[ ] sin ( ) ( ) = 2 2 ( ) ( ) ( ) ˆ Mechanical Spectroscopy II

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1 Solid State Pheomea Vol. 89 (003) pp (003) Tras Tech Publicatios, Switzerlad doi:0.408/ A New Impulse Mechaical Spectrometer to Study the Dyamic Mechaical Properties of Filamets A. Wasilewski *, T. Wi cek * ad L. B. Magalas ** * Departmet of Physics, Rzeszów Uiversity of Techology W. Pola, Rzeszów, Polad ** Uiversity of Miig ad Metallurgy, Faculty of Metallurgy ad Materials Sciece al. Mickiewicza 30, Kraków, Polad Keywords: filamets, mechaical spectroscopy, iteral frictio, dampig Abstract. A ew impulse mechaical spectrometer to measure mechaical dampig of filamets is described ad first experimetal results are briefly reported. The time-domai spectrum of a free decay iduced by a light impulse mechaical excitatio is aalyzed by a laser beam diffractio techique ad umerical aalysis of the data. This techique is successfully used to measure the logarithmic decremet, δ, ad the resoat frequecy, f o, of a bar filamet sample fixed at oe ed. Itroductio ad Descriptio of the Optical Techique Mechaical spectroscopy of filamets yields iformatio about various mechaisms of the dissipatio of mechaical eergy. A optical method is developed for a impulsed mechaical spectrometer to measure decayig oscillatios i the time-domai ad to determie the dyamic mechaical properties of filamets. Fig. Diffractio patter of the slit. I - light itesity, D - detector, d - width of the slit, l - distace betwee a photodiode ad a slit. All rights reserved. No part of cotets of this paper may be reproduced or trasmitted i ay form or by ay meas without the writte permissio of TTP, (ID: , Pesylvaia State Uiversity, Uiversity Park, Uited States of America-03/06/4,09:05:37)

2 344 Mechaical Spectroscopy II The laser beam which passes through the slit of width d geerates a diffractio patter show schematically i Fig.. This slit is created by a stable blade ad a vibratig oe; its width d is about 00 µm. A photodiode located at the cetral frige records variatio i the light itesity. Figure illustrates fluctuatios i the light itesity at the cetral frige geerated by a chage i the slit s width d. The light itesity is calculated from the expressio: I( x, d) A d = π d x λ l π d x λ l si ( ) ( ), () where A deotes the amplitude of the icidet light o a slit, I ( x, d ) is the light itesity ad λ represets wavelegth. The amplitude of oscillatios is directly measured by the diffractio of a laser beam o the slit [, ]. Mechaical loss ca be measured through umerical aalysis of a sigal represetig free decay of damped harmoic oscillatios [3]. The loss of mechaical eergy after impulsed ad/or harmoic excitatio ca be measured from the dimesioless logarithmic decremet δ, that is, the average relative chage of the decayig amplitude A : + da A δ = = l, () A A where is the umber of oscillatios to decay from amplitude A to A +. I this work δ is determied from the height of successively decayig amplitudes accordig to the followig equatio [3]: + B B i i i i i= i= i= i i i= i= δ =, (3) where B i is the atural logarithm of the amplitude or the surface area of the i -th oscillatio. The mechaical loss spectrum i the time domai ca also be aalysed from the evelope a ( t) of the expoetially damped harmoic oscillatios (see Fig. 4). The evelope is give by [ ] ( ) ( ) ( ) ˆ a t = A exp δ f t = A t A ( t), (4) o o + where Â ( t) is the imagiary part of the Hilbert trasformatio of the oscillatio amplitude ad f o is the frequecy of the free decay. The plot of the evelope o a logarithmic scale i the time domai directly yields the logarithmic decremet δ. Sice the evelope is positive for all values it ca be used to detect fie oliearities i the collected free decay data ad to aalyze whether the decay shows a expoetial or stretched expoetial behavior. It ca be cocluded that the mechaical loss spectra i the mechaical time-domai (free decay) ca be aalysed by: () classical measuremets of the logarithmic decremet (see Eqs. (), (3)), () Fourier ad Hilbert trasforms of decayig sigals [3, 4, 5], ad /

