Strategic distribution of resonators’ parameters for broadband vibration mitigation

Felipe Alves Pires 1

DMMS Lab Flanders Make / KU Leuven, Department of Mechanical Engineering, Division LMSD Celestijnenlaan 300 B, B-3001, Heverlee, Belgium

Régis Fabien Boukadia 2

DMMS Lab Flanders Make / KU Leuven, Department of Mechanical Engineering, Division LMSD Celestijnenlaan 300 B, B-3001, Heverlee, Belgium

Elke Deckers 3

DMMS Lab Flanders Make / KU Leuven, Campus Diepenbeek, Department of Mechanical Engineering, Wetenschapspark 27, 3590 Diepenbeek, Belgium

Wim Desmet 4

DMMS Lab Flanders Make / KU Leuven, Department of Mechanical Engineering, Division LMSD Celestijnenlaan 300 B, B-3001, Heverlee, Belgium

Claus Claeys 5

DMMS Lab Flanders Make / KU Leuven, Department of Mechanical Engineering, Division LMSD Celestijnenlaan 300 B, B-3001, Heverlee, Belgium

ABSTRACT Locally resonant metamaterials have recently come to the fore as novel lightweight and compact Noise, Vibration and Harshness (NVH) solutions. They have the potential to create stop bands i.e. frequency regions with superior noise and vibration performance. Nonetheless, these stop bands are typically only e ff ective within a limited frequency region. In order to broaden the frequency range of noise and vibration reduction, an optimization procedure could be applied to obtain the ideal distribution of resonance frequencies and masses to achieve a minimal noise radiation over a frequency range. However, this approach typically requires a considerable computational time. Thus, this paper proposes a set of guidelines that define a strategy to determine resonance frequencies and resonant masses in view of widening the NVH reduction without the use of optimization for systems under point force excitation. The proposed strategy is applied to a finite plate with an a priori determined grid for the resonator positions. The performance of the strategy is numerically analyzed by assessing the plate’s vibration response for multiple excitation locations and within di ff erent frequency ranges containing several modes. The obtained responses are then compared to the plate’s response with an optimized grid of resonators. The study shows that the proposed strategy leads to a reasonable vibration reduction with similar levels with respect to an optimized response.

1 felipe.alvespires@kuleuven.be

2 regis.boukadia@kuleuven.be

3 elke.deckers@kuleuven.be

4 wim.desmet@kuleuven.be

5 claus.claeys@kuleuven.be

a slaty. inter.noise 21-24 AUGUST SCOTTISH EVENT CAMPUS O ¥, ? GLASGOW

1. INTRODUCTION

Due to stricter economic, ecological and environmental requirements [1], the use of lightweight structures and designs has become an important aspect for a variety of engineering applications. Nonetheless, lightweight structures typically exhibit a poor noise and vibration insulation performance. Classical NVH solutions are typically bulky and heavy, which conflicts with the trends towards lightweight designs. In the past years, novel techniques have been developed to achieve a good noise and vibration insulation performance while maintaining a lightweight design. For example, locally resonant metamaterials have recently come to the fore and proven to hold great potential to improve the vibro-acoustic response of engineering applications [2, 3]. Moreover, the addition of single Tuned Vibration Absorbers (TVAs) has been widely used [4,5]. However, such solutions are only e ff ective within a limited frequency range. To broaden the frequency range of noise and vibration attenuation, multiple TVAs can be optimally tuned [6, 7]. However, performing optimizations if often computationally demanding. Thus, this paper presents a set of guidelines to achieve a noise and vibration attenuation over broader frequency ranges which contain several modes without carrying out optimizations. The strategy is developed to target systems under a point force excitation. The reduction performance of the proposed approach is numerically evaluated by analyzing the vibration response of a finite plate for di ff erent point force locations and frequency ranges, which contain di ff erent mode shapes. Subsequently, the responses are compared to the response of an optimized grid of resonators. In order to accelerate the system’s response evaluations, a model order reduction (MOR) scheme is used, namely the modal coordinate reduction [8]. The paper is structured as follows: Section 2 presents the numerical model of the finite structure used as a reference, the method to compute the responses in the study and the reference grid of resonators. Subsequently, the optimization method used to compute the optimized configurations is presented. In section 3, the strategies for widening the mitigation of vibrations due to point force excitation are introduced. Section 4 presents the numerical results obtained by applying the developed guidelines to the reference plate as well as to structures with complex shapes for di ff erent mass additions. Lastly, section 5 summarizes the main conclusions of the paper.

