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

Investigation towards an active barrier for structure borne soundusing structural intensity
Alexander KOKOTT1*; Thomas HAASE1; Hans Peter MONNER1
1

German Aerospace Center, Institute of Composite Structures and Adaptronics, Germany

ABSTRACT
Vibrations of aircraft or vehicle engines, for instance, are often transmitted by structure borne sound, leading
to a significant radiation of noise inside passenger cabins. Current active vibration control concepts use either
velocity or acceleration as a control source. However, these only lead to a local reduction in vibration and notnecessarily to the reduction of the vibration energy flow.
This study presents the implementation of current methods for structural intensity measurement with areal-time control. The work investigates one and two-dimensional structures. A reduction of energy flow in abeam structure is shown experimentally, as well as numerically for plates. The measurements are strongly
influenced by theoretical simplifications concerning the composition of the structure borne waves and thequality of the sensor arrays used, i.e. the sensor spacing and the positioning accuracy. Though, sufficientaccordance between numerically and experimentally estimated structural intensity can be found using
methods with smaller sensor arrays. A barrier effect is shown by numerical investigations and is measured ona beam. Therefore, the control of vibration energy flow is a more effective method for a global reduction if
vibration downstream the control area.
Keywords: active control, structural intensity

1. INTRODUCTION
Vibrating structures leading to noise radiation are often excited by sources distant from the relevant
structural parts. In many cases structure borne sound is a major transmission factor for these vibrations.
In order to reduce the share of structure borne noise transmission, in for instance passenger cabins, thecontrol of the structural intensity (STI) as a measure for the energy transmitted proposes an effective
method.
Hence, this paper addresses measuring techniques for the estimation of structural intensity in orderto implement an active barrier for structure borne sound. First of all these methods are investigat ed
numerically for an unstiffened plate. A control set up is furthermore examined. Experimental research
towards measurement and control of structural intensity is done on a beam structure with different
damping configurations. Furthermore the active control of structural intensity is investigated on an
inhomogeneously damped plate in comparison to a velocity control.

2. MEASUREMENT OF POWER TRANSMISSION
2.1 State of the art methods
The measurement of structural intensity has already been addressed by NOISEUX in 1970 (1). In
the beginning methods neglecting near field components of the bending waves due to inhomogeneity,
i.e. waves decaying exponentially with their distance to the origin, were proposed, enabling theapproximation of STI with two sensors for a beam. Based on this research more complex methods were
established using more sensors and including one or more near field components (2, 3). For one
dimensional STI estimation, a total of four sensors is necessary for the notionally exact estimation,
respectively 8 sensors for the two dimensional case (2). All these methods use finite differences forapproximation, being advantageous for a real-time implementation, as the calculation effort is reduced.
Other methods for example use wave decomposition methods, leading to a higher demand for solving*

alexander.kokott@dlr.de8073

the equation systems (4, 5)
The following derivation is based on (2) using the internal forces of a plate element as described infigure 1.

Figure 1 – Coordinate system and definition of displacements, rotations,internal forces of a plate element (positive influence line)
The structural intensity is a vector describing the vibratory energy flow through the cross section of
the plate. In this case only the energy flow of bending waves is examined, as those are seen to largely
contribute to sound radiation (6). Assuming a homogeneous thickness of the plate, normalizing theenergy to the cross section may be omitted in the following formulas, so only the vibratory power isdisplayed. For plates, the bending wave power flow in x direction can be described as follows:ܲ௫ ൌ ܳ௫ ߟሶ ൅ ‫ܯ‬௫௫ ߱௫ ൅ ‫ܯ‬௫௬ ߱௬(1)

The equivalent formula for the y direction can be derived by interchanging the subscripts x and y.
The out of plane velocity ߟሶ and angular rates ߱, or their time derivatives respectively, can bemeasured by current technology sensors, whereas the direct estimation of the internal forces ܳ andmoments ‫ ܯ‬is impossible. One approach to calculate these measures is the application of constitutive
equations of plate theory. Hence equation (1) can be written as follows߲ଷߟ߲ଷߟ߲ ଶ ߟ ߲ߟሶ߲ଶߟ߲ ଶ ߟ ߲ߟሶሺͳܲ௫ ൌ ‫ ܤ‬൭ቆ ଷ ൅
ቇ
ߟሶ
െ
െ
ߥሻ
െ
ቆ
൅
ߥ
ቇ ൱
߲‫ݔ‬
߲‫ ݕ߲ݔ‬ଶ߲‫ݕ߲ ݕ߲ݔ‬߲‫ ݔ‬ଶ߲‫ ݕ‬ଶ ߲‫ݔ‬(2)only using the flexural stiffness B of the plate, the Poisson ratio ߥ and the normal displacement of
the plate element ߟ and its spatial and time derivatives. The first may be obtained by integration oftime signals from accelerometers. The latter require approximation by finite differences, where local
derivatives are estimated by the difference of measures of an array of accelerometers. For example thethird spatial derivative of the local displacement as well as the mixed derivative is shown below, using
an array of twelve equally spaced sensors (figure 2) based on (2).

