Optimal structural design of helical springs with Ludwik-type elastic–plastic materials

This paper is a comparative between two springs such as helical and wave spring. Wave springs are precise flat wire compression springs that fit into assemblies that other springs cannot. These are used as an alternative spring for helical spring. Wave springs provide 50% reduction in spring height and axial space. They possess the same force and deflection as coil springs. They have reduced material requirements. They provide improved cost reduction. The Wave Spring has been subjected to Compression Test, Modal Analysis and Equivalent Elastic Strain Test and then compared to Helical Spring which again was subjected to the above same tests under the same conditions and parameters. The design of helical spring and wave spring has been done in CREO Parametric 4.0 and analysed in ANSYS R 18.1. The results are then compared.

An optimization algorithm based on SLSQP solver was created to design helical compression springs. The goal is to reach the required behavior even when the standards formula is inaccurate. Indeed, it was already shown in previous works that the classic formula determining the global load-length behavior of the spring is not always accurate enough because it does not consider the effects of the spring’s ends, in particular for springs with low index and low number of coils. Based on a new tri-linear model, this algorithm finds optimized spring design which respects the required operative points. In order to test the algorithm, the first tests were carried out to optimize the design of a PLA 3D-printed spring. It was shown that the initial design stiffness is overestimated by over 28% by the formula extracted from the standards. Then, the design of the spring was implemented in the algorithm to optimize the pitch and the number of active coils. The experimental curve of the optimized design spring goes through the target behavior, with a non-linearity at the beginning of the deflection. The experimental error about the final stiffness is successfully reduced to 1.1%. It is hoped that this work will provide a valuable progress to assist engineers and manufacturers in helical spring design.

Studied the main theory about helical cylindrical spring design, based on the existing theory,found the parameters that affect helical spring's stiffness, size, stroke, and also how much they effect the parameters. Draw the curve which can be a reference for helical cylindrical spring design, put forward a method of cylindrical helical spring design, which is according to the installation dimensions and working stroke and applicable to the design of cylindrical helical spring when is given workspace and stroke. Provided a new reference for the design of cylindrical helical spring, with important significance to engineering design.

A helical spring is a critical part of an independent automobile suspension system. Helical coil-spring is subjected to stress as a result of tire excitation at contact with an uneven road surface. Design of helical spring is very vital in the light of many design parameters in order to improve passenger comfort and car stability on roads. In this paper the most important of parameters have been taken into consideration in the proposed 3D design such as type of spring material, geometrical dimensions and stresses. The proposed design of suspension spring aims to improve the kinetic energy storage capacity, expressed with deformation of the spring when tires are moving on the road surface in order to increase passenger comfort. This has been done to achieve a better safety factor while reducing the weight of a spring made of nanomaterials. In addition, comparisons have been made between the operation accuracy of the two common different FEA packages under fatigue test conditions. The results showed differences due to changing design parameters as well as the types of FEA software packages used.

Design of helical springs used in valvetrains of high revolving internal combustion automotive engines is analyzed and optimized. The object of study is elastic nonlinear helical spring with variable non-circular wire cross-section. Variable radius and pitch angle of the helix are also taken into account. The design problem is to optimize the spring resonance characteristics under the geometrical constraints and the constraint on the maximal stresses. The static loading of the spring is modeled to calculate the nonlinear stiffness matrix. A variational method is employed to solve the static problem for the helical spring under stepwise external loading. A nonlinear condensation procedure is suggested for dynamic solution instead of full simulation to make the numerical procedure more time-effective. Dynamics of the spring is compared to experimental data for several industrial springs showing good matching. It is demonstrated that as the result of the suggested optimization procedure the dynamic characteristics of industrial valve springs can be improved up to 10% while controlling the maximum stresses.

Failure analysis of a helical compression spring with relatively low spring index

Shape memory alloys are smart materials which have remarkable properties that promoted their use in a large variety of innovative applications. In this work, the shape memory effect and superelastic behavior of nickel-titanium helical spring was studied based on the finite element method. The three-dimensional constitutive model proposed by Auricchio has been used through the built-in library of ANSYS® Workbench 2020 R2 to simulate the superelastic effect and one-way shape memory effect which are exhibited by nickel-titanium alloy. Considering the first effect, the associated force-displacement curves were calculated as function of displacement amplitude. The influence of changing isothermal body temperature on the loading-unloading hysteretic response was studied. Convergence of the numerical model was assessed by comparison with experimental data taken from the literature. For the second effect, force-displacement curves that are associated to a complete one-way thermomechanical cycle were evaluated for different configurations of helical springs. Explicit correlations that can be applied for the purpose of helical spring's design were derived.

