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

Measurement of the structural intensity of curved shell structuresby means of 3D laser vibrometry
Nikolai KLEINFELLER1; Joachim BÖS2; Tobias MELZ3
1,2

System Reliability, Adaptive Structures, and Machine Acoustics SAM, TU Darmstadt, Germany
3

Fraunhofer Institute for Structural Durability and System Reliability LBF, Germany

ABSTRACTThe structural intensity describes the energy flow of structure-borne sound within a vibrating structure. For
complex structures, the STI is usually obtained from a finite element analysis. At the present time, if
structures with arbitrary shapes are considered a measurement procedure for validation purposes is not
sufficiently developed. For this reason, a measurement procedure for determining the STI of arbitrarily
shaped thin-walled structures is developed within the framework of this research project. This paper deals
with the measurement of the STI of a curved shell structure by means of 3D laser vibrometry. Therefore, ameasurement procedure is set up that consists of measurement data acquisition, measurement data processing,and a general STI calculation. In this context, the challenges of the measurement procedure should beemphasized, if the structures are not flat but curved. In addition to the flexural vibrations, the extensional
vibrations must also be recorded simultaneously. In the case of curved structures, knowledge of the curvature
properties is necessary in addition to the structural vibrations. Finally, the results are compared to the STI
obtained from the virtual data of a finite element analysis.
Keywords: structural intensity, 3D scanning laser vibrometry, thin-walled shell structures

1. INTRODUCTION
The structural intensity (STI) characterizes the energy propagation of structure-borne sound withina vibrating structure (1). In this way transmission paths of structure-borne sound can be analyzed andinfluenced by structural modifications, e. g., additional masses, to reduce sound radiation at specificfrequencies (2). A study has also shown a strong relation between STI-based scalars and the soundradiation of a plate-rip model (3). Thus, the STI can also be used to rate the value of structuralmodifications compared to an initial state. If the structure is complex, e. g., an oil pan, the STI is
usually obtained from a finite element (FE) analysis (2). Consequently, an experimental validation of
the STI is a mandatory step if meaningful decisions should be made for structural design.
In the last decades the measurement of STI has mainly been done for straight beams, e. g., (4, 5)or flat plates, e. g., (6, 7). Only a few papers show results of measuring STI of beams and shells with
one-dimensional constant curvature, e. g., (8–12). Usually, the progress was delayed by the limitationsof the measurement equipment used. In recent years, new measurement techniques have beendeveloped that allow the recording of three-dimensional vibrations with high accuracy. So it ispossible to obtain the STI of arbitrarily shaped structures from measurement data. Except from somerecently published results by Pires et al. (13) the STI has never been measured for structures with
arbitrary shape. Pires et al. (13) used a FE approach for the STI calculation and the digital image
correlation (DIC) for the measurement data acquisition. Further, Pires et al. (14) published analternative measurement strategy for STI based on the shell theory by considering curvature properties.
A similar approach is used in this study and has already been tested for a curved beam structure (15).
Therefore, the aim of this paper is the application of the measurement procedure to a curved shellstructure. The methodology for determining the STI in curved shells can be divided into three steps
as shown in Figure 1.
1

kleinfeller@sam.tu-darmstadt.deboes@sam.tu-darmstadt.de (now with Fraunhofer IDMT, joachim.boes@idmt.fraunhofer.de)3
info@sam.tu-darmstadt.de28102

measurement dataprocessing

measurement dataacquisition

structural intensity
calculation

Figure 1 – Measurement procedure for the STI
In the first step the measurement data are recorded by means of a 3D scanning laser vibrometer. In
the second step methods of signal processing are used to minimize noise and enable spatial derivative
calculation out of measurement data. Finally, the STI is calculated based on a general STI formulation
with defined assumptions. The method is applied to a thin-walled and curved shell structure.
The paper is structured as follows. In Section 2, the test specimen as well as the measurement setup
and the numerical modelling concept are introduced. A general formulation of the STI for arbitrarily
shaped shells is presented in Section 3. In Section 4, the methods of the measurement data processing
are described. The results for the STI calculation from measurement data are discussed in Section 5.
Section 6 gives a summary and a conclusion.

