A new technique for the characterization of viscoelastic materials: theory, experiments and comparison with DMA
AA new technique for the characterization of viscoelastic materials:theory, experiments and comparison with DMA.
Elena Pierro , Giuseppe Carbone , Scuola di Ingegneria, Universit`a degli Studi della Basilicata, 85100 Potenza, Italy Department of Mechanics, Mathematics and Management,Polytechnic University of Bari, V.le Japigia, 182, 70126, Bari, Italy and Physics Department M. Merlin, CNR Institutefor Photonics and Nanotechnologies U.O.S. Bari,via Amendola 173, Bari, 70126, Italy
Abstract
In this paper we present a theoretical and experimental study aimed at characterizing the hys-teretic properties of viscoelastic materials. In the last decades viscoelastic materials have become areference for new technological applications, which require lightweight, deformable but ultra-toughstructures. The need to have a complete and precise knowledge of their mechanical properties,hence, is of utmost importance. The presented study is focused on the dynamics of a viscoelasticbeam, which is both experimentally investigated and theoretically characterized by means of anaccurate analytical model. In this way it is possible to fit the experimental curves to determine thecomplex modulus. Our proposed approach enables the optimal fitting of the viscoelastic modulusof the material by using the appropriate number of relaxation times, on the basis of the frequencyrange considered. Moreover, by varying the length of the beams, the frequency range of interestcan be changed/enlarged. Our results are tested against those obtained with a well establishedand reliable technique as compared with experimental results from the Dynamic Mechanical Anal-ysis (DMA), thus definitively establishing the feasibility, accuracy and reliability of the presentedtechnique. a r X i v : . [ phy s i c s . i n s - d e t ] F e b art I Introduction
Recent scientific advancements in field of automotive, electronics, micromechanical systems,pipe technologies, have led to new technologies where the use of lightweight, tough, soft andhigh deformable materials has become ubiquitous. In this scenario, viscoelastic materialshave spread in many different contexts, from seals [1] to bio-inspired adhesives [2–5], becauseof their superior damping and frictional properties. For an appropriate use of such materials,however, the proper knowledge of their mechanical properties is a basic requirement. Alongthis line, the most popular technique to characterize the viscoelastic modulus of such materi-als is the Dynamic Mechanical Analysis (DMA) [7, 8], which allows the measurement of theviscoelastic complex modulus depending on both frequency and temperature. In particular,it consists of imposing a small cyclic strain on a sample and measuring the resulting stressresponse, or equivalently, imposing a cyclic stress on a sample and measuring the resultantstrain response. Nevertheless, such an experimental procedure exhibits different limits (e.g.high frequency characterization is considerably difficult [9–11]) and requires expensive testequipment. In this view, several techniques have been proposed, based on the vibrationalresponse of beam like structures. In Ref. [12], the complex modulus of acoustic materialsusing a transfer function method of a lumped mechanical model was utilized. In Ref. [13]the response of the endpoint of an impacted beam was measured in terms of displacements,by means of electro-optical transducers, and then an iterative numerical scheme was consid-ered to retrieve the viscoelastic modulus. Other experimental procedures have been recentlypresented, with the aim of simplifying the setup, as in Ref. [14] where a double pendulumwas utilized to excite a viscoelastic sample without any other source of external excitation,and recorded oscillations were induced by gravity. In Ref. [15] a cantilever beam was ex-cited by means of a seismic force, and curve fitting of experimental data with a fractionalderivative model was employed to characterize the complex modulus. However, in all thepresented works dealing with mechanical characterization of viscoelastic materials, there isno the simultaneous presence of i) a very simple setup, ii) an analytical model to describethe dynamics of the vibrational system considered, and iii) a constitutive model able to ac-curately capture the behaviour of viscoelastic materials in a wide frequency range. Moving2rom these facts, in this paper we present a rigorous easy-to-use approach for determiningthe viscoelastic modulus, based on the experimental vibrational identification of viscoelasticbeams with different lengths. Both a very simple setup is utilized for acquisitions, and anaccurate analytical model of the beam are considered to determine the viscoelastic modulus,which takes into account multiple relaxation times of the material. In particular, by properlychanging the length of the considered beam, it is possible to broaden the frequency rangeunder analysis, and by selecting the appropriate number of relaxation times it is possibleto optimize the fitting procedure. The results presented in this paper show that these twoaspects are of pivotal importance to correctly determine the viscoelastic complex modulus,and they open new paths towards challenging further improvements. The paper is organizedas follows: at first, the analytical model of a viscoelastic beam dynamics is recalled, thenthe experimental setup and the data acquisition are explained in detail. Finally, the curvefitting scheme is defined and results are discussed in depth, with particular emphasis on thecomparison with DMA results.
