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

High Frequency Structure-borne Sound Simulations using a hybrid
Dynamical Energy Analysis / Advanced Transfer Path Analysis
Approach
Gregor TANNER1; Satoshi MORITA2; Timo Hartmann1; Martin RICHTER1
1

University of Nottingham, United Kingdom
2

Yanmar R&D Europe s.r.l, Italy

ABSTRACT
Dynamical Energy Analysis (DEA) is a mesh-based high frequency method modelling structure borne soundfor com- plex built-up structures. This has proven to enhance vibro-acoustic simulations considerably bymaking it possible to work directly on existing finite element meshes circumventing time-consuming andcostly remodeling strategies. In ad- dition, DEA provides detailed spatial information about the vibrationalenergy distribution within a complex structure in the mid-to-high frequency range. DEA was successfully
applied and validated to a structure borne sound calculation of an assembled agricultural tractor. Modelling
solid structures is still a challenge for DEA, however, as it is based on 2D wave transmission calculations.
We propose a novel method to generate DEA elements based on measurement data in order to model solidparts of a complex structure. Advanced Transfer Path Analysis (ATPA) is employed to extract energy
transmission characteristics of a structure. First, Frequency Response Functions are measured between
interface points on a structure. Then a direct transfer function between interface points is calculated by using
ATPA. Finally, DEA elements connecting interface points and representing energy transmission
characteristics of the structure are created based on the ATPA result. Applications of the method will bepresented.

Keywords: High frequency structure borne sound, DEA, ATPA

1. INTRODUCTION
Simulations of the vibro-acoustic performance of cars, trains, airplanes as well as heavy goods vehicles suchas lorries and tractors are routinely carried out at various design stages. To understand the transmission of
structure-borne sound in such mechanical structures, it is necessary to have effective and efficient modelling
tools to support the structural design process, ideally before a prototype vehicle is built. Numericalcalculations are widely used to aid the understanding and estimation of noise and vibrations. In particular,
the Finite Element Method (FEM) is used routinely for noise and vibration simulations in complex structuresin the low frequency regime. It requires extremely fine meshes at high frequencies, however, to capture
shorter wave lengths leading to large model sizes. Additionally, since the structural response at highfrequencies is very sensitive to small variations in material properties and boundary conditions, thesimulation result from FEM become less reliable. Statistical representations such as the Statistical Energy
Analysis (SEA) [1] have been developed, leading to relatively small and simple models in comparison with
FEM. SEA has found widespread applications in the automotive and aviation industry, as well as inarchitectural acoustics. However, SEA is based on a set of often hard to verify assumptions, which effectively
require diffuse wave fields and quasi-equilibrium of wave energy within sub-systems. In addition, SEA gives
results only on a relatively course scale and cannot deal with details of the structure [2,3,4].
Here, we start from a ray-tracing ansatz reformulated in terms of integral equations. This leads to linear flow
equations for the mean vibrational energy density and forms the basis of the Dynamical Energy Analysis
1

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gregor.tanner@nottingham.ac.uk

(DEA) method introduced first in [5]. DEA includes SEA as special case via a low order representation of
the so-called transfer operator. Higher order implementations enrich the DEA model with information fromthe underlying ray dynamics, leading to a relaxation of SEA assumptions. In particular, DEA allows for more
freedom in sub-structuring the total system and variations of the energy density across sub-structures can bemodelled. Hence, DEA can resolve the full geometrical complexity of the structure. An efficient
implementation of DEA on meshes has been presented in [6, 7]. Vibro-acoustic energy densities are computed
for multi-modal propagation, including energy transport over curved surfaces. Connections at materialinterfaces are described in terms of reflection/transmission matrices. DEA has been used to calculate thestructure borne sound of an assembled agricultural tractor and good agreement between measurements and
DEA calculations have been shown [8]. In particular, it has been demonstrated that DEA can model shellstructures accurately. However, it is still difficult to model a solid structure because currently DEA is basedon wave transmission calculations through plate/plate junctions. Additionally, it is often difficult to generate
accurate FE meshes of assembled complex structures because of welds, bolts and rubber bushes between
each of the components. To overcome these limitations, we suggest to integrate measurement data into DEA
to improve the effectiveness of DEA modelling.
In this paper, we propose a novel method to generate DEA elements based on the Advanced Transfer Path
Analysis (ATPA) [11]. ATPA is employed to extract energy transmission characteristics of a structure. Firstly,
Frequency Response Functions (FRFs) are measured between interface points on a structure. Then the directtransfer functions between all interface points are calculated using ATPA. Finally, DEA elements connecting
interface points and representing energy transmission characteristics of the structure are created based on the
ATPA result. The proposed method is verified with a finite element model of a simple structure.

