bulletin of the jsme vol.3, no.1, 2016 mechanical

J-STAGE Advance Publication date: 21 January, 2016Paper No.15-00589

© 2016 The Japan Society of Mechanical Engineers[DOI: 10.1299/mej.15-00589]

Vol.3, No.1, 2016Bulletin of the JSME

Mechanical Engineering Journal

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Abstract

Operational transfer path analysis (OTPA) calculates contributions of reference points to response point

vibration by using only operational data. Through OTPA, effective interior noise and vibration reduction are

achieved by applying intensive countermeasure to the high contributing part. However, it becomes difficult

occasionally when many reference points have similar contributions by a vibration mode. In this case,

obtaining high contributing vibration mode and considering how to reduce the mode become important

information. In this study, we attempted to calculate the vibration mode contribution by modifying OTPA.

Principal component calculated in OTPA procedure is composed of correlated vibration factors among

reference points. We then considered the relationship between the principal component and the vibration mode,

and associated the principal components with the vibration modes of a test structure. As a result, high

contributing vibration modes to the response point could be found. In addition, information about which side of

the structure (response or reference side) had better to be measured intensively was also obtained by evaluating

the influence of each principal component to the response point (principal component transfer function).

Finally, Several countermeasures were applied to the structure considering the principal component and

vibration mode contributions. The result shows effective vibration reduction at the response point could be

carried out. Through these procedures, the modified OTPA became more useful tool for applying effective

countermeasure.

Key words : Transfer path analysis, Principal component, Vibration mode, Transfer function, Contribution

1. Introduction

Obtaining contribution of each vibration source to the response point quantitatively is essential to achieve effective

countermeasure. Transfer path analysis (TPA) has been developed to obtain the contribution and several TPA methods

were proposed until now (Van der Auweraer, 1979; Starkey, 1989; Plunt, 1998; Noumura, 2006; Brandl, 2008;

(OTPA) is one of the TPA methods recently developed (Noumura and Yoshida, 2006). This method informs us the

contributions with smaller man-hour using only operational data for calculating the contributions (Noumura, 2006;

calculated using the operational data and the contribution is obtained by multiplying measured reference signal at the

operational condition with the transfer function (Noumura and Yoshida, 2006). The countermeasure for the reduction of

the response point vibration has been applied typically to the high contributing reference point such as engine mount

attachment point. However, if many reference point vibrations correlate each other strongly by putting the reference

points to close points on a flame, the contributions are occasionally very similar and finding out high contributing part

and applying intensive countermeasure becomes difficult. In this situation, the contribution separation of the reference

point by OTPA becomes ineffective. However, if high contributing vibration mode of the structure could be clarified

instead of high contributing reference point, this information has a possibility to become important information for the

Contribution analysis of vibration mode utilizing operational TPA

Junji YOSHIDA* and Koki TANAKA* *Osaka Institute of Technology

5-16-1 Ohmiya, Asahi-ku, Osaka 535-8585, Japan

E-mail: [email protected]

1

Received 26 October 2015

Lohrmann, 2008; Sonehara, 2009; Tcherniak, 2009; Klerk, 2010; Yoshida, 2013; Yoshida, 2014). Operational TPA

Lohrmann, 2008; Sonehara, 2009; Klerk, 2010; Yoshida, 2013; Yoshida, 2014). In this method, the transfer function is

2

Yoshida and Tanaka, Mechanical Engineering Journal, Vol.3, No.1 (2016)

© 2016 The Japan Society of Mechanical Engineers[DOI: 10.1299/mej.15-00589]2

effective countermeasure.

Meanwhile, principal component mode has been discussed recently as a CAE technique to extract important

vibration modes excited at the operational condition from many vibration modes (Mochizuki, 2015; Kawai and

Yanagase, 2015). This enables us to focus less vibration modes as the reduction target. If we can find out which the

excited vibration mode at the operational condition affects the response signal significantly, the countermeasure

becomes more effective.

In this study, we propose an experimental analysis method of contribution separation using the principal

component through the modification of the present OTPA method instead of obtaining the reference point

contributions. Then, we tried to extract high contributing vibration mode to the response point by associating the

principal component to the vibration mode after consideration of the relationship between principal component mode

and vibration mode. In addition, we considered an evaluation procedure to determine whether the vibration mode at

around the reference points or the resonance at the response point is the main factor of the large vibration peak at the

response point by comparing the characteristics of principal component and the transfer function to make OTPA

method more effective tool.

