PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Renaissance of the Structural Intensity Analysis
Heow Pueh LEE1; Kian Meng LIM2
1,2

National University of Singapore, Singapore

ABSTRACT
Structural intensity represents the flow of energy within a structure, subjected to external excitations. Theinterest in structural intensity comes from its practical applications such as damage detection, structure-borne
noise control and structure optimization. By visualizing the flow of power, it is possible to identify thetransmission paths and sources/sinks of mechanical energy within structure. It is a powerful tool formonitoring the structure behavior under different loading conditions which would result in altering energy
flow path by using dissipative elements or mechanical modifications. We have extended the applications to
not only structural mechanics, but also impact mechanics, fracture mechanics, rotating disk, thermal inducedvibration as well as biomechanics. In recent years, Acoustic Black Hole (ABH) is developed for the flexural
waves in the beam plate structures. It attracts lots of attentions in focusing wave energy and reducing
vibration-acoustic level. It also enhances the efficiency of the passive damping systems, reducing the weightof the damping layers. In this paper, we will first present an overview of the various applications of structuralintensity as well as our more recent work on the use of structural intensity for guiding the design of ABH.
Keywords: Structural Intensity, Energy Flow, Acoustic black hole.

1. INTRODUCTION
Structural intensity is a vector quantity representing the power flow per unit cross -sectional area inan elastic medium [1-5]. Analysis of structural intensity in a complex structure could provide essential
information about the locations of vibration sources and dominant energy transmission path [6 -10].
Reported studies of structural intensity were for various applications ranging from identification of
structural defects, vibration reduction, and design optimization [1-17]. There are both reported
experimental and numerical studies. Numerical simulations such as the Finite Element Method could
be used to generate the flow path of Structural Intensity. The earlier reported works were all related
to the vibration of plates and other structures in the general area of structural mechanics. We have over
the years extended the applications to not only structural mechanics, but also impact mechanics,
fracture mechanics, rotating disk, thermal induced vibration as well as biomechanics. The formulationof Structural Intensity can be found in many papers and will not be presented here. Instead, the various
examples of applications will be highlighted in this paper.

2. EXAMPLES OF STRUCTURAL INTENSITY APPLICATIONS
2.1 A Plate with a Damper Excited by a Point Force
Figure 1 shows the classical example of the structural intensity flow path from the point ofexcitation which is the energy source to a damper, which is the energy sink. The flow of the Structural
Intensity can be presented in both the vector plot and the streamline plot. The analysis can be easily
extended to a plate with multiple excitation point sources and multiple energy sinks in terms ofmultiple dampers. A potential application is the diversion of energy away from a crack tip orquiescence area by placing dampers at suitable locations.

1mpeleehp@nus.edu.sg 2 mpelimkm@nus.edu.sg8069

Figure 1 – The Structural Intensity of a plate with a damper subject to a point force excitation.
2.2 Vibration of a Box Structure
Figure 2 shows the structural intensity of a box structure with the external distributed exciting
forces at the bottom of the structure. The structural intensity vector and the streamline distribution are
also shown in the figure. The box structure could be the cabin of a vehicle or a container.

Figure 2 – The Structural Intensity of a box structure subjected to base excitation.
2.3 A Human Head subject to Impact LoadingHead injury is traditionally analyzed in terms of stress wave propagation. Figure 3 s hows thestructural intensity flow pattern of a human head model subject to an impact loading at the forehead aswell as the back of the head. We can see the energy flow towards the spines.

Figure 3 – The Structural Intensity of a human head model subject to frontal and rear impact.
2.4 A Beam with Acoustic Black Hole
Figure 4 shows the structural intensity of a beam with acoustic black hole. More details can befound in the authors’ earlier work [11]. We can observe the band of in-plane pointing SI vectors atdifferent regions along the beam. Higher SI values can also be seen on the top and bottom edges of theribs connecting the beam to the wedge region. Some of the structural intensity vectors at the ribs arepointing out-of-plane instead of in-plane, signifying that the energy/power flow at these regions are
actually not along the beam axis.

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Figure 4 – The Structural Intensity of a beam with Acoustic Black holes.
2.5 A High Heel Shoe with Reinforcement Bars
The Structural Intensity analysis can also be used for shoe design such as the diversion of energy
from the point of impact. Figure 5 shows the Structural Intensity of the base of a high heal shoe withtwo reinforcement bars to divert the energy from the point of impact at the heel to the front of th e shoe.
Other potential designs such as reinforcement base plate has also been investigated.

Figure 5 – The Structural Intensity of a high heel shoe with two reinforcement bars.
The drastic differences between the structural intensity of a flat shoe and a high heel shoe is shownin Figure 6. We can see the severe stress concentrations in the calcaneus for high heel shoe.

Figure 6 – The Structural Intensity of a flat shoe and a high heel shoe.

3. CONCLUSIONSThe paper has presented a few examples from traditional structural mechanics to biomechanics andalso specific examples related to the analysis of shoe designs. There are many other potential
applications from ships to airplanes which have not been covered in this short presentation. For
example in the field of fracture mechanics, it has been derived theoretically that the normal componentof the structural intensity along crack edge equals J-integral. Structural Intensity is the power flux or8071

vector representation of the J-integral at the crack tip [17].

ACKNOWLEDGEMENTS
This study is supported by the Singapore Maritime Institute (Project ID: SMI -2015-MA-08).

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