PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Low-vibration design for shell structure based on
structural intensity distribution
Takeshi MIYAMA1; Hiroki NAKAMURA2; Toru KIKUCHI2; Toru YAMAZAKI2
1

Graduate School of Kanagawa University, Japan
2

Kanagawa University, Japan

ABSTRACT
Reducing structure-borne sound is an important issue for mechanical industrial products. This paper focuseson countermeasures at the vibration-energy transmission path rather than at the sound source as a potentiallybetter alternative for controlling structure-borne sound. The propagation path can be determined by either
measuring or calculating the structural intensity (SI), which is defined as the instantaneous rate of energy
transfer per unit area in a structure. Although there have been many reports concerning SI, none can be foundthat implemented structural design and modification based on the SI distribution. This paper presents amethod for implementing structural design by using the SI distribution. First, the ability to control vibration
by means of the SI is discussed by comparing the kinetic energy and SI in wave solutions. A shell structureis then designed using the proposed method. The results show that the SI can be used to identify the location
to be modified and changing the structure at that location facilitates low vibration. Also presented are theresults of an experimental application using a sheet sensor developed specially for visualizing the flow ofstructural vibration energy. It is shown that this sheet sensor makes it easy to locate the countermeasure areas
on a shell structure.
Keywords: Transmission, Structural intensity, Power flow

1. INTRODUCTION
Reducing structure-borne sound is an important issue for mechanical industrial products. Structure-borne
sound in manufactured products can be controlled at three locations, namely, the source, the transmission
path, and the receiver, and it is often easiest and most effective to control it at the source as the upstream
origin of noise and vibration. However, there are many cases in which the source properties cannot bechanged without impacting the product performance, and control at the receiver can also be difficult because
noise and vibration spread out, making that approach inefficient. This leaves the transmission path aspotentially the best approach to controlling structure-borne sound. The vibration transmission path can be
determined by either measuring or calculating the structural intensity (SI), which is defined as theinstantaneous rate of energy transfer per unit area in a structure (1). There have been many reports concerningSI, but to the best of our knowledge none have implemented structural design and modification based on the
SI distribution (4–6).
Against this background, we have conducted a study aimed at establishing a method for low-noise andlow-vibration structural design based on the SI (i.e., the flow of vibration energy). In our previous research
on SI (7–9), we derived an SI mode expansion equation and showed that SI can be classified roughly as beingof either translational or rotational flow type. These flow types can be used to promote transmitted power via
translational flow and reduce transmitted power via rotational flow. We also developed a simple andapproximate technique for measuring the propagation of vibration energy (10).
The present paper presents a method for specifying the countermeasure location for low-vibration designby using the SI distribution. First, we discuss the ability to control vibration by means of SI by comparing
the kinetic energy and SI for wave solutions on a beam. We show that the vibration energy and SI are highly
correlated when the progressive wave is dominant, which corresponds to large structural damping, whereas
12r201202408ui@gmail.com
toru@kanagawa-u.ac.jp

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the kinetic energy is small when the progressive wave is small. Therefore, under large structural damping,
the SI can give important information about the behavior of the progressive wave. We then show that the SI
distribution can be used to identify the location to be modified and changing the structure at that location
facilitates low vibration. Experimental results suggest that an SI-based countermeasure is useful for reducing
the vibration to the same extent over a wide range of frequency. We also present the results of an experimental
application in which we use a measurement sheet developed by us to visualize the flow of structural vibrationenergy. We show that our measurement sheet makes it easy to find the countermeasure location on a shellstructure.

