forces and the target response are known. The target is mass element m1, and there is no indicator. In order to determine transfer path contribution, all the responses except the response of mass m4 should be known or measured synchronously. In theory, at least three sets of measurements are needed for solving the equations when using the frequency-by-frequency method such as the mount stiffness method and the matrix inversion method. In order to show the capability and advantage of the proposed strategy, only one set of measurement is utilized to perform the identification of dynamic mount stiffness (operational forces). In this example, the frequency range of interest is [0–250] Hz. The frequency band interval is 1 Hz, and the frequency bandwidth is 6 Hz. Therefore, there are totally six multi-band models in a moving multi-band model, resulting in six estimates at each frequency point. All the estimates are obtained for three sets of parameters, respectively, and the CV and RAE values are also calculated. The results are shown in Figs. 9–11. All the estimated values are illustrated with the amplitude, and the phase is omitted for clarity.

Fig. 10. Results calculated with P2: (a) estimates of K25; (b) estimates of K26; (c) estimates of K78; (d) CV for K25; (e) CV for K26; (f) CV for K78; (g) RAE for K25; (h) RAE for K26; (i) RAE for K78.

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In Fig. 9, subplots (a), (b) and (c) show the estimates of K25, K26 and K78 respectively by applying the parameter set P1, and each curve represents the result solved by an independent multi-band model. There are totally six estimates for each path. Subplots (d), (e) and (f) show the corresponding CV values, and subplots (g), (h) and (i) show the corresponding RAE values. The results in Figs. 10 and 11 are solved using P2 and P3 respectively. From the above results, some meaningful findings are observed: (i) First, the estimated values of the dynamic mount stiffness obtained from each approximate model are not exactly the same in the frequency range of interest, and in some frequency points, the estimated values fluctuate greatly. Fig. 9(b) shows there are six quite different estimate values at 100 Hz. However, there exists at least one good estimate for each frequency point. Fig. 9(a) shows, for example, there is a good estimate value of K25 at 100 Hz. Similar results can be found from the remaining figures. (ii) Second, when the estimated values are equal or close to each other at a frequency point, the CV value and RAE value at this frequency point are both small, and vice versa. When estimated values fluctuate greatly, the CV value and RAE at the corresponding frequency point are both large. It is obvious that there is a good consistency between CV and RAE, which indicates CV is a good quality indicator, playing the role of RAE. Since OCV and RRMSE are formulated respectively based on CV and RAE, it is believed that there is also a good consistency between OCV and RRMSE. (iii) Finally, the determined estimates for the three set of parameters are shown in Fig. 12. Subplots (a), (b) and (c) show the determined estimates of K25, K26 and K78 respectively by applying the parameter set P1, subplots (d), (e) and (f) by applying P2, and subplots (g), (h) and (i) by applying P3. It can be found the determined values obtained by the moving multi-band OPAX method are more accurate than that obtained by the multi-band OPAX method. The moving multiband OPAX method seems like a smoothing technique to a certain degree, making the estimated value curve smoother.

Fig. 11. Results calculated with P3: (a) estimates of K25; (b) estimates of K26; (c) estimates of K78; (d) CV for K25; (e) CV for K26; (f) CV for K78; (g) RAE for K25; (h) RAE for K26; (i) RAE for K78.

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Three kinds of system structures (see Fig. 13) combined with three sets of parameters are employed, thus there are totally nine configurations. By applying the proposed method and the multi-band OPAX, the dynamic mount stiffness is identified, and the RRMSE values for these nine cases are calculated, which are presented in Fig. 14. The symbol P1C1 refer to the structure in Fig. 13(a) combined with P1, and the rest symbols may be deduced by analogy. It is readily observed that the RRMSE values obtained by using the proposed method are much lower than the ones by using multi-band OPAX method, especially in the case of poor estimation.

Fig. 12. MMB analysis results in comparison with MB. The determined estimates: (a) K25 (K1) with P1; (b) K26 (K2) with P1; (c) K78 (K3) with P1; (d) K25 (K1) with P2; (e) K26 (K2) with P2; (f) K78 (K3) with P2; (g) K25 (K1) with P3; (h) K26 (K2) with P3; (i) K78 (K3) with P3.

(a)

(b)

(c)

Fig. 13. Three kinds of system structures: (a) structure C1; (b) structure C2; (c) structure C3.