3 Solid State Pheomea Vol (3) o-liear fittig of decayig harmoic oscillatios to pure ad/or stretched expoetial fuctios. I additio, the spectral cotet of the sigal ca be obtaied from the fast Fourier trasform (FFT) of the sigal while the short-time Fourier trasform ad the wavelet trasform [3, 6] ca be used to fid out how the spectral cotet of the mechaical respose sigal varies with time. The time-frequecy-amplitude represetatio obtaied from the wavelet trasform of a sigal geerated after impulsed mechaical excitatio is a powerful tool to aalyze mechaical respose sigals recorded i a impulse mechaical spectrometer (i.e. strai sigal i the time domai). The wavelet trasform of oscillatios ca also be used to compare the quality of the experimetal results to the ideal decay of moochromatic oscillatios. Experimetal I order to ivestigate purely logitudial vibratios of a filamet ay trasverse vibratios i a mechaical spectrometer must be effectively elimiated. For this reaso a mechaical set-up of a impulsed spectrometer is costructed with air ad frictioless bearig to allow oly vertical motio of the vibratig filamet, as show schematically i Fig.. Fig. Schematic of the mechaical set-up of a impulse mechaical spectrometer: - the filamet, - coectig frame, 3 - air bearig, 4 - bearig movable rod, 5 - movable blade, 6 - statioary blade, 7 - support table with adjustable positio, 8 laser beam light, 9 fie metallic odule, 0 - set up for droppig of a metallic odule, - holder, - supply of compressed air.

4 346 Mechaical Spectroscopy II The filamet sample is fixed i a holder ad weighted by the elemets, 4 ad 5. The sample is excited by a light mechaical impulse caused by droppig a metallic odule at the poit of the filamet s coectio with the coectig frame. Air bearigs (elemets 3 ad 4, Fig. ) are used to limit vibratios oly alog the sample axis. Decayig oscillatios iduce a chage i the slit s width, which iflueces the diffractio patter. The iitial width of the slit determies optimum coditios for light diffractio. The fluctuatios i the light itesity at the cetral frige are trasformed by a photodiode ito a electrical sigal. This sigal is amplified, digitalized by a -bit A/D coverter, recorded i computer memory, ad processed by custom software. It is iterestig to ote that the impulse mechaical spectrometer described i this work shows high sesitivity ad is almost uiertial [7-0]. A block diagram of the ew impulse mechaical spectrometer operatig with the laser beam diffractio techique is show schematically i Fig. 3. LASER MECHANICAL CELL DETECTOR 3 AMPLIFIER A / C P C START 7 Fig. 3 Block diagram of a measuremet set-up: - He-Ne laser, - cell for mechaical measuremet, 3 - detector, 4 - amplifier, 5 - bit A/C coverter, 6 - computer, 7 - start butto ad delay lie. Experimetal Results I this sectio we report selected experimetal results obtaied for a bar specime fixed at oe ed. Bars made of several materials (metal, glass, ad polymer) are ivestigated. Figure 4 shows a typical time-domai free decayig oscillatios obtaied through the laser diffractio techique. The mechaical loss spectra i the time domai (the free decay sigal) are aalysed as a fuctio of time. The oscillatios are aalysed by a liear ad o-liear fittig of experimetal results to a mathematical fuctio g(t) as illustrated i Fig. 4. It is foud that for the ivestigated materials the decay of the amplitude is purely expoetial or is very close to expoetial. For the assumed sigle pure expoetial decay the dampig coefficiet β ( β = δ f ) is calculated. I additio, the logarithmic decremet δ is directly computed through the fittig procedure ad is compared to δ values obtaied through classical techiques (see Eqs. () (4)) ad Fourier ad Hilbert trasforms.