2. IDENTIFICATION OF COUNTERMEASURE LOCATION BASED ON
STRUCTURAL INTENSITY DISTRIBUTION
We have previously investigated structural design methods to reduce vibration and radiated noise of
mechanical structures using the SI distribution. From this, we clarified that the vibration can be reduced by
the vibration energy flow to the place is blocking or flowing to other places. Being similar to controlling
water flow, this idea is easily understood. The main energy flow for reducing vibration is that represented by
the progressive wave. If the reflected wave also becomes influential, then it can be difficult to determine theflow of the progressive wave, but it remains possible to identify the countermeasure location from the energy
flow including the reflected wave. Therefore, we describe a method for identifying the countermeasurelocation based on the SI distribution.
2.1 Correlation between Kinetic Energy and Structural Intensity
First, consider the bending vibration in the far field on a uniform beam along the x axis. The vibration
displacement ζ(x) is expressed as a progressive wave with complex amplitude A and a backward wave withcomplex amplitude B, that is,] x

Ae jkx  Be jkx(1)

and the kinetic energy KE(x) and the SI I(x) are evaluated as
2

2

2

KE x

m jZ]

mZ 2 A  B  2 Re ª¬ AB*e  j 2 kx º¼

(2)

I x

§ w] *
w] ·
2
2
3
] *
jZ EIk 2 ¨ ]¸ 2Z EIk A  B
w
w
x
x
©
¹

(3)

where k is the bending wavenumber, m is the structural mass density, ω is the angular frequency, EI is thebending stiffness, j is the imaginary unit, and Re[] represents the real part.
The kinetic energy is the squared sum of the amplitudes of the progressive wave (A) and backward wave(B) as shown in Eq. (2), whereas the SI is the difference between them as shown in Eq. (3). In the case of astanding wave, the amplitudes of both waves are the same and the SI is zero because A = B. In the absenceof a backward wave, only the progressive wave is obtained, and the kinetic energy and SI are each
proportional to A2. In other words, the correlation between kinetic energy and SI increases as the progressive
wave becomes more dominant. It is therefore important to focus on the progressive wave of SI. In numericalanalysis such as the finite-element method (FEM), if the attenuation of the object is increased, then the effectof the progressive wave can be intensified.
2.2 Test Structure
In this section, a steel-panel structure comprising eight flat plates as shown in Fig. 1 is used to describe
the advantages of focusing on the progressive wave. Excitation is applied to subsystems 1 and 2, and thevibrations of subsystems 3 and 4 are to be reduced. Consider the problem in which the vibration energy ofsubsystems 3 and 4 is reduced to be less than the power input into subsystems 1 and 2. Consider also changing
the energy propagation paths through subsystems 5–8. The initial thickness of each subsystem is set as
follows: 2 mm (subsystems 1 and 2), 4 mm (subsystems 3 and 4), and 1 mm (subsystems 5–8).

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Kinetic energy normalized by Pin

Figure 1 – Test panel comprising eight steel-plate subsystems [thickness of subsystems: 2 mm (1 and2), 4 mm (3 and 4), and 1 mm (5–8)]

10

2

10

0

10

-2

10

-4

10

-6

䢢

InitialCountermeasure

䢢

0

50

100
Frequency [Hz]

150

200

Figure 2 – Vibration energy of subsystem 3 in test structure shown in Fig. 3
The black line (labeled “Initial”) in Fig. 2 shows the frequency distribution of the kinetic energy ofsubsystem 3 normalized by the power input to subsystems 1 and 2 under an external force of 1 N obtainedusing FEM calculation. This result is for the lowest loss factor of 0.01. The energy is calculated from thespatial average throughout the system. The response is larger around 130 Hz, so we focus on this frequency
range.
2.3 Advantages of Focusing on Structural Intensity
In this section, we explain the advantages of focusing on the progressive wave. Consider the response
around 130 Hz as shown in Fig. 2 with the black line (Initial). In the frequency range around 130 Hz, Fig. 3
shows typical examples of the forced vibration shapes at 131 Hz and 135 Hz. It can be seen that the vibration
shapes differ drastically despite the frequency difference of only 4 Hz. Also, it is difficult to identify
appropriate countermeasures from the vibrational displacement distribution. Fig. 4 shows the SI
distributions at these frequencies with the lowest loss factor (0.01). It is immediately clear that these SI
distributions are dissimilar, as for the vibration shape. This is because the backward wave due to reflection
at the boundary edge is strong. Comparing the SI distributions shown in Fig. 4 in more detail reveals certain