The basic information required to utilize one of possible computation tools/algorithms (mainly the evolution strategy) to solve a wide class of real practical engineering optimization problems is presented and discussed in the present paper. The effectiveness of the considered method is demonstrated by the possibility of the use of different form of objective functions, various and numerous nonlinear constraints and different types of design variables (continuous, discrete, real, integer). The sensitivity of the algorithm to the choice of the evolution strategy parameters is also discussed herein. The generality of the evolution strategy is illustrated by the analysis of three examples dealing with: the design of helical springs, the buckling of cylindrical composite panels and the buckling of pressure vessels with domed heads.

Helical springs are indispensable elements in mechanical engineering. This paper investigates helical springs subjected to axial loads under different dynamic conditions. The mechanical system, composed of a helical spring and two blocks, is considered and analyzed. Multibody system dynamics theory is applied to model the system, where the spring is modeled by Euler–Bernoulli curved beam elements based on an absolute nodal coordinate formulation. Compared with previous studies, contact between the coils of spring is considered here. A three-dimensional beam-to-beam contact model is presented to describe the interaction between the spring coils. Numerical analysis provides details such as spring stiffness, static and dynamic stress for helical spring under compression. All these results are available in design of helical springs.


 
 
 
 Optimization is one of the important process in solving engineering problems. Regrettably, there are numerous problems in practical optimization that cannot be solved flawlessly within reasonable computational effort. Thus, metaheuristic approach is often useful to get near-optimal solution when the best solution is not achievable. This paper demonstrates the usefullness of a metaheuristic algorithm called single-solution simulated Kalman filter (ssSKF) in helical spring design, which is an example of structural engineering design problem. The ssSKF is a single agent-based optimization algorithm based on the Kalman filtering. The solution obtained by the ssSKF is compared againsts the genetic algorithm, co-evolutionary particle swarm optimization, co-evolutionary differential evolution, bat algorithm, and artificial bee colony.
 
 
 

In the design of helical springs where the load, deflection, allowable stress, and material are specified, there are an infinite number of solutions. In this paper, equations and graphs are presented for the selection of a spring index that will result in a spring of minimum weight, volume, or length. If, an addition to these requirements, the inside or outside diameter of the spring is fixed, there is only one solution. Equations and graphs are included for the selection of the spring index which will satisfy this additional requirement.

