The dynamic coupling of torsional and flexural strains in cylindrical helical springs

Summary We have examined, both theoretically and experimentally, the coupling between torsional and flexural strains in cylindrical helical springs axially excited. The analytical investigation, developed on the basis of the theory till now at our disposal, has shown the existence of two series of resonance conditions: the first one corresponds to the one already known for the axial vibrations, the other one is relative to the vibrations constituted by the rotations of the coils around the spring axis. The experimental research, developed into two springs having different characteristics and subjected to different assembly preloadings, has shown that the coupling between torsional and flexural strains is much more complex than the one indicated by theory, because of the presence of lateral deformations of the spring. The different tests, effected with an exciting variable frequency in a field below the second natural frequency of the axial vibrations, have shown the existence of more than two resonances.

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Sommario Viene esaminato, sia teoricamente che sperimentalmente, l'accoppiamento tra le deformazioni torsionali e flessionali in molle ad elica cilindrica sottoposte ad eccitazione secondo il proprio asse. L'indagine analitica, svolta sulla base della teoria finora disponibile, ha messo in evidenza l'esistenza di due serie di condizioni di risonanza: la prima corrisponde a quella ben nota per le vibrazioni assiali, la seconda è relativa alle vibrazioni costituite da rotazioni delle spire intorno all'asse della molla. L'indagine sperimentale, svolta su due molle aventi diverse caratteristiche e sottoposte a precarichi di montaggio diversi di volta in volta, ha mostrato che l'accoppiamento fra le deformazioni torsionali e flessionali è ben più complesso di quello indicato dalla teoria, a causa della presenza di deformazioni laterali della molla. Le diverse prove, effettuate con frequenza eccitante variabile in un campo al di sotto della seconda frequenza naturale delle vibrazioni assiali, hanno messo in evidenza l'esistenza di più di due risonanze.

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Research supported by the Italian Research Committee (C.N.R.).     

Consideration of the masses of helical springs in forced vibrations of damped combined systems

The common point about many systems modeled as Bernoulli-Euler beams with attachments is that the own masses of the helical springs are neglected. Some researchers accounted for the masses of the springs during free vibrations of those systems. Further to these studies, the present study deals with the investigation of the effect of not taking into account the masses of the helical springs in damped combined systems during their forced vibrations. It is shown that the spring mass effect may be important in regions near to the resonance frequencies. Further, this effect influences the phase angles more than the amplitudes.     

Delayed presetting of helical springs

Abstract

This article establishes the calculation of remaining deformation and residual stress for helical springs after long-lasting presetting process. The article extends the model for the immediate presetting process accounting the creep deformation of the spring. The method is based on plasticity theory for the instant flow overexposed by the relaxation over the long-term presetting. In this article, the following method is used. The plastic deformation of the helical spring with the circular cross section occurs instantly. If the shortening of the spring in the tool holder persists, the relaxation of stresses occurs and the force of the spring reduces. As the consequence, after the elastic unloading of the long-time presetting, the residual stresses spring reduce gradually with the squeezing time as well. The final length of the springs considerably shortens with the increasing preset duration. The advantage of the discovered closed form solutions is the calculation without the necessity of complex finite-element simulation of spring length loss and residual stresses after presetting process. The analytical expressions are proposed and the exact calibration applied for evaluation of factors for presetting processes.     

Transverse vibrations of arbitrary non-uniform beams

Free and steady state forced transverse vibrations of non-uniform beams are investigated with a proposed method, leading to a series solution. The obtained series is verified to be convergent and linearly independent in a convergence test and by the non-zero value of the corresponding Wronski determinant, respectively. The obtained solution is rigorous, which can be reduced to a classical solution for uniform beams. The proposed method can deal with arbitrary non-uniform Euler-Bernoulli beams in principle, but the methods in terms of special functions or elementary functions can only work in some special cases.     