2. Calculation procedure of OTPA

In OTPA, the individual contributions are calculated through two-step processes. In the first step, the transfer

function from the reference point to the response point is calculated using various operational data measured at the

reference and response points. In the second step, each contribution is obtained by multiplying the reference point

signal by the calculated transfer function. The transfer function in this method is calculated using a principal

component regression method (Noumura and Yoshida, 2006). The calculation procedure is shown in Fig. 1.

Fig. 1 Calculation image of transfer function of OTPA using principal component regression method.

1) At first, principal component analysis (PCA) is applied to the reference signal matrix [Ain] by the singular

value decomposition (SVD) to remove correlation among reference signals as shown in Eq. (1) and (2). The calculated

uncorrelated signals were named as principal component [T].

Tin VSUA (1)

SUVAT in (2)

2) Noise component in the uncorrelated signal was eliminated using the size of each singular value (Noumura

and Yoshida, 2006).

3) Multiple regression analysis is applied between the remained (signal) principal component [T] and the

response signal [Aout] to obtain the influence [B] of each principal component to the response signal as shown in Eq. (3)

and (4).

BTAout (3)

out

TTATTTB

1

(4)

Fig. 1 Operational TPA using principal component regression.

Principal component analysis

T1Ain1

Ain2

Ain3

Ain4

Ain5

T2

T3

T4

T5

Aout

Referencesignal

Principalcomponent

Siz

e o

f p

rin

cip

al

com

po

nen

t

Responsesignal

Regression analysis

Noisereduction

2



4) Transfer function from the reference signal to the response signal [H] is calculated by multiplying the

coefficient matrix [V], that connects the reference signal to the principal component, and the regression coefficient [B],

that connects the principal component and the response signal as shown in Eq.(5).

out

TTATTTVH

1

(5)

In the second step, the contribution from the reference point is obtained by multiplying the transfer function with

the reference signal.

This is the outline to obtain contributions of reference points. Countermeasure to reduce the response vibration is

typically applied to the high contributing reference point. In this study, we calculated the principal component

contribution as shown in Eq. (3) by multiplying the principal component [T] and the principal component transfer

function [B] instead of the reference point contributions to obtain additional useful information by OTPA.

3. OTPA application to a simple structure

As the experimental structure on which reference point signals are supposed to correlate each other, two flat plates

connected by four rubber bushes were used as shown in Fig. 2. Material of these plates was Aluminum and thickness of

them was the same at 3 mm. The weight of the lower plate (supporting plate) and that of the upper plate (response

plate) were 1240 g and 320 g, respectively. Each edge of the supporting plate was hung by a soft rubber cable. A swept

sinusoid force from 10 to 1000 Hz was given at the input point on the supporting plate by an electrodynamic shaker

(Modalshop K2007E01).

Fig. 2 Two plates structure for application of OTPA and force input and vibration measurement points.

Four reference points (from Ref. 1 to Ref. 4 in Fig. 2) for applying OTPA were set on the rubber bush attachment

points on the supporting plate, and the response point (Res. in Fig. 2) was set on the center of the response plate as

shown in Fig. 2. These vibrations at the reference and response points were measured simultaneously using

accelerometers (PCB C35265) weighting 2 g each, when the shaker was activated in the operational condition. The

contribution from each reference point was calculated according to the OTPA procedure. Figure 3 shows the average

spectrum of each reference point contribution to the response point.

Fig. 3 Contribution of each reference point to the response point.

300 mm

200 mm

Ref.1 Ref.2

Ref.3 Ref.4

P.1 P.2 P.3

P.4 P.5 P.6

P.9

P.10 P.11 P.12

Input

P.7 P.8

Res.

30

50

70

90

110

130

0 100 200 300 400 500

Ref1Ref2Ref3Ref4

Freq. (Hz)

Acc

eler

atio

nle

vel

(d

B)

2



Horizontal and vertical axes indicate frequency and the vibration acceleration level, respectively. The result shows

contributions of most reference points had vibration peaks at the same frequency and the contribution levels were also

very similar. The vibrations at the reference point seemed to have high correlation each other and depended on the same

vibration mode in this situation because the reference points were set at very close points on the same supporting plate.

This result indicates that there was no dominant reference point affecting the response point. The obtained contribution

of reference point by OTPA is not significant information for conducting effective countermeasure. We then tried to

separate contributions of principal component to the response point instead of the reference point contribution by

modifying the OTPA procedure.

4. Principal component obtained in OTPA

4.1 Principal component mode at the operational condition

Principal component at the operational condition [T] is obtained by Eq. (1) and (2) in the OTPA procedure. The

principal component is composed of correlated factors in the measured reference signals and the component (column

element in principal component matrix [T]) has no correlation with the other component. In addition, the reference

point signal could be obtained by multiplying the principal component matrix [T] with the inverse of the matrix [V] as

shown in Eq. (6). Here, the inverse matrix of [V] could be expressed as the transpose of matrix [V] because this matrix

is unitary matrix obtained by SVD. In case the number of the reference point is only two as an instance for the

explanation, the matrix equation could be developed as Eq. (7).

