JournalofSoundand < ibration (2002) 256(4), 787 } 790 doi:10.1006/jsvi.2001.4224, available online at http://www.idealibrary.com on CALCULATION OF NON-LINEAR FUNDAMENTAL FREQUENCY OF A CANTILEVER BEAM USING NON-LINEAR STIFFNESS C. PANY AND G. V. RAO StructuralEngineeringGroup, < ikramSarabhaiSpaceCentre, ¹rivandrum 695022, India. E-mail: gv } rao@vssc.org (Received 13 November 2001, andin ,nalform 8 January 2002) 1. INTRODUCTION Large amplitude vibrations of cantilever beams are of interest to engineers in many "elds of engineering. The linear theory of vibrations predicts the frequencies of natural vibration to be independent of the amplitude. In many instances, if the amplitude of the vibrations is large, then the above statement is not justi"ed due to the non-linear e!ects involved. Wagner [1] has obtained approximate solutions for free non-linear oscillations of an initially straight, uniform elastic bar with clamped}free and free}free end conditions. The problem of large amplitude vibrations has been presented for clamped}free and free}free uniform beams [2] and a tapered cantilever beam [3]. Recently, a simple relationship has been presented [4] to determine the "rst mode linear natural frequency of linearly tapered cantilever beam as a function beam sti!ness (small deformation theory), the beam mass, and a mass distribution parameter. In this note, the fundamental frequency, when the amplitudes are large, has been evaluated for a uniform cantilever beam undergoing large amplitude utilizing the methodology of reference [4]. The present results compare well with those of Wagner [1] and Rao and Rao [2]. However, these are in better agreement with those obtained by Wagner [1]. 2. METHODS OF ANALYSIS 2.1. LOADS AND NON-LINEAR DEFLECTION FUNCTION The approximate formula for the large de#ection of a cantilever beam by linearizing the elliptic integral solution are presented explicitly in reference [5]. For the case of a cantilever beam of length ¸ and with a vertical tip load P (Figure 1), the non-linear solution is expressed in terms of  the tip slope of the beam. The expressions useful to the present study are reproduced below from reference [5]. B "(2sin )  1# 1 2 # 2 3   1!  2 # 17 120   :(2sin )  1# 4 15   , (1) 0022-460X/02/$35.00  2002 Elsevier Science Ltd. All rights reserved.