[attachment=8916]
Deflection of beams
Abstract
The basic aspects of Probe Beam Deflection techniques, used to detectconcentration gradients in electrochemical systems, are described.
The different techniques is first discussed. Then, a critical review of published applications of the techniques to various electrochemical systems is performed. Among the systems whom these techniques has been applied are: electrochromic oxides, conducting polymers, redox polymers, underpotential deposition, surface electrochemical reaction on solid electrodes, oscillatory phenomena, double layer charging and metal deposition.
Introduction
The deflection of a spring beam depends on its length, its cross-sectional shape, the material, where the deflecting force is applied, and how the beam is supported.
The equations given here are for homogenous, linearly elastic materials, and where the rotations of a beam are small.
In the following examples, only loads applying at a single point or single points are considered - the application point of force F in the diagrams is intended to denote a model locomotive hornblock (or vehicle axlebox) able to move vertically in a hornguide, and acting against the force of the spring beam fixed to or carried by the locomotive or vehicle mainframes. The proportion of the total weight acting on each axle of a loco or vehicle will depend on the position of its centre of gravity in relation to the axle (or the chassis fixing points of equalising beams where these are used).
Application to model locomotive hornblocks
As can be seen from the equations, the thickness of the material (h or d) is very critical, and hence the incremental sizes in the range of guitar strings available make them very attractive for use as spring beams. There is also a considerable difference in the deflection of a beam, for a given force, depending on how it is supported and fixed and whether it is supported at one end only or at both ends.
It is suggested that design should be based on a given deflection of a hornblock, and then determine what length, thickness and style of beam is most suitable for the specific force intended to be supported by each axle.
For locos weighted to be between 4 and 6 grams per prototype ton, the masses to be supported by each individual locomotive hornblock are likely to fall within the range 30 to 60 grams (equating to a prototype loading of between 14 and 20 tons per axle).
Choosing a deflection value
For reasonable 4mm scale finescale track, a recommended value for hornblock deflection, δ, under the final load of a locomotive, is 0.5mm.
The above recommendation is known to be an oversimplistic and possibly incorrect assumption on what the design value for the deflection should be, and has given rise to considerable debate. Any experience on applying this recommendation to real chassis modelling practice is welcomed - the purpose of this article is a starter for discussion rather than a conclusion of it
Deflection equations and diagrams
Note on diagrams and equations. The diagrams given here have been inverted from their normal textbook presentation, to reflect their application for model locomotive and vehicle axleboxes. However, whilst the equations for deflection have been kept consistent with their textbook presentation, the normal sign convention (+ or -, to indicate deflections in the vertical y axis from the beam datum line) has been ignored, as we are concerned here only with the absolute value of a beam deflection.