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4. Automotive example In this section, a passenger car model is used to investigate and validate the benefits of the proposed method by finite element simulations. The simulation model has been validated by several physical tests. All the relative data, including FRFs and all the accelerations, are calculated by the commercial finite element code MSC.NASTRAN.

4.1. Simulation model The well correlated simulation model consists of a power plant, three engine mounts, an auto-body structure, and an enclosed acoustic cavity, which is shown in Fig. 15. The power plant including an I4 engine is modeled as rigid body with its mass, moment inertia characteristics, and main geometry information: a point mass element and three rigid beam ele-

Fig. 14. The RRMSE values of nine cases for: (a) K25 (K1); (b) K26 (K2); (c) K78 (K3).

Fig. 15. Finite element model of a passenger car: (a) auto-body structure model; (b) interior acoustic cavity model.

Table 2 Power plant parameters. Mass (kg)

XC 157.0

Moment of inertia (kg mm2)

Mass center location (mm)

96.0

YC 49.3

ZC 256.1

Im,xx 13.052  10

Im,yy 6

3.80  10

Im,zz 6

7.80  106

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ments. The parameters of the power plant are measured experimentally and listed in Table 2. The rubber mount is modeled by the generalized spring-damper element CBUSH, which has the frequency-dependent complex-valued dynamic stiffness. In this study, a set of 9 structural paths (three directions of each rubber mount), 1 sound pressure point (target response) and 9 indicators are considered, which is illustrated in Fig. 16. The frequency range of interest is [10–200] Hz, since it covers the typical running frequency range of the power plant. The operational data is described in accordance with orders from 1 to 10. The data includes 9 active-side mount accelerations, 9 passive-side mount accelerations, a 10*9 FRF matrix, the target sound pressure and 9 indicator accelerations. Some noise was introduced to simulate a real-life situation. 4.2. Analysis and results The moving multi-band OPAX method is used to identify the dynamic mount stiffness with a subset of 2 indicators and all 10 orders components. The frequency bandwidth is defined as 5 Hz and the frequency band interval is 1 Hz. The CV is

Fig. 16. The MMB model with 9 structural paths, 1 sound pressure point and 9 indicators. Ki (i = 1, 2,. . ., 9) are the dynamic stiffness values of three rubber mounts corresponding to 9 structural paths respectively and Fi (i = 1, 2,. . ., 9) are the interface forces. aai (i = 1, 2,. . ., 9) denote 9 active-side mount accelerations and api (i = 1, 2,. . ., 9) 9 passive-side mount accelerations. Ui (i = 1, 2,. . ., 9) denote 9 indicator accelerations and Y1 the target sound pressure. Hqi is a FRF between the acceleration or sound pressure and the interface force.

Fig. 17. The results calculated by MMB based OPAX method: (a) the estimated value of the y-direction dynamic stiffness of the right mount, Kry; (b) CV for Kry; (c) RAE for Kry; (d) the estimated value of the x-direction dynamic stiffness of the back mount, Kbx; (e) CV for Kbx; (f) RAE for Kbx.

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selected as the uncertainty metric. When the CV value of a path is less than 5%, the dynamic stiffness of this path is considered to be accurate and selected as the priori known parameter in the next loop. The estimated value of the y-direction dynamic stiffness of the right mount is shown in Fig. 17(a). The dynamic stiffness estimated by the proposed method (red line) is in comparison with the one estimated by the multi-band OPAX method (black line) and the true value (blue line). Fig. 17(b) and (c) shows the CV value and RAE value respectively. It is found that the proposed method gets a better estimate than the multi-band OPAX method. The CV curve captures well the critical peaks of the RAE curve, and displays clearly the frequency range of poor estimation. For example, estimates at 56 Hz, 109 Hz, 154 Hz and 180 Hz are not good according to the CV curve, and this is confirmed by estimates shown in Fig. 17(a) and the RAE curve in Fig. 17(c). The results of the x-

Fig. 18. The mean values of the RRMSE metric against number of indicators and number of orders in the case of the frequency bandwidth: (a) 7 Hz; (b) 13 Hz; (c) 19 Hz; (d) 33 Hz; (e) 45 Hz; (f) 55 Hz.