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(a) 131 Hz(b) 135 Hz

Figure 3 – Vibration displacement distributions close to 130 Hz

9

9

8

8

7

7

6

6
5

4

4

2

2

0

0

(a) 131 Hz(b) 135 Hz

Figure 4 – Calculated structural intensity (SI) distributions with lowest loss factor (ηn = 0.01)
similarities between them, such as a vortex flow distribution on subsystems 3 and 4 and a translational flow
on subsystem 5. Adopting countermeasures for these similar flows would reduce the vibration over a wide
frequency range because the progressive wave does not change over a wide frequency range (11).
2.4 Structural Change Based on Structural Intensity
In this section, SI-based design is applied in the vicinity of 130 Hz as shown in Fig. 2. As shown in Fig. 4,
the SI distribution near 130 Hz is distributed across subsystem 5. Energy is transmitted along the main flow
to subsystems 3 and 4, therefore suppressing the energy flow to subsystems 3 and 4 is considered by changing
the structure to inhibit the main flow. In this case, the energy transferred from subsystems 1 and 2 tosubsystem 5 propagates to subsystems 3 and 4 so as to traverse the center of the structure. Therefore, thecountermeasure is to increase the plate thickness for the purpose of suppressing both the flow fromsubsystem 5 to subsystems 3 and 4 and the flow in the central part of subsystem 5.
The reduction in the vibration of subsystems 3 and 4 is confirmed by FEM (loss factor: 0.01). The redline in Fig. 4 shows the frequency distribution of the summed kinetic energies of subsystem 3 and 4.
Fig. 5 shows the post-countermeasure SI distribution at 135 Hz. The prominent flow across the bottomof subsystem 5 is believed to be due to vibration suppression. Propagation within the region in which the

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1

1

9

9

8

8

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0

(a) 131 Hz

0
0

02

04

06

08

(b) 135 Hz

1

12

14

Figure 5 – Structural intensity distributions after countermeasure with loss factor of ηn = 0.01
plate thickness of the upper part of subsystem 5 is increased, with propagation being promoted to the lower
part instead. In other words, when the structure is changed based on propagation, focusing not only the partto be interrupted but also the part promoted by the structural change can greatly affect the vibration level ofthe receiver.
In this paper, we discuss partial structural change based on the SI distribution, but we do not identify themost efficient place to implement structural change for an arbitrary frequency. In future work, we will clarify
how much structural change is necessary, which range of frequency to target, and whether vibration
reduction is possible over a full order of magnitude of frequency.

3. IDENTIFICATION OF COUNTERMEASURE LOCATION BASED ON
STRUCTURAL INTENSITY MEASUREMENT
We aim to develop a system that can plan vibration-reduction measures from a measured SI distribution.
In Section 2, we discussed how to identify the vibration-reduction countermeasure location from the SI
distribution. In this section, we use a specially developed sheet sensor to measure the SI distribution. By
doing so, we identify the vibration-reduction countermeasure location and confirm vibration reduction andchange of the SI distribution.
3.1 Test Structure and Method
The target structure used for this experimental measurement is shown in Fig. 8(a), where the numbers
labeled “#” are the plate numbers. Consider reducing the vibration of plate 3 when the power input is toplate 1. The ideal energy flow for low vibration is transmitted to plate 2 without being transmitted to plate 3.
As an initial structure, each flat plate is made of steel and is 2 mm thick. Fig. 8(b) shows the state inwhich the visualization sheet is placed on the test plate. The sheet measures the SI distribution at seven points
on a straight line using 16 microelectromechanical sensors. The test structure is bolted at one point on plate 1,
and the end of plate 2 is placed on the damping material as shown in Fig. 9; all other places are free. Whenthe edge of plate 1 is excited with an impulse hammer, the accelerance of one point on plate 3 is measured.
3.2 Identification of Countermeasure Location
The red line in Fig. 10 shows the results of measuring the accelerance in the initial state (when each flat plateis 2 mm thick). The vibration is clearly large around 100 Hz, therefore the results of measuring the SI in thatstate are shown in the lower part of Fig. 9(a). The SI distribution is toward plate 3, from which we conclude
that the vibration of plate 3 is large. It is therefore possible to identify the vicinity of the attachment point ofplate 3 as the countermeasure point so as to inhibit the flow to plate 3.