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Figure 2 – Sensor array for two dimensional STI estimation of a plate (sensor spacing ȟ)ߟଽ െ ߟଵ଴ െ ߟଵଵ ൅ ߟଵଶ߲ଶߟൎ߲‫ݕ߲ݔ‬
߂ଶ(3.1)

߲ ଷ ߟ ߟସ െ ͵ߟଷ ൅ ͵ߟଶ െ ߟଵ
(3.2)
ൎ

߲‫ ݔ‬ଷ
߂ଷ
Applying finite differences to equation (2) and omitting derivatives in y-direction (i.e. assuming abeam), the following simplified equation can be derived for the STI calculation for a beam with thefour point method (4FD):
ܲସி஽ ൌ

‫ܤ‬ሾߟሶ ሺʹߟସ െ Ͷߟଷ ൅ ʹߟଶ ሻ ൅ ߟሶ ଷ ሺെʹߟଷ ൅ Ͷߟଶ െ ʹߟଵ ሻሿ
ȟଷ ଶ

(4)

However even these relatively simple equations show the importance of the exact sensor spacing
for a sufficient approximation, as errors in phase and amplitude rise with the order of the derivative.
Furthermore a time synchronous data acquisition is necessary, avoiding ph ase errors between thesensors. Since finite differences are quite vulnerable to these kinds of errors, simplifications may bemade to reduce the order of finite differences needed and therefore the amplification of errors. As
already introduced, another approach is proposed by NOISEUX’s method for one dimensional STI
calculation, assuming a far field (1). By separating the wave equation and omitting the time part aswell as the near field components, two sensors are sufficient to estimate the STI. The derivation of thetwo point method (2FD) results in the following equation, using the wave number ݇:ܲଶி஽ ൌ‫ ݇ܤ‬ଶ
ሾሺߟଵ ൅ ߟଶ ሻ െ ሺߟሶ ଶ െ ߟሶ ଵ ሻሿ
ȟ(5)

2.2 Comparison of methods
Numerical and experimental investigations towards the accuracy and robustness of the two and four
sensor measurement were already done for the STI estimation on a beam (one dimensional) in (7). One
major issue of finite difference approaches is the sensitivity to sensor spacing. The theoretically exact
estimation of spatial derivatives demands an infinitesimal spacing between the sensors. The size of
this systematic error caused by enlarging the spacing is dependent on the wavelength, i.e. larger
wavelengths with slower changes in amplitude over distance allow larger sensor distances.
On the other hand, small sensor spacing leads to small differences in amplitudes. This occurs to bea problem when the signal to noise ratio is small, as mainly the local derivative of the noise iscalculated instead of the signal, leading to a random error in STI estimation.

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Further numerical investigations on an aluminum plate model were done to estimate the error
sensitivity of the two and four point STI estimation method as well as the theoretically exact eight
position method (assuming infinitesimal sensor spacing and no additional noise). The system wasmodelled in ANSYS® using SHELL281 elements.
A virtual minimum sensor spacing of ͷ݉݉ was initially assumed to minimize the systematic errorof finite differencing. However, the sensor spacing in a practical implementation will be larger. For
estimating the systematic error, the STI was calculated with twelve equally distanced virtual sensors
for a frequency range between ʹͲǤ ǤͳͲͲͲ‫ݖܪ‬. The following figure 3 shows the error due to finite
differencing for the twelve sensor method as an example. The error level was calculated in respective
to the calculated STI using the internal forces of the finite elements.

Figure 3 – Finite differencing error for approximating shear forces(i.e. spatial derivative of third order) with variable sensor spacing
As already presumed, the twelve position method is highly sensitive towards the increase of sensor
spacing. With decreasing wavelengths the sensor distance has to decrease too in order to avoid a large
systematic error.
This error was also found to be less when using finite differences of lower order. The two point
method appeared to be the most robust. Nevertheless, the simplified methods include systematic errors
due to neglecting parts of the energy transmission (e.g. twisting moments or near field). These errors,
however, are potentially smaller than the finite differencing errors with large spacing, depending on
the distance to structural discontinuities.
Apart from the systematic error at larger sensor distances, a random error occurs with decreasing
values. As already presumed, a small sensor distance leads to a very small change in amplitude
between the signals. Hence even numerical noise leads to a severe error in approximating thedeviations.
Furthermore the sensitivity towards phase errors of the array sensors was investigated. Those canoccur for example due to different signal transit times or misplaced sensors. It was observed that theuse of finite differences of higher order for the twelve and four point method led to larger amplificationof phase errors in comparison to the two point method, what makes it vulnerable for a real time control
application.
The numerical investigations were examined experimentally on a beam. It could be shown that thetwo position method delivers adequate results even in the near field, as the four point method lacked
accuracy due to noise and phase errors. Therefore the two point method proposes to be preferable fora practical implementation of a real time control, as the main part of the structural intensity can beaddressed.

