Hindawi Publishing Corporation Advances in Materials Science and Engineering Volume 2013, Article ID 329530, 6 pages http://dx.doi.org/10.1155/2013/329530 Research Article Flexural Vibration Test of a Beam Elastically Restrained at One End: A New Approach for Young’s Modulus Determination Rafael M. Digilov and Haim Abramovich Faculty of Aerospace Engineering, Technion—Israel Institute of Technology, 32000 Haifa, Israel Correspondence should be addressed to Rafael M. Digilov; eduraf@technion.ac.il Received 1 May 2013; Accepted 11 July 2013 Academic Editor: Xing Chen Copyright © 2013 R. M. Digilov and H. Abramovich. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A new vibration beam technique for the fast determination of the dynamic Young modulus is developed. Te method is based on measuring the resonant frequency of fexural vibrations of a partially restrained rectangular beam. Te strip-shaped specimen fxed at one end to a force sensor and free at the other forms the Euler Bernoulli cantilever beam with linear and torsion spring on the fxed end. Te beam is subjected to free bending vibrations by simply releasing it from a fexural position and its dynamic response detected by the force sensor is processed by FFT analysis. Identifed natural frequencies are initially used in the frequency equation to fnd the corresponding modal numbers and then to calculate the Young modulus. Te validity of the procedure was tested on a number of industrial materials by comparing the measured modulus with known values from the literature and good agreement was found. 1. Introduction Te Young modulus is a fundamental material property and its determination is common in science and engineering [1, 2]. It is a key parameter in mechanical engineering design to predict the behavior of the material under deformation forces or more to get an idea of the quality of the material. Young’s moduli are determined from static and dynamic tests. In static measurements [3, 4] such as the classical tensile or compressive test, a uniaxial stress is exerted on the material, and the elastic modulus is calculated from the transverse and axial deformations as the slope of the stress-strain curve at the origin. Dynamic methods [5–12] are more precise and versatile since they use very small strains, far below the elastic limit and therefore are virtually nondestructive allowing repeated testing of the same sample. Tese include the ultrasonic pulse-echo [6, 7] or bar resonance methods [4, 8–14]. In the sonic pulse technique, the dynamic Young modulus is determined by measuring the sound velocity in the sample. In the resonance method, the linear elastic, uniform, and isotropic material of density  usually in the form of a bar of known dimensions is subjected to transverse or fexural vibrations, the natural frequency of th mode of which   related to Young’s modulus  by the relation [15, 16]   =  2  2 2 √   , =1,2,3,... (1) can be accurately measured. In (1)   is the modal eigenvalue that depends on boundary conditions,  is the vibrating length of the bar,  is its cross-sectional area, and  is the second moment of the cross-section, equal to  4 /4 for a rod of radius  and ℎ 3 /12 for a rectangular beam with width  and depth ℎ. Knowing the modal numbers, by simply measuring the resonance frequencies, geometry, and density of the specimen, the Young modulus can be determined from (1) as = 4 2  2   4   4   . (2) Te test sample is usually arranged in a manner to simulate free-free or clamped-free end conditions [10–12], when   , associated with the th fexural mode is a constant.