RAYLEIGH-BASED ANALYSIS OF TRANSVERSE VIBRATIONS IN A MANIPULATOR BOOM

Abstract

The article addresses the problem of determining the fundamental natural frequency of transverse vibrations of a manipulator boom modelled as an elastic Euler–Bernoulli beam supported on two hinged bearings. The aim of the study is to enhance the accuracy and engineering reliability of evaluating the dynamic characteristics of such structures by applying the Rayleigh energy method. To solve this problem, a set of approximate trial functions was formulated to satisfy the boundary conditions of zero deflection at the ends and to reflect the actual mode shape of the vibrations. Eight variants of trial functions of varying complexity are examined, ranging from the classical sinusoidal form to polynomial, exponential and combined expressions with variational parameter selection. Integral relations for the bending and kinetic energies were evaluated, enabling the determination of the natural frequencies of vibration and their comparison with the analytically exact solution. The analysis revealed that employing a sinusoidal trial function provides an almost exact value of the fundamental natural frequency, while approximations based on the static deflection shape and the potential function produce an error of less than one per cent. Parabolic and linear approximations exhibit an acceptable, though noticeably larger, error of about eleven per cent. In contrast, cubic and exponential forms proved to be insufficiently adequate, which is confirmed by a significant overestimation of the calculated frequency relative to the exact solution. A comparative assessment of the different trial functions justified the choice of the most effective options for engineering calculations and emphasised the importance of a preliminary analysis of the trial function curvature, on which the accuracy of the energy method depends. The findings have practical value for the design and optimisation of lifting and transport machinery, robotic manipulators and crane systems, where the influence of dynamic loads must be considered and resonant operating conditions avoided. The use of the Rayleigh method with an appropriately selected trial function ensures a rapid and reliable evaluation of natural frequencies without recourse to complex numerical models, which is particularly important at the early stages of design and structural verification.