Fig. 19. The mean values of the OCV metric against number of indicators and number of orders in the case of the frequency bandwidth: (a) 7 Hz; (b) 13 Hz; (c) 19 Hz; (d) 33 Hz; (e) 45 Hz; (f) 55 Hz.

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direction dynamic stiffness of the back mount are presented in Fig. 17(d), (e) and (f). It shows that the dynamic stiffness value obtained by the proposed method fits the true value better than the one obtained by the multi-band OPAX method. Estimates in the frequency range of [150–200] Hz are well improved. Although there are some differences, the changing trends of the CV curve in Fig. 17(e) and the RAE curve in Fig. 17(f) are somehow consistent, both indicating that poor estimation appears at frequency points 63 Hz, 150 Hz, 156 Hz and 191 Hz. The estimation accuracy of the proposed method depends on several factors, such as the amount of input data, the number of transfer paths, the frequency bandwidth, the frequency band interval and the priori known parameters or relations among parameters. In order to study the effects of these factors, some combinations of (i) number of indicators (1,2,. . .,10 indicators), (ii) the frequency bandwidth (7,13,. . .,55 Hz), (iii) number of orders (1,2,. . .,10 orders) were tested. OCV is selected as the estimation quality indicator, and RRMSE the estimation error metric. The frequency band interval is defined as ceil(bw/10). The mean values of the OCV metric and RRMSE metric for all transfer paths are plotted in Figs. 18 and 19 respectively. They are all in function of the frequency bandwidth, the number of orders and the number of indicators for a large number of scenarios. First, it can be seen from Fig. 18 that each subplot has a large area (the blue area in the first five subplots and almost the whole area in the last subplot) with small estimation errors (RRMSE less than 20%). The more orders and indicators are used, the smaller the RRMSE value is. Fig. 20(d) presents this, for example, for points G (7 indicators, 3 orders) and M (6 indicators, 1 order) of Fig. 18(a). With the increase of the frequency bandwidth, the area of small RRMSE value becomes larger, and this is also illustrated in Fig. 20(c) for points M, N and O (all having 4 indicators, 3 orders) of Fig. 18. From the above analysis, it can be concluded that estimates can be further improved by more orders, indicators and larger frequency bandwidth. This is because more available data can increase the number of equations and improve the solving conditioning of the system of equations to be solved. Second, it can be observed from Fig. 19 each subplot has a large area with low uncertainty (OCV less than 30%). Except for the ones with the frequency bands of 45 Hz and 55 Hz (in Fig. 19(e) and (f) respectively), the OCV value decreases as the orders, indicators and the frequency bandwidth increase. However, for the estimates with the frequency bands of 45 Hz and 55 Hz, the OCV value does not decrease as the indicators or orders increase, and what is even worse is that the uncertainty metric may become higher when more orders or indicators are added (e.g. the OCV value with 6 indicators and 9 orders in Fig. 19(f)). Finally, by comparing the values in Fig. 19 with Fig. 18, it can be found that there is clearly a good agreement between the OCV and RRMSE. But the OCV values in the latter two subplots do not necessarily coincide with the RRMSE values. Given that

Fig. 20. The results in different scenarios.

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the mount dynamic stiffness is a constant complex value within a frequency band, it is considered wise to use a small frequency band rather than a large frequency band. This is because a large frequency band gives very few interpolation points in the frequency range of interest, resulting in a straighter dynamic stiffness curve, which may deviate from the original assumption. This is illustrated in Fig. 20(c), for example, for points M, N and O. When using a large band (55 Hz) the dynamic stiffness curve is much straighter than the curve with a small frequency band (7 Hz and 33 Hz). This explains why the RRMSE values are so small though the corresponding OCV values are large, since the true mount dynamic stiffness curve used in this simulation model is a simple increasing function close to a straight curve. Fig. 20(a) and (b) illustrate this for twelve points in Figs. 18 and 19. It is obvious that the OCV values become larger when applying frequency bands of 45 Hz and 55 Hz than the ones with small frequency bands. Yet a too small frequency bandwidth should also be avoided to ensure an adequate num-

(a) Target

Indicators

Vibration-active

SEAT

Vibration-passive

Engine

(b) Fig. 21. Schematic representation of the experimental setup: (a) accelerometers on the passive and the active side; (b) measurement setup.

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Fig. 22. Experimental setup for FRF measurements.

