Bending vibrations of stepped rods

Abstract

Manufacturing stepped rods with segments of sizes proportional to stresses induced in operation is often used to reduce the material consumption in various fields of technology, for example, in the aircraft industry, where the requirements for the weight of structural elements are high. The vibration problems of continuous systems, i.e. systems which masses are considered distributed, are close to the resistance of materials and elasticity theory problems. They are described by partial differential equations. In this case, we consider a homogeneous isotropic material, obeying Hooke’s law. Of all the vibration problems of continuous systems, the transverse vibration problems of shafts and beams is of greatest practical importance. The simplest examples of vibrations of prismatic rods were studied in the 18th century in works on acoustics. But before solving problems of practical importance, the problems of stepped beams, it had taken another two hundred years and the development of approximate methods of solving differential equations. The paper presents a solution to the problem of determining the fundamental frequencies of bending vibrations of two-stepped rods with various boundary conditions, using the approximate Lagrange-Ritz method. The calculation error does not exceed 2.6 %. The fundamental frequency of vibration is defined considering different lengths and stiffness ratios of stepped rod segments. The obtained results can be used in solving practical problems in various fields of technology.