On elastic wave propagation in helical springs

Wittrick, W. H.

doi:10.1016/0020-7403(66)90061-0

Cited by 135 publications

(59 citation statements)

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“…[21], p. 80). This model was used and developed by Wittrick [13], who took into account couplings between longitudinal and torsional spring vibrations. He directed his attention to different wave propagation velocities along the helix line related to the wire twisting and bending.…”

Section: Introductionmentioning

confidence: 99%

Natural transverse vibrations of helical springs in sections covered with elastic coatings

Michalczyk

2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. It has been demonstrated in previous studies that local elastomer coatings covering the end coils of helical springs can efficiently reduce the amplitudes of circum-resonant vibrations in such springs. The analysis of the influence that elastic coatings have on the frequencies and modes of natural transverse vibrations of springs is presented in this paper. The concept of the equivalent beam of the Timoshenko type is utilized in calculations of the frequencies and modes of transverse vibrations. The model developed allows users to determine the frequencies and modes of symmetric as well as antisymmetric vibrations of axially loaded springs with elastic coatings of arbitrary length. A comparison of the results obtained using FEM analysis, in which the model represented the actual spring geometry, with the results obtained by means of the presented model indicates its high accuracy. model of a spatially curved rod, the model of an equivalent rod or beam, and the model in which a continuous system, such as a spring, is substituted by a discrete-periodic system consisting of concentrated masses joined by a mass-less stiffness, jointly representing one spring coil or its fragment. Equations of motion are written for an elementary fragment of a spring wire in the first model. Such a formulation of the problem [12, 13] allows arbitrary modes and frequencies of spring vibrations to be determined, however, solving it by analytical methods is very difficult and therefore such a solution is not suitable in practical applications. These difficulties mean that in several studies concerning the analysis of spring vibrations treated as a spatially curved rod, the authors use numerical methods. New finite elements, capable of modeling spring coils or their fragments were proposed in [14]. They are suitable for application in static problems as well as when looking for natural frequencies of helical springs. Mottershead provided, in the above-mentioned paper, experimental results later utilized by other authors. A similar approach to the problems of helical springs was presented in [15][16][17]. Pearson expanded -in [18] -equations provided by Wittrick [13], taking into account the influence of a static axial force. For determining natural vibrations, he applied the transfer matrix method. The same method was used in [19], concerning the analysis of parameters influencing natural frequencies of helical springs (utilizing the Cayley-Hamilton theorem), and in [20]. In this last paper, the authors compared their results with the results obtained on the basis of the Haringx model [21]. They pointed out that, in the case of helical springs with geometrical parameters, typical of springs used in machine buildings, the conformity between the results obtained by them and the results obtained using the Haringx model is high. In the numerical example quoted by them, for which the first 16 natural frequencies were determined, the maximum difference between their results and the ones from the Haringx model did not exceed 1.3%. In ...

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Section: Introductionmentioning

confidence: 99%

Natural transverse vibrations of helical springs in sections covered with elastic coatings

Michalczyk

2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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“…The solution of the three-dimensional problem in elastodynamics is the natural reference for such an assessment-exactly as the Rayleigh-Lamb solution of the problem of wave propagation in an elastic layer is used to assess validity ranges of elementary theories of straight beams. However, this paper is concerned with wave propagation in a helical spring at relatively low frequencies and the reference is chosen as Timoshenko-type theory, which incorporates shear deformation and rotary inertia as was formulated by Wittrick (1966). As long as all branches of dispersion curves are adequately described by any reduced theory, there is no reason to employ this one.…”

Section: Introductionmentioning

confidence: 99%

“…This author has employed a semi-analytical finite-element method to study wave propagation in a spring with a large pitch. The Timoshenko-type theory has been introduced by Wittrick (1966) and used by Mottershead (1980), Yildirim (1996), Lee & Thompson (2001), Becker et al (2002), Telem et al (2004) and Lee (2007). Simplification of this model by neglecting rotary inertia and shear deformation is a standard way to proceed from the Timoshenko model to the classical Bernoulli-Euler model.…”

Section: Introductionmentioning

confidence: 99%

“…Simplification of this model by neglecting rotary inertia and shear deformation is a standard way to proceed from the Timoshenko model to the classical Bernoulli-Euler model. However, in the literature (see Wittrick 1966;Pearson & Wittrick 1986), the Bernoulli-Euler model is introduced with an additional assumption that deformation due to axial load may be neglected. As is shown in §2, this simplification is dangerous because it dramatically narrows the frequency range, where this 'standard' model is adequate.…”

Section: Introductionmentioning

confidence: 99%

“…An analysis of dispersion curves for a reduced model of spring has been performed by Guido et al (1978), whereas the general Timoshenko-type theory has been treated by Lee & Thompson (2001) and Lee (2007). Although the exact formulation of dispersion equation has been provided by Wittrick (1966), its solution is elaborated in that reference either for the limiting case of zero helix angle or under assumptions referred to as Bernoulli-Euler model. The danger of these assumptions has been recognized in the concluding section of the paper by Wittrick (1966), but no quantification has been provided.…”

Section: Introductionmentioning

confidence: 99%

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Linear dynamics of elastic helical springs: asymptotic analysis of wave propagation

Sorokin

2009

Proc. R. Soc. A.

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Helical springs serve as vibration isolators in virtually any suspension system. A variety of theories to describe the dynamic behaviour of these structural elements, which involves interaction of flexural, torsion and longitudinal waves, can be found in the literature. Alongside this, various approximate methods are employed to determine the eigenfrequencies of vibrations of springs. In this paper, the validity ranges of alternative theories are assessed by comparison of the location of the dispersion curves. This paper also contains a rigorous asymptotic analysis of the exact dispersion equation with two small parameters being employed. It allows for the identification of significant regimes of linear wave motion in a helical spring. In each of these regimes, simple formulae for wavenumbers are obtained by the dominant balance method and their validity ranges are checked against direct numerical solution. Mode shapes associated with each wavenumber are also analysed.

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Investigation of Parameters Affecting Free Vibration Frequency of Helical Springs

Yıldırım

1996

Int. J. Numer. Meth. Engng.

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On elastic wave propagation in helical springs

Cited by 135 publications

References 0 publications

Natural transverse vibrations of helical springs in sections covered with elastic coatings

Natural transverse vibrations of helical springs in sections covered with elastic coatings

Linear dynamics of elastic helical springs: asymptotic analysis of wave propagation

Investigation of Parameters Affecting Free Vibration Frequency of Helical Springs

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