Development of energy propagation analysis methods for low-frequency
phenomena at BMW
Peter GROBA(1) , Johannes EBERT(1) , Torsten STOEWER(1) , Joachim BÖS(2) , Tobias MELZ(2)(1) BMW Group, Department of Structural Dynamics, Structure-borne Sound and Vibrations, Germany, peter.groba@bmw.de(2) TU Darmstadt, Research group System Reliability, Adaptive Structures, and Machine Acoustics SAM, Germany,boes@sam.tu-darmstadt.de

ABSTRACTIn recent years, a rise in computing power and an enhanced understanding of energy propagation processes have
led to an increased use of the numerical simulation of energetic quantities in vibroacoustical problems bothincluding and excluding the coupled fluids. These energy-based quantities have proven themselves very useful
in delivering information on the behavior of vibrating structures.
Over the last years the BMW Group has invested in research on energetic quantities to evaluate their potential
to support the acoustic engineer in the early stages of the vehicle development process. This paper presents an
overview of the development regarding the structural intensity (STI) and its usage as a tool for the analysis of
low-frequency energy phenomena. It focuses on car structures in steady state vibration with and without fluid
coupling.
Starting with the work of S TOEWER, who conducted numerical simulations in the time-domain to perform
design modifications, the overview continues with analyses conducted by E BERT who linked structure-bornewith air-borne energy propagation. The paper closes by introducing the work of G ROBA who analyzed energy
propagation using power-based quantities calculated from the STI, allowing for a simplified quantification anddescription of energy propagation phenomena.
Keywords: Energy Flow, Power Balance, Structural Intensity, Vibratory Energy

1

INTRODUCTION

The car manufacturer’s efforts to achieve an optimized design regarding noise, vibration, and harshness (NVH)of their products lead to a broad variety of approaches to understand a vehicle’s vibrational behavior. The usedmethods include the estimation of local and global stiffness and the identification of global dynamics via modal
analysis, transfer path analysis, laser-scanning vibrometry, use of acoustic cameras and so forth. While thesemethods deliver information on the structures’ properties as well as a specific vibration behavior, they hold no
information about the mechanisms that lead to the occurrence of a particular vibration phenomenon resulting
from an existing excitation.
This paper focuses on the analysis of energy propagation based on acoustic intensities. While analyzing energy
fields in structures via their mathematical representation by means of vector fields it is necessary to be aware of
the topic’s complexity. Though the information density is one of the major advantages of energetic examinations,
it can be a disadvantage as well. The user faces the following difficulties:• vector fields deliver information on the absolute value and the direction of a given quantity for every
point of the field, i.e. the structure,• within the vibrational energy flow in structures exists a distinction between flexural and longitudinal components,• the high modal density in a given frequency band for structures with a high number of degrees of freedom(DOF),

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• the intrinsic, complex patterns of vibration energy fields make their assessment time consuming and designmodifications are difficult to compare against one another and• so far no consistent relations between design modifications and their influence on the acoustic intensities
could be specified.
The present research projects aim to achieve a better understanding of the acoustic chain of effects, as depicted
in Figure 1. Most of the current methods used to describe the acoustic behavior of a structure solely connect
different points of the acoustic chain without delivering further information about the processes that lead tothe establishment of a specific vibration state. In order to gain further insight into the processes that lead toan acoustic phenomenon in a structure the introduced research aims at finding the links from the excitation(that ultimately leads to a noticeable sound event) to the coupling points between subsystems and finally tothe recipient. The chosen tools for all analysis derived from the works of S TOEWER (1), E BERT (2; 3; 4) andG ROBA (5; 6; 7) are the acoustic intensities, i.e the structural intensity (STI) in solids and the sound intensity(SI) in fluids, respectively. They are introduced in Section 2.

Figure 1. Acoustic chain of effects, from structural excitation to noise input at the recipient’s position

2

ENERGY FLOW

The first law of thermodynamics is the starting point for energy conservation and propagation analysis. For
an elastic medium in a closed system and negligible heat transfer in a steady-state, i.e. with constant energy
density, the first law of thermodynamics yields the relation
ZZ

Is ( f )ns dA = Ps,in ( f ) − Ps,diss ( f )

(1)

As

between the structural intensity, Is , and the input power, Ps,in , as well as the dissipated power, Ps,diss , (8; 4).
Equation (1) does not take a connected fluid into account, which would have the effect of an additional damping
on the structure. In this case it would be necessary to extend Equation (1) by a fluid related term (4). In steadystate dynamics it is feasible that all field quantities are described in the frequency and thus the complex domain.
The active part of the complex intensity, in the frequency domain,1
T →∞ T

