Acoustical characteristics of segmented plates with contact interfaces
AAcoustical characteristics of segmented plates withcontact interfaces
Srinivas Varanasi a , Thomas Siegmund a, ∗ , J. Stuart Bolton b a School of Mechanical Engineering, 585 Purdue Mall, West Lafayette, IN 47907-2088,Phone: (765) 494-9766, U.S.A. b Ray W. Herrick Laboratories, School of Mechanical Engineering, Purdue University,177 S. Russell St., West Lafayette, IN 47907-2099, Phone: (765) 494-2132, U.S.A.
Abstract
The possibility of shifting sound energy from lower to higher frequencybands is investigated. The system configuration considered is a segmentedstructure having non-linear stiffness characteristics. It is proposed here thatsuch a frequency-shifting mechanism could complement metamaterial con-cepts for mass-efficient sound barriers. The acoustical behavior of the ma-terial system was studied through a representative two-dimensional modelconsisting of a segmented plate with a contact interface. Multiple harmonicpeaks were observed in response to a purely single frequency excitation,and the strength of the response was found to depend on the degree ofnon-linearity introduced. The lower and closer an excitation frequency wasto the characteristic resonance frequencies of the base system, the strongerwas the predicted higher harmonic response. The broadband sound trans-mission loss of these systems has also been calculated and the low frequencysound transmission loss was found to increase as the level of the broadbandincident sound field increased. The present findings support the feasibil-ity of designing material systems that transfer energy from lower frequency1 a r X i v : . [ phy s i c s . c l a ss - ph ] F e b ands, where a sound barrier is less efficient, to higher bands where energyis more readily dissipated. Keywords:
Sound transmission, Non-linear stiffness, Contact interaction,Harmonics, Temporal energy transfer
Nomenclature c : Sound speed in air ρ : Mass density of air d cl : Clearance between segment interfaces D i : Positive-traveling wave amplitude in the downstream duct D r : Negative-traveling wave amplitude in the downstream duct δ : Contact overclosure f : Frequency of sound waves f ex : Excitation frequency of a single frequency sound source f s : Sampling frequency for time domain data f m : Characteristic modal frequencies I : Intensity of the sound field on the incident side I : Net + ve Intensity of the sound field on the incident side I t : Intensity of the sound field on the transmitted side I t + : Net + ve Intensity of the sound field on the transmitted side I s : Sound source IntensityIL: Sound Intensity Level ∗ Corresponding author
Email addresses: [email protected] (Srinivas Varanasi), [email protected] (Thomas Siegmund), [email protected] (J. Stuart Bolton)
Preprint submitted to Elsevier February 10, 2021 : Wavenumber in the ambient fluid L : Edge length of the panel overall L p : Distance between the outer ends of the segments ω : Circular frequency in radians/s P , P , P and P : Complex pressure amplitudes at the ’virtual’ microphonelocations 1, 2, 3 and 4, respectively P | x =0 : Pressure at the incident side of the plate P | x = t p + t s : Pressure on the transmitted side of the plate P i rms : Root mean square sound pressure acting on the incident face of theplate p c : Contact pressure s : Slope relating the contact pressure with overclosure σ : Mass per unit area P w : Lowpass Gaussian white noise sound sourceSPL: Sound Pressure LevelSTL: Sound Transmission LossSTL limp : Analytical sound transmission loss of a limp panel for a givenmass per unit area σ powerShiftFrac: Fraction of net incident power shifted to frequencies otherthan the excitation frequency on the transmitted side in % t p : Thickness of the segments t s : Thickness of the skin u and u : Displacements of the two-dimensional model in x and x u rot : Rotational displacement of the two-dimensional model in x direction3 i : Positive-traveling wave amplitude in the upstream duct U r : Negative-traveling wave amplitude in the upstream duct V | x =0 : Acoustic particle velocity at the incident side of the plate V | x = t p + t s : Acoustic particle velocity at the transmitted side of the plate x and x : Model coordinates
1. Introduction
Metamaterial-based sound barrier solutions [1–5] hold promise as lightweight material alternatives for low frequency noise control. Yet, such ap-proaches generally suffer from poor performance at their resonance frequen-cies which is an important issue that needs to be addressed. Potential so-lutions which mitigate that drawback include: incorporation of dampingmechanisms, stacking multiple panels in sequence [6] to expand the benefit-region frequency-span, use of multi-celled arrays [7] mutually compensatingfor the poor performance of individual cells, disrupting the in-phase motionof the cells at their resonance frequencies, thereby reducing the correspond-ing radiation efficiency, or transferring the energy from the problematic fre-quency regions to higher frequencies where the barrier materials are moreefficient [8].The latter approach, i.e., the shift of energy to higher frequencies, canpotentially be realized by using material systems having non-linear responsecharacteristics. Lazarov and Jensen [9] have shown that the presence ofnon-linear oscillators allows for spreading the local resonances of a peri-4dic structure over a wider frequency range when compared to structureshaving locally-attached linear oscillators. Forced-excitation studies of mate-rials having a bilinear stiffness character revealed their rich sub- and higherharmonic response [10–12]. Work has been performed to understand the re-sponse of such material systems when subject to forced harmonic excitationloads [10–14] for