Adámek, V.

Kroft, R.

Šulda, J.

DSpace at University of West Bohemia

2022 November 7 - 9, 2022Srní37   conference with international participation  thIdentification of dispersion and attenuation curves ofthin non-prismatic heterogeneous viscoelastic rodsJ. Šuldaa, R. Kroftb, V. AdámekaaDepartment of Mechanics, Faculty of Applied Sciences, University of West Bohemia, Univerzitnı́ 8, 301 00 Plzeň, Czech RepublicbNTIS – New Technologies for the Information Society, Faculty of Applied Sciences, University of West Bohemia,Technická 8, 301 00 Plzeň, Czech RepublicThis work deals with the identification of the dispersion and attenuation properties of a thinheterogeneous viscoelastic non-prismatic rod. The first two parts of the text will discuss ap-plications of these properties and the possible identification method. Next, the experiments aredescribed, which are performed on several thin homogeneous prismatic rods but also on het-erogeneous and non-prismatic rods. The experimental results are compared with analyticallyobtained results in the following section. At the end of this work, the results are discussed.The reason for the identification of dispersion and attenuation properties of viscoelastic ma-terials is mainly to solve the problems of passive vibration damping, and for the SHPB test,which consists of a thin input and output rod, between which a sample of similar impedanceis placed (see [3]). Previously, typically elastic thin rods have been used in the SHPB test,which was based primarily on investigating the propagation of longitudinal elastic waves. Overtime, during the development of plastic materials, it was also necessary to investigate wavephenomena in rods with significantly lower impedance, for which the theory of elastic wavepropagation was no longer sufficient. For the description of wave phenomena in viscoelasticmaterials, discrete rheological models are advantageously used, which can be drawn schemati-cally as a connection of elastic springs and viscous dashpots.Each homogeneous part of the layered rod is described by a different dashpot viscosityand spring stiffness. Due to this and the variable cross-section of the rod, different attenuationand dispersion behaviour occurs in these parts. To determine the properties of the layered rod,measuring after the wave has passed through the entire rod under investigation is necessary.Several identification methods can be found that meet these requirements. A method describedby Blanc (see [2]) will be used in this work. This method requires a Fourier transform of theacceleration measured at two different locations (x1 < x2) on the rod caused by the passageof the wave packet. According to Blanc, the relations for calculating the phase velocity c(ω)and the wave number κ(ω) or the attenuation α(ω) can be written as [2]c(ω) =ωκ(ω)= −ω x2 − x1θ(x2, ω)− θ(x1, ω),α(ω) = − 1x2 − x1lnϑ(x2, ω)ϑ(x1, ω),(1)where ω is the angular frequency, θ(x, ω) and ϑ(x, ω) are the phase and the modulus of theFourier transform of the acceleration, respectively.0 2 4 6 8 10 12t [s] 10-50123456F [N](a) Excited pulse0 0.002 0.004 0.006 0.008 0.01t [s]-2000-1500-1000-500050010001500a [m/s2]Accelerometer 1Accelerometer 2(b) Acceleration responseFig. 1. Experimental data for the heterogeneous non-prismatic rodAcceleration was measured by two accelerometers of brand Brüel & Kjær Miniature Delta-Tronr Accelerometer – Type 4519 at the ends of the investigated rods. Rods were excited atone end by an impact hammer Brüel & Kjær Miniature Impact Hammer – Type 8204. Thesignals from the accelerometers and the hammer were processed by an eight-channel analyserOROS OR35 with a sampling rate of 100 kHz. The measurement was performed on four ho-mogeneous prismatic rods of approximate length 1m made of polypropylene (PP), polyvinylchloride (PVC), polyethene terephthalate (PET), polylactic acid (PLA). Also on a homoge-neous non-prismatic rod made of PLA of a length 1m and on a heterogeneous non-prismaticrod, which was composed of 4 homogeneous prismatic parts of different cross-sections andmade of acetal heteropolymer (POM-C), aluminium (AL), polycarbonate (PC1000) and PP inthis order with a total length of 4m. A smooth cosine pulse of length 100− 200µs was gener-ated for all rods using the mentioned impedance hammer. This pulse excited the heterogeneousrod at the end of the POM-C segment, and the acceleration response to this pulse can be seenin Fig. 1b. The amplitude of each excited pulse depends proportionally on the weight of theexamined rod, so Fig. 1a shows the highest measured amplitude. From Fig. 1b, it is possible toobserve a significant decrease in the acceleration amplitude, i.e., the energy dissipation due tothe viscosity and various cross-sections of the rod segment.The measured phases and modulus of the transformed acceleration were used in relation(1), and the frequency dependences of longitudinal wave phase velocity and attenuation wereobtained for each rod. These curves were compared with analytically calculated curves forhomogeneous prismatic rods whose material is characterised by Zener rheological model. An-alytical relations for the evaluation of these curves are provided, for example, by Ahonsi in hiswork [1]. In his relationsκ2(ω) =ρω22EE(H1 +H2) , α2(ω) =ρω22EE(H1 −H2) , (2)the author considers transverse contractions during the propagation of the pulse, i.e., the non-zero Poisson ratio ν and the inertia of the rod elements in the radial direction characterised bythe radius of gyration r. The density of the rod is denoted by ρ, and EE is the modulus of0 1 2 3 4 5f [Hz] 104150016001700180019002000c [m/s]AhonsiBlanc - experiment(a) Dispersion curve of the homogeneousPVC rod1 2 3 4 5f [Hz] 104-20246 [m-1]AhonsiBlanc - experiment(b) Attenuation curve of the homogeneousPLA rodFig. 