2. DESIGN OF EXPERIMENTAL AND NUMERICAL MODEL
The measurement procedure for the STI is validated based on a thin-walled spherical shell structure.
First, an experimental and a numerical model are built up and are introduced in this section. In thispaper the results obtained from the numerical model are called virtual data whereas results from theexperimental model are called real measurement data.
2.1 Geometry
Figure 2a shows the geometry of the spherical shell structure considered. The shell structure ismade out of steel and was manufactured by a hydroforming process at the Institute of Production
Engineering and Forming Machines (PTU) at TU Darmstadt (16). The design parameters are listed inTable 1.

ܴ୤

‫ݕ‬

‫ݐ‬

‫ݖ‬

‫ݔ‬

݄

ܴ୸୩

ܵ
Figure 2 – Curved thin-walled shell structure; a) design parameters (16); b) experimental setup
2.2 Experimental setup
All the measurements were taken at an experimental setup shown in Figure 2b. The test specimen
is attached by wires to a frame to achieve free boundary conditions. For harmonic excitation anelectrodynamic shaker with a premounted impedance head is used. The measurement data acquisition
is done by a 3D scanning laser vibrometer (Polytec PSV-500-3D). So, a simultaneous and complete
measurement data acquisition of the three-dimensional structural vibrations (‫ݑ‬௫ , ‫ݑ‬௬ , ‫ݑ‬௭ ) of the shell
as well as a measurement of scan point coordinates (‫ݔ‬, ‫ݕ‬, ‫ )ݖ‬is done. The calibration of the laser
vibrometer has been done with a high accuracy (<ͲǤʹ mm) of the 3D-alignment. First, the vibrations
are recorded on a coarse grid using periodic chirp excitation. In this way, the resonance frequenciesof the structure are identified. Next, the structure is excited at specific single resonance frequencies.
Here, a refined circular measurement grid with a scan points distance of approximately 8 mm in global‫ݔ‬-direction as well as in global ‫ݕ‬-direction is used. This leads to a total number of scan points of 3201

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for the complete surface of the spherical shell structure. For every scan point a complex averaging isdone over 16 measurements using the excitation force as the reference signal.
Table 1 – design parameters of the curved thin-walled shell structure
parameter

nomenclature

value

unit

diameter

ܵ

620

mm

height

݄

66.8

mm

radius

ܴ୤

750

mm

radius

ܴ୸୩

35

mm

thickness

‫ݐ‬

1

mm
11

Youngs’s modulus‫ܧ‬

2.1·10Poissons’s ratio

ߥ

0.3

mass density

ߩ

7850

N/m2
kg/m3

2.3 Numerical modelling
The shell structure is built up in ANSYS 19.2 by using eight-node shell elements (SHELL281).
The wires and the electrodynamic shaker are considered in the model by stiffness (COMBIN14) andmass properties (MASS21). Subsequently, a model-updating is performed. Based on the scanningvibrometer measurements the model parameters are changed to reduce the derivations of the numericalmodel in relation to the experimental model. This procedure leads to a maximum deviation of ʹ% forresonance frequencies up to ͷͲͲ Hz. Further, the structural loss factor (ߟ ൌ ʹ ή ͳͲିସ ) is identified
using the ͵ dB interpolation method. The model is excited by a discrete excitation force (‫ ܨ‬ൌ ͲǤͳͶ N).
In this way a sufficient agreement between the results of the numerical model and measured
displacements is achieved. Figure 3 shows a comparison of magnitude and phase between thesimulated (a–d) and the measured (e–h) displacements at a resonance frequency of ʹͲͻ Hz. On theleft hand side (a, b and e, f) are the displacements in global ‫ݔ‬-direction ‫ݑ‬௫ and on the right hand side(c, d and g, h) are the displacements ‫ݑ‬௭ in global ‫ݖ‬-direction. The following conclusions can be drawn.
First, there is an angular rotation between the results of both models, which can especially be seenfrom the magnitude of ‫ݑ‬௭ (c and g). Second, the amplitudes of the experimental model areapproximately six times higher than the amplitudes of the numerical model. It should be noted thatfor the study in this paper a qualitative agreement is sufficient. Here, the results of ‫ݑ‬௭ show aqualitatively good agreement between the numerical and the experimental model.