2. Theory
2.1

Dynamical Energy Analysis on meshes

We will give a brief account of the major ideas behind DEA here – for a detailed description of the theory
and the implementation on 2D meshes, see [6, 8]. In the high frequency regime, the dynamics described bylinear wave equations can be approximated using semi-classical or ray-tracing methods describing thetransition from wave acoustics to ray acoustics; this serves as a starting point for DEA. By neglecting phase
information, DEA approximates the transport of wave energy by a Hamiltonian flow [5] described by the
Liouville equation. The flow is phase space volume preserving and can be formulated in terms of trajectories
or rays. In contrast to ray tracing, DEA does not work with individual rays, but describes how ray densities
are transported along the flow. In this work we are primarily interested in the stationary solution of transport
problems corresponding to wave problems in the frequency domain with time-harmonic driving terms.
Interference (or phase) effects are neglected.
The ray density can be related to a wave energy density and is defined on a phase space, that is, it depends
both on position and direction. For two-dimensional systems (as considered in this work), the phase space isfour dimensional and is parametrized by two position and two momentum coordinates. The momentum
coordinates are equivalent to the wave vector. For fixed frequency ω and a given wave mode, the modulusof the wave vector, that is, the wavenumber, is fixed. Hence, there is only one free parameter necessary todescribe the direction.
The transport problem can be conveniently solved on a mesh as described in [6, 8]. In particular, it isadvantageous to restrict the Hamiltonian flow to sections on the union of all the edges of the mesh describing
the two-dimensional structure in question. The coordinates used to describe this restricted two-dimensional
phase space, the so-called Surface of Section (SoS), are the arc-length along each edge i, si, and thecorresponding component of the wave vector along the direction of the edge, pi. The continuous Hamiltonian
flow is thus transformed into a discrete map, also called the Discrete Flow Map (DFM) [6].
DEA describes the transport of ray densities from the SoS to itself using a linear transfer operator which is
now discretised and represented as a matrix with entries of the from [6, 8]6291

2m + 1T(i,l0 ,m0 )0 (j,l,m) = p
2 Al 0 Al

Z Al Z c j 1

cj 1

0

0

wrt (p0i ) e ⌫L(sj ,si ) Pm0 (p0i ) Pm (pj )dpj dsj ,where Pm is a scaled Legendre polynomial of order m as detailed in [8], and the energy is propagated along
a trajectory through the mesh element j with propagation speed cj. This trajectory starts at position sj on edge
l of element j and is transported to position s’i on the common edge l’ of the boundary of element i. The
tangential component of the momentum (or slowness) of the trajectory at sj is denoted pj and likewise for p’i
at s’i. Note, that p’i and s’i are uniquely determined by pj at sj. Also L(sj,s’i) is the length of the trajectoryfrom sj to s’i, η is a (viscous) damping parameter. Furthermore, Al is the length of edge l from element j, andlikewise A’l is the length of edge l’ from element i. The weight function wrt gives the reflection/transmission
coefficients which are obtained from wave scattering theory [9]. Mode conversion between in-plane andflexural waves at boundaries can be included in the treatment. Shell effects leading to curved rays [10] areincluded by treating the meshed structure as set of plate-like elements, see [6-8] for details.
Once the matrix B has been constructed, the energy density on the boundary phase-space of each element isgiven by the solution of a linear system of the form(I

T)⇢0 = ⇢where I is the identity matrix and T has entries as defined by equation (1). Here, ρ0 is the initial density
produced by a source term. Once ρ has been computed, the energy density at any location inside the structure
can be computed in a post-processing step.
2.2