2. PROBLEM DEFINITION

This section introduces the numerical model of the host structure and the grid of resonators used in the investigation. In addition, the optimized configurations to benchmark the non-optimization strategies is presented.

2.1. Host structure In this investigation, the host structure is a steel flat plate with dimensions 297 × 420 × 1 mm, whose material properties are shown in Table 1. The finite element (FE) model of the structure is built by using the commercial FE software NX Nastran [9] and composed of 2520 CQUAD4 linear shell elements. Clamped boundary conditions (BCs) are applied along the boundaries of the plate.

Table 1: Material properties of the steel plate.

Young’s Modulus Density Poisson’s Ratio Structural damping

210 GPa 7800 kg / m 3 0.3 1%

The structure is excited by a harmonic point force of 1 N of magnitude applied to the out-of-plane z direction for three di ff erent force locations at coordinates F 1 (63 mm, 63 mm), F 2 (127 mm, 63 mm)

and F 3 (85 mm, 161 mm), as illustrated by the red dots in Figure 1 a). These are used in order to verify the robustness of the guidelines with respect to excitation locations since this work develops strategies for broadening the vibration reduction for point force excitation. Due to symmetry of the host structure, the point force locations are chosen within the same quadrant, namely the bottom left quadrant of the plate. For this investigation, mainly two frequency ranges are of interest, which contain the mode shapes depicted in Figure 1 b): (i) 50 Hz - 250 Hz, with five modes and (ii) 50 Hz - 350 Hz, with seven modes.

Figure 1: Illustration of the a) force locations on the structure and b) mode shapes of interest and their respective eigenfrequencies.

Structural response evaluation

In order to speed up all the computations in the paper, the modal coordinate reduction MOR scheme is applied [8]. The technique is chosen due to its simplicity and accuracy when evaluating global vibration responses. The method uses a modal transformation strategy by using a modal transformation matrix, T , which is formed by the modal matrix Ψ = [ ψ 1 , ψ 2 , ψ 3 ... ψ e ], which consists of e mass-normalized eigenvectors ψ of the system under the chosen BCs. To ensure that the dynamic responses computed from the method have enough accuracy, all the modes whose frequencies are up to three times the highest frequency of interest are retained, as specified in [8]. Thus, for this investigation, modes up to 1050 Hz i.e. 30 modes in total, are used to model the dynamic behavior of the plate. The response of the structure is predicted in terms of Root Mean Square (RMS) velocity assessed in the out-of-plane direction over all of the nodes of the model which are not along the BC edges, for the considered excitation cases. The RMS velocity, v z , RMS , is calculated as

v u t

N X

1 N

j = 1 | v z , j | 2 , (1)

v z , RMS =

where N is the total number of nodes and v z , j the velocity at the j -th node in the out-of-plane direction

As an example, Figure 2 a) compares the response of the bare plate for F 1 calculated using the full and reduced models. A frequency resolution of 0.25 Hz is applied and will be used in all results shown in this work. A good correlation is noticed between the two spectra. Besides, the computational time to obtain the responses are 203s and 2s for the full and reduced models, respectively. Figure 2 b) illustrates the bare response of the plate for the three excitation locations, which are utilized in the following study as a target to achieve broad vibration reduction within the desired frequency ranges.

2.2. Grid of resonators A fixed grid of resonators is used for the development of the guidelines in this study. The grid is composed by a uniform set of TVAs distributed in a 5 × 7 pattern with dimensions 63 × 63 mm i.e. the distance between each TVA is 63 mm. The grid is created to have three resonators per wavelength with respect to the maximum frequency of interest, which serves as a benchmark to develop the strategies. In this case, a total of 35 resonators are added to the host structure, as illustrated in Figure

397 mm 1420 mm ») 120He 1891206122302 20762

Figure 2: Comparison of RMS velocity of the bare panel a) between the reduced and full models for F 1 and b) for the three excitation locations computed by the reduced model.