I. THEORETICAL MODEL OF THE VISCOELASTIC BEAM DYNAMICS
In this section, we derive the analytical formulation of the viscoelastic beam vibrationalresponse. The main purpose is to get a simple-to-use formula, which can be utilized tocharacterize the viscoelastic modulus, by fitting the experimental acquisitions. It is known,in particular, that for viscoelastic materials, the stress-strain relation is governed by thefollowing integral [6] σ ( x, t ) = (cid:90) t −∞ G ( t − τ ) ˙ ε ( x, τ ) d τ (1)where ˙ ε ( t ) is time derivative of the strain, σ ( t ) is the stress, G ( t − τ ) is the so called relax-ation function. The viscoelastic complex modulus E ( s ) is closely related to the relaxationfunction G ( t ), and the simple equality E ( s ) = sG ( s ) exists in the Laplace domain. In thisdomain, in particular, it is possible to represent the complex modulus E ( s ) as the followingseries E ( s ) = E + (cid:88) k E k sτ k sτ k (2)which derives from the generalized Maxwell model, consisting of a spring with elastic con-stant E , that represents the elastic modulus of the material at zero-frequency, and k Maxwell3
IG. 1: The schematic of the test rig. elements connected in parallel, i.e. spring elements characterized by both the relaxation time τ k and the elastic modulus E k . By considering the above Eqs.(1)-(2) in the theoretical modelutilized to fit the experimental responses of the vibrating beam, it is possible to establishthe optimal number of relaxation times to better capture the viscoelastic behaviour in acertain frequency range.The geometrical characteristics of the beam considered in the present paper are chosen inorder to follow the Bernoulli theory of flexural vibrations, i.e. L (cid:29) W , L (cid:29) B , being L thelength of the beam, W and B respectively the width and the thickness of the rectangularcross section.In particular, by assuming that the transversal displacement | u ( x, t ) | (cid:28) L , it is possibleto neglect the contribute of shear stress, which is always very low at the first resonances.The equation of motion can be therefore written as [16] J xz (cid:90) t −∞ E ( t − τ ) ∂ u ( x, τ ) ∂x d τ + µ ∂ u ( x, t ) ∂t = f ( x, t ) (3)being µ = ρA , ρ the bulk density of the material, A = W B the cross section area, J xz =(1 / W B the moment of intertia, and f ( x, t ) the generic force acting on the beam. Thesolution of Eq.(3) can be formulated in terms of the eigenfunctions φ n ( x ) by means of the4 IG. 2: The experimental setup, consisting of i) the suspended viscoelastic beam (length L = 60[cm]) with a PCB 333B30 ICP accelerometer, glued on the upper surface, ii) the cDAQ-9184 NIdata acquiring, iii) the PCB 086C03 ICP Impact Hammer and iv) a portable pc. decomposition of the system response u ( x, t ) = + ∞ (cid:88) n =1 φ n ( x ) q n ( t ) (4)It can be shown that the eigenfunctions φ n ( x ) do not change if we consider a viscoelasticmaterial instead of a perfectly elastic one. The eigenfunctions φ n ( x ), in particular, can becalculated by solving the homogeneous problem J xz (cid:90) t −∞ E ( t − τ ) ∂ u ( x, τ ) ∂x d τ + µ ∂ u ( x, t ) ∂t = 0 (5)with the opportune boundary conditions. In the Laplace domain, by considering the initialconditions equal to zero, Eq.(5) becomes φ xxxx ( x, s ) − β eq ( s ) φ ( x, s ) = 0 (6)( φ x ( x, s ) = ∂φ ( x, s ) /∂x ) having defined β eq ( s ) = − µ s J xz E ( s ) = − µ s J xz C ( s ) (7)5 a) (b) FIG. 3: The measured frequency response functions H m (˜ x, ˜ x f , ω ) (solid lines) and the coherence(dashed lines), in the range 0 −
