Towards the placement of actuators and sensors for an active control of structure-borne sound in a stiffened rectangular panel Alexander Kokott 1 Hans Peter Monner 2 German Aerospace Centre – Institute of Composite Structures and Adaptronics Lilienthalplatz 7, 38108 Braunschweig, Germany

ABSTRACT The active control of structure-borne sound proposes an alternative approach towards the global reduction of vibration and noise. By actively reducing or blocking the energy flow from upstream noise sources, e.g. engines or APUs of an aircraft, inside a structure, acoustically relevant down- stream section of a system, e.g. the passenger cabin, may be attenuated. This work deals with a numerical analysis towards the placement of actuators and sensors on a sim- ple stiffened structure in order to achieve a reduction in energy flow, continuing recent research on the topic for simpler plates, where advantages of the approach could be shown already. A set of virtual error sensors is used to measure the components of structural intensity on either the shell or stiffening elements. A feed-forward control is used to drive pairs of control forces to minimize those components and, hence, overall energy transmission. It can be shown that an inclusion of internal forces can achieve a greater global reduction in vibra- tion compared to a simple velocity control also for a stiffened plate. By using structural intensity or its components, a greater reduction of downstream vibration can be achieved using a less dense array of sensors respective to the sound wavelength.

1. INTRODUCTION

The location of noise emission and its sources are not collocal in many cases. This poses the issue of large radiating surfaces compared to the comparably smaller source of vibration. Rear mounted counter-rotating open rotors, for instance, would cause high amplitudes of vibration being transmitted to the aircraft’s front and, hence, the passenger cabin. There, sidewall panels and ceilings, for exam- ple, act as large radiating surfaces all over the cabin. To attenuate this radiation, multiple approaches exist already. Active noise control (ANC) uses the destructive interference of airborne sound waves, whilst active vibration control (AVC) has a comparable approach, but is targeted on the structural vibration itself. Both approaches have in common that they yield a local attenuation only. This loca- tion can be the passenger ’ s ear or head position or certain areas of the mentioned radiating surfaces. A global attenuation of the whole cabin, however, would need a very large number of sensors (and actuators) or would be even unfeasible. The approach of Active Structural Acoustics Control (ASAC), though, can achieve global attenuation by targeting the air-borne sound power. This idea

1 alexander.kokott@dlr.de

2 hans.monner@dlr.de

was adapted to structure-borne sound. Therefore, the structure-borne sound power is used as the con- trol target. The benefit of this approach is the reduction or even blockade of energy transmission to downstream areas. Consequently, this would lead to a global reduction of noise due to simply lacking vibratory energy. The idea itself is not new. Literature reveals several examples of sound power con- trol, e.g. [1 – 3]. However, modern signal processing capabilities bear good prospects for the applica- tion in more complex structures, as for example stiffened curved shells as they appear in an aircraft fuselage. Previous studies of the author confirmed the advantage of this approach on simple structures as beams and plates. This work deals with the next step towards the targeted structures by adding stiffening elements to a rectangular plate. Here, the same approaches are investigated in order to reduce downstream vibration globally.

2. NUMERICAL SET-UP

2.1. Measurement of structure-borne sound

The control of structure-borne sound requires a mathematical definition of the latter in order to define a respective control algorithm. Hence, structure borne sound can be described by the sound power estimate or its areic version, structural intensity. The latter is comparable to sound intensity, which, on the other hand, describes the air-borne sound power per area. Sound power and structural intensity were already investigated in the early 1970s for simple beams and plates. The structural intensity for flexural waves in a single direction can be derived by the following equation [4]:

P x = 𝑄 𝑥 ⋅𝜂ሶ+ 𝑀 𝑥𝑥 ⋅𝜔 𝑥 + 𝑀 𝑥𝑦 ⋅𝜔 𝑦 , (1)

where 𝑄 𝑥 defines the shear force, 𝜂ሶ the element velocity, 𝑀 𝑥𝑦 the bending moment, 𝑀 𝑥𝑥 the twisting moment and 𝜔 𝑥/𝑦 the respective angular rates, as depicted in figure 1.

Figure 1: Coordinate system and measures of positive influence line of a plate element It shall be stated here that this equation as well as the general approach assume the kinematic conditions of Kirchhoff-Love plate theory. This assumption is necessary in order to allow the approx- imation of structural intensity, or its components respectively, by application of available sensors (accelerometers or strain gauges) and finite differencing in multiple orders. This approximation is valid for unstiffened flat plates. For the current case of a stiffened plate, these assumptions are not met anymore. However, it shall be investigated, how the mismatch of conditions and assumptions

still allows a better performance considering a global vibration reduction. Hence, several simplifica- tions and adaptations can be considered. NOISEUX [4] already proposed the assumption of free-field vibration, i.e. vibrations far away (in terms of wavelength) from discontinuities. Hereby, the number of required accelerometers for finite differencing could be reduced to four for a single direction (com- pared to eight for a near-field approximation [5]) by assuming the equality of shear force and moment components of equation 1 and, thus, omitting the shear force summand.