3. In addition, as studied in [5], an important parameter that is directly connected to the reduction performance of TVAs is the amount of damping present in the resonators. In this paper, 5% of structural damping is utilized for the resonators, which is a typical value found in realized resonant additions [10].

Figure 3: Reference grid of resonators. Each black dot represents one added TVA. A representation of a TVA is also shown, where m , k and c are the mass, the sti ff ness and damping of the resonators.

44 RMS Veocty tos] % a bo Frequency{H2)

2.3. Optimized configurations The main goal of the paper is to develop strategies for widening the vibration reduction over a frequency range with several modes without the use of optimization. However, in this work, the reduction performance of the non-optimization strategies of all test cases is compared with an optimized grid of resonators. The optimal tuning frequency of the resonators are found by optimizing as design variables their masses, m i , and sti ff ness k i , for m i ≥ 0, k i ≥ 0 and i = 1,..., n , where n is the total number of added resonators. To this aim, the hybrid Genetic Algorithm (GA) + fmincon optimization function within the commercial software MATLAB R ⃝ is utilized. In this case, GA is used in order to cover a wide range of possibilities until it reaches a local minimum. Subsequently, fmincon starts to refine this local minimum. In fact, when performing an optimization, an objective function to be minimized is used. This work proposes as an objective function, f( f , v z , RMS ), the sum of v z , RMS computed from the reduced FE model of the structure for the frequency range of interest, and is chosen here as a representative of the vibrational energy in the system. In addition, the optimization terminates when the average relative change in the objective function is less than 10 − 6 .

The optimizations are then constrained by Σ m i ≤ m % , where m % is the total relative mass addition, which is represented as a fraction of the mass of the host structure, M . The optimization problem is formulated as follows

f upper X

Minimize f( f , v z , RMS ) =

f = f lower v z , RMS ( f ) (2)

where f lower and f upper are the lower and upper bounds of the desired frequency range in Hz. This objective function is also used in this paper to have a qualitative evaluation of the vibration reduction

420 mm

performance of the proposed non-optimization strategies with respect to the bare and optimized responses of the structure. In order to have a clearer view of the optimization workflow to achieve the optimized results, a summary of the main steps of the process is shown in Figure 4.

Figure 4: Overview of the main steps to build and optimize a numerical model for broadening the vibration reduction over a frequency range.

3. STRATEGIC DISTRIBUTION OF RESONATORS’ PARAMETERS

This section introduces the strategic approach for widening the vibration reduction over di ff erent frequency ranges. First, the approach to distribute the masses within the grid of resonators is presented. Second, the scheme to tune the resonators is introduced. Subsequently, the design guidelines are summarized.

3.1. Strategy for mass distribution As indicated in the previous section, amongst others, the masses of each resonator within a grid can be optimized to reduce the vibrational response of a system. In this paper, the strategies are developed for di ff erent values of m % , namely from 5% to 40% with respect to the mass of the host structure, M .

As a step towards the strategy, this work proposes a distribution approach for the resonators’ masses proportionally to the static displacement vector, {Q} = [ q 1 , q 2 , q 3 ,..., q u ] [11], where u is total number of degrees of freedom (DoFs) of the system for the applied BCs, which can be calculated as follows

[ K ] u × u { Q } u × 1 = { F } u × 1 , (3)

where [ K ] and {F} are the sti ff ness matrix and load vector, respectively. In this work, only the absolute values of {Q} are used. Furthermore, once {Q} is computed, this work applies an additional step to the calculation, that is {Q} = {Q} 2 , which is done in order to highlight larger values and restrain smaller ones within {Q} . This enables that heavier resonators are placed strategically according to the force location, which allows a more e ffi cient way of blocking the vibrational energy. Assuming that the TVAs are attached to certain DoFs of the structure in the out-of-plane direction, d i , for i = 1,..., n , where n is the total number of resonators with d i ∈ [[1, u ]], the masses of each resonator are calculated proportionally to the value of {Q} 2 at their respective d i . The vector of proportional masses, { m } TVAs , at each d i is calculated as follows