Figure 3 – Magnitude and phase of displacements ‫ݑ‬௫ and ‫ݑ‬௭ ; a–d virtual data; e–h measurement data8104

In contrast, the displacements ‫ݑ‬௫ show major deviations. The results of ‫ݑ‬௬ are similar to ‫ݑ‬௫ sothey are not shown here. Summarized, the agreement is sufficient so that the virtual data as well as
the real measurement data are taken to obtain the STI based on equations described in Section 3.

3. STRUCTURAL INTENSITY OF SHELLS
A formulation of the STI in thin-walled shells with arbitrary shape was published by Pavić (17)for the first time. The approach is based on the theory of thin shells, which is explained in detail by
Qatu (18) and Ventsel and Krauthammer (19). Generally, the spatial intensity vector (1)்(1)ࡵ ൌ ൣ‫ܫ‬௫ ǡ ‫ܫ‬௬ ǡ ‫ܫ‬௭ ൧ ൌ െࡿ ή ࢜
is calculated from the product of the stress tensor ࡿ and the vibrational velocity ࢜. Equation 1 is valid
for any elastic medium. In the case of thin-walled structures, the component of the STI in thickness
direction can be neglected (‫ܫ‬௭ ൎ Ͳ). If the shell has an arbitrary shape, it is useful to define a new local
coordinate system (ߙ, ߚ, ߛ) that is curvilinear and body-fitted and makes mathematical operations
easier to perform (17–19). The resulting intensity at the shell’s middle surfaceା௧Ȁଶܰఈ ‫ݑ‬ሶ ఈ ൅ ܰఈఉ ‫ݑ‬ሶ ఉ ൅ ‫ܯ‬ఈ ߣሶఈ ൅ ‫ܯ‬ఈఉ ߣሶఉ ൅ ܳఈ ‫ݑ‬ሶ ఊ‫ܫ‬ᇱ ఈ‫ ܫ‬ᇱ ൌ  න ࡵ ‫ ݖ‬ൌ  ൤ ᇱ ൨ ൌ െ ቈ቉‫ܫ‬ఉܰఉ ‫ݑ‬ሶ ఉ ൅ ܰఉఈ ‫ݑ‬ሶ ఈ ൅ ‫ܯ‬ఉ ߣሶఉ ൅ ‫ܯ‬ఉఈ ߣሶఈ ൅ ܳఉ ‫ݑ‬ሶ ఊ

(2)

ି௧Ȁଶ

can be obtained by integrating the intensity vector across the shell thickness ‫ݐ‬. The internal forces
and moments of an arbitrary shell in Equation 2 are defined as (17–19)ܰఈǡఉ ൌ

‫ݐܧ‬൫ߝ ൅ ߥߝఉǡఈ ൯ǡͳ െ ߥ ଶ ఈǡఉܰఈఉǡఉఈ ൌ‫ܯ‬ఈǡఉ ൌ‫ݐܧ‬ߝǡʹሺͳ ൅ ߥሻ ఈఉǡఉఈ‫ ݐܧ‬ଷ൫ߢ ൅ ߥߢఉǡఈ ൯ǡͳʹሺͳ െ ߥ ଶሻ ఈǡఉ‫ ݐܧ‬ଷ
‫ܯ‬ఈఉǡఉఈ ൌߢǡʹͶሺͳ ൅ ߥሻ ఈఉǡఉఈ
ܳఈ ൌ