Tin VTVTA 1

(6)

222211122111

2222212112221121

2212211112121111

21

2221

1211

vtvtvtvt

vtvtvtvt

vtvtvtvt

aa

aa

aa

nnnnnn

(7)

The left side box by solid line denotes principal component 1 (PC 1) included in reference signal 1 (Ref. 1), and the

right side box by solid line denotes PC 1 in the reference signal 2 (Ref. 2). Each PC 1 element has complex value

(having amplitude and phase information) and composes the reference signal. Accordingly, we can separate the

vibration behavior at the reference signals to each principal component behavior using the information. The separated

component is named as the principal component mode in this study.

4.2 Relationship between principal component mode and vibration mode

The principal component is obtained by extracting correlated factors in the reference signals and each component is

no correlated to the other components by PCA. In addition, measured reference signal at each point is composed of the

superposition of all principal component modes. On the other hand, behavior of a vibration mode in a reference point

synchronizes (having perfect correlation) at the other reference points. Vibration mode also has orthogonality to the

other mode. Hence, each principal component mode has been regarded as the vibration mode in some previous studies

(Yoshida, 2013; Yoshida, 2014). However, there is almost no discussion about the relationship theoretically in detail

between them. In this section, we discuss the relationship.

As described in Eq. (2), the principal component [T] is obtained by multiplying the reference signal matrix [Ain]

with the matrix [V]. The principal component mode could be expressed in the element of the multiplied matrix of [T]

and [V]T as shown in Eq. (8).

222211122111

2222212112221121

2212211112121111

21

2221

1211

vtvtvtvt

vtvtvtvt

vtvtvtvt

aa

aa

aa

nnnnnn

(8)

4

2



In the principal component mode, the PC 1 vector included in Ref. 1 {t11v11, t21v11 ,,, tn1v11} (left solid box) has a

perfect correlation with the PC 1 vector in Ref. 2 {t11v21, t21v21 ,,, tn1v21} (right solid box) because the relationship

between PC 1 in Ref. 1 and PC 1 in Ref. 2 is always v11:v12 in each row. The PC 1 vector in Ref. 1{t11v11, t21v11 ,,, tn1v11}

(left solid box) and the PC 2 vector in Ref. 1 {t12v12, t22v12 ,,, tn2v12} (left dotted box) has no correlation each other

vector {t12, t22 ,,, tn2}, respectively, which are no correlated by PCA. Therefore, a PC mode has a perfect correlation

with the same PC mode and does not have any correlation with the other PC mode. This is the characteristic of PC

modes composes the reference signals.

On the other hand, the acceleration at the two reference point could be expressed by using eigenmode vector [φ]

and modal coordinate [ξ] as follows.

2221

1211

2121

aa (9)

When the input forces are continuously applied to the system, the above relationship is developed as Eq. (10)

according to the FFT repetition.

2221

1211

21

2221

1211

21

2221

1211

nnnn aa

aa

aa

(10)

The above equation could be developed as follows.

222121212111

2222122121221121

2212121121121111

21

2212

2111

nnnnnn aa

aa

aa

(11)

The solid boxes and the dotted boxes are the first and second vibration modes excited in the operational condition,

respectively. Through the comparison of Eq. (8) and (11), we can see both PC modes and vibration modes consist of

reference signal. The reference point vibration is composed of the superposition of PC modes or vibration modes.

In the upper Eq. (11), if the 1st vibration mode vector in Ref. 1 {ξ11φ11, ξ21φ11,,, ξn1φ11} (left solid box) and the same

mode in Ref. 2 {ξ11φ12, ξ21φ12,,, ξn1φ12} (right solid box) have a perfect correlation, and the 1st vibration mode in Ref. 1

{ξ11φ11, ξ21φ11,,, ξn1φ11} (left solid box) does not have any correlation with the 2nd mode in Ref. 1 {ξ12φ21, ξ22φ21,,,

ξn1φ21} (left dotted box), these relationships are exactly same as the PC modes. And then, the PC modes calculated in

the procedure of OTPA are regarded as the vibration modes.