600 [Hz], for the beams of length L = 60 [cm] (a) and L = 40 [cm](b). (a) (b) FIG. 4: Master curves of LUBRIFLON (Dixon Resine [19]) at 20 ◦ C (dashed lines), fitted by meansof Eq.(2) by considering 50 relaxation times (solid lines), as real part Re[ E ( ω )] (a) and imaginarypart Im[ E ( ω )] (b) of the viscoelastic modulus E ( ω ). with C ( s ) = 1 /E ( s ) the compliance of the viscoelastic material. In the present study, wedetermine experimentally the beam response when it is suspended at a fixed frame, in theso called ”free-free” boundary condition. From a mathematical point of view, this set-up6 a) (b) (c) FIG. 5: The comparison between the measured FRF H m (˜ x, ˜ x f , ω ) (black dashed lines) and thetheoretical FRF H th (˜ x, ˜ x f , ω ), obtained through beam-best fitting (red lines) and DMA data (blacksolid lines), for the beam of length L = 60 [cm]. Curves are shown in terms of real part (a),imaginary part (b) and absolute value (c) of the FRFs. corresponds to the following mathematical conditions φ xx (0 , s ) = 0 (8) φ xxx (0 , s ) = 0 φ xx ( L, s ) = 0 φ xxx ( L, s ) = 0From the solution of Eq.(6), which is of the following type φ ( x, s ) = W cos [ β eq ( s ) x ] + W sin [ β eq ( s ) x ] + W cosh [ β eq ( s ) x ] + W sinh [ β eq ( s ) x ] (9)it is possible to derive the well known equation[1 − cos ( β eq L ) cosh ( β eq L )] = 0 (10)by simply forcing equal to zero the determinant of the system matrix obtained from Eqs.(8).Let us observe that the solutions β n L = c n of Eq.(10) are the same of the perfectly elasticcase, and can be substituted in Eq.(7) to calculate the complex conjugate eigenvalues s n corresponding to the n modes of the beam, and the real poles s k related to the materialviscoelasticity [17, 18]. Furthermore, by means of the solutions β n L = c n of Eq.(10), it ispossible to derive the following eigenfunctions φ n ( x ) φ n ( x ) = cosh ( β n x ) + cos ( β n x ) − cosh ( β n L ) − cos ( β n L )sinh ( β n L ) − sin ( β n L ) [sinh ( β n x ) + sin ( β n x )] (11)7 a) (b) FIG. 6: The real part (a) and the imaginary part (b) of the viscoelastic modulus E ( ω ), obtainedby fitting the vibrational response of the beam with L = 60 [cm] (dashed line) and by DMA (solidline), in the frequency range 30 −
150 [Hz]. which have the same analytical form of the eigenfunctions of a beam made of an elasticmaterial. In particular, these functions follow the orthogonality condition1 L (cid:90) L φ n ( x ) φ m ( x ) d x = δ nm (12)being δ nm the Kronecker delta function, as well as from Eq.(6) one gets1 L (cid:90) L ( φ n ) xxxx ( x ) φ m ( x ) d x = 1 L (cid:90) L φ n ( x ) β n φ m ( x ) d x = δ nm β n (13)Through the above Eqs.(12)-(13), and by defining the projected solution u m ( t ) on the m th eigenfunction φ m ( x ) as u m ( t ) = (cid:104) u ( x, t ) φ m ( x ) (cid:105) = 1 L (cid:90) L u ( x, t ) φ m ( x ) d x (14)it is possible to rewrite Eq.(3), after simple calculations, as following µ ¨ q n ( t ) + J xz β n (cid:90) t −∞ E ( t − τ ) q n ( τ ) d τ = f n ( t ) (15)where f n ( t ) = L (cid:82) L f ( x, t ) φ n ( x ) d x is the projected force. The Laplace Transform ofEq.(15), with initial conditions equal to zero, is µs Q n ( s ) + J xz β n E ( s ) Q n ( s ) = F n ( s ) (16)and therefore the system response, defined in Eq.