Hilbert-Twin – A Novel Hilbert Transform-Based Method To Compute Envelope Of Free Decaying Oscillations Embedded In Noise, And The Logarithmic Decrement In High-Resolution Mechanical Spectroscopy HRMS

Open access


In this work, we present a novel Hilbert-twin method to compute an envelope and the logarithmic decrement, δ, from exponentially damped time-invariant harmonic strain signals embedded in noise. The results obtained from five computing methods: (1) the parametric OMI (Optimization in Multiple Intervals) method, two interpolated discrete Fourier transform-based (IpDFT) methods: (2) the Yoshida-Magalas (YM) method and (3) the classic Yoshida (Y) method, (4) the novel Hilbert-twin (H-twin) method based on the Hilbert transform, and (5) the conventional Hilbert transform (HT) method are analyzed and compared. The fundamental feature of the Hilbert-twin method is the efficient elimination of intrinsic asymmetrical oscillations of the envelope, aHT (t), obtained from the discrete Hilbert transform of analyzed signals. Excellent performance in estimation of the logarithmic decrement from the Hilbert-twin method is comparable to that of the OMI and YM for the low- and high-damping levels. The Hilbert-twin method proved to be robust and effective in computing the logarithmic decrement and the resonant frequency of exponentially damped free decaying signals embedded in experimental noise. The Hilbert-twin method is also appropriate to detect nonlinearities in mechanical loss measurements of metals and alloys.

[1] K.L. Ngai, Y.N. Wang, L.B. Magalas, Theoretical basis and general applicability of the coupling model to relaxations in coupled systems, J. Alloy Compd. 211/212, 327-332 (1994).

[2] L.B. Magalas, Snoek-Köster relaxation. New insights – New paradigms, J. Phys. IV 6, 163-172 (1996).

[3] M.S. Blanter, L.B. Magalas, Strain-induced interaction of dissolved atoms and mechanical relaxation in solid solutions. A review, Sol. St. Phen. 89, 115-139 (2003).

[4] A.S. Nowick, B.S. Berry, Anelastic Relaxation in Crystalline Solids, Academic Press, 1972.

[5] L.B. Magalas, Mechanical spectroscopy – Fundamentals, Sol. St. Phen. 89, 1-22 (2003).

[6] J.S. Bendat, A.G. Piersol, Analysis and Measurement Procedures, Wiley-Interscience, 1986.

[7] A.D. Poularikas (ed.), The Transforms and Applications. Handbook, CRC Press Inc., 1996.

[8] L.B. Magalas, Determination of the logarithmic decrement in mechanical spectroscopy, Sol. St. Phen. 115, 7-14 (2006).

[9] L.B. Magalas, A. Stanisławczyk, Advanced techniques for determining high and extreme high damping: OMI – A new algorithm to compute the logarithmic decrement, Key Eng. Materials 319, 231-240 (2006).

[10] L.B. Magalas, M. Majewski, Ghost internal friction peaks, ghost asymmetrical peak broadening and narrowing. Misunderstandings, consequences and solution, Mater. Sci. Eng. A 521-522, 384-388 (2009).

[11] L.B. Magalas, M. Majewski, Recent advances in determination of the logarithmic decrement and the resonant frequency in low-frequency mechanical spectroscopy, Sol. St. Phen. 137, 15-20 (2008).

[12] L.B. Magalas, M. Majewski, Toward high-resolution mechanical spectroscopy HRMS. Logarithmic decrement, Sol. St. Phen. 184, 467-472 (2012).

[13] M. Majewski, A. Piłat, L.B. Magalas, Advances in computational high-resolution mechanical spectroscopy HRMS. Part 1 – Logarithmic decrement, IOP Conf. Series: Materials Science and Engineering 31, 012018 (2012).

[14] M. Majewski, L.B. Magalas, Critical assessment of the issues in the application of Hilbert transform to compute the logarithmic decrement, Arch. Metall. Mater. 60, 1103 (2015).

[15] C.A. Von Urff, F.I. Zonis, The square-law single-sideband system, IRE Trans. on Communications Systems 10, 257-267 (1962).

[16] C.B. Smith, N.M. Wereley, Composite rotorcraft flexbeams with viscoelastic damping layers for aeromechanical stability augmentation, in M3DIII: Mechanics and Mechanisms of Material Damping, American Society of Testing and Materials, ASTM STP 1304, A. Wolfenden and V.K. Kinra, Eds., American Society for Testing and Materials, 62-77 (1997).

[17] D.S. Laila, M. Larsson, B.C. Pal, P. Korba, Nonlinear damping computation and envelope detection using Hilbert transform and its application to power systems wide area monitoring, Power and Energy Society General Meeting, 2009. PES ’09. IEEE (2009).

[18] X.J. Shi, X.J. Zhao, G.H. Xiao, Boxed milk metamorphism detecting method based on wavelet and Hilbert transform, 2009 IEEE International Conference on Automation and Logistics (ICAL 2009), August 05-07, 2009, Shenyang, China. New York: IEEE, 1-3, 1454-1458 (2009).

[19] I. Yoshida, T. Sugai, S. Tani, M. Motegi, K. Minamida, H. Hayakawa, Automation of internal friction measurement apparatus of inverted torsion pendulum type, J. Phys. E: Sci. Instrum. 14, 1201-1206 (1981).

[20] M. Feldman, Non-linear system vibration analysis using Hilbert transform – I. Free vibration analysis method ‘FREEVIB’, Mechanical Systems and Signal Processing 8, 119-127 (1994).

[21] M. Feldman, Non-linear system vibration analysis using Hilbert transform – II. Forced vibration analysis method ‘FREEVIB’, Mechanical Systems and Signal Processing 8, 309-318 (1994).

[22] M. Feldman, Non-linear free vibration identification via the Hilbert transform, Journal of Sound and Vibration 208, 475-489 (1997).

[23] M. Feldman, Considering high harmonics for identification of non-linear systems by Hilbert transform, Mechanical Systems and Signal Processing 21, 943-958 (2007).

[24] Ž. Nakutis, P. Kaškonas, Bridge vibration logarithmic decrement estimation at the presence of amplitude beat, Measurement 44, 487-492 (2011).

[25] E. Bonetti, E.G. Campari, L. Pasquini, L. Savini, Automated resonant mechanical analyzer, Rev. Sci. Instrum. 72, 2148-2152 (2001).

[26] S. Amadori, E.G. Campari, A.L. Fiorini, R. Montanari, L. Pasquini, L. Savini, E. Bonetti, Automated resonant vibrating-reed analyzer apparatus for a non-destructive characterization of materials for industrial applications, Mater. Sci. Eng. A 442, 543-546 (2006).

[27] X. Zhu, J. Shui, J.S. Williams, Precise linear internal friction expression for a freely decaying vibrational system, Rev. Sci. Instrum. 68, 3116-3119 (1997).

[28] L.B. Magalas, M. Majewski, Free Decay Master Software Package, 2014.

Archives of Metallurgy and Materials

The Journal of Institute of Metallurgy and Materials Science and Commitee on Metallurgy of Polish Academy of Sciences

Journal Information

IMPACT FACTOR 2016: 0.571
5-year IMPACT FACTOR: 0.776

CiteScore 2016: 0.85

SCImago Journal Rank (SJR) 2016: 0.347
Source Normalized Impact per Paper (SNIP) 2016: 0.740

Cited By


All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 129 129 10
PDF Downloads 39 39 2