About the relationship among the vibration modes at the operational condition, the 1st vibration mode in Ref. 1

(left solid box) apparently has a perfect correlation with the 1st mode in Ref. 2 (right solid box) because the

relationship among them at each row is always φ11 : φ12. The relationship between the 1st mode in Ref. 1 (left solid

box) and the 2nd mode in Ref. 1 (left dotted box) depends on the excitation condition to the two vibration modes. In

other words, the relationship depends on the relationship between the 1st vibration mode vector {ξ11, ξ21,,, ξn1} and the

2nd vibration mode vector {ξ12, ξ22,,, ξn2} under the operational condition. Because the 1st mode in Ref. 1 is the result

of the multiplication of the 1st vibration mode vector with φ11, and the 2nd mode in Ref. 1 is the result of the

multiplication of the 2nd mode with φ21.

However, there is a special case at around the natural frequency of the vibration mode regardless the relationship

among the vibration modes. If the frequency of the input force closes at the natural frequency of the 1st vibration mode

in the operational condition, the amplitude of the 1st mode becomes much larger than the 2nd mode. In this situation,

the 1st vibration mode becomes dominant at both reference points as shown in Eq. (12).

because these vectors are just the result of the multiplication of v11 and v21 with PC 1 vector {t11, t21 ,,, tn1} and PC 2

2



121111

12211121

12111111

222121212111

2222122121221121

2212121121121111

21

2212

2111

nnnnnnnn aa

aa

aa

(12)

In the above situation, vibration behavior of the reference signals are determined mostly by the 1st vibration mode

(solid boxes) and each reference point vibrates with high correlation each other by the dominant vibration mode.

On the other hand, PC mode in a reference point also vibrates having perfect correlation with the same PC mode in

the other reference points. And the PC mode does not have any correlation with the other PC modes as described

before. In addition, the PC number is determined according to the size of each PC in PCA procedure, therefore, the

reference point vibration dominated by the 1st vibration mode (may include very small 2nd mode influence) is

calculated as the PC 1 mode in the PCA procedure theoretically in this situation. In this case, PC 2 includes very small

non-correlated component to PC 1 like measurement errors or 2nd vibration mode.

This is the basic theory background in which the PC 1 mode indicates the dominant vibration mode at around the

natural frequency of the vibration mode. The above described special cases are expected to occur at around peak

frequencies of averaged acceleration level in the operational condition as shown in the following image.

Fig. 9 Image of the average spectrum of a reference point vibration at an operational condition.

5. Obtaining principal component and vibration modes, and the relationship of them

5.1 Obtaining principal component mode

In this section, we actually obtained the principal component mode from the measured operational data of the

employed simple two plate structure. We verified the relationship between the calculated principal component mode

and the vibration mode of the structure. To obtain the principal component mode and compare the principal component

mode shape with the vibration mode shape, we increased the number of reference point from four to twelve (P. 1 to P.

12 in Fig. 2). Figure 10 shows the average spectrum of the principal component calculated from the 12 reference point

signals. Here, 12 principal components were calculated theoretically by using 12 reference point signals, and principal

component (PC) 1, 2, 3 and 4, those were relatively larger components, are shown in Fig. 10.

40

60

80

100

120

0 100 200 300 400 500Freq. (Hz)

Acc

eler

atio

nle

vel

(d

B)

2



Fig. 10 Calculated principal component of PC 1, 2, 3 and 4 at the operational condition.

Seven vibration peaks of the principal component 1 (PC 1) indicated by arrows were observed. This indicates

seven vibration mode are supposed to be excited by the actual input force in this operational condition if the principal

component expresses the vibration mode correctly.

5.2 Obtaining vibration mode

Subsequently, we analyzed vibration modes of the supporting plate experimentally by conducting an impact

measurement test to evaluate the relationship between the principal component mode and the vibration mode. Seventy

seven points on the supporting plate were employed in the test for obtaining vibration modes experimentally. Table 1

(a) shows the extracted vibration modes (from 1st vibration mode to 15th vibration mode) under 550 Hz and (b) shows

the frequencies where PC 1 has the peak (named as 1st PC 1 to 7th PC 1) for the comparison.

Table 1 Obtained vibration modes and principal component 1.

As the result, 15 vibration modes were found on the supporting plate, and some natural frequencies of the vibration

modes were similar with the vibration peak frequencies of the principal component. However, the number of the

vibration peaks of the principal component was less than the number of the vibration modes. This difference is

considered to be generated by the difference of the input force condition. In the test for obtaining the vibration modes,

more vibration modes of the supporting plate were excited by conducting the impact measurement at all 77 points on

the plate. Oppositely, the input force was given at a fixed point and the frequency range of the force was increased

constantly by the shaker in the operational condition. Therefore, only some vibration modes of the supporting plate are

considered to be excited in the operational condition. In other words, excited vibration modes at the operational

condition could be extracted in many vibration modes of the structure utilizing the principal component. This leads to

conduct effective countermeasure by focusing only the excited vibration modes at the operational condition if we can

associate the vibration mode with the principal component 1 that has the largest size among all principal components.