(4), becomes in the Laplace domain U ( x, s ) = + ∞ (cid:88) n =1 φ n ( x ) Q n ( s ) = + ∞ (cid:88) n =1 φ n ( x ) F n ( x, s ) µs + J xz β n E ( s ) (17)8 a) (b) (c) FIG. 7: The comparison between the measured FRF H m (˜ x, ˜ x f , ω ) (black dashed lines) and thetheoretical FRF H th (˜ x, ˜ x f , ω ), obtained through beam-best fitting (red lines) and DMA data (blacksolid lines), for the beam of length L = 40 [cm]. Curves are shown in terms of real part (a),imaginary part (b) and absolute value (c) of the FRFs. (a) (b) FIG. 8: The real part (a) and the imaginary part (b) of the viscoelastic modulus E ( ω ), obtainedby fitting the vibrational response of the beam with L = 40 [cm] (dashed line) and by DMA (solidline), in the frequency range 150 −
550 [Hz].
For the scope of our investigation, we need to further modify Eq.(17). Indeed, we exper-imentally excite the beam (see Section II) by means of an impact hammer, in the section x = x f , at the instant t = t . Analytically, this condition is equivalent to consider asforcing term, a Dirac Delta of constant amplitude F , in both the time and the spatialdomains f ( x, t ) = F δ ( x − x f ) δ ( t − t ), i.e. in the Laplace domain the projected force is F n = (cid:82) L F δ ( x − x f ) φ n ( x ) d x = F φ n ( x f ). Hence, the analytic response of the beam canbe rewritten as 9 ( x, x f , s ) = F ∞ (cid:88) n =1 φ n ( x ) φ n ( x f ) µs + J xz β n E ( s ) (18)The beam response, derived in the above Eq.(18), can be utilized to determine the viscoelas-tic modulus E ( s ), previously defined by Eq.(2). In particular, the following theoreticalfrequency response function (FRF), can be defined in terms of inertance H th ( x, x f , i ω ) = A ( x, i ω ) F = (i ω ) ∞ (cid:88) n =1 φ n ( x ) φ n ( x f ) µ (i ω ) + J xz β n E (i ω ) (19)being the acceleration A ( x, i ω ) = U ( x, i ω ) (i ω ) . In this way, Eq.(19) can be utilized to fitthe experimentally acquired FRF, as discussed in the next Section. II. EXPERIMENTAL TEST
Two viscoelastic beams made of LUBRIFLON (Dixon Resine [19]), with thickness B = 1[cm], width W = 10 [cm], lengths L = 60 [cm] and L = 40 [cm], were suspended at a fixedframe through soft ropes. Different lengths, in particular, enable to enlarge the frequencyrange of interest, thus resulting in a better characterization of the material damping prop-erties, as it will be thoroughly discussed in the next Sections. Furthermore, by consideringbeams with different lengths, it is possible to survey potential peaks suppression or mitiga-tion, according to the theoretical studies previously presented in Ref.[17, 18]. The schematicof the test rig is drawn in Figure 1. The basic experimental setup (laboratory of AppliedMechanics, University of Basilicata, Potenza, Italy), is shown in Figure 2. This kind ofsetup, which represents the free-free boundary condition, is suitable in order to avoid exter-nal influences on damping due to constraints [20], as it happens for example when the beamis clamped. The cDAQ-9184 CompactDAQ (National Instruments) data acquiring has beenutilized to collect the time histories, through the NI Sound and Vibration Toolkit includedin LabVIEW (National Instruments). The slender beam has been excited in the z -directionthrough the PCB 086C03 ICP Impact Hammer, and the accelerations have been