Preface. About the Authors. Foreword. Prologue. Part I: Fundamentals. 1. Sampling Plans. 1.1 The 'Curse of Dimensionality' and How to Avoid It. 1.2 Physical versus Computational Experiments. 1.3 Designing Preliminary Experiments (Screening). 1.3.1 Estimating the Distribution of Elementary Effects. 1.4 Designing a Sampling Plan. 1.4.1 Stratification. 1.4.2 Latin Squares and Random Latin Hypercubes. 1.4.3 Space-filling Latin Hypercubes. 1.4.4 Space-filling Subsets. 1.5 A Note on Harmonic Responses. 1.6 Some Pointers for Further Reading. References. 2. Constructing a Surrogate. 2.1 The Modelling Process. 2.1.1 Stage One: Preparing the Data and Choosing a Modelling Approach. 2.1.2 Stage Two: Parameter Estimation and Training. 2.1.3 Stage Three: Model Testing. 2.2 Polynomial Models. 2.2.1 Example One: Aerofoil Drag. 2.2.2 Example Two: a Multimodal Testcase. 2.2.3 What About the k -variable Case? 2.3 Radial Basis Function Models. 2.3.1 Fitting Noise-Free Data. 2.3.2 Radial Basis Function Models of Noisy Data. 2.4 Kriging. 2.4.1 Building the Kriging Model. 2.4.2 Kriging Prediction. 2.5 Support Vector Regression. 2.5.1 The Support Vector Predictor. 2.5.2 The Kernel Trick. 2.5.3 Finding the Support Vectors. 2.5.4 Finding . 2.5.5 Choosing C and epsilon. 2.5.6 Computing epsilon : v -SVR 71. 2.6 The Big(ger) Picture. References. 3. Exploring and Exploiting a Surrogate. 3.1 Searching the Surrogate. 3.2 Infill Criteria. 3.2.1 Prediction Based Exploitation. 3.2.2 Error Based Exploration. 3.2.3 Balanced Exploitation and Exploration. 3.2.4 Conditional Likelihood Approaches. 3.2.5 Other Methods. 3.3 Managing a Surrogate Based Optimization Process. 3.3.1 Which Surrogate for What Use? 3.3.2 How Many Sample Plan and Infill Points? 3.3.3 Convergence Criteria. 3.3.4 Search of the Vibration Isolator Geometry Feasibility Using Kriging Goal Seeking. References. Part II: Advanced Concepts. 4. Visualization. 4.1 Matrices of Contour Plots. 4.2 Nested Dimensions. Reference. 5. Constraints. 5.1 Satisfaction of Constraints by Construction. 5.2 Penalty Functions. 5.3 Example Constrained Problem. 5.3.1 Using a Kriging Model of the Constraint Function. 5.3.2 Using a Kriging Model of the Objective Function. 5.4 Expected Improvement Based Approaches. 5.4.1 Expected Improvement With Simple Penalty Function. 5.4.2 Constrained Expected Improvement. 5.5 Missing Data. 5.5.1 Imputing Data for Infeasible Designs. 5.6 Design of a Helical Compression Spring Using Constrained Expected Improvement. 5.7 Summary. References. 6. Infill Criteria With Noisy Data. 6.1 Regressing Kriging. 6.2 Searching the Regression Model. 6.2.1 Re-Interpolation. 6.2.2 Re-Interpolation With Conditional Likelihood Approaches. 6.3 A Note on Matrix Ill-Conditioning. 6.4 Summary. References. 7. Exploiting Gradient Information. 7.1 Obtaining Gradients. 7.1.1 Finite Differencing. 7.1.2 Complex Step Approximation. 7.1.3 Adjoint Methods and Algorithmic Differentiation. 7.2 Gradient-enhanced Modelling. 7.3 Hessian-enhanced Modelling. 7.4 Summary. References. 8. Multi-fidelity Analysis. 8.1 Co-Kriging. 8.2 One-variable Demonstration. 8.3 Choosing X c and X e . 8.4 Summary. References. 9. Multiple Design Objectives. 9.1 Pareto Optimization. 9.2 Multi-objective Expected Improvement. 9.3 Design of the Nowacki Cantilever Beam Using Multi-objective, Constrained Expected Improvement. 9.4 Design of a Helical Compression Spring Using Multi-objective, Constrained Expected Improvement. 9.5 Summary. References. Appendix: Example Problems. A.1 One-Variable Test Function. A.2 Branin Test Function. A.3 Aerofoil Design. A.4 The Nowacki Beam. A.5 Multi-objective, Constrained Optimal Design of a Helical Compression Spring. A.6 Novel Passive Vibration Isolator Feasibility. References. Index.

: In this mathematical study on helical spring design, three basic types of load requirements are distinguished and treated individually: (1) the load at assembled height; (2) the load at minimum compressed height; and (3) the energy content of the spring. Conventional load and stress deflection formulas are modified by the replacement of dependent variables with independent values. The ratio of the final spring deflection over the working stroke is formulated to show the variation of the final stress with various required load-space conditions. Optimum design parameters are established to minimize the final operating stress value. Direct and simplified analytical design procedures are developed for round wire and rectangular wire compression springs. Also presented are nomographs for use as design aids and detailed numerical design examples. This study combines design characteristics and stress advantages of nested spring systems versus a single spring for equivalent load conditions.

Exact determination of the global tip deflection of both close-coiled and open-coiled cylindrical helical compression springs having arbitrary doubly-symmetric cross-sections

The helical spring is a widely used element in suspension systems. Traditionally, the springs have been wound from solid round wire. Significant weight savings can be achieved by fabricating helical springs from hollow tubing. For suspension systems, weight savings result in significant transportation fuel savings. This paper uses previously published equations to calculate the maximum shear stress and deflection of the hollow helical spring. Since the equations are complex, solving them on a computer or spreadsheet would require a trial-and-error method. As a design aid to avoid this problem, this paper gives nomograms for the design of lightweight hollow helical springs. The nomograms are graphical solutions to the maximum stress and deflection equations. Example suspension spring designs show that significant weight savings (of the order of 50% or more) can be achieved using hollow springs. Hollow springs could also be used in extreme temperature situations. Heating or cooling fluids can be circulated through the hollow spring.