Abstract
The structural elements supporting motors or engines are frequently seen in technological applications. The operation of machine may introduce additional dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko single-span beams carrying a number of spring-mass system and multi-span beams carrying multiple spring-mass systems are plenty, but the free vibration analysis of Reddy-Bickford multi-span beams carrying multiple spring-mass systems has not been investigated by any of the studies in open literature so far. This paper aims at determining the exact solutions for the natural frequencies and mode shapes of Reddy-Bickford beams. The model allows analyzing the influence of the shear effect and spring-mass systems on the dynamic behavior of the beams by using Reddy-Bickford Beam Theory (RBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1–4 span beams are studied. The natural frequencies of Reddy-Bickford single-span and multi-span beams calculated by using the numerical assembly technique and the secant method are compared with the natural frequencies of single-span and multi-span beams calculated by using Timoshenko Beam Theory (TBT); the mode shapes are presented in graphs.
Highlights
The analysis of beams has been performed over the years mostly using Bernoulli-Euler beam theory (BET)
The frequency values obtained for the first five modes are presented in (Table 3) by comparing with the frequency values obtained for Timoshenko beam and mode shapes for the model with one spring-mass system of Reddy-Bickford beam are presented in (Fig. 6)
The frequency values obtained for the first five modes are presented in (Table 4) by comparing with the frequency values obtained for Timoshenko beam and mode shapes for the model with three spring-mass systems of Reddy-Bickford beam are presented in (Fig. 8)
Summary
The analysis of beams has been performed over the years mostly using Bernoulli-Euler beam theory (BET). The classical Bernoulli-Euler beam is well studied for slender beams, where the transverse shear deformation can be safely disregarded. This theory is based on the assumption that plain sections of the cross-section remain plain and perpendicular to the beam axis. The cross-sectional displacements of Bernoulli-Euler beam theory are shown in (Fig. 1.a) [1]. The cross-sectional displacements of Timoshenko beam theory are shown in (Fig. 1.b). Han et al presented a comprehensive study of Bernoulli-Euler, Rayleigh, Shear and Timoshenko beam theories [6]
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