Is,a ( f ) = Re (Is ( f )) = hIs (t)i = lim

Z T
t=0

Is (t) dt(2)

is related to the time-averaged value of the STI in the time domain, hIs (t)i. For three dimensional structuresin steady-state, the cross-spectral density of the complex stress tensor, S, and the conjugate of the complex
velocity vector, v∗ , yields the complex STI
1
Is ( f ) = − S( f ) · v∗ ( f ).
2

(3)

In thin-walled structures the complex STI in shell elements, I0 ( f ), is calculated by means of section forces N,
Q and section moments M and their multiplication with the respective translational and rotational velocities v,
ϕ̇ (8)
"
#
∗
∗
∗
∗
∗
1 N x vx + N xy vy + M x ϕ̇ y − M xy ϕ̇ x + Qx vz
0
I (f) = −.
(4)
2 N y v∗y + N yx v∗x − M y ϕ̇ ∗x + M yx ϕ̇ ∗y + Qy v∗z
The connection between Equation (3) and (4) is given through the integration of the STI in the local z-direction
I0 ( f ) =

Z h

2

− h2

Is ( f ) dz.

(5)

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This yields the energy flow over the thickness, h, of the plate. A simplification is made for thin-walled structures. Under the assumption of a negligible energy flow in the z-direction the two dimensional problem inEquation (4) is obtained.
In order to conduct the analogous calculation of the energy flow in a structure the complex pressure, p, islinked to the complex sound particle velocity, v, as follows:
If ( f ) =1
p( f ) · v∗ ( f ).
2

(6)

Using Equation (6), the sound intensity (SI) is obtained, which constitutes the equivalent quantity to the STI ina fluid.

3

ENERGY FLOW IN VEHICLE STRUCTURESS TOEWER evaluates whether or not the STI can be used as a tool in an effective structural design process byassessing the energetic transfer paths from excitation to radiation, i.e. the energy flow in the machine parts
as depicted in Figure 1. In (1) the analysis of the STI is conducted in time and frequency domain. Throughhis assessment of the energy flow in the time domain S TOEWER shows that the inclusion of in-plane-waves
delivers further information on the energy propagation in structures. He is able to explain the occurrence ofout-of-plane vibrations at a structure’s discontinuities in a measurement set up, even before the initial wave isable to propagate into the specified region. An additional simulation verifies that the in-plane-waves transportenergy, which is converted into out-of-plane waves at the discontinuity and thus create a ’new’ vibration source.
Via examinations of test structures S TOEWER shows the various benefits of analysis conducted in time andfrequency domain, herein either for singular frequencies or frequency ranges.
In a next step, a more realistic structure is analyzed. Figure 2 shows the result for the energy flow in threedifferent set-ups for the excitation frequency of 120 Hz. The structure is excited on its right hand side, as
indicated by the red dot. This analysis’ aim is the reduction of the vibration on the left side via suppression of
the energy flow over the tunnel. By fixating all translatory DOFs and the in-plane DOFs of the marked nodes(pink) in Figure 2 b) and c), respectively, it is possible to reduce the magnitude of the displacement on theleft side by a factor of 2 and 100, respectively. The changes have been chosen to affect the paths with thehighest energy density. However, while it is viable to completely block the energy from propagating to the left
side for state c) (connected to an higher magnitude of the vibration on the ride side) the energy chooses a newpath in case of modification b). This is a general observation when manipulating energy paths. Although it ispossible to efficiently influence the energy flow by imposing an impedance discontinuity, the energy often finds
new paths to propagate on, thus making the effect of a structural change difficult to predict.

Figure 2. Underbody panel in different states at an excitation frequency of 120 Hz: a) original state, fixed nodesat b) top side of tunnel and c) at right hand side of the structure; excitation marked by red dot, fixed nodes via
pink dots (1)Via analysis of the depicted body in white in Figure 3 and a number of variations the following observations
are derived:• components with high, homogeneous stiffness tend to be the main paths of the energy flow,• increased stiffness in energy conducting components lead to an increased energy density at the interior
confining panels, i.e. the roof, windows, underbody and so forth, whereas increased stiffness of theconfining panels leads to a decreased energy density and• the absolute value of the STI is not related to the structural sound level.
In the presented case the highest reduction of the structural sound level is obtained via increasing the thickness of the car’s roof as shown in Figure 3, which constitutes an increased stiffness at an interior confining

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panel. S TOEWER concludes that the STI is a valuable quantity in describing the energetic behavior of complex
structures and in understanding the formation of a certain state energy-wise. However, the effect of a certain
design variation on the energy flow has proven to be predictable with difficulties and even then, since there is
no relationship between the STI and the sound level, it is difficult to evaluate such a measure in terms of its
acoustic impact.

