2024 | Book

# Fundamentals of Springs Mechanics

Author: Vladimir Kobelev

Publisher: Springer Nature Switzerland

2024 | Book

Author: Vladimir Kobelev

Publisher: Springer Nature Switzerland

This book highlights the mechanics of the elastic elements made of steel alloys with a focus on the metal springs for automotive industry. The industry and scientific organizations study intensively the foundations of design of spring elements and permanently improve the mechanical properties of spring materials. The development responsibilities of spring manufacturing company involve the optimal application of the existing material types. Thus, the task entails the target-oriented evaluation of the mechanical properties and the subsequent design of the springs, which makes full use of the attainable material characteristics. The themes about the new design of disk springs and the hereditary mechanics—namely creep and relaxation resistance—were extended. The fatigue life diagrams were reconsidered, and the relations between the traditional diagrams revealed. The book stands as a valuable reference for professionals in practice as well as an advanced learning resource for students of structural and automotive engineering.

The former editions were known as "Durability of Springs”. Reflecting the substantial enlargement of the discussed themes, starting with this 3rd Edition the book entitled as "Fundamentals of Springs Mechanics”.

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Abstract

Calculation formulas for linear coil, or helical springs with inconstant wire diameter and with variable mean diameter of the spring are presented. These formulas are used to optimize the spring for a given spring rate and wire strength. The principles of design of optimal leaf springs are briefly presented. This chapter is the introductory section of the first part, which studies springs from the point of view of design.

Abstract

This chapter examines the stress distribution across the wire cross-section of helical springs. For simplicity, the pitch of the helical spring is neglected and the traditional representation of a coil as an incomplete torus is used. This model generalizes StVenant's torsion problem of an elliptical straight rod, considering the curvature of the rod. The closed-form solution for the torsion problem of an incomplete torus is discussed. This chapter is the next section of the first part, which studies the springs from the point of view of design.

Abstract

In this chapter, the coil spring is replaced by a flexible rod located along the axis of the coil. This rod has the same mechanical properties as the spring itself. Its bending, torsion, and compression stiffnesses are equal to the corresponding stiffnesses of the helical spring. This rod is known as the “equivalent column” of the coil spring. The “equivalent column” equations are much easier to handle than the original helical spring equations. The integral properties of the spring, such as axial and lateral stiffness, buckling loads, and natural frequencies, can be determined directly using the equivalent column equations. On the other hand, the local properties, such as stresses in the wire or contact forces, could only be evaluated with the more complicated equations of the helical elastic rod. In this chapter, the stability and transversal vibrations of the spring are studied from the unified point of view based on the “equivalent column” concept. Buckling refers to the loss of stability up to the sudden and violent failure of straight bars or beams under the action of compressive forces whose line of action is the column axis. This concept is applied to the stability of helical springs. An alternative method of approach is based on the dynamic criterion of spring stability. The equations for transverse (lateral) vibrations of compressed helical springs have been derived. This solution expresses the fundamental natural frequency of the lateral vibrations of the column as a function of the axial force, as well as the variable length of the spring. This chapter is the third section of the first part, which studies the springs from the design point of view.

Abstract

In the current chapter, we study the disk spring using the models of thin and moderately thick isotropic shells. The equations developed here are based on common assumptions and are simple enough to be applied to the applied analysis. The analysis of thin-walled disk springs could be performed using basic spreadsheet tools, eliminating the need for cumbersome finite element analysis. The theory of linear and progressive disk wave springs is also presented. This chapter is the fourth section of the first part, which deals with the design and modeling of springs.

Abstract

In the current chapter, the calculation of disk springs is studied with free sliding edges and edges with constrained radial motion. The disk springs with constant material thickness are studied using the models of thin and moderately thick isotropic shells. The variation formulations are used to derive load–displacement formulas for the disk springs with different constraints on radial movement on the inner and outer surfaces. The kinematic hypothesis is used for the shell models of conical shells. The motivating feature of the presented theory is its ability to calculate the cup springs with free gliding edges and the edges with constrained radial motion. The equations developed here are based on general assumptions and are suitable for disk springs made of isotropic materials, such as spring steel and light metal alloys. The advantage of the methodology is the derivation of closed-form solutions for several common restrictions on the radial motion of the inner and outer edges. The developed formulas are recommended for industrial calculations of free and restricted disk springs.

Abstract

This chapter concludes the treatment of disk springs. In the current chapter, we study the variable thickness cup spring. The thickness is assumed to be variable along the meridional and parallel axes of the conical coordinate system. The calculation of the disk springs includes the cases of free gliding edges and edges on cylindrical curbs, which constrain the radial movement. The equations developed here are based on common assumptions and are simple enough to be applied to industrial calculations.

Abstract

The semi-open cross sections are made from standard tubes by flattening them. Due to the possibility of significant variation of mechanical parameters along the length, the flattening could be easily adjusted for optimal design. From the design point of view, the semi-open cross sections fill the gap between the classic closed and open single-wall sections. Due to the high variability of the section geometry, the simple analytical model is essential for primary design purposes and estimation of numerous opposing static effects. The effective torsional stiffness \({r}_{t}\), effective bending stiffness \({r}_{z}\) and effective bending spring \({r}_{c}\) of the twist beam in terms of section properties of the twist beam with semi-open cross section have been expressed with the analytical formulas. This chapter is the last section of the first part, which studies the elastic elements from the point of view of modeling and design.

