Continuous springy beams
by Russ Elliott
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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.
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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.
To allow rotation of the beam about the fulcrum pivot points, the beam is not constrained along its longitudinal axis by any additional longitudinal force, i.e. it is not fixed to any of the fulcrum pivot points nor is it fixed to the chassis. In the CSBs shown on this page, there are no additional rotational moments applied to the outermost fulcrum points.
Although the principle of using a single wire across several hornblocks is not new, there being a number of locomotives marketed in the USA in the 1950s with tension-assisted devices, Ted Scannell is regarded (from a CLAG perspective) as having the brainwave of making a spring beam continuous over two (and later, three, etc) axles. The first implementation of this type of continuous spring beam is thought to have been in Bill Bedford's coach bogies.
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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 each of the 3 hornblocks is to be kept reasonably equal.
The sum of the hornblock reaction forces equals the sum of the weight forces of the chassis/body.
In the CSBs shown on this page, there are no additional rotational moments applied to the outermost fulcrum points, i.e. the beam is not constrained to a particular angle at those points.
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Although the diagrams on this page show overslung CSBs, they can of course be implemented in underslung mode.
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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 (the 'Moment distribution method' *) to determine the optimum combination of fulcrum position points, beam diameter and beam deflection.
* There are other methods of analysing continuous beams, notably the 'moment-area method' and the 'method of superposition', but the differences in these methodologies are not within the scope of this page.
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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 here are incorrect. A more equitable loading of axles requires a larger middle span.
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photo courtesy Ted Scannell
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Some worked examples
Note: The following examples apply irrespective of overall weight and spring diameter. Unless otherwise indicated, the diagrams do not take into account the effect of any unsprung mass (gearboxes or motors).
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If you want a CSB plot for a particular wheelbase or chassis, please feel free to contact me. |
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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.
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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
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In this example, the length of the span between the inner fulcrum points is:
2x + 2(0.37x) - 2(0.82x) = 1.1x
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In this example, the length of the span between the inner fulcrum points is:
2x + 2(0.423x) - 2(0.854x) = 1.138x
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In this example, the length of the span between the inner fulcrum points is x
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Pannier The common Churchward/Collett large Pannier 0-6-0 has a wheelbase of 7'3" + 8'3". The plots assume a prototypical centre of gravity. The frontmost fulcrum point has been positioned to avoid the front axle brake hangar pivot.
In the upper plot, 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.)
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The lower plot has a more symmetrical rear axle span, and provides approximately 11% more deflection on the middle axle compared to the outer ones. If the 13.1mm and 19.4mm dimensions are changed to 13mm and 19.5mm respectively, the middle axle deflection is approximately 17% more than the outer ones.
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Collett standard The common Collett wheelbase for the majority of his larger 6-coupled engines was 7' + 7'9".
The plot loads the middle axle deflection approximately 3% greater than the outer ones - to all intents and purposes, these axles are equally loaded. |
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56xx This large tank has a wheelbase of 7'3" + 8'.
The plot loads the middle axle deflection approximately 5% greater than the outer ones - to all intents and purposes, these axles are equally loaded. |
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45xx This small prairie tank has a very short driver wheelbase of 5'6" + 6'.
The plot loads the middle driver axle approximately 8% softer than the outer ones.
The comparative shortness of the CSB exacerbates the criticality of the fulcrum widths. |
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LNER 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. |
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GER N7 This wheelbase was 7'6" + 8'9", and the plot is similar to the LNER V1/V3.
The plot loads the axles to be nearly identical. |
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GER J15 This wheelbase was 7'7" + 8'6". In this case, there is not much room between the rear driver and the end of the frame, and the rearmost fulcrum point has been pulled in to allow a feasible fulcrum point implementation.
The plot loads the axles to be nearly identical, within a few percent. |
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Midland 8' + 8'6" A common Midland/LMS wheelbase was 8' + 8'6".
The plot loads all axles equally.
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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. |
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Black 5 The LMS Black 5 has two driver wheelbase variants: 7' + 8', and 7' + 8'3".
The plots load the deflections of 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. |
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Ivatt 2MT The Ivatt 2MT Mogul has a driving wheelbase of 6'9" + 7'.
The plot unloads the centre axle deflection by about 5% less than the front and rear ones, so to all intents and purposes the axles are equally loaded.
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3.5mm scale continental 0-6-0 This small tank loco has a prototype wheelbase of 1900mm + 2000mm, with plenty of room fore and aft of the front and rear axles respectively for fulcrum location.
