An alternative solution for the determination of elastic parameters in free–free flexural vibration of a Timoshenko beam
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An alternative inversion solution is presented to compute the elastic parameters based on the Timoshenko equation of motion in free flexural vibration. The interest of this solution was its relatively simple formulation and its validity domain, which was not restricted to the initial frequencies and a certain range of length-to-height ratio. The uniqueness of the solution was verified numerically, and the best optimization strategy was deduced. It was underlined that the shear modulus determination depended on the length-to-height ratio coupled with the number of resonance frequencies involved in the computation. The accuracy of the computed elastic modulus values was in the same scale as that for the measurement errors of frequency. However, the sensitivity of the shear modulus was found to be very high and depended on the number of frequencies taken into account. The theoretical model was found to be more accurate and to better estimate the frequency values than a classic solution of the Timoshenko equation. Furthermore, the inversion procedure gave equivalent elastic property values to those obtained with a classic solution (in the validity range of the classic solution) and was more robust when the number of frequencies taken into account was low.
KeywordsShear Modulus Vibration Frequency Experimental Frequency Inverse Solution Flexural Vibration
Equation (3) was first devised in the year 2000; this paper therefore represents the original findings of this research. The approach has been mentioned, though not fully presented in two previously published books (Brancheriau 2011a and b). I am very grateful to my colleague Yves DUMONT (applied mathematics) who generously shared his ideas and interest with me. In a sense, this work was more of his initiative than of mine. I express here my sincere thanks to him. I also thank Nick ROWE who carefully read the initial document.
- Bordonné PA (1989) Dynamic modulus and internal friction of wood—assessments on beams in free vibration condition. (In French) PhD Thesis, Institut National Polytechnique de Lorraine, FranceGoogle Scholar
- Brancheriau L (2002) Characterization of structural timber by vibration analysis in the acoustic domain. (In French) PhD Thesis, University of Aix Marseille II, FranceGoogle Scholar
- Brancheriau L (2011a) Vibrations of beam. Editions Universitaires EuropéennesGoogle Scholar
- Brancheriau L (2011b) Corrections for Poisson effect in longitudinal vibrations and shearing deformations in transverse vibrations applied to a prismatic orthotropic body. In: Mechanical vibrations: types, testing and analysis. Nova Science Publishers, pp 205–223Google Scholar
- Graff K (1975) Wave motion in elastic solids. Courier Dover Publications, Mineola, pp 180–187Google Scholar
- Kubojima Y, Yoshihara H, Ohta M, Okano T (1996) Examination of the method of measuring the shear modulus of wood based on the Timoshenko theory of bending. Mokuzai Gakkaishi 42(12):1170–1176Google Scholar
- Perstorper M (1994) Dynamic modal tests of timber evaluation according to the Euler and Timoshenko theories. 9th International Symposium on Non-Destructive Testing of Wood. 45–54Google Scholar
- Sobue N (1986) Instantaneous measurement of elastic constants by analysis of the tap tone of wood, application to flexural vibration of beams. MokuzaiGakkaishi 32(4):274–279Google Scholar
- Timoshenko S (1921) On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos Mag J Sci. XLI - Sixth Series: 744–746Google Scholar
- Timoshenko S (1922) On the transverse vibrations of bars of uniform cross-section. Philos Mag J Sci. XLIII - Sixth Series: 125–131Google Scholar
- Yoshihara H (2011) Measurement of the Young’s modulus and shear modulus of in-plane quasi-isotropic medium-density fiberboard by flexural vibration. BioResources 6(4):4871–4885Google Scholar