ͳ ߲
ͳ ߲
߲‫ܤ‬
‫ܯ‬ఉ ൅
൫‫ܣ‬ଶ ‫ܯ‬ఈఉ ൯൨ǡ൤ ሺ‫ܯܤ‬ఈ ሻ െ‫ߙ߲ ܤܣ‬‫ߚ߲ ܣ‬
߲ߙ

ܳఉ ൌ

ͳ ߲
߲‫ܣ‬
ͳ ߲
‫ܯ‬ఈ ൅
൫‫ܤ‬ଶ ‫ܯ‬ఉఈ ൯൨Ǥ൤ ൫‫ܯܣ‬ఉ ൯ െ‫ߚ߲ ܤܣ‬߲ߚ‫ߙ߲ ܤ‬(3)

In this case the kinematics of the shells are described by the Kirchhoff’s assumptions. Furthermore,
the material properties are assumed to be linear and isotropic. The midsurface strains (ߝఈ , ߝఉ , ߝఈఉ ,ߝఉఈ ), curvatures and twist (ߢఈ , ߢఉ , ߢఈఉ ) as well as rotations (ߣఈ , ߣఉ ) are obtained from
ߝఈ ൌ

‫ݑ‬ఊ
ͳ ߲‫ݑ‬ఈͳ ߲‫ܣ‬
൅‫ݑ‬ఉ ൅ ǡ‫ߚ߲ ܤܣ ߙ߲ ܣ‬
ܴଵ

ߝఉ ൌ

‫ݑ‬ఊ
ͳ ߲‫ݑ‬ఉͳ ߲‫ܤ‬
൅‫ݑ‬ఈ ൅ ǡ‫ߙ߲ ܤܣ ߚ߲ ܤ‬ܴଶߝఈఉ ൌ ߝఉఈ ൌ‫ݑ ߲ ܣ‬ఈ‫ݑ ߲ ܤ‬ఉቀ ቁ൅ቀ ቁǡ‫ܣ ߚ߲ ܤ‬‫ܤ ߙ߲ ܣ‬

ߢఈ ൌ

ͳ ߲ߣఈ
ͳ ߲‫ܣ‬൅ߣ ǡ‫ ߚ߲ ܤܣ ߙ߲ ܣ‬ఉ

ߢఉ ൌ

ͳ ߲ߣఉ
ͳ ߲‫ܤ‬൅ߣ ǡ‫ ߙ߲ ܤܣ ߚ߲ ܤ‬ఈ

(4)

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ߢఈఉ ൌ‫ߣ ߲ ܤ‬ఉ‫ߣ ߲ ܣ‬ఈቆ ቇ൅൬ ൰ǡ‫ܤ ߙ߲ ܣ‬‫ܣ ߚ߲ ܤ‬ߣఈ ൌ‫ݑ‬ఈ ͳ ߲‫ݑ‬ఊെǡܴଵ ‫ߙ߲ ܣ‬ߣఉ ൌ‫ݑ‬ఉ ͳ ߲‫ݑ‬ఊെǤܴଶ ‫ߚ߲ ܤ‬

Therefore, the displacements of the midsurface in the local curvilinear coordinate system (‫ݑ‬ఈ , ‫ݑ‬ఉ ,‫ݑ‬ఊ ) as well as their spatial derivatives are necessary. ‫ ܣ‬and ‫ ܤ‬are called the Lamé parameters of thesurface. The Lamé parameters are obtained from (17, 19)‫ܣ‬ǡ ‫ ܤ‬ൌ ට‫ ݔ‬ଶ ఈǡఉ ൅ ‫ ݕ‬ଶ ఈǡఉ ൅ ‫ ݖ‬ଶఈǡఉ

