An analytical model for stamped ring wave springs is proposed, based on the theory for circular beams loaded normal to the plane of curvature. The model profits from the adoption of Castigliano’s theorem as a tool for mathematical treatment. Closed-form expressions for both load4eflection and load-stress relationships are presented. Because of the particular shape of the spring in the undeformed configuration, the load-deflection curve is found to be appreciably bilinear in character. A similar but less pronounce4 behaviour is displayed also by the relationship between load and internal stresses. The analytical results are compared to earlier theoretical findings and are shown to correlate well with experimental measurements.
1 INTRODUCTION
In many mechanical installations where radial and/or axial space is limited, wave springs represent a valid alternative to more traditional springs (e.g., helical or Belleville) to provide appreciably large deflections or preloads. Representative applications of these springs include such devices as pneumatic clutches, bearing supported gears, hydraulic cylinders, and adjustable-speed belt drives. Wave spring washers either can be fabricated by stamping or can be formed by wire bending (I)?. These two manufacturing processes lead to the basic configurations illustrated in Fig. l(a) (closed-type) and (b) (opentype), respectively. Over their stamped counterparts, edge-wound wave springs have an advantage in the capability for being coiled with multiple turns in prestacked configurations. So, they can be easily piled up in series without the need of special locating devices. Nevertheless, since the stamping method is by far more common, the following development relates strictly to closed-type wave springs. A detailed study of open-type washers is deferred to a future paper. A number of theoretical models for wave springs is found in the engineering literature. Nikitina (2) proposes an empirical expression, derived from statistical processing of comprehensive laboratory data, correlating the spring thickness to its load capacity. Wells (3) analyses stress and deflections occurring in a wave spring by means of equations based on the theory for simple beams. The spring portion between two adjacent lower crests of the wave is likened to a rectilinear beam with fixed ends, loaded at the centre of the span. Well’s assumption forms the premise of the pragmatic design procedure suggested in wave spring producers’ catalogues. A refinement of this model was put forward by Pinskii (4) who described a wave spring washer as a circular ring loaded normal to its plane of curvature by forces located at the wave crests. Torsional effects are neglected in the calculation of stresses and deflection. A thorough finite element analysis of both stamped and The MS. of this paper was received at the lnstitution on 29 October 1987 and t References are given in the Appendix. JOURNAL OF STRAIN ANALYSIS VOL 23 NO 3 1988 0 IMechE 1988 accepted for publication on 10 February 1988 edge-wound wave springs has been carried out by Cannarozzi and Molari (5). Contrary to the above studies, their numerical results predict closely the bilinear behaviour typical of such machine elements (see section 2 below). Their investigation, however, appears lacking from a practical standpoint as it does not lend itself to be summed up in the form of expressions, diagrams, or tables of general validity and easy access. This paper addresses the development of an analytical model for wave spring washers, conceived for the purpose of combining theoretical rigour with ease of use

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