fault diagnostic applications.The idea of shifting energy to higher frequencies where energy absorp-tion is more efficient was studied by Denis et al. [15] in the context ofvibration mitigation. The authors studied plates with tapered power-lawwedges possessing a flexural wave trapping effect, also referred to as theacoustic black hole (ABH) effect, and examined the geometric non-linearityaspect of the tapered wedges in shifting energy to higher frequencies. Theauthors showed through numerical simulations and experiments that theresponse characteristics of such plates have a significant non-linear signa-ture and energy is transferred to higher frequencies. Also, the characterof the response power spectral density was dependent on excitation ampli-tude. Energy absorbing visco-elastic layers were used in the tapered regionsto dissipate the trapped energy, thereby resulting in vibration mitigation.Touz´e et al. [16] demonstrated numerically that effective vibration mitiga-tion can be achieved by using a combination of ABH power-law wedges andcontact non-linearity. The mechanism of vibration mitigation in the lowfrequency band was attributed to transfer of energy to higher frequenciesby contact non-linearity and the more efficient damping of energy achievedat the higher frequencies through the use of ABH wedges combined with5iscoelastic damping material.However, the prospect of using material systems with non-linear re-sponse characteristics for sound barrier applications was investigated onlyrecently by Rekhy et al. [17]. The authors studied both numerically andexperimentally the characteristics of plate structures with a mass-loadedmembrane and an impactor; thus creating a non-linear system, and the fea-sibility of shifting energy from lower frequencies to higher frequencies wasdemonstrated. The Sound Transmission Loss (STL) was measured acrossa frequency band of white noise by using a four-microphone standing wavetube procedure [18]. An improvement in the STL at the lower frequencieswas attributed to the up-frequency shift of the incident energy. However,the measurement of the inter-microphone transfer functions in that pro-cedure requires that the upstream and downstream signals be linearly re-lated at each frequency, which is unlikely to be the case when the incidentbroadband random signal passes through a nonlinear system. Further, theauthors used a one-load measurement, which is not appropriate for asym-metrical samples like theirs. Thus, their results are interesting but requirefurther verification.Here, we consider a planar cellular panel with its unit cells as shown inFig. 1. Unit cells can either comprise a cell frame and a cell-filling mono-lithic plate [4], Figs. 1(b) and (d) or a cell-filling segmented plate, Figs. 1(c)and (e). A planar cellular panel with its unit cell comprising segmentedplates with contact interfaces can possess linear, bilinear or nonlinear stiff-nesses. Although there are a number of variations in possible contact be-6avior, the qualitative nature of their harmonic responses to a pure toneexcitation was expected to be similar [11, 12, 19, 20]. The behavior hasbeen studied extensively in the context of fault diagnostics, but the mainobjective of the present study was to explore the feasibility of employingcontact non-linearity in the design of efficient sound barriers. Therefore, asimple contact scenario of frictionless hard contact was studied in this work:i.e., the examination of the sound transmission characteristics of segmentedplates possessing bilinear or non-linear stiffness behavior due to the pres-ence of contact interfaces. Here, numerical models were employed to studythe characteristics of representative two-dimensional models comprising asegmented plate having a contact interface, Fig. 2(b), and connected to eachother with an adhesive skin having a very small stiffness.
2. Model Definition
Two-dimensional models, Fig. 2, based on the three-dimensional em-bodiment of segmented metamaterial panels, Fig. 1, were studied. Thetwo-dimensional models consisted of two plate segments attached to a skinthat provides an integrating support to the structure. The distance betweenthe outer ends of the segments, L p , Fig. 2(a), was chosen to be 63 . t s , was taken to be 50 µ m. The thickness of theplate segments was set to t p = 1 .
27 mm, equivalent in stiffness at its centerto a three-dimensional plate with its in-plane dimensions aspect ratio beingequal to 1 and having a thickness of 1 .
00 mm (assuming the same elasticmodulus for the three-dimensional and two-dimensional cases). The choice7f L p was inspired by the duct dimensions of an available standing wavetube apparatus [21] with a square cross-section of 63 . × . t s , was representative of a typical adhesive tape. The choiceof the clearance, d cl , magnitudes was determined by the possible interfacefineness that can be achieved with machined acrylic plates.Two models based on the configurations shown in Fig. 2(b) are referredto here as M1 and M2; they differed from each other in the clearance sep-arating the segments. While M1 had a clearance ( d cl ) of 10 µ m, M2 wasgiven a clearance of 100 µ m. A third segmented plate model, M3, with askin thickness t s of 500 µ m and zero clearance ( d cl ) between the segmentswas also studied. All the other dimensions characterizing the model re-mained the same as specified above. An unsegmented equivalent of modelsM1 and M2, Fig. 2(a), representing a structure with linear response, wasalso used in this study for comparative analysis, and is referred to as modelM0. Model M0 includes the segments and the skin with all of them bondedto each other. The dimensions of the various models are summarized inTable 1. Typical acrylic material properties, i.e., E = 3 . ν = 0 .