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(a) Test T-shaped panel structure(b) SI measurement sheet on test structure

Figure 8 – Test T-shaped structure and apparatus for measuring SI

2

Excitation

2

2

4

Excitation

2

Damp.
sheet

2

1

2

Excitation

Damp.
sheet

2

䢰䢲䢺

䢺

䢺

䢰䢲䢹

䢹

䢹

䢰䢲䢸

䢸

䢸

䢰䢲䢷

䢷

䢰䢲䢶

䢶

䣏䣇䣏䣕䢼䢳䢴䢷䣊䣼

10mm

1

2

Damp.
sheet

䢷
䢶

䣏䣇䣏䣕䢼䢳䢴䢷䣊䣼䣏䣇䣏䣕䢼䢳䢴䢷䣊䣼

䢰䢲䢵

䢵

䢵

䢰䢲䢴

䢴

䢴

䢰䢲䢳

䢳

䢳

䢲

䢲

䢲

(a) Initial

4(b) Countermeasure 1 (CM1)(c) Countermeasure 2 (CM2)

Figure 9 – Measurement conditions and SI results for T-shaped structures with initial configuration and two
types of countermeasure (the numbers on the upper figures show the panel thicknesses in millimeters)

3.3 Countermeasures and Their Effects
Of the various possible countermeasures, we consider the two shown in Fig. 9(b) and (c). In the first plan,
plate 4 is 1 mm thick and plate 3 is 4 mm thick. In the second plan (termed CM2), the 1-mm-thick portion ofplate 4 is extended by 10 mm in the direction of plate 3.
The SI distribution with CM1 is shown in the bottom part of Fig. 9(b), and the results of measuring theaccelerance are shown by the blue line (labeled CM1) in Fig. 10. The energy flow to plate 3 is suppressed,
and the vibration of plate 3 is reduced. Similarly, the SI distribution with CM2 is shown in the bottom partof Fig. 9(b), and the results of measuring the accelerance are shown by the green line (labeled CM2) in Fig. 10.
The energy flow to plate 3 is further reduced, as is the acceleration of plate 3.
Although the countermeasure location is clearly identifiable in this case, we have shown that it can befound from the SI distribution. Also, as in Section 2, Fig. 10 confirms that the vibration is reduced over awide frequency range.
The purpose of this research is to identify the countermeasure location, and various countermeasures canbe considered. In addition, because the SI distribution differs depending on the frequency, thecountermeasures using SI are different from those in other frequency ranges. In other words, it cannot be
avoided that specific countermeasures bring about to a trade-off between frequency ranges.
From the above discussion, we have confirmed that it is possible to identify the countermeasure location
using the SI measurement sheet. In future work, we will assess the validity of using measurements of the SI
distribution to identify the countermeasure location for structures that are not uniform flat plates.

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Accelerance [dB ref 1.0]

100

Initial

80

CM1

CM2

60

40
20
0
-20
-40
0

100

200
300
Frequency [Hz]

400

500

Figure 10 – Comparison of measure targeted for reduction, namely accelerance of plate 3 of T-shaped structure

4. CONCLUSIONSIn this paper, we discussed how to identify countermeasure locations based on SI distributions andpresented numerical examples. We also confirmed the feasibility of the method by means of experiments
using an SI measurement sheet that is under development. We summarize our findings as follows.
1) SI is highly correlated with kinetic energy for structures that are either heavily damped or away fromresonance. Even when lightly damping or at resonance, reducing the progressive wave lowers both the
kinetic energy and the SI.
2) There are some advantages in SI-based design, which can work well to reduce vibration over a wide
frequency range, including the frequency of interest.
3) The countermeasure location for reducing vibration can be identified using the developed SI
measurement sheet, and the reduction of vibration at the target location and the change of the SI
distribution have been confirmed

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