acquired,in the same direction, by means of a PCB 333B30 ICP Accelerometer. The beam sectionschosen for the impacting excitation and for the acceleration acquisitions were respectively˜ x f = 0 . L and ˜ x = 0 . L . The motivation behind this choice lies in the fact that, in thisway, by properly avoiding the nodal points, the first vibrational modes φ , ( x f ) should allbe present in the measures FRFs, in the frequency range under analysis. However, it is10 a) (b) FIG. 9: The real part (a) and the imaginary part (b) of the viscoelastic modulus E ( ω ), in thefrequency range 30 −
550 [Hz], obtained from DMA (solid line) and by fitting the two modulidetermined through the vibrational responses of the two beams with different lengths (dashedline). expected a mitigation of both the first and fifth peaks, since the section of the input force˜ x f is close to nodal points for the corresponding two mode shapes.We acquired a group of 10 time histories, each lasting 1 [s], with sampling frequency f s = 25600 [Hz]. It should be observed that, because of the heavy damped material, thesignal decreased to zero at about 1 / A (˜ x, ω ) and the impacting forces F (˜ x f , ω ). Finally, the H estimator [21] has been considered to determine the measuredfrequency response functions H m (˜ x, ˜ x f , ω ), which are shown in Figure 3, for the beams oflength L = 60 [cm] (Figure 3-a) and L = 40 [cm] (Figure 3-b), in terms of absolute value ofthe function H m (˜ x, ˜ x f , ω ) (solid lines), in the frequency range 0 −
600 [Hz]. Moreover, thecoherence function [21] (dashed lines) for each acquisition is shown, for the same frequencyrange. It is possible to observe that, as expected, same resonances moves forward higherfrequencies, by decreasing the beam length L . However, in contrast to a perfectly elasticbeam, the amplitude of such peaks changes. In particular, the second peak, which is at ω (cid:39)
60 [Hz] for the beam of length L = 60 [cm] (Figure 3-a), moves to ω (cid:39)
120 [Hz] for thebeam with L = 40 [cm] (Figure 3-b) and its amplitude increases. This circumstance suggeststhat the transition region of the material, where damping effects are more significant, shouldbe found at lower frequencies. For both the material and the beam geometry considered in11he present study, a slight mitigation of the resonances can be observed, and not a completepeak suppression [17, 18]. This fact helps in interpreting the nature of the peaks in thefrequency range considered, and thus enables us to perform a correct viscoelastic modulusfitting. III. VISCOELASTIC PARAMETERS IDENTIFICATION
By observing the coherence functions in Figure 3, it is possible to notice that someproblems occurred before the second and after the fourth peaks, for both the tests. Aspreviously discussed, this condition could be related to the selected impact section ˜ x f , whichis near to the nodal points of both the first and the fifth mode shapes φ , ( x f ). Therefore,in order to get the correct information from the experimental data in the fitting procedure,we have considered only the frequency range with maximum coherence, i.e. 30 −
250 [Hz]for the beam with L = 60 [cm], and 150 −
500 [Hz] for the beam with L = 40 [cm]. In thefirst case ( L = 60 [cm]), we have excluded the first peak at ω (cid:39)
20 [Hz], since it is too nearto the zone with low coherence. In the latter case ( L = 40 [cm]), we have excluded both thefirst ( ω (cid:39)
