Open AccessDevelopment of Polynomial Mode Shape Functions for Continuous Shafts with Different End Conditions
Author(s) -
Mahesh Chandra Luintel
Publication year - 2021
Publication title -
journal of the institute of engineering
Language(s) - English
Resource type - Journals
ISSN - 1810-3383
DOI - 10.3126/jie.v16i1.36653
Subject(s) - trigonometric polynomial , trigonometric functions , mathematics , polynomial , vibration , boundary value problem , transcendental function , normal mode , mathematical analysis , galerkin method , transcendental equation , trigonometry , finite element method , geometry , numerical analysis , structural engineering , engineering , acoustics , physics
Common methods used to determine the solutions for vibration response of continuous systems are assumed mode method, Rayleigh-Ritz method, Galerkin Method, finite element method, etc. Each of these methods requires the shape functions which satisfy the boundary conditions. Shape functions derived in most of the classical textbooks are simple trigonometric functions for some end conditions but are very complex transcendental functions for many end conditions. It is very difficult to determine the vibration response of a continuous system analytically by using such transcendental shape functions. Hence this paper presents a method to develop polynomial shape functions required to solve the vibration of continuous shafts with different end conditions. The natural frequencies obtained from the developed polynomial shape functions are compared to those obtained from the classical transcendental shape functions and found very close for the first three modes.