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3. ACTIVE CONTROL OF POWER TRANSMISSION
3.1 Beam
Investigations towards the control of structural intensity were done for a cantilever in the beginning.
A steel beam of ͳͲͲͲ‫ݔ‬ͶͲ‫͵ݔ‬ሺ݉݉ሻଷ was investigated, being excited on the free end by a single point
force. An optimal control was calculated according to (8), using the force excitation as reference signal.
The secondary controlling force was also assumed to be a point force at ͹ͷͲ݉݉ and a measuring
point ͸ͲͲ݉݉ from the bearing.
The structural intensity according to equations (4) and (5) however cannot be used as a costfunction for the control, as the relationship between excitation and STI is non -linear. Hence theintensity is split into its multipliers, delivering two components linear dependent from the reference
for the two sensor method and four components for the four sensor method, as proposed in (9). For thefour sensor method for example, the components can be written as‫ ܣ‬ൌ ߟሶ ଶ

(6.1)

‫ ܤ‬ൌ ʹߟସ െ Ͷߟଷ ൅ ʹߟଶ

(6.2)

‫ ܥ‬ൌ ߟሶ ଷ (6.3)‫ ܦ‬ൌ െʹߟଷ ൅ Ͷߟଶ െ ʹߟଵ

(6.4)

஻

and equation (4) can be abbreviated to ܲସி஽ ൌ య ሾ‫ ܤ ڄ ܣ‬൅ ‫ܦ ڄ ܥ‬ሿ . Preliminary numericalଶ୼
investigations were done using tonal and random excitation. For the tonal excitation, four frequencies
were chosen: Two eigenfrequencies ( ͷ͹‫ݖܪ‬ǡ ͸ʹͲ‫ݖܪ‬ሻ and two frequencies between resonances(͹͸‫ݖܪ‬ǡ ͸͹͸‫ݖܪ‬ሻ. These two types of frequencies were used to examine the efficiency with standing
waves on the one hand and a higher ratio of travelling waves on the other hand. In order to compare theeffectiveness of the methods, a simple control using the velocity at measuring point as error signal was
calculated.
As an example, the structural intensity and velocity of the beam elements are shown in figure 4 forthe uncontrolled vibration, a velocity control (VEL) and the four point STI control (4FD) at ͹͸‫ݖܪ‬.

Figure 4 – velocity (left) and STI (right) versus beam position at ͹͸‫ݖܪ‬The velocity control achieves a nearly total attenuation of velocity at the measuring point, however
there is nearly no effect on the downstream structure. This can also be seen in the structural intensity,
as there is only a local reduction. Using the STI (or its components respectively) as error signals showsadvantages in the global attenuation of travelling waves. The STI itself is reduced by about ͳͲ݀‫ܤ‬
globally, also leading to a global reduction of velocity. The same effect could be observed for thesecond non-resonant frequency.
In the resonant cases on the other hand, the STI control showed no advantage compared to a one
point velocity control, as there is no change in the phase of the beam elements when purely standing
waves are observed.

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Based on the numerical investigations, a beam structure, clamped on one side, was investigatedexperimentally. To estimate the difference between the control of standing and travelling waves, twoconfigurations of boundary conditions were set, as seen in figure 5.

Figure 5 – experimental setup for control of flexural waves, cantilever (left, configuration I) and cantileverwith inhomogeneous damping (right, configuration II); disturbance (P), control force (S) and measuring
point (M); sensor spacing ȟ ൌ ʹܿ݉ for all arrays
The second beam setup is terminated in a sand box to achieve inhomogeneous damping and avirtually non-reflecting boundary. According to the pre-investigations, an optimal control was
calculated using measured transfer paths and reconstructing intensity time signals. The beam was
excited by an electrodynamic shaker. The control force was induced by an inertial shaker. The STI was
measured using four PCB 352A24/NC accelerometers, enabling the calculation using the two and four
position method with the same sensor spacing. The velocity of the beam along its length was measured
with a laser scanning vibrometer (POLYTEC OFV 055), also providing an error signal for the one
point velocity control. The beam in configuration I was excited in a resonance frequency (ͳʹ͹ǡͷ‫ݖܪ‬ሻ
and a non-resonant frequency (ʹͳͲ‫)ݖܪ‬. Configuration II was excited using a frequency with mainly
travelling waves (͹͸͵ǡͷ‫ݖܪ‬ሻ.

