References related to the equations and calculations used by Sonelastic® Software

For bars and cylinders:

- S. P. Timoshenko; On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars. Phil. Mag. Ser. 6 [41] 774-746 (1921).

- S. P. Timoshenko; On the Transverse Vibrations of Bars of Uniform Cross Section. Phil. Mag. Ser. 6 [43] 125-131 (1922).

- S. P. Timoshenko; Vibration Problems in Engineering. 2nd Ed. D. Van Nostrand Co., New York, 337-342 (1937).

- F. Z. Forster; Ein neues Me verfahren zur Bestimmung des Elastizitäts-moduls und der Dämpfung. Zeitschrift Für Metallkunde. v. 29, n. 109 (1937).

- G. Pickett; Equations for Computing Elastic Constants from Flexural and Torsional Resonant Frequencies of Vibration of Prisms and Cylinders; Proceedings ASTM, 45 846-865 (1945).

- S. Spinner, R. C. Valore; Comparisons between the Shear Modulus and Torsional Resonance Frequencies for Bars and Rectangular Cross Sections. Journal of Research, NIST, JNBAA, 60 RP2861, p. 459 (1958).

- T. Kaneko; Relation between Flexional Resonant Frequency Equations for the Flexional Vibration of Cylindrical Rods. J. Res. Natl. Bur. Stand., v. 64B, p. 237 (1960).

- Resonance Frequencies of Uniform Bars. J. Res. of the National Bureau of Standards-A. Physics and Chemistry, 64A [2] 147-155 (1960).

- S. Spinner, W. E. Tefft; A Method for Determining Mechanical Resonance Frequencies and for Calculating Elastic Moduli from these Frequencies. Proceedings ASTM, 61 1221-1239 (1961).

For discs:

- J. A. Salem, A. Singh; Polynomial Expressions for Estimating Elastic Constants from the Resonance of Circular Plates. Materials Science and Engineering, A, 422 [1] 292–297 (Apr 2006).

- G. Martincek; The Determination of Poisson's Ratio and the Dynamic Modulus of Elasticity from the Frequencies of Natural Vibration in Thick Circular Plates. J. Sound Vib., 2 [2] 116-127 (1965).

For plates:

- A. W. Leissa, Y. Narita; Vibrations of Completely Free Shallow Shells of Rectangular Planform. J. Sound &Vib. 96 [2] 207-218 (1984).

- A. A. Wereszczak, R. H. Kraft, J. J. Swab; Flexural And Torsional Resonances Of Ceramic Tiles Via Impulse Excitation Of Vibration; Ceramic Engineering and Science Proceedings, 24 (2003).

- T. Lauwagie, H. Solb, G. Roebbenc, W. Heylena, Y. Shib, O. V. der Biest; Mixed numerical–experimental identification of elastic properties of orthotropic metal plates. NDT&E International, 36 487–495 (2003).

- M. Alfanol L. Pagnotta; An Inverse Procedure for Determining the Material Constants of Isotropic Square Plates by Impulse Excitation of Vibration. Appl. Mech. Mat., 3-4 287-292 (2005).

- M. Alfanol L. Pagnotta; Measurement of the Dynamic Elastic Properties of a Thin Coating. Review of Scientific Instruments, 77 056107 (2006).

For discs and rings (grinding wheels):

- N. Raju; Vibrations of Annular Plates. J. Aeron. Soc. India, 14 [2] 37-52 (1962).

- J. Peters, R. Snoeys; The E modulus , a suitable characteristic of grinding wheels. Revue M, II [4] 1-11 (1965).

- S. M. Vogel, D. W. Skinner; Natural Frequencies of Transversely Vibrating Uniform Annular Plates. J. Appl. Mech., 32 926-931 (1965).

- R. D. Blevins; Formulas for Natural Frequency and Mode Shape. Publ. Krieger Publishing Company (1979).