The plots unloads the centre axle deflection by approximately 14% compared to the outer ones. This results in the outer axles carrying about 35% of the overall sprung weight, and the middle carrying about 30% of the overall sprung weight. The builder intends to drive on the middle axle, and the comparatively high extent of the centre axle unloading is to make some allowance for the extra unsprung weight on that axle, namely a proportion of the weight of the motor/gearbox. For a small-ish 3.5mm scale tank engine, the unsprung proportion of the weight of the motor/gearbox is going to be significant in relation to the total weight.
For an all-up weight of between 150g and 180g, I suggest using 10 or 11 thou steel wire for the CSB.
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LSWR G6 0-6-0 This small tank loco has a prototype wheelbase of 6'10" + 7'5".
The plot increases the centre axle deflection by approximately 4% compared to the outer ones. This results in each of the outer axles carrying about 33.8% of the overall sprung weight, and the middle carrying about 32.5% of the overall sprung weight.
The overall weight of this loco is intended to be 220g, and at this weight a 0.013" diameter steel CSB will give 0.5mm deflection on the axles.
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LNER J72 0-6-0 This small tank loco has a prototype wheelbase of 6'8" + 7'.
The plot increases the centre axle deflection by approximately 9% compared to the outer ones.
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Class 59 bogie This 3-axle bogie has a prototype wheelbase of 6'7 ¼" + 6'11 ¾".
The frame length forward of the front axle of the bogie is restricted, and alternative plots are offered.
The plots load the axles equally.
For an overall loco weight of c 500g, a 0.013" diameter steel spring will give nearly 0.5mm axle deflection, 0.012" will give just over 0.6mm deflection, and 0.011" will give approx 0.85mm deflection.
An alternative, but very different, 'inside the wheelbase CSB', is given further down this page.
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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. |
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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).
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10-coupled locos The prototype BR Standard 9F 2-10-0 has a 5'5" + 5'5" + 5'5" + 5'5" driving axle wheelbase, and the WD 2-10-0 has driving axles spaced at 5'3". It is debatable whether a CSB is the optimum solution for a 10-coupled; the outermost fulcrum points can be close to their axles, and it is uncertain whether a CSB would give enough axle deflection unless a small diameter beam is used. The frictional effects over the many fulcrum points could become significant.
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A larger and more predictable deflection would be given by two pairs of equalising springy beams, one pair between the front pair of drivers and other pair between the rear pair of drivers, with a separate (cantilever beam or coil) spring on the middle driver. CSBs can be used instead of these springy equalising beams - the springing characteristic will be equivalent.
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Load distribution relationship to deflection relationship
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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.
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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.
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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 location of the maximum deflection of the outer spans becomes further away from the axle.
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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.
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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.
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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%.
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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.
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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:
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CSBs without outer fulcrums
All the above plots have outer fulcrums. The deflection of all the axles is therefore constrained by a similar principle - that the slope of the curve of the CSB at each axle is nominally zero. For some 3-axle drive bogies however, there is not always adequate length to have outer fulcrums. In theory, it is possible to have only two fulcrum points for a 3-axle case. The 2-fulcrum model for the 3-axle case is where the outer axles are deflected by the principle of the free cantilever. The deflection characteristic of the outer axles is therefore radically different compared to the case where outer fulcrums are present. As can be seen above for the outer fulcrum model, slight movements of fulcrum points have dramatic implications for springrate; these implications become acute where no outer fulcrums are present, because of the exceptionally high sensitivity of the equalisation.
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The diagram adjacent shows the general case for the near-equal loading of axles in the symmetrical 3-axle case.
In the context of an example of a 27mm + 27mm bogie (6'9" + 6'9" in 4mm scale), moving each of the fulcrum points 0.1mm outwards leads to a situation where the middle axle is 33% softer than the outer ones. Conversely, moving each of the fulcrum points 0.1mm inwards leads to the middle axle being 31% harder than the outer ones. The positioning of these fulcrum points is therefore exceptionally critical and arguably far too sensitive, given the practical constraints that real fulcrum points have finite widths, of up to an order of magnitude greater than 0.1mm.
I draw attention to the possiblity of CSBs without outer fulcrums only to dismiss them from further consideration.
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CSBs inside the total wheelbase
The first constraint in the CSB design is c, the clearance needed from the end of the CSB to the outer axle bearings/hornblocks.
