A Discrete Model for the Natural Frequencies and Mode Shapes of Constrained Vibrations of Beams with Various Boundary Conditions

¹ Laboratoire de Mécanique Productique et Génie Industriel (LMPGI) Université Hassan II Ain Chock, Ecole Supérieure de Technologie, KM 7 Route El Jadida, B.P 8012, Oasis Casablanca Maroc
² Laboratoire des Etudes et Recherches en Simulation, Instrumentation et Mesures (LERSIM) Université Mohammed V - Ecole Mohammadia des Ingénieurs, Avenue Ibn Sina, Agdal, Rabat, Maroc

^a e-mail: abd.rahmouni@gmail.com
^b e-mail: Zbeidouri@gmail.com
^c e-mail: rbenamar@emi.ac.ma

Abstract

The purpose of the present paper is the development of a physically discrete model for free transverse constrained vibrations of beams. The discrete model consists on an N-degree of freedom system made of masses placed at the end of solid bars connected with spiral springs. The calculations made involve two tensors, namely the mass tensor [m_ij] and the linear rigidity tensor [k_ij]. The results obtained by the physically discrete model show a good agreement and a quick convergence to the equivalent continuous beam for various end conditions for both the natural frequencies and the corresponding mode shapes. The model proposed in the present paper, which has been validated here using classical cases, may be easily applied to the flexural vibration of beams with various types of discontinuities, and to beams carrying concentrated masses.