35 and a mass density of ρ = 1160 kg/m , were applied to boththe skin and the plate segments in all models.The contact interaction between the two plate segments was modeledusing a solid-to-solid contact described by a linear pressure over-closurerelationship, as illustrated in Fig. 3. Such a model is available in the FE-code ABAQUS [23]. The contact pressure ( p c ) - overclosure ( δ ) relationshipis p c = sδ , with s = 10 Pa/m for M1 and M2, and s = 10 Pa/m forM3. The contact constraint was enforced using a Direct method [23]. TheDirect method strictly enforces a given pressure-overclosure behavior foreach constraint, without approximation or use of augmentation iterations:this is akin to enforcing a hard contact with a penalty constraint. Thedisplacements u and u in the x and x directions were monitored at twonodes located on the far right end on each of the contact interfaces.
3. Analysis Methods
Two-dimensional finite element (FE) models were built for a static anal-ysis to evaluate the stiffness characteristics of the plate systems. The analy-sis was performed with FE-code ABAQUS [23]. The present model employsa combination of solid elements for the plate components as well as beamelements for the skin component. The use of plate elements was not possi-ble in this study since the face-edge-to-face-edge contact conditions betweenthe segmented plates cannot be enforced in a model built with plate ele-ments. The plate segments were discretized using two-dimensional plane9train quadratic elements (CPE8R in ABAQUS). The segments were mod-eled by using a biased mesh with finer resolution towards the contact inter-face: the smallest element size was ∼
1% of L p and the largest element sizewas ∼
3% of L p . The thickness dimension of the segments was discretizeduniformly with an element size of 5% of t p . The skin was modeled byusing one-dimensional linear structural beam elements (B21 in ABAQUS)which are based on Timoshenko beam theory with a uniform element sizeof 0 . ∼ . L p . The choice of analytical beam elements over con-tinuum solid elements to model the skin greatly reduced the computationalexpense. The out-of-plane depth of the plate segments and the skin weretaken to be unity.Figures 4(a) and (b) show the loading and boundary conditions. Auniform static pressure load of 100 Pa was applied consecutively to the theskin in the + ve x direction, Fig. 4(a), and the segment faces in the − vex direction, Fig. 4(b). The load applied to the skin was released while thesegment faces were loaded. The ends of the beam extending beyond theplate segments were fully constrained: i.e., u = u = u rot = 0, where u and u are the displacements in the x and x directions, respectively, and u rot was the rotation about the x axis. Two-dimensional FE models were created to evaluate the natural fre-quencies of the models M0, M1, M2 and M3. The natural frequency ex-traction methods in the FE-code ABAQUS were employed with the detailsof this procedures described in [24–26]. Characteristic eigenmodes of the10odels were determined by modal analyses performed by using the Lanc-zos eigensolver method [23]. The domain discretization and the boundaryconditions remained the same as in the static analysis. Note that a linearprocedure cannot account for contact non-linearity. Thus, the character-istic modal frequencies ( f m ) for M1, M2 and M3 were determined for thetwo following configurations: (1) the open configuration, in which the platesegments were coupled only by the thin skin so that the plate segmentsinteracted with each other without any contact, i.e., representing a config-uration with a wide gap and small amplitude vibrations; and (2) the closed configuration in which the plate segments were fused to each other to form aconventional monolithic plate. The characteristics of bilinear materials [27]are determined by their bilinear frequencies, which are the harmonic meansof the corresponding modal frequencies in the open and closed configura-tions. By using those modal frequencies along with those of M1 and M2 intheir open configuration, the corresponding first bilinear frequencies weredetermined. Similarly, for M3, its first bilinear frequency was determinedby using its modal frequencies in the open and closed (interface tied) config-urations. The various natural frequencies are listed in Table 2 The bilinearfrequencies of the models were used when choosing the excitation frequen-cies of the sound sources in the subsequent acoustical analyses. A two-dimensional FE model was also built to study the acoustical prop-erties of the models, Fig. 5. Again, the FE-code ABAQUS [23] was em-11loyed. The model was built to reproduce the standing wave tube setupthat is typically used for characterizing barrier materials [18, 21, 22]. Thenumerical model involved a coupled structural-acoustic procedure basedon implicit dynamic analysis using implicit time integration methods [23].That procedure allows for simulating the non-linear behavior of contactinteractions. The contact interface was modeled following the same rulesdescribed above in the context of the static analysis. The model consistedof two acoustic domains (upstream and downstream) and a solid domainconsisting of the segmented plate structure separating the two acoustic do-mains. The lengths of the up- and downstream acoustic domains were takento be 180 mm each with their width equal to 63 . A sound source radiating a single frequency ( f ex ) sound pressure wassimulated at the upstream end of the acoustic domain by prescribing thenormal derivative of the acoustic pressure per unit density. In addition,the upstream termination was modeled as non-reflecting by imposing thespecific acoustic impedance of air ρ c ( ρ is the density of air and c is12he speed of sound in air). Similarly, the downstream end of the acousticdomain was also modeled as an anechoic (non-reflecting) termination. Eachof the models M0, M1, M2 and M3 was studied for their behavior at