50 [Hz]) and the second peak ( ω (cid:39)
120 [Hz]), because of the heavy drop of thecoherence in correspondence of the first resonance.The measured frequency response function H m (˜ x, ˜ x f , ω ) has been fitted by means of thetheoretical FRF H th (˜ x, ˜ x f , ω ) defined in Eq.(19), in which only the viscoelastic modulus E ( ω ) is unknown. Hence, we have defined the cost function (cid:15) k as the squared differencebetween the real and imaginary parts of the theoretical H th (˜ x, ˜ x f , ω ) and the measured H m (˜ x, ˜ x f , ω ) FRFs: (cid:15) k = m (cid:88) i = n (cid:2) (Re[ H th (˜ x, ˜ x f , ω i )] − Re[ H m (˜ x, ˜ x f , ω i )]) + (20)+ (Im[ H th (˜ x, ˜ x f , ω i )] − Im[ H m (˜ x, ˜ x f , ω i )]) (cid:3) (21)The best fit of the theoretical model has been performed by minimizing the above de-fined cost function (cid:15) k , which depends on the number k of relaxation times considered tocharacterize the viscoelastic modulus E ( ω ). In this manner, the viscoelastic modulus E ( ω )(see definition in the Laplace domain in Eq.(2)), can be determined in terms of i) the elasticmodulus at zero-frequency E , ii) the relaxation times τ k , and iii) the correspondent elasticmoduli E k . The fundamental novelty of the presented approach, with respect to the other12 a) (b) (c) FIG. 10: The comparison between the measured FRF H m (˜ x, ˜ x f , ω ) (black dashed lines) and thetheoretical FRF H th (˜ x, ˜ x f , ω ), obtained through beam-best fitting (red lines) and DMA data (blacksolid lines), for the beam with L = 60 [cm], in the range 30 −
550 [Hz], in terms of real part (a),imaginary part (b) and absolute value (c) of the FRFs. similar vibration-based procedures presented in literature (e.g. [15]), consists in the fittingmethod, which can be optimized by properly choosing the number k of relaxation times, tocorrectly fit the beam dynamic response. This number, in particular, is influenced by thewidth of the frequency band considered, and by the amount of damping present in a certainfrequency range.With the aim of assessing the presented technique, we have also experimentally char-acterized our viscoelastic material through a Dynamic Mechanical Analyzer - MCR 702MultiDrive - Anton Paar GmbH (Tribolab, Politecnico di Bari, Bari, Italy). However, theDMA approach is different, since the viscoelastic modulus E ( ω ) is directly measured byconsidering the stress - strain relation shown in the Eq.(1). Therefore, in order to define afrequency response function based on DMA results, we need an analytical form of the vis-coelastic modulus E ( ω ) to be considered in Eq.(19). Hence, we have fitted the experimentalviscoelastic modulus E ( ω ) measured with DMA, by means of Eq.(2). In the frequency range10 − − [Hz], 50 relaxation times have been utilized. In Figure 4 it is shown the goodcorrelation between the experimental master curve and the fitted complex modulus, for boththe real part (Figure 4-a) and the imaginary part (Figure 4-b).13 a) (b) (c) FIG. 11: The comparison between the measured FRF H m (˜ x, ˜ x f , ω ) (black dashed lines) and thetheoretical FRF H th (˜ x, ˜ x f , ω ), obtained through beam-best fitting (red lines) and DMA data (blacksolid lines), for the beam with L = 40 [cm], in the range 30 −
550 [Hz], in terms of real part (a),imaginary part (b) and absolute value (c) of the FRFs.