{ m } TVAs = m % × { Q (d i ) } 2

P { Q (d i ) } 2 , (4)

Fe made ofthe + Optimization [Model Order procedures reduction Reduced FE model of the structure ‘Resonator gd (sesign variables) + Dynamic anatss + Evaluate the objective function t "just the resonator parameters to inimie the objective function

Although realizing resonant additions is out of the scope of the study, but in order to be as close as possible to reality, a filter is applied to remove the TVAs with masses smaller than 0.0004 kg given the authors’ experience on the manufacturability di ffi culties of such resonators. The removed masses are then uniformly re-introduced into the grid.

As an example, Figure 5 illustrates the mass distribution for the reference 5 × 7 grid of resonators for the three force location cases. A test case of mass addition m % = 20% is here used. It can be seen for the three cases that not all resonators within the original grid are utilized and most of the resonators near the BCs are filtered out. It is worth pointing out that the resonators near the point force locations are the heaviest within the grid. The step of filtering out small masses is, however, not crucial for the development of the guidelines and is done here for practicality purposes.

Figure 5: Mass distribution within the 5 × 7 grid of resonators for the di ff erent force locations a) F 1 , b) F 2 and c) F 3 , for m % = 20%. The color bars illustrate the values of the masses within the grid scaled from lightest (blue) to heaviest (red). The size of the dots representing each resonator is scaled relatively to its mass value within the grid.

3.2. Frequency allocation approach Subsequently, a strategy is presented to tune the resonators using the pre-allocated masses from the previous step. This work proposes a strategy which uses the modal information given by Ψ = [ ψ 1 , ψ 2 , ψ 3 ... ψ h ], where h is the number of modes within the desired frequency range. The frequency strategy is applied in di ff erent stages as follows: • One step is performed to tackle modes individually by using a TVA e ff ect approach, as similarly described in [12]. In this case, all resonators except the three heaviest ones are tuned such that an e ff ect is achieved only on the mode they are tuned to, which is the frequency of the structural mode that has the highest value of modal displacement within Ψ at their respective location d i . For example, assume h = 5, such that Ψ u × 5 = [ ψ 1 , ψ 2 , ψ 3 , ψ 4 , ψ 5 ]. The tuning frequency of each resonator, f res , i , is determined via max [ Ψ u × 5 (d i )] e.g. if for the resonator attached at the center of the structure in Figure 3 at (148.5 mm, 210 mm), ψ 4 has the highest value at its respective d i within Ψ , then f res , i = 206 Hz. This approach is referred to here as the frequency picking strategy. • Another step is performed to tune the three heaviest resonators to cause additional e ff ects at three specific frequency ranges: (i) frequency range containing all modes except the first two, (ii) frequency range containing only the first mode, (iii) frequency range containing only the second mode. One resonator per frequency range is chosen in this regard in order to be economical in the number of resonators for these separate e ff ects. The resonators are tuned to a lower frequency according to a percentage with respect to the frequency region where the e ff ect is intended:

- The heaviest resonator among the three i.e. near the point force location, is used to block the energy that excites all frequencies within the desired frequency range excluding the first two modes, as a higher modal mass is needed for this purpose. This resonator is tuned to f force according to a percentage X% with respect to the frequency of the last mode present within the desired frequency range such that f force = f last mode - X% f last mode . The best value of X% is found such that a minimum f( f , v z , RMS ) value is obtained within the desired frequency range by applying varying values of X% from 1% to 40%.

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- The two remaining resonators among the heaviest are used to target modes one and two. The choice of resonators to tackle each mode is determined according to Ψ = [ ψ 1 (d i ) , ψ 2 (d i )], in which the resonator located at d i with the highest modal displacement for the first mode is picked to target the first mode and, in turn, its counterpart chosen to target the second mode. These resonators are tuned to f TVA 1 and f TVA 2 according a percentage Y% with respect to the frequency of the modes to be tackled such that f TVA 1 = f mode 1 - Y% f mode 1 and f TVA 2 = f mode 2 - Y% f mode 2 . For the sake of simplicity, Y% is considered the same for modes 1 and 2. As for X%, the best value of Y% is obtained via finding which Y% between 1% and 40% leads to a minimum f( f , v z , RMS ) within the desired frequency range.

