A Modal Solution for Finite Length Rods with Non-Uniform Area

Andrew Hull

doi:10.3390/app8010094

Abstract

This paper derives a modal solution to the displacement field of a finite length rod whose area varies with respect to its length. This new method facilitates a solution to any problem where the area and derivative of the area can be represented as analytical functions. The problem begins by writing the longitudinal displacement of the non-uniform area rod as a series of indexed coefficients multiplied by the eigenfunctions of the uniform area rod. This series solution is inserted into the non-uniform area rod equation and multiplied by a single p-indexed eigenfunction. This equation is then integrated over the interval of the rod. Although the resultant expressions are not orthogonal, they can be written as a set of linear algebraic equations which can be solved to yield the unknown coefficients. Once these are known, the displacement of the system can be calculated. Five example problems are included: the first one has a non-uniform area that corresponds with a known analytical solution, the second has an area that can be represented by a Fourier series, the third and fourth have areas that do not have a known analytical solution and the fifth is a generic second order non-constant coefficient differential equation. Four of these problems are verified with other methods. Convergence of the series solution is discussed. It is shown that this new model is almost two orders of magnitude faster than corresponding finite element analysis.

Highlights

Rods are common mechanical elements in the automotive, aerospace, construction and maritime industries
Modeling allows an understanding of the vibrational response of these systems before they are built so that costly prototyping can be minimized or avoided altogether. Increased knowledge of these systems allows designers to incorporate features and characteristics that might not be understood and omitted from the initial design. Modeling of these systems started with the basic rod equation [1], which is sometimes referred to as elementary theory, and this governing second order differential equation is the focus of this work
This system can be modeled by differential equations that contain higher order effects and this usually includes some form of transverse motion or rotation. These models include Love theory [2], Mindlin-Herrmann theory [2], three mode theory [2], Rayleigh-Bishop theory [3] and fully elastic theory [4]. These higher order models are typically used for rods with relatively large cross-sectional areas or higher frequency ranges where additional dynamic effects need to be included in the differential equations to properly capture the dynamics of the rod

Summary

Introduction

Rods (or bars) are common mechanical elements in the automotive, aerospace, construction and maritime industries. Modeling allows an understanding of the vibrational response of these systems before they are built so that costly prototyping can be minimized or avoided altogether Increased knowledge of these systems allows designers to incorporate features and characteristics that might not be understood and omitted from the initial design. These models include Love (or Rayleigh-Love) theory [2], Mindlin-Herrmann theory [2], three mode theory [2], Rayleigh-Bishop theory [3] and fully elastic theory [4] These higher order models are typically used for rods with relatively large cross-sectional areas or higher frequency ranges where additional dynamic effects need to be included in the differential equations to properly capture the dynamics of the rod. These higher order models are not discussed further in this paper

Methods

Results

Conclusion