threeexcitation frequencies, which were selected based on their correspondingfirst characteristic bilinear frequencies.The predicted sound pressures were recorded by using virtual ’micro-phones’ at two axial locations both in the upstream and downstream acous-tical domains at time intervals of 1 × − sec: i.e., at a sampling rate, f s ,of 10 kHz, the Nyquist frequency thus being 5 kHz. The upstream ’micro-phones’ were located at x = − . − . x = 151 . . f s , was at least 5 times higher than the maximum frequency ofinterest. Note that higher order acoustic modes would begin to propagateat approximately 2700 Hz in a standing wave tube having the cross-sectionaldimension considered here. The pressure measurement locations were cho-sen so that they were at least a distance L p away from the unit cell oneither side to avoid near-field effects. The results are presented in terms ofsound intensity spectra which were calculated as described in Section 3.3.3below. Apart from recording the sound pressures at the virtual ’micro-phone’ locations, the displacements u and u were monitored at the farright ends of the segments’ contact interfaces. The latter information wasused to ascertain the occurrence of contact and to correlate the observationsof displacements and sound pressure.13 .3.3. Analysis of the Microphone Data An acoustic layer having linear behavior can be characterized by a 2 × ve x and − vex directions in the upstream and downstream ducts. That information wasfurther used to determine the pressures and acoustic particle velocities onboth sides of the test specimen [18]: i.e., on the incident and transmittedsides, Fig. 5. From that data, the time-averaged sound intensity spectraon both sides of the test specimen were computed following the proceduredescribed in Appendix A. The time-averaged sound intensity spectra werefurther resolved into + ve x and − ve x traveling components based ontheir algebraic sign. The spectrum with + ve valued intensity on the up-stream side of the specimen was the net incident intensity, here referred toas I , and the + ve valued intensity on the downstream side corresponded14o the net transmitted intensity, here referred to as I t + . The fraction ofsound power shifted to frequencies other than the incident frequency wasquantified by summing the corresponding intensities in I t + , dividing the re-sult by the sum of the net transmitted intensities I t + , and expressing thatratio in percentage terms, i.e., (cid:80) f (cid:54) = f ex I t + (cid:80) I t + × While the harmonic sound source excitation studies helped to illustratethe qualitative response characteristics, a broadband sound source excita-tion study was conducted to quantitatively study the STL characteristics ofthe segmented plates. A lowpass Gaussian white noise time-domain signal( P w ) was generated by applying a low-pass FIR filter with a cutoff fre-quency of 1000 Hz to a Gaussian white noise signal with a mean value of0 Pa and a standard deviation of 1 Pa. The signal was generated for atotal time length of 10 sec with a sampling frequency of 10000 Hz. Signalswith standard deviations of 10, 25, 50 and 100 Pa were then created byscaling the standard deviation to 10, 25, 50 and 100 Pa, respectively. Thesignal with a standard deviation of 10 Pa was used for the comparativestudy of models M0 − M3 while all the signals were used to study the effectof sound source amplitude on M1. The modeling of the sound source end asnon-reflecting helped to improve the signal-to-noise ratio by minimizing re-flections at the source end. The predicted sound pressures were recorded by15sing virtual ’microphones’ at two axial locations in both the upstream anddownstream acoustic domains at time intervals of 1 × − sec, as previouslynoted. Sound pressures were also recorded at two additional locations; onecloser to the sound source in the up-stream duct, and the second in thedown-stream duct nearer to the sample compared to the other two micro-phones to serve as reference signals, respectively. The upstream reference’microphone’ was located at x = − . x = 139 . The time-domain non-deterministic signals from the six microphones de-scribed above were processed using the procedure laid out for the standardtest method to measure normally incident sound transmission of acousti-cal materials [21]. The complex acoustic transfer functions between thereference and the other two microphones in each of the up-stream anddown-stream ducts were determined by taking the ratio of the one-sidedcross-spectral density of each of the microphone signals with the referencemicrophone one-sided power spectral density. The physical process was as-sumed to be stationary and ergodic in nature and time-domain averagingwas used to evaluate the cross-spectral densities. The complex transfer func-tions were in-turn used to resolve the incident and reflected sound waves as16 function of frequency in the up-stream and down-stream ducts, separately.Sound intensities on the incident ( I ) and transmitted ( I t ) side of the samplewere evaluated in the frequency domain by using Eqs. (A.2) and (A.3) givenin Appendix A. The STL in the frequency band of excitation (0 − I t + )to sound source intensity ( I s ) and converting the result to decibels: i.e.,STL = −
10 log ( (cid:107) I t + /I s (cid:107) ) . (1)Note that the transfer matrix method described in [21] could not be usedto compute STL since the plate response was non-linear in nature. Instead,the STL measure defined above is in-line with the method typically usedto characterize a sound barrier in a reverberation room test setup [5]. Thepredictions were compared against an analytical limp panel STL. For a limppanel with a mass per unit area of σ , the STL for a normal-incidence soundfield (cid:0) STL limp (cid:1) was calculated as [8]:STL limp = 10 log ρ c + ω σ ρ c . (2)Here, ρ is the density of air, c is the speed of sound in air, ω = 2 πf where f is the frequency in Hz, and σ is the areal mass.