2. Comparison of analytically obtained and experimentally measured dispersion and attenuationcurves for rods made of homogeneous materialelasticity of the alone-standing spring in the Zener material model. Real functions H1 and H2in (2) are given byH1 =√√√√√ 1 + ω2t2R[(1 +E1EE)− ρν2r2ω2EE]2ω2t2R +(ρν2r2ω2EE− 1)2 ,H2 =[(1 +E1EE)− ρν2r2ω2EE]ω2t2R −(ρν2r2ω2EE− 1)[(1 +E1EE)− ρν2r2ω2EE]2ω2t2R +(ρν2r2ω2EE− 1)2 .(3)Symbols E1 and tR denote the modulus of elasticity of the spring and the relaxation time cha-racterising the Maxwell branch in the Zener model, respectively.When comparing the analytically obtained dispersion curves for all homogeneous materials,it was possible to achieve agreement within 20 to 50 kHz. For example, the comparison fora rod made of PVC material is shown in Fig. 2a. For the attenuation curves, it was possible toachieve a much smaller agreement with the analytics (curves marked as Ahonsi), i.e., from 8up to 25 kHz, see Fig. 2b showing the attenuation of the PLA rod. The accuracy of the results,especially the attenuation, was influenced by the relatively low sampling rate of the signal. Thetransverse contraction of the rod in the relation (2) positively influenced the agreement of theresults. In the case of a non-prismatic and heterogeneous rod, the comparison of the curvesidentified from the excitation at both ends of the rod was used to verify the results. In thisway, it was possible to verify the dispersion curve up to 40 kHz for the non-prismatic rod (seeFig. 3a), and the attenuation curve only up to 35 kHz (see Fig. 3b). For the heterogeneous rod,dispersion and attenuation have been verified up to 18 kHz as shown in Fig. 3c and 3d.The identified dispersion and attenuation curves in Fig. 2 are similar to the curves presentedin [1] and [2]. The phase velocity and the attenuation for a homogeneous viscoelastic rod in-crease with higher frequencies. For thicker rods, where the measured acceleration is more influ-enced by transverse contractions, it was shown that with increasing frequency, the wave speeddecreases but the attenuation increases. The dispersion curve of homogeneous non-prismaticand heterogeneous non-prismatic rods is constant on the identified part, and the attenuation0 1 2 3 4 5f [Hz] 104010002000300040005000c [m/s]Excitation on acc. 1Excitation on acc. 2(a) Dispersion curve for non-prismatic rod0 1 2 3 4f [Hz] 10400.511.522.5 [m-1]Excitation on acc. 1Excitation on acc. 2(b) Attenuation curve for non-prismatic rod0 1 2 3 4 5f [Hz] 10401000200030004000c [m/s]Excitation on acc. 1Excitation on acc. 2(c) Dispersion curve for heterogeneous rod0 1 2 3 4 5f [Hz] 1040.511.5 [m-1]Excitation on acc. 1Excitation on acc. 2(d) Attenuation curve for heterogeneous rodFig. 3. Experimentally measured dispersion and attenuation curves for homogeneous non-prismatic andheterogeneous non-prismatic rodcurve is increasing (see Fig. 3). The maximum frequency to which these properties can be de-termined is mainly influenced by the weight of the accelerometers, the low sampling frequencyand the shape of the excited cosine pulse.In this work, a method of identifying the dispersion and attenuation properties of homo-geneous prismatic as well as heterogeneous non-prismatic thin viscoelastic rods was described.Using this method, it was possible to determine these properties from the measured accelerationand verify them up to tens of kHz.AcknowledgementsThe publication was supported by the project SGS-2022-008 and by the grant GA 22-00863K.References[1] Ahonsi, B., Harrigan, J. J., Aleyaasin, M., On the propagation coefficient of longitudinal stresswaves in viscoelastic bars, International Journal of Impact Engineering 45 (2012) 39-51.[2] Blanc, R. H., Transient wave propagation methods for determining the viscoelastic properties ofsolids, Journal of Applied Mechanics 60 (1993) 763-768.[3] Hopkinson, B., A method of measuring the pressure produced in the detonation of high explosiveor by the impact of bullets, Philosophical Transactions of the Royal Society A 213 (1914) 437-456.

Identification of dispersion and attenuation curves of thin non-prismatic heterogeneous viscoelastic rods

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Identification of dispersion and attenuation curves of thin non-prismatic heterogeneous viscoelastic rods

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