The main problem of approximating structural intensity is the estimation of shear forces and mo- ments, as they require the application of finite differences of up to third order. This amplifies approx- imation errors additionally to the general error posed by plate theory assumptions. In order to counter this differencing error to a major extent, the application of strain gauges poses a promising option. Hereby, the use of finite differencing schemes can be avoided by using structural intensity in its stress formulation [6]:

P x = 𝜎 𝑥𝑥 ⋅𝜂ሶ 𝑥 + 𝜏 𝑥𝑦 ⋅𝜂ሶ 𝑦 + 𝜏 𝑥𝑧 ⋅𝜂ሶ 𝑧 , (2)

The stresses, in turn, can be transformed to strains by Youngs- and Shear-Modulus.

2.2. Finite Element modelling

The FE model examined consists of a rectangular steel baseplate of area 1200 𝑚𝑚 𝑥 1000 𝑚𝑚 (see figure 2). The stiffening is applied by two stringers, i.e. rectangular plates in x-direction, as well as three spars, in y-direction respectively. All boundaries are assumed to be unsupported. The areas are discretised using SHELL281 elements in ANSYS® with a shell thickness of 7 𝑚𝑚 and an element size of 10 𝑚𝑚 for the baseplate as well as 5 𝑚𝑚 thickness and 10 𝑚𝑚 element size for the stiffening panels. Material properties for all elements are defined in table 1.

Figure 2: Geometric model of the investigated stiffened panel

Table 1: Material properti es for the model of a stiffened panel

Young ’ s modulus

Poisson

Density Material damping

ratio

1.82e11 0.3 7953.82 0.1

N/m² - kg/m³ %

All forces are applied as unit nodal forces normal to the respective surface of application in order to estimate the transfer functions to the virtual error sensors. A mono-tonal disturbance force is ap- plied in z-direction at ሾ𝑥, 𝑦ሿ= ሾ0,28,0,05ሿ 𝑚 . The secondary actuators to control structural intensity components are applied as nodal force pairs (in x-direction) in order to apply both, shear forces and bending moments. For each configuration, a different density of control force pairs (in y-direction) is chosen to estimate the density necessary for a barrier effect for power transmission. The particu- lar control force pairs are positioned in an equidistant manner. Additionally, single control forces are located equidistant on the first spar to investigate the possibility of the actuation of the stiffening elements.

The target functions for the control algorithm are derived from virtual error sensors. Virtual, in this case, refers to the fact that it is possible to directly calculate the measures needed in FEM, e.g. shear forces, moments, stresses, without any approximation error by finite differencing. For shear forces and moments, however, this error will occur in an experiment, as there are no sensors availa- ble which are capable of directly measuring these in plane measures. This analysis, though, shall deal with the theoretically necessary density of sensors and actuators for the “ optimal ” case. The error sensors are again positioned equidistant in y-direction at 𝑥= 0,6 𝑚 in different numbers to investigate different error sensor densities.

2.3. Feedforward controller

As formulae 1 and 2 display, structural intensity is non-linearly related to its base measures, as e.g. velocity and, hence, also to exciting force. In order to use a linear feedforward filter for controller design, these measures, being linearly dependent on excitation force, are controlled separately, as also suggested in [7]. The filter weights for the optimal Wiener filter are calculated as presented in [8] with the disturbing force as reference signal.

Multiple target functions are evaluated in order to investigate different control approaches. Firstly, a simple velocity control (AVC) is designed to give a reference control approach with the least num- ber of sensors to be applied. Subsequently, different configurations of virtual error sensors are used. Here, simplifications as well as non-simplified stacks of measures are considered. For the non-sim- plified case, as presented in [5], shear forces, moments, angular velocities and normal velocity are used as a control target. For the simplified case after [4], only moments and angular velocities are considered. Furthermore, an additional use of angular velocities and normal velocities is investigated, proposing a simplification basing on the assumption, that a control of one factor of the summands of equation 1 is sufficient for a global reduction whilst keeping the number of sensors needed to a min- imum. Finally, different combinations of strain gauges are used as target functions. All controllers are calculated for mono-tonal excitations up to 500 𝐻𝑧 .