The next condition is to choose a lever ratio for the rigid beams: if F is the force we want on each axle, the reaction force needed to balance the rigid beam at the point where it bears up against the CSB is Fd1/d2
This greater reaction force on the outer spans requires a different set of CSB inner fulcrum point positions (compared to a conventional CSB) design to keep the axle deflections consistent.
Typically, the relationship of d1 and d2 will be in the region of 1.5d2 = d1
The CSB length is 2(d1 - c)
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For some bogies, frame lengths forward of the front axle and rearward of the rear axle are often restricted, making the incorporation of fulcrum points outside the wheelbase difficult or impossible. It is feasible to bring the CSB's outer fulcrum points inside the wheelbase, and use two separate rigid beams to link each outer axle to the middle axle. Each rigid beam impinges up against the CSB at a chosen point along the length of the rigid beam.
The CSB length is short. The longitudinal distance between between the outer fulcrum and the rigid beam fulcrum point is extremely short. Constructionally, there is a degree of complexity in the design, and a non-CSB alternative using four springy equalisers might be considered to be a simpler strategy.
For 3-axle drive bogies of between 70g and 80g per axle weight, 0.009" or 0.010" diameter steel will give an axle deflection of approximately 0.5mm
The high degree of equalisation brings into question the '0.5mm' value of axle deflection - a greater deflection, perhaps of 0.75mm or more, given by a smaller diameter of spring, might be worth trying.
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6'9" + 6'9" This 'inside' CSB is 48mm long, with c being set at 3mm, and a d1 to d2 ratio set at 1.5. The plot loads the axles equally.
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Class 59 This 'inside' CSB is asymmetrical, for the prototype wheelbase of 6'7 ¼" + 6'11 ¾", with c being set at 3mm, and a d1 to d2 ratio set at 1.5. The plot loads the axles equally.
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Vertical adjustability
For a given diameter of spring and set of fulcrum points, the springrate for each span is set. The only ways of altering the ride height of a CSB loco are:
- to change the diameter of the spring;
- insert packing between the chassis and the loco body;
- change the weight of the loco body.
Changing the spring diameter will change the deflection of each axle. The change in deflection is proportional to the ratio of the wire diameters to the fourth power. For example, if a 0.33mm wire diameter is chosen to provide a nominal 0.5mm axle deflection, changing the wire to 0.36mm diameter will give a deflection of (0.33/0.36)4 x 0.5 = approximately 0.35mm. This lesser axle deflection will of course affect the chassis' ability to cope with undulating track - the harder the spring, the more the chassis will act as a 'fixed axle' one.
If you want to keep your nominal chosen axle deflection of say 0.5mm, and you do not want to alter the weight of the loco body, the only way to alter the ride height is to raise or lower the chassis in relation to the body. Lowering the body on the chassis is usually difficult, but raising it (with packing) is usually easy. It is therefore sensible to err on the low side, and reference should be made to the considerations in the vertical position of the fixed fulcrums on a CSB chassis.
Horizontal forces on the beam
As already stated, a CSB of the type described on this page is not intentionally constrained along its longitudinal axis by any additional longitudinal force, and, in operation, the beam will rotate around each fulcrum point and will move along the fulcrum point to a small degree. This movement is subject to the friction between the beam and the fulcrum point. Because the whole weight of the loco bears on the fulcrum points, the magnitude of this undesirable frictional force, which is proportional to the vertical reaction force at each fulcrum, can be considerable. These undesirable frictional forces can be minimised by keeping the fulcrum points lightly oiled.
to be continued
© Russ Elliott
first issued 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
Continental 0-6-0 plot added 10 September 2008
Adrian Cherry's RMweb 7mm Jinty build reference added 11 October 2008
Reference note to vertical positioning page added 16 October 2008
9F and WD 2-10-0 plots, and Collett standard plot, added 22 October 2008
Extra configurations for 10-coupled locos added 1 December 2008
Ivatt 2MT plot added 8 Febuary 2009
Section for 'CSBs without outer fulcrums' started 12 February 2009
LSWR G6 plot added 31 May 2009
Class 59 plot added 12 June 2009
'CSBs inside the total wheelbase' section added 8 July 2009
Typo amended 17 July 2009
Introduction expanded, and spreadsheet method clarified, 29 July 2009
Pannier variation added 31 July 2009
J72 and 56xx plots added 5 August 2009
N7 plot added 16 October 2009
J15 plot added 31 December 2009
45xx plot added 20 June 2010
sections on vertical adjustability and horizontal forces on the beam added 20 July 2010
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