Free Vibration of a Beam Having a Rotational Restraint at One Pinned End and a Support of Variable Abscissa

To carry out a dynamic analysis of vibrating systems, different methods exist and numerical methods take a large place. In this study, a beam with one overhang, having a rotational restraint at one pinned end and a support of variable abscissa is investigated. Many situations are studied using the finite element method based on the Euler-Bernoulli assumptions to analyze beams vibration, without elastic restraints. The purpose of this investigation consists of treating similar beam cases with and without a rotational restraint at one node. The first step of validation is relative to the extreme situations where either the overhang or the intermediate span length is zero (span length approx. zero). In these two studied cases, the rotational restraint is not considered. It is compared to theoretical, energetic Rayleigh method results and those due to the finite element method for the simply supported beam, pinned-clamped and free-fixed beam (cantilever beam). When the rotational restraint value increases the beam is considered first pinned-pinned and then pinned-fixed behavior is obtained. The results as found are in agreement with analytical ones and after introducing the rotational restraint at the right-hand end, an intermediate behavior is initiated by varying the s/L (span length/beam length) ratio, allowing a simplified extraction of the fundamental vibration frequency for various values of K_r (rotational restraint).