4. Results
The skin and the plate-plate-face-edge contact are an integral part of thesegmented plate configuration of interest. Contact and skin deformation are17he key factors in the deformation response of the system. Figure 6(a) showsthe resultant force versus u (deflection in the x direction) for models M0,M1 and M2, all of which have the same thickness for the beam representingthe skin. The resultant force was calculated by multiplying the applied uni-form pressure by the area of the plate segments normal to the x direction.First, studying the stiffness behavior when a load was applied to thesegment faces in the − ve x direction, it can be seen that M1, d cl = 10 µ mclearance, shows a sharp change in slope due to the closing of the contactinterface at an axial displacement of u ≈ µ m. The displacement u atwhich the plates come in contact was analytically estimated to be equal to L p d cl t p (when it is assumed that the skin kinks at its center point and that thecurvature at the remaining locations is negligible - see Appendix B). Thelatter value was 125 µ m in the present case, the same as the numericallypredicted value. The ratio of the high to low stiffnesses in this phase was127. The estimated displacement u after which M2 with d cl = 100 µ mcomes in contact was 1250 µ m. The contact does not close even for themaximum static load applied, but its behavior becomes nonlinear due tothe geometric nonlinearity of the unsupported segment of the skin at itscenter portion. Here, it can also be seen that the deflection curve for M2traces that of M1 until the point where the contact occurs for M1 but thencontinues in the same trajectory since contact did not occur in the M2 case.For M0, the deflection behavior is linear with its slope being equal to thatof M1 in the closed configuration.Next, studying the characteristics of loading on the skin in the + ve x − ve to + ve . The force magnitudefor the same deflection magnitude was slightly above 3 N for loading onthe skin in the + ve x direction when compared to the corresponding forcemagnitude for loading on the segment faces in the − ve x direction, whichwas slightly below 3 N. In the case of M0, the stiffness remains linear andwas the same as seen in the − ve x loading direction.Figure 6(b) shows the force-deflection behavior for M3 which has athicker beam representing the skin (10 times thicker than that of M0, M1and M2) and with a zero thickness clearance. Here, it can be seen thatthe curve segments are linear in the − ve and + ve loading directions witha slope change occurring at x = 0 with the curve in the third quadranthaving a higher slope. The ratio of the high and low stiffness values in thiscase was 1 . The first mode of the equivalent unsegmented plate M0, Fig. 7(a), occursat 429 Hz. Figures 7(b,c) show the first mode shapes for the models M1,M2 and M3, qualitatively, with the details of their occurrence dependingon the specific model stiffness and mass characteristics. The occurrences ofthe modes for M1 and M2 were very close since the only difference betweenthe two was the contact interface clearance. Hence, their equivalent in19he closed configuration can be taken to be the same: i.e., M0. The firstmodal frequencies in the contact-open, contact-closed configurations, andthe corresponding bilinear frequencies are listed in Table 2. With the ideaof having excitation frequencies for the acoustical analysis that bracket thefirst bilinear frequency for each case, excitation frequencies of 20 ,
80 and320 Hz were chosen for M1 and M2. The same excitation frequencies werechosen for the unsegmented M0 case to allow for a comparative analysis.The excitation frequencies were chosen to be 80 ,
320 and 640 Hz for the caseof M3 since its first bilinear frequency is 451 . Figures 8(a-f) show the net + ve sound intensity spectra ( I and I t + ) onthe upstream and downstream sides of the unsegmented plate (M0) for ex-citation ( f ex ) frequencies of 20 ,
80 and 320 Hz, respectively, at an amplitudeof 100 Pa (strength of the incident sound pressure wave at the source end ofthe up-stream duct). In these plots, the intensity is expressed in a decibelscale with a reference intensity of 10 − W/m , and in that form is furtherreferred to as the Sound Intensity Level (IL). From Figs. 8(a-f), it can be ob-served that harmonics were predicted in the transmitted spectra ( I t + ) withdecreasing orders of magnitude compared to I . The last significant har-monic was around 20 dB while I was around 80 and 100 dB for f ex = 20and 80 Hz, respectively. The magnitude of the lowest level harmonic in I t + was around 20 dB compared to 120 db of I for f ex = 320 Hz. An in-significant response was also predicted for 320 Hz excitation at non-integral20ultiples of the corresponding excitation frequency. Table 3 summarizespowerShiftFrac for all the cases studied in this work and Table 4 lists thecorresponding SPL x =0 on the incident side of the panel. For 20, 80 and320 Hz excitations, powerShiftFrac for M0 was 0 . .
11% and 0 . I and I t + of the segmented plate model M1 for f ex = 20 ,
80 and 320 Hz, respectively, at a sound source amplitude of 100 Pa.Figures 10(a-b) show the zoomed-in views of I t + for f ex = 20 ,
80 Hz exci-tation frequencies. The occurrence of upper-harmonics for excitations of20 and 80 Hz extended to 3900 and 5000 Hz, respectively, with the lowestlevel harmonic of magnitude 0 db and their corresponding I at 120 dB.For f ex = 320 Hz, there were only three upper-harmonics in I t + with thefarthest at 960 Hz and around 30 dB in magnitude against an I of 120 dB.No subsequent upper-harmonics were predicted. There were some upper-harmonics at approximately the same order of magnitude as I for f ex = 20and 80 Hz. The magnitudes of the upper-harmonics decreased steadily asthe frequency increased for all the cases of f ex . No sub-harmonics werepredicted for any of the three excitation frequencies, Figs. 10(a-b), in I t + .Some sub-harmonics were predicted for the excitation frequency of 80 Hzin I with an average magnitude of 20 dB. The powerShiftFrac for 20, 8021nd 320 Hz excitations were 66 . .