5.3 Relationship between principal component mode and vibration mode

Principal component 1 (PC 1) at the operational condition was found to have vibration peaks at the similar

frequencies with the natural frequency of some vibration modes. Then, we investigated the relationship between the

PC 1 mode and the obtained vibration mode by comparing the mode shapes. Figure 11 (a) and (b) shows the mode

shapes of the PC 1 at 59 Hz and 205 Hz, respectively. The vibration mode shapes at 60 Hz (1st mode) and 225 Hz (6th

mode) are also shown in Fig. 12 (a) and (b). Noting that the difference of the measurement point number between them;

the number of principal component was 12 according to the number of the reference point and the number of the

Freq. (Hz)

Acc

eler

atio

nle

vel

(d

B)

50

70

90

110

130

150

0 100 200 300 400 500

PC1

PC2

PC3

PC4

(a) Obtained vibration modes (b) Obtained PC1peaks

Vib. Mode 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th PC. Mode 1st 2nd 3rd 4th 5th 6th 7th

Freq. (Hz) 60 71 153 171 181 225 268 271 323 374 459 484 502 535 543 Freq. (Hz) 59 162 205 252 348 428 479

Damp. (%) 2.7 4.8 1.5 1.4 2.7 1.9 1.3 1.6 1.1 1.3 1.4 0.6 1.2 1.2 1.5

2



vibration mode was 77.

(a) PC mode at 59 Hz (b) PC mode at 205 Hz

Fig. 11 Principal component mode shapes at 59 and 205 Hz.

(a) Vibration mode at 60 Hz (b) Vibration mode at 225 Hz

Fig. 12 Vibration mode shapes at 60 and 225 Hz.

As shown in these figures, the principal mode shapes were observed to be very similar with the vibration modes

each other. Then, we tried to calculate the relationship quantitatively between the PC 1 mode and the vibration mode by

using information of the amplitude and the phase in each mode shape. The relative amplitude and phase ratio of each

principal component could be obtained by using the row elements in Eq. (8) and the ratio of the vibration mode at the

same reference points with the principal component were also extracted from the row elements in Eq. (11). After then,

the correlation coefficient of the relative amplitude and phase among the reference points between each principal

component mode shape and the vibration mode shape was calculated. Table 2 shows the obtained absolute of the

correlation coefficient.

Table 2 Correlation coefficient between PC 1 mode shape and the vibration mode shape.

As shown in the bold underlined number in the above table, each PC 1 mode excited at the operational condition

was found to have very high correlation with one of the vibration modes. On the other hand, some vibration modes

(2nd, 3rd, 5th, 8th, 9th, 11th, 13th and 15th) did not have high correlation. This reveals that the PC 1 expresses

vibration mode excited at the operational condition, and some vibration modes were not excited at the condition. In

addition, the frequency of the PC 1 modes was generally lower than the natural frequency of the associated vibration

mode. This tendency is considered to be caused by the additional weight by the attached 12 accelerometers and the

shaker on the supporting plate in the operational condition. However, even though the frequency of the peak PC mode

at the operational condition is slightly different from the natural frequency of the vibration mode, we will be able to

find the associated vibration mode using the above mentioned method by searching the mode at wide frequency range

in case the mode shapes between them have consistency.

In addition, we discussed the PC 1 mode could be expected to be almost the same as the dominant vibration mode

No 1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th 11th 12th 13th 14th 15th

No Freq. 60 71 153 171 181 225 268 271 323 374 459 484 502 535 543

1st 59 0.88 0.13 0.18 0.17 0.02 0.20 0.24 0.16 0.17 0.79 0.19 0.03 0.11 0.17 0.17

2nd 162 0.20 0.25 0.26 0.85 0.29 0.23 0.05 0.07 0.21 0.15 0.19 0.28 0.08 0.12 0.10

3rd 205 0.16 0.09 0.14 0.10 0.15 0.94 0.39 0.50 0.08 0.18 0.08 0.47 0.14 0.12 0.13

4th 252 0.13 0.32 0.08 0.05 0.09 0.37 0.76 0.53 0.06 0.13 0.25 0.28 0.17 0.28 0.24

5th 348 0.81 0.20 0.12 0.10 0.07 0.26 0.16 0.20 0.31 0.84 0.25 0.34 0.06 0.12 0.13

6th 428 0.24 0.09 0.27 0.23 0.16 0.42 0.16 0.21 0.23 0.28 0.50 0.89 0.19 0.20 0.17

7th 479 0.09 0.44 0.42 0.15 0.17 0.19 0.11 0.08 0.21 0.23 0.26 0.53 0.71 0.74 0.26

Vibration mode

PC

1 m

od

e

8

2



at the resonance frequency. Nevertheless, some correlation coefficients between the associated PC 1 and the vibration

modes were not very high (<0.8). One of the reasons is considered to relate how degree the single vibration mode

dominates to the other modes in the resonance frequency band in addition to the influence of the measurement error.

