Continuous springy beams

by Russ Elliott

News from London Road Models: At Railex 2008, I had a first look at some CSB components recently introduced by London Road Models. The fulcrum and bearing carrier tag etch is designed to accompany LRM's cast-brass hornguides and 3/32" square bearing blocks, and produce units similar to the High Level arrangement. Details of London Road Models products can be found here, and LRM has produced a very useful pdf instruction sheet for their CSB/springing system.

Introduction

A continuous springy beam supports several hornblocks and has more than one span. The use of a continuous springy beam in a chassis or drive bogie has the practical advantage of being changeable, so that an optimum ride characteristic can be achieved.

The way in which the continuous beam bends however is significantly different compared to the use of separate springy beam sections for each hornblock. Although in both cases there is no constraint on the rotation of the beam about the fulcrum pivot points, the deflection of a hornblock in the case of the continuous springy beam influences the deflection of the adjacent one, and therefore a degree of longitudinal equalisation is present.

Diagram showing the forces of reaction of the hornblocks deflecting the beam against the weight forces of the chassis and body imposing on the fulcrum points. Owing to the influence of the bending moments at the intermediate fulcrum points on the middle hornblock, the span for the middle axle hornblock needs to be larger than that of the outer spans if the loading on the 3 hornblocks is to be kept reasonably equal.

The downloadable Excel spreadsheet (29KB) for the 3-axle case, produced by Roger Wyatt, analyses the continuous beam situation by turning the chassis upsidedown. The spreadsheet uses an iterative method to determine the optimum combination of beam deflection, fulcrum position points and beam diameter. This is an initial version of the spreadsheet, and updates of it will be notified here.

Ted Scannell's experimental sideframe for a class 31 drive bogie. Handrail knobs act as the fulcrum points for the continuous beam and as the loading point for the hornblocks. It is important that the beam is a loose fit in the handrail knobs. For this frame, the middle axle hornblock handrail knob is packed to cater for the smaller diameter middle axle wheel of the prototype. The positions of the handrail knobs are now being altered to produce a more equitable loading of axles.

photo courtesy Ted Scannell

Some worked examples

Note: The following examples apply irrespective of overall weight and spring diameter. The diagrams do not take into account the effect of any unsprung mass (gearboxes or motors).

If you want a CSB plot for a particular wheelbase or chassis, please feel free to contact me.

Symmetrical 2-axle case For the symmetrical 3-fulcrum 2-axle case, deflections will be equal for any axle positions symmetrically positioned about a centrally-located fulcrum. Within the limits reasonably imposed on the position of the outer fulcrums, the position at which the deflection of the beam is maximised is shown in the diagram.

Symmetrical 3-axle case There are many solutions* to the 4-fulcrum 3-axle configuration, and the adjacent diagrams show examples of the model for the symmetrically-spaced, symmetrically-weighted, 3-axle case. For good pitch stability, it may be desirable to reduce the springrate of the middle axle of the 3-axle configuration.

* strictly speaking, an infinite number of solutions

In this example, the length of the span between the inner fulcrum points is:
2x + 2(0.37x) - 2(0.82x) = 1.1x

In this example, the length of the span between the inner fulcrum points is:
2x + 2(0.423x) - 2(0.854x) = 1.138x

In this example, the length of the span between the inner fulcrum points is x.

Stanier 8F Two plots are shown for the driving axles of this 2-8-0 engine, which has a 5'6" + 5'6" + 6'3" driving axle wheelbase. The first (upper) plot has the end supports a little too close to their axles, and the second (lower) plot slackens off the springrates on the middle two drivers, thus enabling a corresponding better spread of the end supports. The middle two axles deflect slightly more (approx 8%) than the outer ones.

The centre of gravity has been assumed to give a symmetrical loading over the group of the 4 driving axles: in practice, the centre of gravity will need to be somewhat in advance of this point, as the pony truck will need to carry a proportion (comparatively small) of the loco weight, but the drawbar pull effect will bring the centre of gravity backwards somewhat to restore the loading close to the optimum state assumed by this fulcrum plot.

All dimensions in millimetres (for 4mm scale).

Pannier The common Churchward/Collett large Pannier 0-6-0 has a wheelbase of 7'3" + 8'3".