IV. RESULTS AND DISCUSSIONS
The first experimental data set considered is related to the beam with L = 60 [cm]. Fromthe first iterations, we found that, in order to obtain the best results, it is preferable toconsider two peaks at a time, i.e. the second and the third resonances in the frequencyrange 30 −
150 [Hz] (see Figure 3-a). The best fitting of the theoretical model (Eq.20)has been achieved by means of 11 relaxation times, and it is shown in Figure 5, where themeasured FRF H m (˜ x, ˜ x f , ω ) (black dashed lines) is compared with the theoretical FRFs H th (˜ x, ˜ x f , ω ), obtained by utilizing Eq.(19), and by considering the viscoelastic moduluscalculated by means of both the beam-fitting procedure (red lines) and the DMA-fitteddata (black solid lines), in terms of real part (a), imaginary part (b) and absolute value(c) of the FRFs. Interestingly, it is possible to observe a very good overlapping betweenthe measured curves and the theoretical FRF obtained with our proposed method. Theviscoelastic modulus E ( ω ) calculated by minimizing the cost function (cid:15) k Eq.20 is shown inFigure 6 (dashed lines), where it is compared with the viscoelastic modulus measured withDMA (solid lines), in the frequency range 30 −
150 [Hz].The higher frequency range, i.e. 150 −
500 [Hz], has been studied by investigating thedynamic response of the beam with smaller length, i.e. L = 40 [cm]. In this case, weobtained the best fit with 8 relaxation times. Let us notice that, despite of the broaderfrequency range now considered, the number of relaxation times in this case is less than that14ne utilized in the previous case (i.e. 11 relaxation times). The reason is probably relatedto the fact that the transition region of LUBRIFLON is found at low frequency (see Figure4-b), and therefore the more the frequencies are low, the more the viscoelastic modulus mustbe characterized through a higher number of relaxation times to properly describe the fastincrease of damping, i.e. of the imaginary part Im[ E ( ω )]. In Figure 7, we compare themeasured FRF H m (˜ x, ˜ x f , ω ) (black dashed lines) with the theoretical FRFs H th (˜ x, ˜ x f , ω )(Eq.(19)), calculated by considering the viscoelastic modulus experimentally obtained bymeans of the beam dynamics (red lines) and through DMA (black solid lines), in terms ofreal part (a), imaginary part (b) and absolute value (c) of the FRFs. Also in this case, theproposed approach turns out to be very suitable for the viscoelastic material characterization.Indeed, it is possible to observe in Figure 8 a good overlapping, in the frequency range150 −
550 [Hz], between the viscoelastic moduli E ( ω ) experimentally obtained by DMA(solid lines), and by the proposed approach (dashed lines), both for the real part (Figure8-a) and the imaginary part (Figure 8-b).At last, the viscoelastic moduli E ( ω ), characterized by means of the vibrational analysison the beams with lengths L = 60 [cm] (30 −
150 [Hz]) and L = 40 [cm] (150 −
550 [Hz]),previously shown respectively in the Figures 6-8, have been fitted through Eq.(2) in thewhole frequency range 30 −
550 [Hz]. In Figure 9, we show the viscoelastic modulus E ( ω )determined by means of the beam dynamics (dashed lines) and the one measured with DMA(solid lines). For both the real part (Figure 9-a) and the imaginary part (Figure 9-b) of theviscoelastic modulus E ( ω ), we obtained a fine matching, thus finally assessing the methodproposed in this paper. In Figure 10, we compare, in the range 30 −
550 [Hz], the measuredFRF H m (˜ x, ˜ x f , ω ) (black dashed lines) with the theoretical curves H th (˜ x, ˜ x f , ω ), obtainedthrough the so calculated viscoelastic modulus E ( ω ) (Figure 9) (red lines) and by means ofDMA data (black solid lines), for the beam with L = 60 [cm]. It is important to highlightthat the two theoretical FRFs H th (˜ x, ˜ x f , ω ) are overlapped in all the frequency range, whilethe experimental FRF follows the theoretical curves only in the range 30 −
150 [Hz], wherewe obtained maximum coherence (see Figure 3-a). The ”non-dectected peak” at 300 [Hz] inthe experimental acquisitions, is strictly related to the drop in the coherence function, andcould be the origin of the non perfect overlapping between the theoretical and experimentalFRFs in the range 150 −
550 [Hz].Similar reasonings can be made for the results obtained from the beam with L = 40 [cm].15n Figure 11 we report the theoretical and the experimental FRFs in this case, where it isevident the good correspondence between the theoretical functions H th (˜ x, ˜ x f , ω ), obtainedthrough our proposed procedure (red lines) and by means of DMA data (black solid lines),in all the frequency range 30 −
550 [Hz]. However, also in this case, the experimentalcurve follows the theoretical ones in a limited range, i.e. 200 −
550 [Hz], which is far fromthe presence of a ”non-dectected peak” at around 65 [Hz] (see Figure 3-b), that probablycaused a drop in the coherence function. Moreover, at very small frequencies ( ∼
10 [Hz]),in both the experimental acquisitions (Figure 3-a,b) it should be observed that coherencetends to decrease below limit values, i.e. < .