Figure 3. Energy distribution of a body in white and a modified version with increased thickness of the roof
(right side) (1)4

INSIGHTS INTO FLUID-STRUCTURE INTERACTION

E BERT starts his considerations by tackling the matter of time and memory requirements necessary for energy
flow calculations. Tied to frequency band analysis conducted by S TOEWER, an approach is introduced to reduce
the number of frequencies necessary to adequately describe a frequency band solution of a structure. Therefore,in (2) three frequency selection criteria are derived and compared. The most reliable criterion is a local selection
criterion based on the excitability of the structure. With this, it is possible to show that in order to accurately
describe the energy flow in a complex structure in a given frequency band, only 10% of the contained resonancefrequencies need to be superimposed. One conclusion is, that the greater the complexity of a structure, the lessthe number of frequencies necessary for the calculation of the frequency band solution.
E BERT extends the research on energy flow by taking the surrounding fluid into account in order to createa more holistic understanding of the acoustic chain (Figure 1). For a simple coupled system as depicted in
Figure 4 he concludes, that the wave propagation depends mainly on the dominant subsystem. In the upper
pictures the system is dominated by the behavior of the structure. In these cases a closed loop of active energy
exists and a transport of energy between the structure and the fluid and vice versa takes places, i.e. the fluid
establishes a connection between different parts of the structure and contributes to their active energy supply. In
the pictures in the bottom row, showing the energy flow for the sixth excitation frequency, the system’s behavior
is dominated by the fluid. The energy flow in the fluid clearly varies from the energy flow in the structure.
There is no clear connection between either the SI in the coupling surface and the STI or the SI in the whole
fluid and the STI. The structure is nonetheless responsible for the energy supply of the fluid, but the re transferfrom fluid to structure cannot be detected.
Combining the ideas behind the approaches in (2) and (3), the efficient handling of energy flow simulation andthe description of fluid-structure interaction, E BERT develops a method for determining the main supply path ofenergy for a given problem in (4). In this approach the energy flow is analyzed in a way that allows for theestimation of the most important frequency in terms of energy supply for the sound radiation in a target region.
This idea relates to the well known principle that it is most efficient to tackle the highest peaks in a sum
spectrum. Here, the aim is to reduce or eliminate the main energy path between excitation and energy radiation
by applying local design changes. The first steps of the approach are depicted in Figure 5. For the given
problem that consists of a plate and a coupled fluid it is estimated that the 11th and the 55th frequency are
the main contributors in terms of energy supply to the sound radiation in the system. By deriving a structuraldesign variation based on the energy flow it is possible to reduce the STI by a factor of 30 and the SI at the
coupling surface by a factor of 87.
The approach aims at reducing the energy flow between structure and fluid, which results in reduced energy andsound radiation. An occurring problems is the complex energy flow. The energy tends to split up into several
sub-paths, thus making the effective application of structural modifications complex. Tackling only one sub-path
will lead to a shift in the energy flow towards other sub-paths making the outcome of a design modification
hardly predictable.

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Figure 4. Coupled system consisting of a plate and a fluid cavity and the respective energy flows for various
resonance frequencies of the system as viewed from the top (tv) and the side (sv) (3)Figure 5. Analysis steps to identify the main contributing resonance frequencies to the energy flow between astructure and the coupled fluid (4)5Energy propagation analysis by means of power balance calculations

Thus far, the analyses conducted have concerned the energy flow between linking points in the chain of effects(see Figure 1). To gain further insight into the energetic behavior of structures a method to quantify the energy
transmission at the linking points (excitation, energy transmission and sound radiation) is derived. (5) introduces
an approach to identify whether a linking point is an energy source or sink based on the divergence of the STI’s
active part∂div(Is ) = ∇ · Is =
Ii .
(7)
∂ xi
For a quantitative description of the energy transmission it is necessary to derive scalar quantities. This is doneby calculating power balance values based on the first law of thermodynamics. Therefore, in (6) it is shownthat the integration of the STI’s divergence in shell elements
ZZ

div(I0 ( f )) d A = Ps,in ( f ) − Ps,diss ( f )

(8)

As

is equal to the integration of the STI in Equation (1). In a first series of analyses different calculation approaches have been tested and compared with one another. An example is depicted in Figure 6. For a simply

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supported rectangular plate a surface area has been defined so that the smallest area has twice the size of twoelements adjoining the concentrated nodal force. The largest area is when the balance area reaches the boundaries of the structure. At the right side of the picture the curve progression of the power level over the balanced
surface is given as a direct integration of the STI for each element in the defined area, the calculation of thedivergence from the STI followed by the surface integration, and a reference solution calculated via the volume
integration of the strain energy density obtained from the solver itself. The tests show that in the direct vicinity of the concentrated load the power values obtained based on the STI either over or underestimate the real
power value. This changes with growing distance to the excitation force, where the solutions reach comparable
values for each of the methods. The comparison with the reference value shows a difference of 0.2 dB and is
negligible.