Benchmark case for the strategies

To estimate adequate tuning percentages X% and Y%, a study is performed by using the case of F 1 for the frequency range 50 Hz - 250 Hz, as described in Section 2. This case serves as a reference to derive the guidelines in the paper. To find suitable X% and Y%, respectively, this work proposes a strategy that is divided into two steps since each tuning percentage is dedicated to tune separately mainly two sets of resonators, namely, f force and f TVA 1 together with f TVA 2 . This study is carried out for di ff erent values of mass addition m % from 5% to 40% as follows:

X% percentage

This step considers a frequency range between 150 Hz - 250 Hz, where three modes are present i.e. accounting for frequencies excluding modes one and two, as shown in Figure 6 a), dedicated to tune the resonator near the point force location. To this end, the frequency picking strategy is applied concerning modes three, four and five to all resonators except the resonator near the force, which is tuned via f force . In addition, the fact that the resonators in the grid are tuned to di ff erent frequencies in this step allows that a factor of interaction between resonators, which typically present in reality, is taken into account in the analysis. Figure 6 b) illustrates the X% value that leads to a minimum f( f , v z , RMS ) for the test case of m % = 20% i.e. X% = 27%. Since the last mode within the frequency range is the fifth mode at 230 Hz, then f force = 230 - (0.27 × 230) ≈ 167 Hz. For other values of m % , a similar approach is taken to find X% that leads to a minimum f( f , v z , RMS ), as depicted in Figure 6 c). The figure shows a trend towards greater values of X% as m % increases, which can be explained due to the fact that a greater detuning may be needed to cope with the larger shift in frequencies of the modes for larger m % . The resonator detuned via f force is also highlighted.

Figure 6: Illustration of a) frequency range of investigation represented by the black vertical lines, b) X% for a minimum f( f , v z , RMS ) for m % = 20% and c) suitable X% for all m % .

Y% percentage

This step is carried out for a frequency range between 50 Hz - 250 Hz with five modes as shown in Figure 7 a), which is dedicated to tune the resonators for modes one and two. In this case, the heaviest resonator is tuned according to X% and the frequency picking strategy is applied concerning the five modes present to the remaining resonators except the resonators dedicated to tackle modes one and two, which will be tuned according to Y%. Figure 7 b) depicts the Y% value to obtain a minimum f( f , v z , RMS ) for the test case of m % = 20% i.e. Y% = 13%. Consequently, f TVA 1 = 77 - (0.13 × 77) ≈ 66 Hz and f TVA 2 = 124 - (0.13 × 124) ≈ 107 Hz.

a) 1° Zo! F03 J 2 10 b) d i a x al a 2% “ 100 300 400-500 “4p *

A similar trend as for X% is found for Y% when other values of m % are analyzed, whose explanation is discussed previously. The resonators detuned via f TVA 1 and f TVA 2 are also highlighted.

Figure 7: Illustration of a) frequency range of investigation represented by the black vertical lines, b) suitable Y% for a minimum f( f , v z , RMS ) for m % = 20% and c) suitable Y% for all m % .

3.3. Guidelines for widening the vibration reduction Given the above procedures, the guidelines that define the strategy proposed in the paper to design a grid of resonators to obtain vibration insulation over broad frequency ranges are defined as below:

1. Build the FE model of the structure and store the sti ff ness [ K ] and the modal displacement matrix Ψ . 2. Define the point force location and compute the load vector {F} . This will account for the excitation of the system. 3. Define a grid of resonators. The number of resonators in the x − and y − directions is defined by placing three resonators per wavelength with respect to the maximum nominal frequency within the desired frequency range. The amount of damping in the resonators should be defined. 4. Calculate the static displacement vector, {Q} , by using the relation [ K ] { Q } { F } . Additionally, compute {Q} = {Q} 2 . 5. Define the desired amount of mass addition, m % , and distribute the masses of the resonators within the grid proportionally to {Q} 2 considering the values at the corresponding DoF, d i , where each resonator is attached to, for i = 1,..., n , in which n is the total number of resonators. 6. Rank the resonators according to the value of their masses in order to localize the three heaviest resonators within the grid. 7. For the pre-allocated masses from previous step, the tuning frequency, f res , i , of all resonators except the three heaviest are determined according to the frequency picking strategy max [ Ψ u × h (d i )], where u is total number of degrees of freedom (DoFs) of the system for the applied BCs and h the number of excited modes within the frequency range. The three heaviest resonators are tuned di ff erently to produce additional e ff ects in separate frequency ranges according to the next steps. 8. Tune the heaviest resonator to f force by a tuning percentage X% with respect to the frequency of the last mode within the desired frequency range such that f force = f last mode - X% f last mode . The value of X% depends on the chosen m % , as proposed in Figure 6 c). 9. Among the two remaining heaviest resonant elements, choose the resonator located at the point with the highest modal displacement for the first mode to tackle it. In turn, the other resonator is assigned for the second mode. Tune these to f TVA 1 and f TVA 2 by a percentage Y% with respect to the frequency of the modes to be tackled such that f TVA 1 = f mode 1 - Y% f mode 1 and f TVA 2 = f mode 2 - Y% f mode 2 . The value of Y% depends on the chosen m % , as proposed in Figure 7 c).

4. NUMERICAL RESULTS

This section shows the results obtained by applying the developed strategies to achieve vibration reduction over broad frequency ranges for the steel plate and for structures with complex shapes.

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4.1. Application of the guidelines for vibration reduction of a rectangular plate The guidelines developed in the previous section are first applied for the test case of a rectangular A3 steel plate, excited at di ff erent point force locations, as shown in Figure 1. Figures 8 and 9 illustrate the comparison between the vibration reduction obtained by an optimized 5 × 7 grid of resonators and by the proposed approach in this paper for m % = 20%. They depict the cases for the frequency ranges 50 Hz - 250 Hz and 50 Hz - 350 Hz, respectively. The black vertical lines represent the frequency range limits. All the cases are compared with the response of the plate with equivalent mass addition. On the right-hand side of each graph, the values of the objective function of each case are compared in bar plots. It can be seen that, the optimization cases are generally more e ffi cient in terms of vibration attenuation, leading to smaller objective function values within the di ff erent frequency ranges and excitation locations. However, all cases of the proposed approach lead to a similar vibration reduction over the frequency ranges of interest and objective function values with respect to their optimization counterparts. The strategy also shows robustness for di ff erent excitation locations, in which a better performance with respect to the equivalent mass cases is obtained.

Figure 8: Comparison of responses for the frequency range 50 Hz - 250 Hz a) F 1 b) F 2 and c) F 3 .

Figure 9: Comparison of responses for the frequency range 50 Hz - 350 Hz a) F 1 b) F 2 and c) F 3 .

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4.2. Application to complex structures In the previous subsection, the derived guidelines were tested for a rectangular steel plate with clamped BCs for a mass addition m % = 20%. A further step is taken here to check the robustness of the strategies for structures with more complex shapes under di ff erent BCs. Two aluminum plates (material properties shown in Table 2) with applied simply supported BCs are proposed, which are here referred to as A and B , as shown in Figure 10 a) and Figure 11 a), respectively. They both have a thickness t = 0.5 mm. This thickness is chosen in order to ensure that the plates have enough dynamic behavior at low frequencies and to represent a di ff erent case with respect to the A3 steel structure. The FE model of structures A and B are composed of 1289 and 1628 CTRIA3 linear shell elements, respectively. In this study, the desired frequency range for each structure is chosen such that eight modes are present, namely, 70 Hz - 650 Hz for plate A and 20 Hz - 250 Hz for plate B , where the mode shapes of interest are shown in Figure 10 b) and Figure 11 b), respectively. Both structures are excited by a point force, which is located at F A (155 mm, 93 mm) and F B (14 mm, 282 mm). The grid of resonators for plates A and B are created such that three resonators per wavelength with respect to the maximum nominal frequency of interest are present. In this case, a total of 51 resonators are added to the former while 37 to the latter, as shown in Figure 10 c) and Figure 11 c), respectively. The chosen mass addition for plate A is m % = 40% whereas m % = 15% for plate B . The mass distribution throughout both grids is done as indicated in step 5 of the developed guidelines in Section 3. Furthermore, a filter is applied to filter out resonators with masses smaller than 0.0004 kg and re-introduce them through the remaining resonators, as previously done for practicality purposes. Thus, the mass distributions for plates A and B are shown in in Figure 10 d) and in Figure 11 d), respectively.