DEA-TPA modelling

In many practical applications, providing meshes for some parts of the structure may be very time-consuming
and thus costly due the internal complexity of the parts. Typical examples are the engine, the gear-box or thedrive train as a whole and adjacent parts. In these circumstances, it would be easier to do local measurements
and include effective transmission parameters into the DEA calculation. In order to extract the energy
transmission parameters, the Transfer Path Analysis (TPA) approach is taken here [11]. The DEA-TPA
method is based on the same principle as the RBE-patch method introduced in [8]. The difference to theRBE-patch method is that we use frequency dependent transition rates obtained from measurements. Energy
transitions are assumed to take place between connecting interfaces. Labelling the interfaces with k, k´ = 1,n, we define a transition matrix Sk,k’ between these interfaces. An interface is normally made up of different
edges of the mesh. Let us assume the edge Eb’’ is part of interface k and the edge Eb’ is part of interface k’which is shown in Figure 1. We now allow a transition between Eb and Eb’ via the edge Eb’’ using the equation,(1)Here, Ab , Ab’ are the lengths of the edges Eb, Eb’, β, β’, the indices for the momentum basis and T(b’,β’),(b,β) is
the transfer operator matrix element between two edges not including weighting factors. The first term in (1)represents overall damping, the fraction in the middle gives the relative transition rates for a single edge. The
denominator represents the total outgoing energy flux from interface k.
In [8], the RBE-patch method was introduced to account for coupling between different sub-meshes provided
by an FE-modeler. For combining measurement results with DEA, we determine the rates for the transition
matrix S using the ATPA method which is ideal to derive local transition rates from global response functions.
The basic theory of ATPA is described in the next session.

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b’’

b’

b

Figure 1: Schematic sketch of the connection between mesh edges using patch elements
2.3

Advanced Transfer Path Analysis

ATPA is a variant of the transmission path analysis method using so-called Direct Transfer Functions (DTFs)
calculated from a Global Transfer Functions (GTFs) [11]. The GTFs correspond to the common concept of ameasurable transfer function. The GTFs represent transitions between measured points via all transmission
path. The DTF elements represent the quotient between two specific points where all the other paths remain
blocked.
The GTF Ggij from point i to j is defined as the scenario where one applies an external excitation fext (forexample a force) at point i and the external excitation at all other points is set to zero [11], that is,&𝐺"#$ = &' with 𝑓 /01 (0, ⋯ ,0, 𝑓"/01 , 0, ⋯ ,0)7((2)

where si and sj represent a measured signal which can describe quantities such as acceleration, velocity,
displacement or energy. The DTF TDij from i to j is defined as the scenario where one applies an external
excitation at point i with vanishing external excitation at point j and vanishing displacements at any point
other than i or j, that is&𝐺"#8 = ' with 𝑆: = 0, 𝑘 ≠ 𝑖, 𝑗.
&((3)

As already mentioned, the DTF between two points gives the quotient of their signals when all the otherpoints in the system are blocked. Consequently, the DTF is normally difficult to measured but it can becalculated by using the GTF as follows:@

8𝐺::= [(BC DE]
)

(4)

GG

H[(BC )DE ]8𝐺#:= [(BC )DE] 'G for
GG

𝑗≠𝑘 .

(5)

In this paper, an energetic approach of the ATPA is employed as we focus here on the mid-high frequency
regime. In the energetic approach, the measured signal si represents the energy at point i. Therefore, theenergy density is used which is calculated using𝑠" = 𝜌ℎ𝑣"M(6)

where ρ is the material density, h is the thickness of the plate and vi is the velocity at point i.

3. FE model for DEA-TPA
In order to validate the DEA-TPA approach, the method is tested with an FE model of a simple structure
made up of three plates connected via 4 connecting beams. We assume that the middle plate, plate 2,6293

represents the complicated part and is modelled as the DEA-TPA element. Four interface points are definedat the boundaries of the connecting beams as shown in Figure 2. The GTFs is calculated here using FEM
instead of measuring a real structure, validation is also performed doing an FE calculation of the full structure.
To define the transition matrix described in Eq. (1), several steps are introduced as follows:•

We first normalise the outgoing flow to 1, that isDG̃Dij = Gij /

X

GD
ij

j

•

We then Sji = G̃Dij set in the case that interface i and j are not on the same plate.