- R. L. Smith; The Evaluation of NDT Techniques for Abrasive Wheels. British Journal of Non-Destructive Testing; vol. 28, no2, pp. 73-79 (1986).

Some references related to the techniques employed by Sonelastic® Systems based on natural frequencies of vibration:

- N. Suansuwan, M. V. Swain; Determination of elastic properties of metal alloys and dental porcelains. J. Oral Rehabilitation, 28 133-139 (2001).

- H. D. Tietz, M. Dietz, L. Bühling, B. May; Non-Destructive Testing of Green Ceramic Materials. NDT.net 3 [11] 1-7 (1998).

- W. T. Chu; A Comparison of Two Test Methods for Measuring Young's Modulus of Building Materials. Canadian Acoustics, 24 [3] 11 (1996).

- A. S. Maxwell, S. Owen-Jones, N. M. Jennett; Measurement of Young's modulus and Poisson's ratio of thin coatings using impact excitation and depth-sensing indentation. Rev. Sci. Instrum. 75 [4] 970-975 (2004).

- J. Schrooten, G. Roebben, J. A. Helsen; Young's Modulus of Bioactive Glass Coated Oral Implants: Porosity Corrected Bulk Modulus Versus Resonance Frequency Analysis. Scripta Materialia, 41 [10] 1047-1053 (1999).

- C. Chiu, E. D. Case; Elastic Modulus Determination of Coating Layers as Applied to Layered Ceramic Composites. Materials Science and Engineering, A132 39-47 (1991). C. Y. Wei, S. N. Kukureka; Evaluation of damping and elastic properties of composites and composite structures by the resonance technique. J. Mat. Sci., 35 3785-3792 (2000).

- B. Christaras, F. Auger, E. Mosse; Determination of the moduli of elasticity of rocks. Comparison of the ultrasonic velocity and mechanical resonance frequency methods with direct static methods. Materials and Structures; Volume 27, n4, pp. 222-228 (1994).

- A. FAWZY, C.E. SEMLER; Prediction of Refractory Strength Using Nondestructive Sonic Measurements, Am. Ceram. Soc. Bull., v. 64, n. 12, p. 1555-1558 (1985).

- T. Tonnesen, R. Telle; Thermal Shock Damage in Castables: Microstructural Changes and Evaluation by a Damping Method. Ceramic Forum International, v. 84, n. 9, p. E132-E136 (2007).

- R. Zhang, J. Perez, E. J. Lavernia; Documentation of damping capacity of metallic, ceramic and metal-matrix composite materials. Journal of Materials Science, v. 28, n. 9, p. 2395-2404 (1993).

- R. Morrel; Measuring Elastic Properties of Advanced Technical Ceramics – A review. UK National Physical Laboratory Report, n. 42 (1996).

- R. Morrel; NPL Measurement Good Practice Guide - Elastic Module Measurement. UK National Physical Laboratory Report, n. 98 (2006).

- T. Akashi; On the Measurement of Logarithmic Decrement of Concrete. General Meeting Reviews, Cement Association of Japan. p. 103-104 (1960).

- R.N. Swamy; Damping Mechanisms in Cementitious Systems. Proceedings of a Conference on Dynamic waves in civil engineering, University College, Swansea, July 1970; Wiley-Interscience, p. 521-542 (1971).

- R.N. Swamy, G. Rigby; Dynamic properties of hardened paste, mortar and concrete. Materials and Structures: Research and Testing. v. 4, n. 19, p. 13-40 (1971).

- R. Dieterle, H. Banchmann; Experiments and Models for the Damping Behaviour of Vibrating Reinforced concrete Beams in the Uncracked and Cracked Condition.

- International Association for Bridge and Structural Engineering Report of the working comissions, v. 34 (1981).


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Sonelastic® Systems are modular and customizable instruments for fast, precise and non-destructive elastic moduli and damping characterization of materials using the Impulse Excitation Technique.