On the concept of an equivalent column in the stability problem of compressed helical springs

Krużelecki, Jacek; Życzkowski, M.

doi:10.1007/bf00542566

Cited by 18 publications

(24 citation statements)

References 5 publications

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“…The distance between neighbouring coils in the sectors near supports of a spring is larger than in the middle sector. Hence, the actual compression and bending rigidities are larger in the middle part of the spring than in regions of support (Kru~elecki and Zyczkowski 1990), but a bending rigidity changes very slightly. So, it can be expected that the profit of optimization is rather small.…”

Section: Radius R0(~) Described By Conlinuous Functionsmentioning

confidence: 99%

“…(6) The ratio R(~)/RO = p(~) is described by cos 2 a(1 + v + tg2a) = 0 (7) F2 = P(~) --cos 2 aO(1 + v + tga 0 tga) and (6) and (7) are discussed by Kru~elecki and Zyczkowski (1990). The initial number of coils no = ~Pmax/2"x,…”

Section: Formulation Of the Optimization Problem -State Equationsmentioning

confidence: 99%

“…Optimization of compression helical springs with respect to their stability was initiated by Kru~.elecki (1993Kru~.elecki ( , 1995Kru~.elecki ( , 1996. Using the Pontryagin maximum principle the optimal distribution of a helix angle c~0(z ) and radius Ro(z) of a spring was looked for.…”

Section: Introductionmentioning

confidence: 99%

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Parametrical optimization of compression helical springs against instability

Krużelecki

1997

Structural Optimization

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In the present paper, the parametrical optimization problem of helical springs against instability is considered. Such an initial distribution of a helix angle and radius of a spring is sought in the assumed class of functions which ensure maximal value of the critical axial compressive force under equality and inequality constraints. The volume of the spring material, the initial compression rigidity and also the slenderness ratio of the spring are considered as the equality constraints. The inequality constraints are connected with: a strength condition, a condition of closing up of neighbouring coils, and geometrical conditions in the initial and compressed state.

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Section: Radius R0(~) Described By Conlinuous Functionsmentioning

confidence: 99%

Section: Formulation Of the Optimization Problem -State Equationsmentioning

confidence: 99%

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Parametrical optimization of compression helical springs against instability

Krużelecki

1997

Structural Optimization

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“…The Haringx model was also used in study [34] to determine the natural frequencies of transverse vibrations of a simply supported spring. It was pointed out, in [35], that the above-mentioned equivalent rod concept, proposed in [22] for the analysis of elastic stability of helical springs, can also be applied in the analysis of natural transverse vibrations of helical springs. Its application, after modification resulting from different boundary conditions, provides results nearer to FEM results and the results obtained by other authors [18,26] (treating the spring as a spatially curved rod) than the standard concept of Timoshenko.…”

Section: Introductionmentioning

confidence: 99%

“…The authors noticed that more accurate dependencies, describing equivalent bending and shear stiffness, and applied in relation to the Haringx concept, would be able to explain this effect. Such dependencies, taking into account, among others, the non-zero and variable (together with an axial load-) values of the helix angle, in relation to the elastic stability problem of helical springs, were derived in [22]. A new numerical method of determining natural frequencies of helical springs by means of the stiffness method was proposed in [23].…”

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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Equivalent Beams of Helical Anisotropic Springs

2019

Design and Analysis of Composite Structures for Automotive Applications

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On the concept of an equivalent column in the stability problem of compressed helical springs

Cited by 18 publications

References 5 publications

Parametrical optimization of compression helical springs against instability

Parametrical optimization of compression helical springs against instability

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

Equivalent Beams of Helical Anisotropic Springs

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