Meccanica 36: 525–553, 2001. c  2002 Kluwer Academic Publishers. Printed in the Netherlands. On the Optimal Design of Composite Material Tubular Helical Springs M. GOBBI and G. MASTINU Politecnico di Milano, Dipartimento di Meccanica, Via G. La Masa 34; 20158 Milan, Italy (Received: 17 October 2000; accepted in revised form: 3 October 2001) Abstract. A method is presented for the design of helical springs, with particular reference to those made from composite material and having a hollow circular section. Given the technical specifications, (e.g. stiffness, max- imum deflection, ...), the method allows to define the spring geometrical and mechanical parameters in order to get the best compromise among spring performances (minimum mass, maximum strength, ...), with constraints on local and global stability, on resonance frequency, ... The method is based on Multi-Objective Programming (a branch of Operations Research), which provides a theoretically correct way for defining the values of many design variables when many objectives (performance indexes) have to be taken into account. Mathematical mod- els are developed for describing the mechanical behaviour of the spring. The models have been validated with satisfactory results. The design solutions coming from the application of the method suggest the best parameter setting for obtaining the desired spring performances. Sommario. E’ presentato un metodo per il progetto di molle ad elica, con particolare riferimento a quelle a sezione tubolare, realizzate in materiale composito. Il metodo consente, assegnati i dati di progetto (rigidezza, massima deflessione in corrispondenza del massimo carico, ...), di definire i parametri della molla, assicurando le migliori prestazioni (minima massa, massima resistenza, ...), vincolando il valore della prima frequenza propria di risonanza, garantendo la stabilit` a locale e globale ed il rispetto di altre specifiche tecniche. Il metodo ` e basato sulla Programmazione a Molti Obiettivi (una branca della Ricerca Operativa), e rappresenta un modo teoricamente corretto di trattare problemi definiti da molte variabili di progetto e molti obiettivi. Sono sviluppati e verificati sperimentalmente alcuni modelli matematici descriventi differenti aspetti del comportamento meccanico della molla ad elica di compressione. Applicando il metodo di progetto ` e stato determinato l’insieme dei parametri che ottimizza le prestazioni di una molla. Key words: Helical spring, Hollow circular section, Composite material, Optimisation, Machine design, Machine elements. 1. Introduction Helical springs are usually made from a metal wire of circular cross section. The adop- tion of both a hollow circular section and composite material could reduce the spring mass substantially 1 . The design of a composite material tubular helical spring is not straightfor- ward. The definition of many parameters (design variables) is required in order to get both the desired technical specifications and the best compromise among spring performances. In other words, given the technical specifications, (e.g. stiffness, maximum deflection, max- imum acting load ...), the designer is charged to find the spring geometrical and mechanical parameters (diameter of the helical coil, number of coils, number of laminae of the tubu- lar element, orientation of fibres...) in order to gain the best compromise among the many 1 In fact a hollow circular section is particularly suited for torsional loads and a tubular shell element can be easily made from composite material.