Figure 6 – measured velocity versus beam position for configuration I (left, ͳʹ͹ǡͷ‫)ݖܪ‬and configuration II (right, ͹͸͵ǡͷ‫)ݖܪ‬For the resonant excitation there is no remarkable benefit using the structural intensity as error
signal. A single point velocity control is also able to reduce to overall vibration by about ͻ݀‫ܤ‬. Whenobserving the control of travelling waves in the second configuration, again a global reduction ofvelocity can be achieved by a simple velocity control. A four point intensity control shows no benefit
in attenuation. The two point intensity control however shows slightly better results compared to thefour point method, probably due to the higher robustness against finite differencing errors.
3.2 Plate
A numerical plate model with inhomogeneously damped boundaries was used to estimate theattenuation of an intensity control using different methods. The aluminum plate with the dimensionsͳʹͲͲ݉݉‫ ݉݉͵ݔ݉݉ͲͲͲͳݔ‬was modeled in ANSYS® using SHELL281 elements (element lengthͳͲ݉݉) with free boundaries. The setup conditions can be obtained from figure 7.

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Figure 7 – model of inhomogeneously damped plate for intensity control; disturbance (P), control force (S)and measuring point (M); sensor spacing ȟ ൌ ʹܿ݉ for all arrays
The disturbance is modeled as single point forces, the secondary forces are independently driven
force pairs. The measuring points represent the locations of virtual measuring arrays. For this research,four measuring concepts were chosen: velocity control (single point per array) two point STI control (x direction)
 four point STI control (x direction)
 twelve point intensity control (two dimensional)
The velocity control is again used for comparison. The two and four point method are used under
the assumption that firstly the energy flow in y-direction does not contribute to vibrations downstream
the plate and secondly twisting moments can be omitted, so the plate is represented by beam segments.
The twelve point method is derived from (2), virtually combining the two eight sensor arrays for the
calculation of the additional twisting moments. Any of the above arrays is used at five evenly spaces
measuring points at ‫ ݔ‬ൌ Ͳǡ͸݉.

Figure 8 – numerical velocity distribution of inhomogeneously damped plate at ͹ͻʹ‫;ݖܪ‬from left to right: uncontrolled reference, velocity control, 12 point intensity control
Figure 8 shows the velocity distribution for a simple velocity control and, exemplarily, a twelve
point structural intensity control in the corresponding measuring points in comparison to theuncontrolled reference. A tonal excitation with ͹ͻʹ‫ ݖܪ‬has been chosen, which occurred to be afrequency with mainly travelling waves. A velocity control leads to a merely local reduction ofvelocity in the corresponding measuring points, but only to a comparably smaller global reduction in
the barrier area at ‫ ݔ‬൐ Ͳǡ͸݉. The intensity control on the other hand reduces the velocity globally.

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Using the 2D intensity as error signal, the root mean square reduction of velocity in the barrier areais െ͵Ͳ݀‫ܤ‬, whilst using a velocity control the achievable attenuation is െͳͺܾ݀. Further investigations
using different frequencies have shown a higher performance in terms of global velocity attenuation
for mainly travelling waves. As well as in the one dimensional case, an intensity control shows noadvantage for resonant excitation. Furthermore, simplifying the plate to beam segments using four or
two sensors per measuring point appeared to be less performant. However, noise related errors havenot been taken into consideration yet, so the performance of the twelve point intensity control is
expected to be less in practice.

4. CONCLUSIONSThe experimental investigation towards the active control of structural intensity on a beam partiallyconfirmed the preliminary numerical calculations. There is a small advantage in global attenuation ofvibration for systems with dominant travelling waves, if the structural intensity is used as a control
target. A significant improvement compared to a single point velocity control could not be shown
clearly, especially due to the investigated error sensitivity of the four point method. Generally a global
attenuation could also be achieved using a single velocity sensor.
The numerical results of an actively controlled inhomogeneously damped plate could also show a
reduction in energy transmission and hence a global reduction of velocity in the barrier area, especiallyfor travelling waves. A velocity control on the other hand could only achieve a local reduction. Further
investigations have also shown the importance of measuring the energy components contributed bytwisting moments. In order to make a practical implementation feasible, especially regarding
robustness with respect to phase errors and noise, simplifications have to be taken into account.
Subsequently to this research, the measurement of structural intensity using finite differencing
schemes shall be evaluated experimentally. In this case, especially the influence of sensor spacing inregards to measuring noise shall be investigated numerically and experimentally. Furthermore theseconcepts have to be applied for more complex structures (supported, curved). For an active barrier for
structure borne sound, further investigations towards sensor and actuator placing and numbers have to
be carried out.

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