5 Solid State Pheomea Vol The time-domai mechaical loss spectra obtaied for a bar of polystyree with a 40% admixture of chalk are usually sampled with 8000 poits ad are fitted to the damped siusoid ( βt) k g( t) = A e si( ω t + ϕ ) by the least squares o-liear method. The fitted value of the k parameter was foud to be.0, which is very close to pure expoetial decay. 0.4 Y(t) t g( t) = A e β si( ω t + ϕ ) t [ms] Fig. 4 Mechaical loss spectrum i the time domai recorded by optical techique i the impulsed mechaical spectrometer. Y(t) selected experimetal poits, g(t) fitted curve. Polystyree bar filamet with 40% of chalk. For the assumed value of k =.0 the fittig procedure yielded the followig results: A = 0.450, β = , ω = 0.779, ad ϕ =.67 (samplig rate, f s = 0 khz). The results of o-liear fittig of the experimetal free decay data show whether oscillatios ca be described by purely expoetial or stretched expoetial fuctios. It should be poited out, however, that deviatios from the expoetial decay were frequetly detected i the ivestigated samples. Deviatios from expoetial decay ca also be aalyzed from the Hilbert trasform of decayig sigal. Further developmet of the mechaical spectrometer described i this work ad umerical software is i progress. Coclusios A ew type impulse mechaical spectrometer is costructed to study mechaical loss pheomea of filamets. The mechaical spectrometer described i this work is virtually uiertial. Free decayig oscillatios are recorded by a high-precisio laser beam diffractio techique to calculate the logarithmic decremet ad to aalyze the mechaical loss spectra i the time domai. Applicatio of the optical method allows umerical aalysis of the mechaical loss spectra i the time domai ad to aalyze whether the decay shows a expoetial or stretched expoetial behavior. Ackowledgmets This work was partially supported by the Polish State Committee for Scietific Research (KBN) uder grat No. 7 T08E 0388 ad grat No 7 T08B 035.

6 348 Mechaical Spectroscopy II Refereces. R. Pryor, O. L. Hageiers, W. P. T. North, Appl. Opt., 308 (97).. A. Wasilewski, T. Wi cek, A. K. Joscher, Testig of mechaical properties of materials by meas of o optical method based of light diffractio, Optica Applicata, Vol. XXIX, No. 3, (999). 3. L. B. Magalas, Measuremet of Mechaical Loss Spectra, i Mechaical Spectroscopy, Kluwer Academic Publishers (003), (i press). 4. L. B. Magalas, J. de Physique IV, 6, C8-63 (996). 5. L. B. Magalas, J. of Alloys ad Compouds 30, 69 (000). 6. L. B. Magalas, J. Kwa iewski, Selected Applicatios of the Wavelet Trasform, i this volume. 7. T. Wi cek, Proc. SPIE 340, 449 (994). 8. J. Goodma, Itroductio to Fourier Optics, McGraw-Hill, New York (968); J. Goodma, Itroductio to Fourier Optics, Mir, Moskwa (970) (i Russia). 9. D. Gaskill, Liear Systems, Fourier Trasforms, ad Optics, Wiley, New York (978). 0. R. G. Wilso, Fourier Series ad Optical Trasform Techiques i Cotemporary Optics, A Wiley-Itersciece Publicatio, Joh Wiley & Sos, Ic. (995).

7 Mechaical Spectroscopy II 0.408/ A New Impulse Mechaical Spectrometer to Study the Dyamic Mechaical Properties of Filamets 0.408/

Damped Vibration of a Non-prismatic Beam with a Rotational Spring

Damped Vibration of a Non-prismatic Beam with a Rotational Spring Vibratios i Physical Systems Vol.6 (0) Damped Vibratio of a No-prismatic Beam with a Rotatioal Sprig Wojciech SOCHACK stitute of Mechaics ad Fudametals of Machiery Desig Uiversity of Techology, Czestochowa,