(5)with the derivative of the scan point coordinates with respect to the local coordinate directions ߙ andߚ. Further, ܴଵ and ܴଶ are the radii of principal curvature and are related to the change in directionof the normal vectors of the surface. The determination of principal curvatures based on discrete
coordinate data is explained by (17, 19) and will not be described here. In this study a spherical shellis considered, which leads to equal principal curvatures at any point on the surface. If monofrequent
steady-state signals are assumed, the complex STI ࡵሺ݂ሻ in the frequency domain yields
ͳࡵሺ݂ሻ ൌ െ ή ࡿሺ݂ሻ ή ࢜‫ כ‬ሺ݂ሻʹ

(6)by using peak values of the complex stress tensor ࡿ and the conjugate complex vibrational velocity࢜‫כ‬. The real part of the complex STI, ࡵୟ ൌ ൛ࡵሺ݂ሻൟ, is called active STI, whereas the imaginary partof the complex STI, ࡵ୰ ൌ ൛ࡵሺ݂ሻൟ, is called reactive STI. In the framework of this paper the STI isalways analyzed in the frequency domain.

4. MEASUREMENT DATA PROCESSING
It is shown in Section 3 that the STI can be calculated based on measured discrete threedimensional vibrations and on related scan point coordinates. Thus, a measurement procedure is tested
by using real measurement data recorded by a 3D scanning laser vibrometer. The measurement
procedure shown here has also been published by Pires et al. (14). The approach is based on a complete
and general description of the STI in a thin-walled shell structure, see Equations 2–4. The entire STI
can be obtained for the grid of defined scan points if the coordinates with respect to a global
measurement coordinate system (‫ݔ‬, ‫ݕ‬, ‫ )ݖ‬as well as the three-dimensional structural vibrations (‫ݑ‬௫ ,‫ݑ‬௬ , ‫ݑ‬௭ ) are known. The complete measurement procedure is verified by using virtual data obtained
from an FE analysis.
Measured data of three-dimensional vibrations and coordinates must be filtered to reduce noise.
Here, a spatial filter of the Polytec Strain Processor (PSV 9.3) is used, which produced satisfactory
results. An alternative approach can be a Savitzky-Golay filter (20) used frequently for dynamic strain
measurements, e. g., by Wildy (21).
Subsequently, the spatial derivatives of the local displacements are obtained from the data. In thefirst step, a local curvilinear coordinate system must be identified for each scan point. This can bedone by calculating the tangential vectors with respect to the ߙ- and ߚ-direction for each scan point
and assuming that the direction of vectors fits with the direction of a local curvilinear coordinate
system. If the local vector base is known, the related transformation matrix is calculated. Additionally,
the curvature properties of the underlying shell surface are obtained. The left hand side of Figure 4 (a,b and e, f) shows a selection from two out of nine transformation coefficients calculated by virtual
coordinates (a, b) and measured scan point coordinates (e, f). In the first case (a and e) a quantitative
agreement is achieved except for single deviations located around the circle center. In the second case(b and f) results are worse. The deviations are caused due to noise of the scan point coordinate
measurement data. Summarized, over all nine transformation coefficients a qualitative agreement isachieved.

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In the next step, the measured three-dimensional vibrations are transformed to the local coordinate
systems by using the transformations calculated before. Subsequently, the derivatives of the local
structural vibrations up to the third order are calculated. For the calculation of the spatial derivatives
the finite difference scheme is applied. So, the first order spatial derivate of a scan point with number
݅, e .g., of the displacement in ߙ-direction, is obtained from߲‫ݑ‬ఈ‫ݑ‬ఈǡ௜ାଵ െ ‫ݑ‬ఈǡ௜ିଵ
൬
൰ ൎ
Ǥ
߲ߙ ௜
ο‫ݏ‬

(7)