Table 2: Material properties of the aluminum.

Young’s Modulus Density Poisson’s Ratio Structural damping

69 GPa 2700 kg / m 3 0.33 1%

Figure 10: Illustration of plate A a) dimensions b) mode shapes c) grid of resonators and d) mass distribution for m % = 40%. The color bars illustrate the values of the masses within the grid scaled from lightest (blue) to heaviest (red). The size of the dots representing each resonator is scaled relatively to its mass value within the grid.

First, the frequency picking strategy is applied on both cases, as proposed in step 7 of the guidelines. Second, according to Figure 6 c) and Figure 7 c), the detuning percentages for plates A and B for the considered mass additions m % should be X% = 32% and Y% = 21% for the former while X% = 23% and Y% = 11% for the latter. By applying steps 8 and 9 of the guidelines: (A) f force − A ≈ 413 Hz, f mode 1 − A ≈ 82 Hz and f mode 2 − A ≈ 141 Hz and (B) f force − B ≈ 180 Hz, f mode 1 − B ≈ 29 Hz and f mode 2 − B ≈ 64 Hz. The corresponding resonators for each case are highlighted in Figure 12 a) for plate A and Figure 13 a) for plate B .

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Figure 11: Illustration of plate B a) dimensions b) mode shapes c) grid of resonators and d) mass distribution for m % = 15%. The color bars illustrate the values of the masses within the grid scaled from lightest (blue) to heaviest (red). The size of the dots representing each resonator is scaled relatively to its mass value within the grid.

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It can be seen that, as noticed for the steel plate in the previous section, having an optimized grid of resonators is still slightly more e ff ective with regard to vibration reduction at the targeted frequency range. Nonetheless, the proposed approach achieves similar vibration reduction for all cases when compared with the optimization results over the desired frequency ranges. This shows that the developed strategy to design a grid of resonators for widening the vibration mitigation over di ff erent di ff erent frequency ranges with several modes is robust with respect to di ff erent mass additions and BCs. In addition, the strategy also has a strong potential to be applied to structures with di ff erent shapes, in which complex mode shapes can be encountered. In this case, both also outperform their respective equivalent mass cases.

Figure 12: Illustration of a) the detuned resonators and b) comparison of responses for the frequency range 70 Hz - 650 Hz, for plate A with m % = 40%.

Figure 13: Illustration of a) the detuned resonators and b) comparison of responses for the frequency range 20 Hz - 250 Hz, for plate B with m % = 15%.

5. CONCLUSIONS

The work presented a set of guidelines that define a strategy for widening the vibration attenuation over di ff erent frequency ranges via a grid of resonators. To develop the guidelines, a test case that consists of a rectangular plate excited by a point force with an added grid of resonators is used. The grid is created such that there are three resonators per

wavelength with respect to maximum nominal frequency of interest, in which the strategies are based upon. An optimization procedure is introduced for comparison purposes with respect to the proposed strategy. A MOR scheme is used in order to accelerate the computation of all responses. The use of the strategies can lead to a considerable vibration reduction over di ff erent frequency ranges with similar levels to an optimized grid of resonators, also outperforming the equivalent mass cases. Furthermore, the developed guidelines o ff er robustness regarding di ff erent mass additions, force locations, BCs as well as when applied to structures with more complex shapes, showing strong potential as an alternative to performing optimizations.

ACKNOWLEDGEMENTS

The research of F. A. Pires is funded by an European Union’s Horizon 2020 grant within the Project SILENTPROP (GA 882842). This research was partially supported by Flanders Make, the strategic research centre for the manufacturing industry. The Research Fund KU Leuven is gratefully acknowledged for its support.

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