The correlation coefficient between the PC 1 mode at 252 Hz (4th PC 1 mode) and the associated vibration mode at 268

Hz (7th vibration mode) was 0.76. The averaged vibration peak level at 252 Hz at the operational condition (Fig. 10)

was not so high comparing with the other peaks. Hence, the PC 1 mode is considered to be affected (or deteriorated) by

the other vibration modes slightly in addition to the dominant vibration mode (7th vibration mode).

About the 7th PC 1 mode at 479 Hz, the coefficient with the associated vibration mode (14th vibration mode) was

also not so very high at 0.74. The 14th vibration mode was 2nd bending mode along the lateral direction as shown in

Fig. 13 (a) and the associated 7th PC 1 mode did not express the 2nd bending mode well as shown in Fig. 13 (b).

(a) 14th vibration mode at 535 Hz (b) 7th PC 1 mode at 479 Hz

Fig. 13 Associated 14th vibration and 7th PC 1 modes.

Because the number of the reference signals along the lateral direction for the PC mode was only three and

describing the 2nd bending mode accurately was difficult using only the three points. The insufficient number of nodes

for representing the 14th vibration mode is one of the main reasons of the correlation decrease. Similar phenomena

(fault high or low correlation) caused by the difference of the node number were observed in the other combination

such as the combination of the 10th vibration mode at 374 Hz and the 1st PC 1 mode at 59 Hz. Although we will be

able to find out the fault combination by checking the wide frequency difference and the comparing the coefficient, this

phenomenon had better to be taken into account for the application of the proposed method.

About the relationship between the 14th vibration mode and the 7th PC 1 mode, even though the number of the

node in the PC mode was not sufficient for describing the vibration mode shape accurately, the correlation coefficient

of them was over 0.7, which is a criterion value regarded as “high correlation” in general, and the PC 1 mode have

already clarified to express the dominant vibration mode at around the natural frequency. From them, the vibration

modes relating the PC 1 mode were considered to be successfully found out at the operational condition in many modes

using the association method.

6. Contribution of vibration mode and transfer function of principal component

If we can extract not only the excited vibration modes at the operational condition but also obtain the contribution

of the each excited modes to the response point, this information becomes more useful for applying effective

countermeasure to reduce the response point vibration. The contribution of principal component to the response point is

obtained by multiplying the principal component [T] with the principal component transfer function [B] as described in

Eq. (3). Figure 14 shows the calculated contributions of PC 1 to 4 having relatively higher contribution in total 12

principal component contributions.

9

2



Fig. 14 Contribution of PC 1, 2, 3 and 4 to the response point vibration.

The contributions of principal component were observed to be separated more clearly than the contribution of

reference point (Fig. 3). In addition, PC 1 contribution was observed to have the largest in all contributions. In the PC 1

contribution, four vibration peaks at 162, 205, 348 and 479 Hz indicated by bold arrows were especially high level. In

contrast, the contribution at 59, 252 and 428 Hz indicated by dotted arrows were low level. The vibration modes

excited at the three frequencies seem to be unimportant for the reduction of response point vibration although the

vibration was excited largely before multiplying the transfer function as shown in Fig. 10. The vibration at these

frequencies is considered to be transferred inefficiently between the supporting and response plates comparing with the

other vibration having large peaks at the response point. The PC 1 modes at 162, 205, 348 and 479 Hz, those were

found to be important components, have already known to associate with 4th, 6th, 10th, and 14th vibration modes.

Hence, these analytical results inform us which vibration modes should be measured intensively among many modes of

the structure for the reduction of the response vibration by utilizing principal component contribution in OTPA.

On the other hand, the contribution of principal component at 505 Hz indicating by triangle in Fig. 14 was found to

have high contribution to the response point although the principal component did not have high vibration peak itself at

the frequency as shown in Fig. 10. This phenomenon is considered to be generated by the transfer characteristic from

supporting plate to the response point. Then, we investigated the calculated transfer function of PC 1 (principal

component transfer function) to the response point as shown in Fig. 15.

Fig. 15 Transfer function of PC 1 to the response point.

As shown in this figure, the transfer function had large peak at the frequency (indicated by triangle). This implies

that resonance seems to have occurred at the response plate at the frequency and this is the factor to increase the

contribution of PC 1. Subsequently, we obtained a point inertance of the response point and the vibration mode of the

response plate by an impact measurement test to obtain the vibration characteristic of the response plate. Figure 16 (a)

and (b) show the point inertance at the response point and the vibration mode of the response plate at 501 Hz,

respectively.