Figure 6. Simply supported rectangular plate and curve progression of estimated input power values for different
calculation approaches over balanced surface area, power values are averaged over all occurring eigenfrequencies
in a frequency band from 0 to 1000 Hz
Via the analysis of test structures such as the rectangular plate, it is shown that the power calculations based on
acoustic intensities deliver consistent results. Furthermore, this is valid for the calculation of power balances atthe boundaries of a specified region, showing that it is not always necessary to know the behavior of an entire
enclosed area. This is an important outcome because it shows that the description of the coupling points in the
acoustic chain of effects delivers valuable information about the whole system. In order to show the usefulnessof the introduced approach a further quantity, the effective radiated power (ERP), is introduced (1):
ERP( f ) = ρa ca Sṽ2 ( f ).

(9)

The ERP is a widely used quantity since it delivers an approximation for the sound radiation of a structure. As
already stated, there are (so far) no known correlations between the STI and the ERP. Figure 7 shows the resultof an analysis on the behavior of the ERP and power balance values. The structure resembles the rectangularplate depicted in Figure 6 containing gaps to achieve a more concentrated energy flow. The structure is excitedby a concentrated load in the area marked by ’exc’. The ERP values are calculated for the overall structure, theinput power, P, for the excitation area and both values for the right side as the target area. The power values
have been calculated by integrating the STI at the boundaries of the excitation area (marked by the dark blueline) and the boundary connecting the left and right side (marked by the light blue lines). The results show thatthe curve progression over the eigenmodes is similar for all quantities. Furthermore, both values depict that thesound radiation of the analyzed structure is dominated by the right side, since the differences between ERPalland ERPright as well as Pexc and Pright are small.
This analysis is also applied to more complex structures. Figure 8 represents the power distributions in a car’s
underbody panel as the mean level for all eigenmodes in a range between 0 and 150 Hz. The power balancevalues, Pbal , are calculated at the boundaries between each region. The ERP and Pbal values are normalized
to the overall structure and input power values, respectively. The results show that both quantities predict thesame normalized values for each of the identified areas within the structure. The difference between the twoquantities results from the inclusion of the in-plane power in the Pbal value calculation.
The presented examples show that both, ERP and Pbal deliver comparable results regarding the radiation behavior
of defined areas in a structure. This principle can be extended to more complex structures, in that the radiation
tendencies of the parts within those structures can be compared solely by describing the power balance relations
at the part boundaries, and to the evaluation of structural changes based indirectly on the energy flow byderiving power quantities.

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Figure 7. Test structure and results for the ERP and power balance for the first 25 eigenmodes of the structure

Figure 8. Underbody panel divided into balance regions (color reproduction) and their classification in terms ofnormalized power values

6

CONCLUSIONSThe presented research cases show that analyzing the STI results in an improved understanding of the acoustic
chain of effects since it delivers precise information on the energy propagation and the mechanisms that leadto the occurrence of an acoustic phenomenon. There are ways to influence the energy propagation in structuresand design variations have shown to be more effective when positioned in areas with a high energy density.
The introduced approaches show methods that make the calculation of intensity values more efficient and theassessment of the results less time consuming. Furthermore, since there are now methods focusing either on theestimation of energy paths in a structure or the quantification of energy transmission in terms of singular power
values, the user is able to choose the most feasible method for their purpose. Regarding the goal to achievea better understanding of the energetic chain of effects, the introduced methods are feasible to describe certain
points in the chain.
Future research should concern the sound radiation, which already can be estimated by power balance valuesof the structure, and its link to the power input at the recipients position. Here an improved understanding isachievable when comparing the ratio of the dissipated power of the structure with the energy input into thefluid in order to identify the area where a fluid is inclined to absorb the energy. Furthermore for complex
systems, where energy is always given the possibility to choose new paths to flow through, the outcome ofdesign changes is not accurately predictable when design variations are manually introduced. In these cases it
is necessary to apply algorithm based approaches that reduce the power input in the sub-structures, i.e. optimization based on the power input as target value, to further increase the ability to handle the complexity ofenergy propagation phenomena.

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