8, because of the intrinsic problematic of theinstrumentations, especially of the impact hammer.In light of what has emerged from the results shown so far, some remarks should bemade, in order to define guidelines for the procedure proposed in this paper. First, we foundthat for a good fitting of the vibrational response of the beam, the frequency range wherecoherence is not maximum, as well as some peaks near these areas, should be excludedfrom the fitting calculations. Furthermore, it has been shown that, the more we proceedtowards frequencies where damping is high, i.e. versus the transition zone of the viscoelasticmaterial, the more we need to consider a narrow frequency band to fit the beam response,and an increasing number of relaxation times to describe the viscoelastic modulus E ( ω ) isrequired too.In conclusion, the proposed method for the characterization of the viscoelastic materials,has revealed to be very efficient, easy to use, and reliable with inexpensive instrumentation.Moreover, the analytical model here presented and used to fit the experimental responseof the beam, in particular, has proven to be accurate. The comparison between the vis-coelastic moduli E ( ω ) characterized by means of our technique and through DMA, indeed,finally assessed the possibility to retrieve this so important mechanical quantity, by simplyinvestigating the dynamics of a viscoelastic beam. We also found that, the idea to considermore beams with different lengths, is very useful to increase the frequency range of interest,and, in principle, by studying the dynamics of even longer or shorter beams, it is possibleto cover wider frequency ranges. However, it should be highlighted that in order to ob-tain a range comparable with the one usually covered by DMA, a different instrumentationshould be utilized. In particular, the impact hammer represents a limit in this direction,and it should be substituted with an electrodynamic shaker, which enables to investigate16 wider frequency range maintaining high coherence. At last, also by controlling the sur-rounding temperature, is possible to enlarge the range of interest, i.e. by a frequency shiftof the viscoelastic modulus E ( ω ) under study. These last two modifications to the actualexperimental setup, will be object of further investigations. A. Conclusions
In this paper we have presented a very simple and accurate experimental approach fordetermining the complex modulus of viscoelastic materials. By means of the vibrationalbehaviour of suspended viscoelastic beams with different lengths, we have characterized thecomplex modulus of LUBRIFLON by fitting the measured response through an accurateanalytical model of the beam dynamics, which takes into account multiple relaxation timesof the material. In particular, the possibility to properly select the number of relaxationtimes in a frequency range of interest, turned out to be a key factor to obtain very goodresults. The instrumentation utilized in our experiments is inexpensive and easy to use,and it consists of an impact hammer and a suspended beam, instrumented by means of anaccelerometer connected to a data acquiring module. Comparisons with DMA measurementsdemonstrate the validity of the proposed technique on a frequency range which could becomparable with the one usually covered by DMA technique. In conclusion, the proposedprocedure represents a valid alternative approach to DMA, and can be considered as asignificative step forward the improvement of the mechanical characterization of viscoelasticmaterials. [1] Bottiglione F., Carbone G., Mangialardi L., Mantriota G., Leakage Mechanism in Flat Seals,Journal of Applied Physics 106 (10), 104902, (2009).[2] Carbone G., Pierro E., Gorb S., Origin of the superior adhesive performance of mushroomshaped microstructured surfaces, Soft Matter 7 (12), 5545-5552, DOI:10.1039/C0SM01482F,(2011).[3] Carbone G., Pierro E., Sticky bio-inspired micropillars: Finding the best shape, SMALL, 8(9), 1449-1454, (2012).
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