Fig. 23. Comparison of the path contribution results of moving multi-band OPAX and multi-band OPAX.

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ber of multi-band models. Therefore a frequency bandwidth not larger than one-sixth (33 Hz) of the frequency range of interest is recommended, and in such a case, the OCV metric is a reliable quality indicator for estimates. According to the above observations for all scenarios, a rule of thumb can be formulated for the proposed method to obtain a well-conditioned system of equations:

1:3v ceilð0:5 mÞ ceilð0:1 bwÞ P 2n

ð18Þ

Fig. 24. The dynamic mount stiffness test system: (a) Normal dynamic stiffness of the back mount; (b) Lateral dynamic stiffness of the back mount; (c) Normal dynamic stiffness of the left mount; (d) Lateral dynamic stiffness of the left mount.

Fig. 25. Dynamic mount stiffness estimations for the left mount in the Z direction: (a) Dynamic stiffness; (b) Damping Factor.

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where v is the number of indicators, m is the number of orders, bw is the frequency bandwidth, n is the number of transfer paths. When this rule is satisfied, it indicates the estimate is accurate and can be used further in TPA analysis. For example, the condition is fulfilled for points G (7 indicators, 3 orders, 7 Hz, 9 paths, 1 Hz, left term = 18.2, right term = 18) and N (4 indicators, 3 orders, 33 Hz, 9 paths, 4 Hz, left term = 41.6, right term = 18), and the corresponding results are presented in Fig. 20(c) and (d). It is not fulfilled for point M (4 indicators, 3 orders, 7 Hz, 9 paths, 1 Hz, left term = 10.4, right term = 18). It can be found that less orders or indicators are required to fulfill the condition compared with the multi-band OPAX method. Furthermore, in order to get a more robust estimation model, a frequency band of one-sixth of the frequency range of interest is recommended, and the frequency band interval is defined as ceil(0.1bw) in this rule of thumb:

bw ¼ ceilðfr=6Þ;

Df ¼ ceilð0:1bwÞ

ð19Þ

where fr is the length of the frequency range of interest and Df is the frequency band interval. 5. Experimental study In this section, a validation test campaign is carried out on a full vehicle to assess the proposed method. The main focus of the analysis is on the structural vibration transferring from the powertrain through three mounting points to the driver’s seat. Each mounting point is equipped with accelerometers on both the passive and the active side, as depicted in Fig. 21 (a). Three additional accelerometers are also placed on the passive side as indicators. The target response is the acceleration of the driver’s seat mounting point. The measurement setup is shown in Fig. 21(b). Finally the system consists of 9 transfer paths, 9 indicators and 1 target. The following measured data is acquired for the proposed method: (1) FRF data  FRFs between 9 force inputs and 1 target: #=9  FRFs between 9 force inputs and 9 indicators: #=9  9 = 81 (2) Operational data (Steady-run 1400 RPM, fourth gear)  Target acceleration: #=1  Active and passive side mount accelerations: #=9  2 = 18  Nine acceleration indicators: #=9 Measurements of decoupled FRFs of the car body are performed when the engine is removed, as depicted in Fig. 22. A crane and a wood stick are used to hoist the engine to make it completely separate from the car body. An instrumented hammer is used for measuring decoupled FRFs. Operational responses are measured in steady-run condition (1400 RPM, fourth gear). In this study, only three indicators are utilized to perform the identification of operational forces. The path contribution results are presented in Fig. 23. Fig. 23(a) shows the path contribution results of the moving multiband OPAX method and Fig. 23(b) the multi-band OPAX method. It is found that the summed path contributions are in good agreement with the measured values. It is also readily found that the results of the proposed method are similar to those of the multi-band OPAX method. The small discrepancies between the results are mainly due to the fact that enough orders are involved in operational data, since for the proposed method fewer orders in the data are yet sufficient. Beside, both methods are capable to identify the critical path (P6). The dynamic mount stiffness is estimated by the proposed method and the multi-band OPAX method respectively. In order to validate the proposed method, the dynamic mount stiffness is also measured according to ISO10846-1-2008/IS O10846-2–2008. The MTS 322 Test Frame and the corresponding data acquisition and analysis system are utilized in the test. The preload is applied to take into account the effect of the mass of the powertrain. The dynamic mount stiffness test system is shown in Fig. 24. The measured dynamic mount stiffness is available for the frequency range [1–30] Hz, and the frequency resolution is 2.5 Hz. Since the frequency range of interest is [20–200] Hz, a comparison of the dynamic mount stiffness for the frequency range [20–30] Hz is presented, as shown in Fig. 25. It can be found that the identified dynamic mount stiffness is well in line with the measured one. It is clear that a better agreement can be found between the result obtained from the proposed method and the one from the test. 6. Conclusions The research presents an improved TPA method, which is referred to as the moving multi-band OPAX method. The key of this method is to establish the dynamic mount stiffness model by using the moving multi-band model and calculate the CV as the quality indicator playing the role of the RAE metric. The dynamic mount stiffness or operational force is estimated iteratively in a loop with CV as the quality indicator. The benefit of the proposed method is that it provides engineers with a simple means to conduct TPA analysis using smaller amount of operational data than the multi-band OPAX method, at the same time predicting quantitatively the estimation accuracy. Both a numerical example and a vehicle example are conducted to validate the proposed method. A rule of thumb is formulated to obtain good estimation for a typical vehicle case. To demonstrate the method in practice, an experimental case study is carried out. It is shown that compared with the multi-