7% and 0 . f ex = 20 and 80 Hz with the fractionbeing higher for 20 Hz. The influence of the proximity of the excitationfrequency to the first flexural frequency of 55 . f ex = 320 Hz, unlike the other excitationfrequencies, the powerShiftFrac was zero suggesting that contact did not oc-cur at the interface. Figures 11(a-c) show the transverse displacement, u ,versus time of the far right end of segment 1, Fig. 2(b), for f ex = 20 ,
80 and320 Hz, respectively. It can be seen that contact does occur in the casesof 20 and 80 Hz, but not in the case of the 320 Hz excitation. While thedisplacement u was limited to 5 microns in the former cases (the clearancewas 10 microns for M1), the maximum displacement reached for 320 Hzexcitation was below 5 microns.Figure 12(a-f) depicts results for model M2 which only experienced ageometric nonlinearity, unlike M1 which experienced both geometric andcontact non-linearities (Section 4.1). The occurrence of upper-harmonics in I t + extended to 840 Hz for the f ex = 20 Hz excitation with its I at around120 dB. The drop in magnitude of I t + was linear in the dB scale and almostmonotonically decreasing. The frequency range for non-zero I t + extendedto 1280 and 960 Hz, respectively, for the 80 and 320 Hz excitations. Nosub-harmonics were predicted for any of the excitation frequencies eitherin I or I t + . All the observations made in the context of M1 apply tothis case as well. The main difference was that the occurrence of upper-harmonics was less pronounced, with their strength dropping much more22apidly with increasing frequency which was reflected in powerShiftFracdropping to 46 .
2% and 7 .
46% for f ex = 20 and 80 Hz, respectively. Therewas no change in the powerShiftFrac for f ex = 320 Hz from 0 . µ m at all the excitation frequencies.Note that the transverse displacement for the excitation frequencies of 20and 80 Hz does not show a pure harmonic behavior although it is periodic.Figures 14(a-f) show the acoustic response characteristics of model M3at an excitation amplitude of 100 Pa for excitations of 80, 320 and 640 Hz,respectively. Higher excitation frequencies were considered for this caseto bracket the higher flexural bilinear frequency of 451 . I t + for all three excitation frequencies with thefarthest significant harmonic magnitudes of 10, 20 and 40 dB, respectively.The corresponding I were 100, 120, and 120 dB. In contrast to the modelsM1 and M2 which experienced geometric non-linearity, M3 showed only acontact non-linearity. Also, M3 had a lower bilinear stiffness ratio of 1 . .
94% for 80 Hz excitation. The trendof decreasing powerShiftFrac with increasing excitation frequency continuedfor M3. The presence of a flexural resonance frequency of 451 . I t + spectra for 320 and 640 Hz excitations.The upper harmonics were caused by the contact closure, as reflected inthe transverse displacement plots at f ex = 80, 320 and 640 Hz shown inFig. 15(a-c), respectively. All the cases discussed in the previous section were based on the samesource excitation strength of 100 Pa. Figures 16(a-f) show I and I t + spectral characteristics of M1 for an excitation amplitude of 10 Pa. Bycomparing the response for M1 at the two different amplitudes, it can beseen that upper-harmonic response was less pronounced at the reduced am-plitude. Significant upper-harmonics were observed only for f ex = 20 Hzextending to 3840 Hz. The drop in the strength of the upper-harmonics in I t + was steeper compared with the characteristics of M1 for the 100 Pa ex-citation. This was reflected in the powerShiftFrac which was only 26 .
68% asopposed to 66 .
80% for the 100 Pa excitation amplitude at 20 Hz. Similarly,the powerShiftFrac dropped to 0 .
00% at f ex = 80 Hz. Extended contactclosure occurred for the 20 Hz excitation, Fig. 17(a-c), thus supportingthe upper-harmonic acoustic response predictions. The powerShiftFrac for10 Pa amplitude excitation for M2 and M3 presented a similar trend in thedrop of powerShiftFrac, as can be seen from Table 3.24 .3.3. Broadband Excitation Response Characteristics Figure 18(a) shows the STL characteristics of M0, M1, M2 for the samesound source excitation: i.e., the lowpass filtered (0 − .
53 kg/m is overlaidfor comparison. Similarly, Figure 18(b) shows the same result for M3. Theequivalent limp panel for M3 had an areal mass of 2 .
05 kg/m . The STLcharacteristics of all the models primarily followed the trend of a classicedge-constrained plate [28–30] starting with a high STL at very low fre-quency, decreasing steeply until the first flexural resonance frequency, andthen steadily increasing beyond it. The dip in the STL for M1 and M2occurred around 50 Hz, which corresponded to the first flexural resonancefrequency of 55 . . − .
53 kg/m was25verlaid for comparison. It can be seen from this result that the improve-ment in STL becomes increasingly significant in the very low frequencyrange of 0 −
200 Hz as the standard deviation of the white noise increases.A unifying feature for all the plots was that each of them followed the trendof a classic edge-constrained plate [28–30] but with a different frequency forthe dip in STL: i.e., a lower excitation amplitude corresponded to a lowerdip frequency. As the amplitude increased, a noticeable loss in STL was pre-dicted in the frequency range of 200 −
600 Hz. The loss in STL decreasedbeyond 600 Hz eventually converging with the STL prediction for 10 Paamplitude. Figure 19(b) shows the coherence plots of downstream micro-phone m3 and the corresponding input sound source signal for the caseswith standard deviations of 10, 25, 50 and 100 Pa, respectively. It can beseen from this result that the coherence value fell increasingly below unityin the low frequency range of 0 −
600 Hz as the excitation amplitude wasincreased, suggesting an increasing non-linearity effect on the predictions.That observation correlated with the differences in the STL predictions seenin Fig. 19(a).