Here, ‫ݑ‬ఈǡ௜ାଵ, ‫ݑ‬ఈǡ௜ିଵ are the displacements values of neighbor points and ο‫ ݏ‬is the scan point distanceobtained from the magnitude of the distance vector between two scan points. For higher orderderivatives an equal scheme is used. For scan points close to the boundary, a forward or backwards
finite difference scheme is applied. Figure 4 (c, d and g, h) shows a comparison between calculated
first order spatial derivatives of ‫ݑ‬ఊ with respect to ߙ-direction from virtual data (c and d) and frommeasurement data (g and h). Here, a qualitative agreement is also achieved if the results of Section
2.3 are taken into account. It should be mentioned at this point that the results for the first orderdisplacements ‫ݑ‬ఈ and ‫ݑ‬ఉ are worse. This fact can be explained by phase noise occurred due tomeasurement deviations. Finally, after having obtained the higher order derivatives as well, the STI
is calculated with respect to the local coordinate system. Subsequently, the STI with respect to theglobal coordinate system is obtained from inverse transformation.

Figure 4 – Transformation coefficients and magnitude and phase of the first order spatial derivative of ‫ݑ‬ఊ ;a–d virtual data; e–h measurement data

5. RESULTS OF THE STUDY
In this section the active and reactive STI calculated from measurement data are compared with the
STI obtained from virtual data. The STI is composed of four parts (bending moment, twisting moment,
shear force and membrane force). Figure 5 shows the results of the active and reactive intensities ofthe bending moment calculated by using virtual data (a, b) and real measurement data (c, d). For
visualization purposes the STI is only shown in a section of the surface. The magnitude of the STI isvisualized as a contour plot and the spatial STI vectors are displayed as white arrows. It should be
noted that at specific spots of the surface a deterioration of the measurement results occurs due tomeasurement deviations. Figure 6 visualizes the results of the active and reactive intensities of thetwisting moment. On the one hand, there is a qualitatively good agreement between the spatial
distribution of the STI magnitude for both the bending moment and the twisting moment. On the other
hand, the direction of the STI vectors is clearly defined. Even though a change in direction occurs for
active STI the transmission paths are identifiable. Summing up the results, it can be concluded thatfor the bending moment and the twisting moment the measurement procedure is reliable enough anda sufficient agreement has been achieved. In contrast, the results for the STI of the shear forces are

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not usable due to distortions by measurement errors. A successful calculation of intensities due tomembrane forces from measurement data can also not be achieved in this study. In comparison to flat
structures the STI calculation of curved structures is affected by calculation errors resulting fromcoupling of up to six measurement quantities and their spatial derivatives if the method proposed inthis paper is used. In this study the results are strongly influenced by deviations of the scan point
coordinate measurement data. Further investigations will focus on improving the measurement
procedure by implementing additional data filters, e.g. using Savitzky-Golay filter (20).

Figure 5 – ࡵୟ and ࡵ୰ of the bending moment; a–b virtual data; c–d measurement data

Figure 6 – ࡵୟ and ࡵ୰ of the twisting moment; a–b virtual data; c–d measurement data

6. SUMMARY AND CONCLUSIONSThis paper presents a measurement procedure for determining the STI in curved shell structures.
In this context, a spherical shell structure is used as a test specimen. In the first step, the structural
vibrations and the scan point coordinates are recorded by a 3D scanning laser vibrometer. In the second
step, the coordinate transformations and the spatial derivatives of the displacements with respect to a
local curvilinear coordinate system are calculated. Finally, the STI is obtained by equations based on
the shell theory by considering curvature properties.
It is found that the results show a sufficient agreement if the STI for the bending moment and thetwisting moment are of interest. A measurement of the STI for the shear forces and the membrane
forces has not been successful especially due to measurement deviations of scan point coordinates.
Future work will involve the enabling of the STI measurements for the membrane and the shear forces
due to improvements in the measurement data processing. Finally, the method will be applied to amore complex structure, e. g., an oil pan.

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ACKNOWLEDGEMENTS
The authors appreciate the contributions of the student Xiang Liu to this research project. The
project is funded by the Deutsche Forschungsgemeinschaft DFG (BO 3624/1-1 and INST 163/520-1
FUGG). This funding is gratefully acknowledged.

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