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band OPAX method, the dynamic mount stiffness obtained from the proposed method is better in line with the measured one. References [1] G. De Sitter, C. Devriendt, P. Guillaume, E. Pruyt, Operational transfer path analysis, Mech. Syst. Signal Process. 24 (2) (2010) 416–431. [2] H.N. Özgüven, Structural modifications using frequency response functions, Mech. Syst. Signal Process. 4 (1) (1990) 53–63. [3] M.H. Tan, J.K. Hannond, A non-parametric approach for linear system identification using principal component analysis, Mech. Syst. Signal Process. 21 (4) (2007) 1576–1600. [4] Z.W. Wang, P. Zhu, A system response prediction approach based on global transmissibilities and its relation with transfer path analysis methods, Appl. Acoust. 123 (2017) 29–46. [5] Z.W. Wang, P. Zhu, Response prediction for modified mechanical systems based on in-situ frequency response functions: theoretical and numerical studies, J. Sound Vib. 400 (2017) 417–441. [6] A. Oktav, Ç. Yılmaz, G. Anlasß, Transfer path analysis: current practice, trade-offs and consideration of damping, Mech. Syst. Signal Proc. 85 (2017) 760– 772. [7] A.S. Wixom, J.G. McDaniel, Identification and application of dynamic uncoupling between modifications to vibrating systems, Int. J. Numer. Meth. Engng. 109 (2017) 533–554. [8] F.X. Magrans, J. Poblet-Puig, A. Rodríguez-Ferran, A subsystem identification method based on the path concept with coupling strength estimation, Mech. Syst. Signal Proc. 100 (2018) 588–604. [9] M.V. van der Seijs, D. de Klerk, D.J. Rixen, General framework for transfer path analysis: history, theory and classification of techniques, Mech. Syst. Signal Process. 68–69 (2016) 217–244. [10] J. Plunt, Finding and fixing vehicle NVH problems with transfer path analysis, J. Sound Vib. 39 (11) (2005) 12–17. [11] A. Gunduz, A. Inoue, R. Singh, Estimation of interfacial forces in time domain for linear systems, J. Sound Vib. 329 (13) (2010) 2616–2634. [12] A. Diez-Ibarbia, M. Battarra, J. Palenzuela, G. Cervantes, S. Walsh, M. de la Cruz, S. Theodossiades, L. Gagliardini, Comparison between transfer path analysis methods on an electric vehicle, Appl. Acoust. 118 (2017) 83–101. [13] D. de Klerk, A. Ossipov, Operational transfer path analysis: theory, guidelines and tire noise application, Mech. Syst. Signal Process. 24 (7) (2010) 1950– 1962. [14] N.B. Roozen, Q. Leclère, On the use of artificial excitation in operational transfer path analysis, Appl. Acoust. 74 (10) (2013) 1167–1174. [15] W. Cheng, Y.Y. Lu, Z.S. Zhang, Tikhonov regularization-based operational transfer path analysis, Mech. Syst. Signal Process. 75 (2016) 494–514. [16] K. Janssens, P. Gajdatsy, L. Gielen, P. Mas, L. Britte, W. Desmet, H. Van der Auweraer, OPAX: a new transfer path analysis method based on parametric load models, Mech. Syst. Signal Process. 