5. Discussion
Upper-harmonics occur in the acoustical response of structures havinggeometric and contact non-linearities. The presence of contact interfacesparticularly accentuates the non-linear effect due to the steep change inthe stiffness. In fact, it was shown that a bilinear stiffness behavior canbe expressed as a polynomial non-linearity of order 4 [12] in the context26f forced-excitation response studies. The predominant trend that can benoted from the powerShiftFrac for all the cases listed in Table 3 was that thelower the pure tone excitation frequency, the larger was the amount of powershifted to higher frequencies. The only case where the trend was differentwas M0, which only exhibited a mild non-linearity discussed further be-low. In that case, the powerShiftFrac increased as the pure tone excitationfrequency increased with a maximum powerShiftFrac of 0 . . f ex = 320 Hz being the closest to it, and that frequency had the highestpowerShiftFrac of 0 . > M2 > M3 > M0which correlates with the bilinear contact stiffness ratio and geometric non-linearity. M1 had both the highest bilinear contact stiffness of 127 andgeometric nonlinearity reflected in the curved nature of the force-deflectioncurve in the first quadrant. M2 had a similar geometric non-linearity witha not-so-significant bilinear stiffness: the ratio of the tangent stiffness at u = 300 and − µ m was ∼ .
1. M3 represented a scenario primarily ofcontact non-linearity with a lower bilinear stiffness ratio of 1 . onclusions The sound barrier performance of planar cellular metamaterials is poorat their flexural resonance frequencies. The possibility of mitigating thoselimitations of cellular panels by incorporating additional dissipation mecha-nisms was studied here. One way to overcome those limitations is to transferthe energy from the frequency band having low STL to higher frequencieswhere it can be blocked more effectively. A material system having non-linear stiffness behavior introduced through contacts has the potential torealize the above mechanism. The effects of contact nonlinearity, geometricnonlinearity, and both acting together, were examined and compared withthe corresponding linear case through two-dimensional numerical models ofa segmented plate with a contact interface subjected to a normally incidentsound field. Based on the numerical study, upper-harmonic response peakswere observed for single frequency sound excitations, the strength of whichwere found to strongly depend on the degree of non-linearity or bilinear stiff-ness ratio. The response was also found to depend on the relative locationof the excitation frequency with respect to the eigenmodes of the structure,with stronger upper-harmonics occurring for excitation frequencies closer tothe characteristic eigen-frequencies, thus supporting the idea of transferringenergy from the deficit regions to higher frequencies. Broadband responseof the materials was also examined and notable improvement in STL waspredicted at low frequencies. Experimental validation of the observationsmade in this work would be the next logical step in taking the idea further.
Acknowledgment : The authors gratefully acknowledge the financial sup-30ort provided by the United States Air Force Office of Scientific Researchthrough the grant FQ8671-090162. We also thank the reviewers for theirdetailed and helpful comments. 31 able 1: Clearance and the skin thickness of the two-dimensional segmented plate modelsM1, M2 and M3.
Model Clearance - d cl [ µ m] Skin thickness - t s [ µ m] CharacteristicsM0 NA 50 UnsegmentedM1 10 50 SegmentedM2 100 50 SegmentedM3 0 500 Segmented Table 2: Natural frequencies for the two-dimensional segmented plate models M1, M2and M3 and the unsegmented model M0.
Model Contact-open Contact-Closed Bilinear Frequencyfirst mode [Hz] first mode [Hz] [Hz]M0 NA 429 NAM1 29 . . . . . . . able 3: Power shifted to other frequencies on the transmitted side in % of net inci-dent power for the two-dimensional segmented plate models M1, M2 and M3 and theunsegmented model M0. Model Source Amplitude Source Amplitude powerShiftFrac in % at f ex =[Pa] [dB] 20 Hz 80 Hz 320 Hz 640 HzM0 100 131 0 .
08 0.11 0.19 NAM1 100 131 66 .
80 51.70 0.00 NAM2 100 131 46 .
20 7.46 0.00 NAM3 100 131 NA 13 .
94 3 .
54 0 . .
68 0.00 0.00 NAM2 10 111 9 .
96 0.00 0.00 NAM3 10 111 NA 11 .
63 3 .
