Elastic Hinges And Couplings

• Introduction
• The Model
• The Selection
• Postscript
• Further reading
• Related case studies

Introduction

Nature makes much use of elastic hinges: skin, muscle, cartilage all allow large, recoverable deflections. Man, too, designs with flexural hinges and elastic couplings: devices which connect or transmit load between components while allowing limited relative movement between them by deflecting elastically (Figure 10.1). Which materials are good for this?

The design requirements depend on the application. If the hinge carries no (or very little) axial load, then the objective is to maximize its movement, that is, the curvature to which it can be bent without damage (Table 10.1, Option (a)). But if it also carries axial loads, these impose a further constraint and the requirements change (Option (b)).

Figure 10.1 An elastic or 'flexure' hinge. The ligaments must bend repeatedly without failing.

Table 10.1 The design requirements

The Model

Consider the hinge for the lid of a box. The box, lid and hinge are to be molded in one operation. The hinge is a thin ligament of material which flexes elastically as the box is closed, as in Figure M10.1(a), but it carries no significant axial loads. Then the best material is the one which (for given ligament dimensions) bends to the smallest radius R without fracturing or failing by fatigue.

When a ligament of thickness t is bent elastically to a radius R, the surface strain is

,

(M10.1)

and — since the hinge is elastic — the maximum stress is

.

(M10.2)

This stress must not exceed the modulus of rupture, σ_MOR, or the endurance limit σ_e, or the fracture strength σ_fr, whichever is least. We shall use the symbol σ_f to represent the failure stress, by which we mean the least of σ_y, σ_MOR, σ_e and σ_fr. Then the radius to which the ligament can be bend without damage is

.

(M10.3)

The best material is the one that can be bent to the smallest radius, that is, the one with the greatest value of the index

.

(M10.4)

We have assumed thus far that the hinge-thickness, t, is dictated by the way the hinge is made. But in normal use, the hinge may also carry repeated axial (tensile) forces, F, due to handling or due to the weight of the box and its contents. This sets a minimum value for the thickness, t, which is found by requiring that the tensile stress, F/tw (where w is the hinge-width) does not exceed the strength limit σ_f :

Substituting this value of t into equation (M10.3) gives

.

(M10.5)

and the second index

.

(M10.6)

The Selection

Both criteria involve ratios of σ_f and E. By way of example, we identify σ_f with the modulus of rupture, σ_MOR. We therefore create a chart of σ_MOR plotted against E (Figure 10.2). Candidates for unstressed hinges are identified by plotting M₁ as a line of slope 1; it is shown at the position M₁ = σ_f / E = 0.03. The best choices for the hinge are all polymers. The short list (Table 10.2) includes polyethylenes, polypropylene, nylon and, best of all, elastomers; these may be too flexible for the body of the box, but are good for the coupling. Cheap products with this sort of elastic hinge are generally molded from polyethylene, polypropylene or nylon. Spring steel and other metallic spring materials (like phosphor bronze) are less good by the criterion M₁.

Figure 10.2(a) A chart of endurance limit σ_e against modulus E, enabling selection of materials with large values of M₁.

Figure 10.2(b) A chart of endurance limit σ_e against modulus E, enabling selection of materials with large values of M₂.

Figure 10.3(a) The same chart as Figure 10.2(a), showing M₁, using the Polymers record subset

Figure 10.3(b) The same chart as Figure 10.2(b), showing M₂, using the Polymers record subset

Steel, bronze and beryllium-copper come into their own when axial loads exist. Figure 10.2(b) shows M₂, positioned at M₂ = 5; the metals are now better than the polymers.

Which index should we now use? It depends on the magnitude of F. When F is very small, the hinge thickness required to carry it is less than that (t*) set by processing; then the first index, M₁, is the one we want. When F is large, it is M₂. The coupling between them — the intersection of the M₁-M₂ grids — is found by equating the two equations for R, (equations (M10.3) and (M10.5)), using the process-limited minimum value of the thickness t*, in equation (M10.3). This gives what we call the coupling equation:

Specified F, t* and w (determined by the box dimensions) give a value for σ_f* at which the indices M₁ and M₂ intersect; the best choice of material is that which lies along both lines. So, for small F, the original index was the right one; only for large F do we switch to the second.

Table 10.2 Materials for elastic hinges

Postscript

Polymers give more design-freedom than metals. The elastic hinge is one example of this, reducing the box, hinge and lid (3 components plus the fasteners needed to join them) to a single box-hinge-lid, molded in one operation. The elastomeric coupling — a flexible universal joint (Figure M10.1), is another: it allows high angular, parallel and axial flexibility with good shock absorption characteristics and low cost. When axial loads are small, polymers are the best choice; but when axial loads become large, metals are better — they combine usable σ_e/E with high E, giving flexibility with good positional stability (as in the suspensions of relays).

Further Reading

'Tribology Handbook' (1973) Ed. Neale, MJ, Butterworths, London, Section C6.

'Elsevier Materials Selector' (1991) Eds. Waterman, NA and Ashby, MF, Elsevier, Amsterdam, Section 1.5.3, p. 388 et seq.

Related Case Studies

1. Materials for Springs

2. Materials for Static Seals

3. Diaphragms for Pressure Actuators

4. Knife Edges and Pivots

5. Materials to Resist Fracture