•D
The diagonal terms correspond to reflection from the interface and we set Sii = G̃Dii + G̃ij
if i and j are on the same plate.

•

We set Sij = 0, if i and j are different points on the same plate.

The excitation point is on plate 1 (lower right plate) and shown in Figure 2.

(4)

2mm

(3)

0.1mm

2mm

5mm

(a) Original structure

DEA-TPAelement(2)(1)(b) DEA-TPA modelling

Figure 2: The FE models for the full structure and the TPA model. The thicknesses of the plates are shownin the left figure. The orange dot shows excitation point. The blue dots give measurement points, see theresults in Figure 4.

4. Results
4.1

Energy distribution

Entities of the transition matrix are calculated based on Eq. (4) and (5). However, for frequencies below
1000Hz negative value of the calculated GTFs have been observed. Therefore, here, we focus on thefrequency above 1000Hz for the ATPA results. Figure 3 shows a comparison of the acceleration distributions
between averaged FEM result over 1/3 frequency band, a full DEA calculation and DEA-TPA result for 2500
Hz. Figure 4 also shows results for the energy calculated at the points depicted in Figure 2 clearly
demonstrating that both the DEA and the DEA-TPA capture the energy distributions accurately at 2500 Hz.
A comparison between DEA, DEA-TPA results and FEM at selected points and over the full frequency range
from 250Hz to 2500Hz are presented in Figure 5 showing good agreement over the whole frequency range.

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FEM

DEA-TPA

(a)

(b)

FEM

DEA-TPA(c)
Figure 3: Comparison of FEM, DEA and DEA-TPA model at 2500Hz; (a) averaged FEM over 1/3 octave
frequency region at 2500Hz, (b) DEA and (c) DEA – TPA

2500HzVibratonal energy density [J/m2]

1.0E-04
FEM
DEA
DEA-TPA(DTF)
1.0E-05

1.0E-06

1.0E-07
1

2
3
Measurement point No.

4

Figure 4: Comparison of FEM, DEA and DEA-TPA model at 2500Hz

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1817120391
1.0E-01

Vibration velocity [m/s]Vibration velocity [m/s]1.0E-01

FEM

1.0E-02

DEA
DEA-TPA(DTF)
1.0E-03

FEM

1.0E-02

DEA

DEA-TPA(DTF)
1.0E-03

1.0E-04

1.0E-04
0

500

1000

1500

2000

2500

0

3000

500

1500

(1)

(2)

37182

47618

2000

2500

3000

1.0E-01

Vibration velocity [m/s]1.0E-01

Vibration velocity [m/s]1000

1/3 Octave frequency [Hz]1/3 Octave frequency [Hz]FEM

1.0E-02

DEA
DEA-TPA(DTF)
1.0E-03

FEM

1.0E-02

DEA
DEA-TPA(DTF)
1.0E-03

1.0E-04

1.0E-04
0

500

1000

1500

2000

2500

3000

0

500

1000

1500

2000

2500

3000

1/3 Octave frequency [Hz]1/3 Octave frequency [Hz]

(3)

(4)

Figure 5: Comparison of FEM, DEA and DEA-TPA model over the frequency range 250Hz to 2500Hz.
4.2

Energy flow calculations

The ray density ρ(r,p) computed in DEA provides more information than just the energy density. In particular,
it allows one to estimate the momentum density vector I(r) (sometimes called intensity vector) directly, andthus gives information about the direction of mean energy flow at each point on the structure. This can bedone as a post-processing step by computing the vectoral quantity

at each point on the structure. Here r is the position and p is the (two-dimensional) momentum variable.

(a)

(b)

Figure 8: Comparison of calculated energy flow between DEA and DEA-TPA at 1600Hz

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5. Conclusions
In this paper, the DEA-TPA modelling method was proposed and compared with an FEM calculation. It was
shown how ATPA input can be integrated into a transition matrix calculation giving rise to a hybrid DEATPA method. Both the DEA-TPA modelling as well as a DEA computation compares favorably with the FEM
results with DEA-TPA giving a slightly more accurate description as expected.

References
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