97 0 . able 4: Sound pressure level on the incident side (SPL x =0 ) for the two-dimensionalsegmented plate models M1, M2 and M3 and the unsegmented model M0. Model Source Amplitude Source Amplitude SPL x =0 on the incident side[Pa] [dB] 20 Hz 80 Hz 320 Hz 640 HzM0 100 131 137 . . . . . . . . . . . igure 1: (a) A planar cellular panel, (b) a unit cell with a monolithic plate filling thecell, (c) a unit cell with a segmented plate filling the cell, (d) and (e) side views of theunsegmented and segmented unit cells illustrating their periodicity. Figure 2: (a) Two-dimensional monolithic model, and (b) two-dimensional model ofsegmented plates with a contact interface supported by a skin. igure 3: The contact pressure ( p c ) – over-closure ( δ ) relationship describing the contactinteraction. igure 4: (a) Static analysis models for obtaining the system stiffness for loading in (a)+ ve x and (b) − ve x directions. igure 5: A schematic of the two-dimensional FE model used for acoustical characteri-zation of the segmented plates models and the unsegmented plate model. Figure 6: (a) Force versus deflection behavior for models M0, M1 and M2. (b) Forceversus deflection behavior for model M3. igure 7: (a) The first mode shape of the unsegmented plate, (b) the first mode shapeof M1, M2 and M3 with the contact open, and (c) the first mode shape of M1, M2 andM3 with the contact closed. Figure 8: Net incident and transmitted sound intensity spectra of M0 for a sound sourceexcitation of 100 Pa amplitude at 20 Hz, 80 Hz and 320 Hz, respectively.
Figure 9: Net incident and transmitted sound intensity spectra of M1 for a sound sourceexcitation of 100 Pa amplitude at 20 Hz, 80 Hz and 320 Hz, respectively.
Figure 10: Zoomed-in views of net transmitted sound intensity spectra of M1 for a soundsource excitation of 100 Pa amplitude at (a) 20 Hz and (b) 80 Hz.Figure 11: Transverse displacement ( u ) of M1 for f ex = 20 Hz, 80 Hz and 320 Hz. Figure 12: (a-f) Net incident and transmitted sound intensity spectra of M2 for a soundsource excitation of 100 Pa amplitude at 20 Hz, 80 Hz and 320 Hz, respectively. igure 13: Transverse displacement ( u ) of M2 for f ex = 20 Hz, 80 Hz and 320 Hz. Figure 14: (a-f) Net incident and transmitted sound intensity spectra of M3 for a soundsource excitation of 100 Pa amplitude at 80 Hz, 320 Hz and 640 Hz, respectively. igure 15: Transverse displacement ( u ) of M3 for f ex = 80 Hz, 320 Hz and 640 Hz. Figure 16: Net incident and transmitted sound intensity spectra of M1 for a sound sourceexcitation of 10 Pa amplitude at 20 Hz, 80 Hz and 320 Hz, respectively. igure 17: Transverse displacement ( u ) of M1 for f ex = 20 Hz, 80 Hz and 320 Hz, and10 Pa amplitude. Figure 18: (a) STL for M0, M1 and M2 for a lowpass (0 − (a) (b) Figure 19: (a) Comparison of STL for M1 for a lowpass (0 − Figure 20: Net incident and transmitted sound intensity spectra of M0 considering onlythe linear terms in the strain-displacement relation for a sound source excitation of 100 Paamplitude at 20 Hz, 80 Hz and 320 Hz, respectively. ppendix A. Sound intensity calculations on the incident andtransmitted sides of the segmented plate Complex sound pressures obtained from the ’virtual’ microphones at thethe four locations (Fig. A.1), i.e., two each in the upstream and downstreamducts are represented by P , P , P and P , respectively. These pressurescan be expressed as a superposition of the positive- and negative- goingplane waves in the up- and downstream ducts of the standing wave tube asgiven in Eq. A.1[18]: P = (cid:16) U i e − jkd + U r e jkd (cid:17) e jωt , (A.1a) P = (cid:16) U i e − jkd + U r e jkd (cid:17) e jωt , (A.1b) P = (cid:16) D i e − jkd + D r e jkd (cid:17) e jωt , (A.1c) P = (cid:16) D i e − jkd + D r e jkd (cid:17) e jωt , (A.1d)where U i , U r are the positive- and negative- traveling wave amplitudes inthe upstream duct, and D i , D r are the positive- and negative- travelingwave amplitudes in the downstream duct, respectively. Here, k representsthe wave number in the ambient fluid, and the e + jωt sign convention wasadopted, where ω = 2 πf is the circular frequency. The pressure and acous-tic particle velocities on either side of the segmented plate can in turn bedetermined based on a knowledge of the complex amplitude of the positive53nd negative traveling waves in both the ducts, as given below in Eq. A.2: P | x =0 = U i + U r , (A.2a) V | x =0 = U i − U r ρ c , (A.2b) P | x = t p + t s = D i e − jk ( t p + t s ) + D r e jk ( t p + t s ) , (A.2c) V | x = t p + t s = D i e − jk ( t p + t s ) − D r e jk ( t p + t s ) ρ c , (A.2d)where P | x =0 , P | x = t p + t s , V | x =0 and V | x = t p + t s are the acoustic pressuresand the particle velocities on the incident and transmitted sides of the plate,respectively. The sound intensity on the incident ( I ) and transmitted ( I t )sides were then computed by using the following expressions: I = 12 (cid:60) (cid:16) P | x =0 V | x =0 (cid:17) , (A.3a) I t = 12 (cid:60) (cid:16) P | x = t p + t s V | x = t p + t s (cid:17) , (A.3b)where the over-bar denotes the complex conjugate. Also note that the soundsource intensity ( I s ) was evaluated as: I s = G P w ( ω )2 ρ c , (A.4)where G P w ( ω ) is the power spectral density of the lowpass white noise time-domain signal P w , and ω = 2 πf is the circular frequency. 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