Abstract

In this chapter, the method for calculation of residual stresses and plastic bending and torsion moments for combined bending and torsion loading is developed. The Bernoulli hypothesis is assumed for the deformation of the beam. The analysis is performed using the deformation theory of plasticity with a nonlinear stress–strain law describing active plastic deformation (SAE AE-22, §8.3.33). The curvature and twist of the beam increase proportionally during plastic loading so that the ratio of curvature to twist remains constant. The complete solutions based on this approximate material law provide a closed analytical solution. The unloading is linear elastic. This chapter is the first section of the second part, which studies mechanical problems arising during the manufacturing of helical springs.

Abstract

In this chapter, the method for calculating residual stress and permanent deformation of helical springs is developed. The method is based on the deformation formulation of the plasticity theory and general kinematic hypotheses (SAE AE-22, §8.3.33). Two main types of helical springs—compression springs and torsion springs—are studied. For the first type (axial compression or tension springs), the spring wire is twisted. The basic approach neglects the pitch and curvature of the coil and replaces the helical wire with a straight cylindrical rod. The helical spring is in the state of screw dislocation; accordingly, the wire is twisted. The elastic–plastic torsion of the straight rod of circular section is studied. In the second type (torsion helical springs), the helical wire is in the state of edge dislocation, so the wire is in the state of bending. The elastic–plastic deflection of the straight bar with rectangular and circular cross section is studied using Bernoulli's hypothesis. The material for both types of springs is nonlinearly hardening elastic–plastic with elastic unloading. The hyperbolic, Ramberg–Osgood, Ludwik–Hollomon, Swift–Voce, and Johnson–Cook laws for the material are studied. For both problems, the elastic–plastic active deformation and the elastic spring-back allow the closed-form solutions. In addition, this chapter examines the calculation of residual deformation and residual stress for helical springs after a prolonged prestressing process. The article extends the model for the immediate prestressing process by considering the creep deformation of the spring. The method is based on the plasticity theory for the instantaneous flow, which is overexposed by the relaxation over the long-term prestressing. In this article, the following method is used. The plastic deformation of the circular section helical spring occurs instantaneously. As the spring continues to shorten in the tool holder, the stress relaxes and the force of the spring decreases. As a result, after the elastic relaxation of the long-term presetting, the residual stress of the spring also gradually decreases with the compression time. The final length of the springs decreases significantly as the presetting time increases. 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 prestressing. The analytical expressions are proposed and the exact calibration is applied for evaluation of factors for presetting processes. This chapter is the final section of the second part, which studies the manufacturing processes of helical springs.

Abstract

In this chapter, we study the time-dependent behavior of spring elements under constant and oscillating loading. The common creep laws are implemented to describe the material. The models for stress relaxation under constant strain and for creep under continuous load are established. For simple spring elements, closed-form solutions for the creep and relaxation models are found. The introduction of the “scaled” time variable simplifies the presentation. The experimental procedure for the experimental acquisition of creep models is presented. This chapter concludes the second part, which deals with the manufacturing of springs. The highly undesirable creep can occur during the life cycle of mechanical springs, but proper manufacturing reduces or eliminates such phenomena.

Abstract

This chapter explains the common methods for considering stress amplitude in the fatigue life of springs. The fatigue of spring materials under fully reversed uniaxial loading is reviewed. Two different fatigue life estimation methods are briefly discussed. The first method implements the stress-life and strain-life procedures. The second method describes fatigue crack growth (FCG) per cycle. Two analogous unifications of the FCG functions are proposed. The expressions for the spring length over the number of cycles are derived. This chapter is the first section of the final part, which studies the life cycle of elastic elements.

Abstract

This chapter examines the effects of mean stress, multiaxial loading, and residual stresses on the fatigue life of springs. We study the fatigue life of homogeneously stressed material subjected to cyclic loading with non-zero mean stress. Traditional fatigue life estimation methods are based on Goodman and Haigh diagrams. The formal analytical descriptions, namely stress-life and strain-life approaches, are more suitable for numerical methods. The method of fatigue analysis described above, which describes the crack growth per cycle, is extended. The extensions allow for the consideration of the mean stress of the load cycle. Closed-form expressions for the crack length over the number of cycles are derived. The complementary effects on the fatigue life of springs, which ultimately significantly influence the fatigue life of springs, are succinctly pronounced. This chapter is the second section of the third part, which surveys the life cycle of elastic elements.

Abstract

This Chapter begins with the study of statistical effects in failure analysis of springs. The procedure for dimensional analysis of helical spring failures is presented. The dimensional analysis spring element is examined using the weakest link concept. This concept is applicable to the extremely brittle material in which the failure of one link causes the complete destruction of the entire structural element. The stress variation over the wire surface is considered for the analytical calculation of Weibull failure probability. Chapter provides analytical formulas for failure probability of helical springs. The derived solution clarifies the effects of spring index and wire diameter on fatigue life. The methodology is valid for different types of springs or basic structural elements. This Chapter is the third section of the third part, which studies the life cycle of elastic elements.

Abstract

In this Chapter, we continue to study the stochastic influences on the fatigue life of springs. First, we evaluate the probabilistic effects that prevail at low amplitudes of cyclic stresses. The question is how to calculate the failure probability as a function of the stress amplitude. The answer to this question results from the study of experimental fatigue life data. The experimental data show different behavior in the regions of low and high stress amplitudes. To describe this phenomenon, we introduce the randomization of crack propagation, which is accompanied by the random deviation and branching of the crack. The randomization of crack propagation escalates with decreasing stress amplitude. The high inhomogeneity of the polycrystalline structure at the micro level hypothetically causes the random propagation. This hypothesis leads to the mathematical model of random crack propagation. The differential equation with stochastic coefficients describes the randomly propagating crack, which is analogous to the forced Brownian motion equation. Examples of solutions to the stochastic Brownian motion differential equation are presented. This Chapter is the final section of the last part of the book, which studies the life cycle of springs.