On the natural frequencies of helical compression springs

Abstract

The linearized disturbance equations governing the resonant frequencies of a helical spring subjected to a static axial compressive load are solved numerically using the transfer matrix method for clamped ends and circular cross-section to produce frequency design charts. The effect of varying the number of turns of the spring is investigated, and in the limit of large numbers of turns, our results validate earlier work on the vibration of helical compression springs in which the helix was modeled as an elastic beam with rigidities corresponding to those of unclosed circular rings.

Introduction

Helical springs represent a fundamental component of engineering design, and quantification of their resonant frequencies has been an active area of research since at least the work of Michell [1] in 1890. The majority of studies in this area have focused on helical springs without externally applied loads (e.g. [2], [3], [4], [5], [6], [7], [8], [9]); the remainder have dealt mainly with the case of purely axial compression, which is perhaps the most common type of loading in practical applications.

The first and most comprehensive theoretical analysis of the effect of static axial compression on the natural frequencies of helical springs was conducted by Haringx [10], in which the spring is idealized as an elastic prismatic rod with rigidities corresponding to those of unclosed circular rings. In our previous, closely related work on buckling of helical springs under compression and/or torsion [11], [12], [13], [14], we have verified that his rod-model approximation is indeed appropriate for springs with sufficiently large numbers of turns. Given this simplified representation of the spring, Haringx obtained analytically a somewhat complicated set of transcendental equations for the natural frequencies corresponding to transverse (symmetric/asymmetric) modes of vibration amplitude, and he solved these equations by graphical means (around 1947) to produce a single comprehensive design chart for the fundamental frequency. This work, together with another pioneering study on the lateral rigidity of springs, constitutes the second in a series of six ingenious papers by Haringx related to the application of springs for vibration-free mountings.

More recent investigations have sought to compute the resonant frequencies and/or stability of a compressed helical spring treated as a curved and twisted rod under load. Mottershead [15] and Pearson [16] independently obtained different governing equations by summing forces and moments on a perturbed element of a helical compression spring, but their equations do not agree with the preceding rigorous treatise on this general subject in the textbook by Wempner [17], and one may verify that their equations do not reduce correctly to those for simpler rods; discrepancies in their results for buckling of helical compression springs are detailed in Ref. [11]. Most recently, Tabarrok and Xiong [18], [19], [20] derived a finite element formulation for the vibration and buckling of curved and twisted rods under load, the sample results of which agreed well with our previous work on buckling of compression springs (see Ref. [14]).

In this paper, we incorporate inertial terms into our prior formulation for buckling of helical springs, and investigate the effect of static axial compression upon the resonant frequencies. In the following section, we state the equations governing the initial deflections and the small-amplitude, linearized vibrations of a helical compression spring, as outlined by Wahl [21] for the static deflection of helices, and by Wempner [17] for the dynamical equations of curved and twisted rods under load. Analogous to our earlier studies, the resonant frequencies in the loaded state are computed using the transfer matrix method detailed in Ref. [16], and the corresponding initial configurations are determined to produce frequency design charts. This study represents primarily the first detailed examination of the fundamental natural frequencies of helical springs, under static axial load with clamped ends, since the pioneering work by Haringx [10] using the rod-model of a helix, and with our formulation we are able to investigate the heretofore untreated effect of low numbers of turns.