Freq. (Hz)

Acc

eler

atio

nle

vel

(d

B)

30

50

70

90

110

130

0 100 200 300 400 500

PC1

PC2

PC3

PC4

-100

-80

-60

-40

-20

0

0 100 200 300 400 500

Freq. (Hz)

PC

tra

nsf

er f

unct

ion

(dB

)

2



-20

0

20

40

60

80

0 100 200 300 400 500

Freq. (Hz)

Tra

nsf

er f

unct

ion

(dB

)

(a) Point inertance at response point (b) Vibration mode of response plate at 505 Hz

Fig. 16 Point inertance and vibration mode of supportng plate.

From the result, a vibration peak was found at the response plate at around the frequency and the response point

had the highest amplitude at the vibration mode. This result indicates the countermeasure had better to be applied on

the response plate not to the supporting plate because the resonance occurred at the response side. From these analyses,

we could obtain which part should be measured intensively at the input side or the response side in addition to the

important vibration modes to the response point by utilizing principal component in OTPA.

7. Countermeasure utilizing the high contributing principal component

Through the OTPA procedure and evaluation method to associate the PC 1 mode with the dominant vibration

mode, we found out high contributing vibration modes to the response point vibration. In addition, we could determine

which response side (response plate) or reference side (supporting plate) affects largely to the high vibration peak at the

response point by assessing the principal component transfer function. We then tried to reduce the response point

vibration effectively using the analytical results. At first, we conducted a countermeasure to reduce the high vibration

peak of the response point at 162 Hz in two ways. In the frequency band, the PC 1 (2nd PC 1 mode) was calculated to

have highest contribution and 4th vibration mode (Fig. 17) was found to associate with the PC 1 mode. Hence, we

applied the countermeasure considering the vibration mode shape.

Fig. 17 Vibration mode shape of the 4th mode associated with 2nd PC 1 mode.

In this countermeasure, we put six weights (total 150 g) to the separated points on the supporting plate. To verify

the efficiency of the countermeasure considering the vibration mode, the weights were put at the antinode points to

inhibit the mode as shown by open circles in Fig. 18 (a) (Case A). In Case B, the same weights were put at the node

point as shown by open circles in Fig. 18 (b) to compare the vibration reduction effect at the response point. We also

measured the original vibration level at the response point as the basis of the level (Case C).

Response point

2



The acceleration vibration level at the response point in the identical operational condition with the previous test

was obtained in these three conditions. Figure 19 shows the average vibration levels at the response point.

Fig. 19 Average response point vibration level at around 162 Hz and comparison in three conditions. Case A: six weights

were put at the antinodes of the high contributing vibration mode, Case B: the weights were put at the nodes, Case

C: no weights (original condition).

Horizontal and vertical axes indicate the frequency and the average acceleration level at the response point,

respectively. Gray solid, dotted and black solid lines indicate the level in Case A, B and C, respectively. As shown in

the above figure, the vibration levels at around 162 Hz (indicating by the dotted circle) were observed to be reduced by

adding the weights in both cases (Case A and B). However, the decreased level was different. Although the vibration

level was reduced only about 3 dB by adding the weights at node of the mode in Case B, the level was reduced about

8 dB by adding them at antinode points in Case A. This indicates that the vibration level at the response level could be

reduced more efficiency by considering the high contributing vibration mode shape.

In the second example, we tried to reduce the response vibration peak at 206 Hz. In this frequency band, 6th

vibration mode (4th PC mode) as shown in Fig. 20 was found to be the important mode that should be measured for the

reduction of the peak vibration at the response point.

Fig. 20 Vibration mode shape of the 6th mode associated with 4th PC 1 mode.

Then, we added the same six small weights (total 150 g) on the supporting plate in two ways as shown in Fig. 21

and compared the reduction level. In Case A, the six weights were put on the antinode points as indicated by opened

circles in Fig. 21 (a), and the same six weights were put at the node points of the vibration mode as shown in the

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100 150 200 250

Freq. (Hz)

Acc

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B)

CaseA: Antinode

CaseB: Node

CaseC: No weight (Original)

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(a) Case A: weights at antinode points (b) Case B: weights at node points

Fig. 18 Weight attachment points for the vibration reduction of the response point at 162 Hz.

2



opened circles in Fig. 21 (b) in Case B. In Case C, the weights were not put to measure the original vibration level at

the response point.

Figure 22 shows the average response point vibration levels at the operational condition in these three conditions.