3. RESULTS

A variety of frequencies was investigated in this scope. The following example for the controller performance at 353 𝐻𝑧 is shown here. Figure 3 shows the velocity distribution of the base panel as a reference. The associated colour scaling is used for all following figures in order to depict the atten- uation.

Figure 3: Velocity amplitude of uncontrolled base panel at 353 Hz

The following figures show the velocity responses of the different controller configurations. The respective rows correspond to different error sensor densities in y-direction of the panel. From the first to the last row, the following number of error sensors is applied: 2-3-5-10-20. The first four columns correspond to different densities of control force pairs, respectively: 2-3-5-10. The last col- umn, in turn, shows the response when actuating the first spar at six, in y-direction equidistant, nodes.

os os Eos of _ 02 1 = 02 04 0808 dem

Figure 4: Velocity amplitude of controlled base panel at 353 Hz;

left: velocity control, right: velocity + angular velocity control

Figure 4 presents a simple velocity control on the left and a combined normal velocity and angu- lar velocity on the right. It can be shown that an addition of angular velocity to the overall target

ie (es Se Es ae i a a MV ae

function can increase the global (global in terms of downstream the error sensors) reduction perfor- mance of velocity of the controller set-up. Comparing the adaptation of the error sensor with the control force density it becomes apparent that a denser error sensor set-up is favourable. Consider- ing the goal of achieving a barrier effect, both set-ups seem to be able to succeed, however, the per- formance of a combined approach proposes to be beneficial.

Figure 5: Velocity amplitude of controlled base panel at 353 Hz; structural intensity control

TT ire 3 ed a Se

Looking at a controller targeting all structural intensity components, i.e. velocity, angular veloc- ity, shear force and moments, in figure 5, a global reduction can even be achieved with a less dense set of error sensors. In this case, also the density of control forces appears to be less critical. Never- theless, for a practical approach it has to be taken into account that the approximation of shear forces and moments demands a way higher number of sensors per measuring position, diminishing the advantage of needing less concrete measuring positions.

ieee Se (AY AY ERY les Res He

Figure 6: Velocity amplitude of controlled base panel at 353 Hz;

left: strain control, right: strain + velocity control

The final set-up to be shown here is the application of strain sensors with and without additional velocity feed in figure 6. By only using strain as target function, the controller performance when using the least number of sensors is slightly better compared to a pure velocity control. Generally speaking, however, for other configurations the benefit cannot be shown. By adding the velocity signal again, the performance can be improved drastically. The performance can be compared to the full structural intensity control as proposed in the set-up before. Though, the number of sensors needed for a single position is way smaller, as second order spatial derivatives are measured di- rectly by the strain sensors (assuming the validity of plate Kirchhoff-Love plate theory).

A general observation can be made when looking at the results of the actuated spar in the respec- tive last column of figures 4-6. Here, no benefit could be shown actuating the stiffened panel at the spars.

4. CONCLUSIONS

The numerical study of this work shows the advantage of expanding the target function for noise reduction control by more measures than velocity only. By adding angular velocity, an overall per- formance increase in terms of global attenuation could be shown whilst keeping the number of sensors for a practical implementation relatively small. The density of error sensors can be decreased as well. Targeting all measures of structural intensity, an even better performance can be achieved. However, the complexity of sensor arrays is much higher and prone to errors, as previous studies have shown. Therefore, it has to be investigated further how far these errors impact the controller performance. Furthermore, ignoring the overall error sensitivity of the approximation, the latter also requires more sensors (up to eight) per measuring point, perhaps leading to the same or even higher number of sensors compared to AVC.

The application of strain gauges shows to be the best of both worlds. Combining strain and velocity sensors improves the controller performance with respect to global downstream attenuation of the structure. Additionally, it poses the possibility to measure second order derivatives (e.g. needed for moment approximation) with a one or two sensors per measuring point, depending on the occurrence of twisting moments. An additional sensor for velocity feedback, however, is still required for global attenuation.

The positioning of control forces, especially the density of those, appears to be of less importance compared to the density of error sensors. Thus, an application of control forces on the main surface of the structure proves to be more performant compared to the actuation of spars.

Nevertheless, further investigation is required. Compared to the results for unstiffened panels in previous studies of the author, stiffening appears to reduce controller performance. This may be ex- plained by the increasing violation of the basic assumptions of plate theory. An even more severe degradation of the controller performance is expected for bent structures and. A practical implemen- tation has to be investigated further, as the errors emerging from noisy signals or misplaced sensors (in terms of finite differencing) also pose a problem for controller performance.

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