The plot assumes a prototypical centre of gravity, but the middle axle deflection is approximately 10% greater than the outer ones. (The static loading proportion is approx 34% on each outer axle and approx 31% on the centre axle.) The frontmost fulcrum point has been positioned to avoid the front axle brake hangar pivot.

V1/V3 A common Gresley wheelbase was 7'3" + 9'0".

The plot assumes a prototypical centre of gravity, but the middle axle deflects approximately 10% more than the outer ones.

Midland 8' + 8'6" A common Midland/LMS wheelbase was 8' + 8'6".

The plot loads all axles equally.

Hudswell Clarke 6'3" + 7' Small industrial tank locos tend to have relatively short wheelbases compared to the overall loco length. There is therefore usually enough room for the outermost fulcrum points to have an adequate distance from their respective axles.

The plot loads all axles equally.

Black 5 The LMS Black 5 has two driver wheelbase variants: 7' + 8', and 7' + 8'3".

The plots load the axles to within 2% of each other. The front fulcrum point is close to the front brake hangar, but Black 5 brake hangars are pivotted at about axle datum height, and the CSB axis is sufficiently above the axle datum to avoid the brake hangar. The rearmost fulcrum point will probably clash with the frame spacer in some kits in this area.

Load distribution relationship to deflection relationship

Load is directly and inversely proportional to the deflection, e.g. a middle axle deflection 6% greater than each of the outer axles lessens the middle axle load compared to the load on each outer one by 6%. The proportion of each load as a percentage of the total can be calculated: if a is the reaction force (i.e. the load) on each outer axle, and b is the load on the middle axle, then a = 1.06b, and the sum 2a + b = 100%. This sum can be expressed in terms of b alone, i.e. 2(1.06b) + b = 100%, i.e. b = 32% of the total loading and a is therefore (100 - 32)/2 = 34% of the total loading.

Effect of moving fulcrum points on springrate

Slight variations in the positioning of fulcrum points can have dramatic implications for springrate. The following diagrams illustrate comparative deflections in an example of a symmetrically-weighted 3-axle symmetrical wheelbase of 6'6" + 6'6" (26mm + 26mm).

The first deflection plot has the inner fulcrum points midway between their respective axles, and with the outer fulcrum points set 13mm outside the outer axles. This configuration makes the middle axle deflect a lot less less than the outer axles, in this case by a factor of 5.5. Such a setting for a middle axle would make it far too strong, and the chassis would porpoise.
The second deflection plot increases the middle span by 2mm. The deflections of the outer axles are still nearly 2.5 times the deflection of the middle axle, which is therefore still far too strong. Note as the middle span increases, the axle locations get further away from the point of maximum deflection.
The third deflection plot increases the middle span by a further 2mm compared to the previous configuration. Here, the deflections of the outer axles are 25% more than the deflection of the middle axle. This is still not enough to prevent porpoising.
By increasing the middle span by a further 2mm, to 32mm, the deflection characteristic has inverted, with the middle axle deflection now nearly 46% more than the outer axles. This middle span dimension is more than we want for a more equitable loading of the axles.
Setting the middle span to 31mm produces a desirable deflection characteristic, with all axles deflecting reasonably equal, and with the middle axle slightly softer than the outer axles by approximately 8%.
The final deflection characteristic is included to show how sensitive the positions of the inner fulcrum points are. This middle span (of 31.2mm) makes the middle axle 15% softer than the outer ones.

In the above cases, where the outermost fulcrum points are kept at the same positions, it can be seen that lengthening the middle span will increasingly shift the outer axle deflections away from the point of maximum deflection of that part of the beam. This is not necessarily a drawback, but a better characteristic, utilising nearer the maximum deflection of the outer span, can be achieved by pulling in the outermost fulcrum points, as in the following example, which has the middle axle 8% softer than the outer ones:

From this plot can be derived another general case for the symmetrical 3-axle wheelbase:

to be continued

April 2002
symmetrical 3-axle worked example added July 2003
8F and Pannier examples added November 2003
symmetrical 2-axle case added January 2007
V1/V3 plot added July 2007
Midland standard plot added December 2007
Section on effect of moving fulcrum points on springrate plus another general symmetrical 3-axle case, plus explanation of load distribution to deflection relationship, added March 2008
Hudswell Clarke plot added April 2008
LRM CSB instruction sheet reference added 1 June 2008
Third symmetrical 3-axle case added 6 June 2008
Black 5 plots added 22 June 2008

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