Fig. 22 Average response point vibration level at around 206 Hz and comparison in three conditions. Case A: six weights

were put at the antinodes of the high contributing vibration mode, Case B: the weights were put at the nodes, Case C:

no weights (original condition).

As shown in the results, the vibration level at around 206 Hz (indicated by dotted circle) was reduced largely

especially by adding the weight at the antinode points (Case A) about 8 dB although the vibration was not reduced

largely by adding them at the node points (Case B). From these verifications of the countermeasure, the

countermeasure considering the high contributing vibration mode obtained by the modified OTPA was found to realize

effective countermeasure.

As the final countermeasure, we tried to reduce the response point vibration at 505 Hz, where the principal

component transfer function had a peak. From the analytical result, the resonance at the response point was considered

to be a main factor. Then, we put a concentrate mass of 150 g to the response point to reduce the resonance vibration

peak at the frequency in Case A. In Case B, the six weights (total 150 g) were put on the supporting plate as same as

Case A in the countermeasure example at 206 Hz as shown in Fig. 21 (a). In Case C, these weights were not put to

measure the original vibration level at the response point. Figure 23 shows the average response point vibration levels

at the operational condition in these three conditions.

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150 200 250 300Freq. (Hz)

Acc

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CaseA: Antinode

CaseB: Node


13

(a) Case A: weights at antinode points (b) Case B: weights at node points

Fig. 21 Weight attachment points for the vibration reduction of the response point at 206 Hz.

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Fig. 23 Average response point vibration level at around 505 Hz and comparison in three conditions. Case A: concentrate

weight was put at the response plate, Case B: the weights were put on the supporting plate at the antinode points of

6th vibration mode, Case C: no weights (original condition).

As shown in the above figure, the vibration level at around 505 Hz (indicated by dotted circle) could be observed to

be reduced very largely at about 20 dB by adding the concentrate weight at the response point. The reduction level was

much larger than Case B where the six weights were added on the supporting plate even though the total additional

weight was the same as 150 g. Consequently, more effective countermeasure was achieved by putting the weight at the

response side when the principal component transfer function had a peak at the target frequency.

From these countermeasure examples, we could verify the effectiveness of the application of the modified OTPA

utilizing principal component and the association method of the high contribution PC 1 mode with the dominating

vibration mode.

5. Summary

In this study, we carried out vibration tests using two plates structure and analyzed the measured signal by OTPA

utilizing the principal component to obtain useful information for effective countermeasure at the response point. In

addition, we discussed the relationship between the principal component mode and the vibration mode theoretically and

derived the PC 1 mode expresses the dominant vibration mode at around the resonance frequency. Subsequently, we

calculated the contribution of the principal component by multiplying the principal component and the transfer function

through the modified OTPA procedure.

As the result, we obtained which part (supporting plate or response plate) affects the vibration peak largely at the

response point by comparing the principal component level and the transfer function at the frequency bands where the

response point vibration was large. In addition, we developed a method to associate the high contribution PC 1 mode

with the dominant vibration mode by evaluating the mode shape correlation. Consequently, high contributing vibration

mode to the response point could be found out. At last, we actually attempted the vibration reduction at the response

point to verify the effectiveness of the proposed analytical procedures in several cases. The results showed that the

vibration at the response point could be reduced effectively by applying intensive countermeasure according to the high

contribution vibration mode shape in case the principal component level was a main factor. In addition, in case the

transfer function characteristic of the principal component made a vibration peak at the response point, the vibration

was successfully reduced more by applying the countermeasure at the response side comparing with the

countermeasure at the reference side on the supporting plate. These countermeasures were very normal method itself

for the vibration reduction. Accordingly, the association method of the PC 1 mode with the vibration mode does not

relate the vibration reduction method directly. However, the effectiveness of the normal countermeasure is increased

largely by combining the countermeasure method with the association method proposed in this study obtained by the

modified OTPA by focusing the reduction target vibration modes in many modes.

As described above, more informative result could be obtained by the modified OTPA in case where reference

signals have high correlation each other. Furthermore, if the vibration mode is obtained through a simulation technique,

we can also expect to extract high contributing vibration modes in the simulated many vibration modes to the response

point vibration by applying the proposed method. In addition, the countermeasure could be considered utilizing the

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450 500 550 600Freq. (Hz)

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CaseA: Response plate

CaseB: Supporting plate


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simulation technique effectively by focusing only the important (high contributing) vibration mode. This realizes the

combination analysis between the experimental OTPA and the simulation technique for finding out important vibration

mode and the countermeasure to reduce the response vibration more instantly and effectively.

Acknowledgments

The authors thank Mueller-BBM in Germany and TOYO cooperation in Japan for supporting this work.

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