General axial response of stranded wire helical springs

The large static deflection of an axially loaded helical spring formed of a twisted strand of smooth circular wires is considered. Contact between the wires in the strand may or may not be maintained upon loading, depending upon the type of construction and the type of loading. It is found that the making or breaking of wire contact within the strand has a drastic ettect upon the extension and twist of a wire and upon the extension of the strand, but has practically no effect upon the twist of the strand, and only a moderate effect upon the overall response of the spring. Limited experimental data tend to verify the theory. It is found that a good engineering approximation for the axial stiffness of a twisted m-wire spring in which contact is maintained can be made by treating the spring as m untwisted helical wires acting independently, provided the strand twist is not too severe.     

On the dynamic behaviour of axially excited helical springs

Summary In this paper we have used Wittrick's theory on the propagation of deformation waves in a coiled wire — taking only into account the coupling between the displacements of the wire cross-section along the tangent and along the binormal of the average helix — in oder to show how the natural frequencies of the axial and rotational modes of a cylindrical helical spring vary according to a non-linear function of the wave number. Moreover it is here demonstrated that the natural frequencies resulting from the proposed theory for the natural modes to which correspond wave lengths greater than two coils do not differ much from those which can be calculated by considering the spring as a periodic discrete system. Theoretical results are finally tested by an appropriate experimental investigation.

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Sommario Viene utilizzata la teoria proposta dal Wittrick sulla propagazione delle onde di deformazione in un filo avvolto ad elica, tenendo conto del solo accoppiamento tra gli spostamenti delle sezioni del filo lungo la tangente e lungo la binormale all'elica media, per mettere in evidenza come le pulsazioni naturali dei modi assiali e rotazionali di una molla ad elica cilindrica varino secondo una funzione non lineare del numero d'onda. Si dimostra che le pulsazioni naturali che si ottengono dalla teoria proposta, per modi di vibrare cui corrispondono lunghezze d'onda maggiori di due spire, sono poco discoste da quelle prevedibili considerando la molla come sistema discreto periodico. I risultati della teoria surriportata vengono infine controllati a mezzo di una opportuna indagine sperimentale.

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Research supported by the Ministero della Pubblica Istruzione, Italy.     

Transverse vibrations in micro-scale viscothermoelastic beam resonators

In this article, the closed form expressions for the transverse vibrations of a homogenous isotropic, thermally conducting, Kelvin–Voigt type viscothermoelastic thin beam, based on Euler– Bernoulli theory have been derived. The effects of relaxation times, thermomechanical coupling, surface conditions, and beam dimensions on energy dissipation induced by thermoelastic damping in MEMS (micro-electromechanical systems) resonators are investigated for beams under clamped and simply supported conditions. Analytical expressions for deflection, temperature change, frequency shifts, and thermoelastic damping in the beam have been derived. Some numerical results with the help of MATLAB programming software in case of Silicon Nitride have also been presented. The computer-simulated results in respect of damping factor and frequency shift have been presented graphically.     

On the linearized disturbance dynamic equations for buckling and free vibration of cylindrical helical coil springs under combined compression and torsion

In this study a set of twelve linearized disturbance dynamic equations in canonical form is derived systematically and in a comprehensive manner based on the first order shear deformation theory to study the buckling and vibration analysis of helical coil springs made of isotropic linear materials. Those complete equations comprise the axial and shear deformation effects together with rotatory inertia effects. The special case of these equations corresponds also to the equations for straight and circular rods. The main differences among the existing formulations based on the same approach are discussed briefly. The resulting equations are used for numerical buckling and free vibration analyses to show its soundness.     

Low-frequency free vibrations of viscoelastic cylindrical shells

Vitebsk State Pedagogical Institute, Belarus. Translated from Prikladnaya Mekhanika, Vol. 28, No. 9, pp. 50–55, September, 1992.     

Free vibrations of long pretwisted cylindrical shells

Tbilisi Mathematical Institute, Academy of Sciences of the Georgian SSR. Translated from Prikladnaya Mekhanika, Vol. 26, No. 9, pp. 54–60, September, 1990.