25 (4) (2011) 1321–1338. [17] K. Kim, S. Sin, S. Choi, J. Im, S. Kim, Lee, Analysis of the cause of hindrance for ride quality with OPAX and OTPA, Trans. Korean Soc. Noise Vib. Eng. 24 (12) (2014) 935–942. [18] M.V. Rao, S.N. Moorthy, P. Raghavendran, Dynamic stiffness estimation of elastomeric mounts using OPAX in an AWD monocoque SUV, Proceedings of the SAE Noise and Vibration Conference, US, 2015, Paper No. 2015-01-2190. [19] A. Elliott, A. Moorhouse, T. Huntley, S. Tate, In-situ source path contribution analysis of structure borne road noise, J. Sound Vib. 332 (24) (2013) 6276– 6295. [20] D.J. Rixen, A. Boogaard, M.V. van der Seijs, G. van Schothorst, T. van der Poel, Vibration source description in substructuring: a theoretical depiction, Mech. Syst. Signal Process. 60–61 (2015) 498–511. [21] D. Lennström, M. Olsson, F. Wullens, A. Nykänen, Validation of the blocked force method for various boundary conditions for automotive source characterization, Appl. Acoust. 102 (2016) 108–119. [22] F.X. Magrans, Method of measuring transmission paths, J. Sound Vib. 74 (3) (1981) 311–330. [23] Z.W. Wang, P. Zhu, J.X. Zhao, Response prediction techniques and case studies of a path blocking system based on Global Transmissibility Direct Transmissibility method, J. Sound Vib. 388 (2017) 363–388. [24] J. Jové, O. Guasch, Direct response and force transmissibilities in the characterization of coupled structures, J. Sound Vib. 407 (2017) 1–15. [25] O. Guasch, C. Garcia, J. Jové, P. Artis, Experimental validation of the direct transmissibility approach to classical transfer path analysis on a mechanical setup, Mech. Syst. Signal Process. 37 (1–2) (2013) 353–369. [26] L. Keersmaekers, L. Mertens, R. Penne, P. Guillaume, G. Steenackers, Decoupling of mechanical systems based on in-situ frequency response functions: the link-preserving, decoupling method, Mech. Syst. Signal Process. 58–59 (2015) 340–354. [27] Z.W. Wang, P. Zhu, Z. Liu, Relationships between the decoupled and coupled transfer functions: theoretical studies and experimental validation, Mech. Syst. Signal Process. 98 (2018) 936–950. [28] X.H. Liao, S.M. Li, L.Y. Liao, H.D. Meng, Virtual decoupling method: a novel method to obtain the FRFs of subsystems, Arch. Appl. Mech. 87 (9) (2017) 1453–1463. [29] J. Wang, X. Hong, Y. Qian, Z.W. Wang, L.X. Liu, Inverse sub-structuring method for multi-coordinate coupled product transport system, Packag. Technol. Sci. 27 (5) (2014) 385–408. [30] P. Gajdatsy, L. Gielen, K. Janssens, H. Van der Auweraer, P. Mas, W. Desmet, A novel TPA method using parametric load models: validation on experimental and industrial cases, Proceedings of the SAE Noise and Vibration Conference, US, 2009, Paper No. 2009-01-2165. [31] K. Janssens, P. Aarnoutse, L. Britte, F. Deblauwe, P. Gajdatsy, T. Tamarozzi, Full vehicle validation of OPAX in comparison to traditional TPA methods, 15th International Conference on Experimental Mechanics, Porto, Portugal, 2012. [32] H. Van Der Auweraer, P. Mas, S. Dom, A. Vecchio, K. Janssens, P. Van De Ponseele, Transfer path analysis in the critical path of vehicle refinement: the role of fast, hybrid and operational path analysis, Proceedings of the SAE Noise and Vibration Conference, US, 2007, Paper No. 2007-01-2352. [33] P. Zhu, F. Pan, W. Chen, S. Liang, Use of support vector regression in structural optimization: application to vehicle crashworthiness design, Math. Comput. Simulat. 86 (2012) 21–31. [34] L. Shi, R.J. Yang, P. Zhu, A method for selecting surrogate models in crashworthiness optimization, Struct. Multidiscip. O. 46 (2012) 159–170.