# Bellville Washer Help



## native34 (May 8, 2015)

Does anyone know how I can calculate the pull force and throw for Bellville washers? I am in the process of designing a pull stud system for my Charter Oaks Mill, using the existing R8 spindle. I am designing the system with a bastardized style like the one David Decausin did with R8 taper and Cat-v Flange. I am having a problem trying to figure out how to calculate how many and what compression height I need for the washers. If anyone could shed some light on this subject it would be greatly appreciated.

Rod


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## RJSakowski (May 8, 2015)

Bellville washers add force when stacked the same way.  When stacked concave to concave and convex to convex, the force is reduced to 1/n where n is the number of washers.  The travel will be the (free height - thickness) for stacked the same way and n times the (free height - thickness) when stacked in opposition.  

See the McMaster Carr catalog for a description.


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## John Hasler (May 8, 2015)

native34 said:


> Does anyone know how I can calculate the pull force and throw for Bellville washers? I am in the process of designing a pull stud system for my Charter Oaks Mill, using the existing R8 spindle. I am designing the system with a bastardized style like the one David Decausin did with R8 taper and Cat-v Flange. I am having a problem trying to figure out how to calculate how many and what compression height I need for the washers. If anyone could shed some light on this subject it would be greatly appreciated.
> 
> Rod


http://www.engineersedge.com/belleville_spring.htm


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## randyc (May 8, 2015)

RJSakowski said:


> Bellville washers add force when stacked the same way.  When stacked concave to concave and convex to convex, the force is reduced to 1/n where n is the number of washers.  The travel will be the (free height - thickness) for stacked the same way and n times the (free height - thickness) when stacked in opposition.
> 
> See the McMaster Carr catalog for a description.



I thought that Belleville washers were similar to coil springs, i.e. the mechanical analog of resistors.  Series springs = parallel resistors and parallel springs = series resistors.  If that is true then the load at a fixed deflection for series washers would be the products of the various load characteristics divided by their sum or:

(F1 x F2 x F3 x Fn)/(F1 + F2 + F3 + Fn)  (Using this expression, one could estimate the load for washers with different spring constants, which is sometimes convenient.)

Of course I've been wrong many times and relying on a 70 year old's memory is a risky proposition at best


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## RJSakowski (May 8, 2015)

randyc said:


> I thought that Belleville washers were similar to coil springs, i.e. the mechanical analog of resistors.  Series springs = parallel resistors and parallel springs = series resistors.  If that is true then the load at a fixed deflection for series washers would be the products of the various load characteristics divided by their sum or:
> 
> (F1 x F2 x F3 x Fn)/(F1 + F2 + F3 + Fn)  (Using this expression, one could estimate the load for washers with different spring constants, which is sometimes convenient.)
> 
> Of course I've been wrong many times and relying on a 70 year old's memory is a risky proposition at best


I could be wrong as well.  If the force is proportional to the displacement, stacking n in opposition means each will have to displace 1/n times as far so the force required should be 1/n times as much. 
To quote McMaster Carr, "Springs stacked inverted increase the deflection of the spring by the number of springs in the stack while retaining the load of only one across the span of the stack."


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## randyc (May 8, 2015)

RJSakowski said:


> I could be wrong as well.  If the force is proportional to the displacement, stacking n in opposition means each will have to displace 1/n times as far so the force required should be 1/n times as much.
> To quote McMaster Carr, "Springs stacked inverted increase the deflection of the spring by the number of springs in the stack while retaining the load of only one across the span of the stack."



Yes, I understand, but that approximation only works if all of the washers have the same load/deflection characteristics.  That probably covers 80%-90% of applications and I suppose that's why the relationship was simplified (and proliferated, LOL).

In many applications, progressively increasing/decreasing force curves are desired and this requires using washers/springs of different load/deflection characteristics.  One may want a gradually increasing load curve - a shock absorber for example - so a number of washers/springs could be cascaded with progressively increasing load/deflection.

Determining the load/deflection curve wouldn't be possible using the simplifications published in McMaster Carr and other distributors of these devices.  I'm sure that the OP has no intention of designing a progressive load pull stud system, I just thought that a clarification might be helpful to anyone considering a project where non-linear load conditions might be desirable.

P.S.  When considering replacing the rear suspension on my '93 Sportster, I found that all of the after-market products that would improve handling used progressive springs.  There would be two ways to design these springs, the simplest probably being finite element analysis (FEA).

That's a viable method now since there are low-cost FEA applications for PCs - I use "LISA", for example, and I love it.  But if FEA is unavailable, the design technique would be to consider the progressive-wound spring as a number of springs of differing characteristics, stacked in series.

Combining the calculated spring characteristics of all the springs, an accurate load curve could be obtained.  The simplification found in catalogs clearly is not appropriate for applications like this.

Sorry if I'm being pedantic, I don't have anything else to do, LOL


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## RJSakowski (May 8, 2015)

randyc said:


> Yes, I understand, but that approximation only works if all of the washers have the same load/deflection characteristics.  That probably covers 80%-90% of applications and I suppose that's why the relationship was simplified (and proliferated, LOL).


I hadn't considered different spring constants in my evaluation.  Yes, if you take the more general case of differing spring constants, it is fairly easy to demonstrate that 1/k  = 1/k1 +1/k2 +....1/kn, where ki is the spring constant for each spring and k is the spring constant for the stack,  which is the same equation as the parallel resistor equation.  When the individual springs are all the same, this reduces to 1/k = n/kn or k = kn/n.


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## randyc (May 8, 2015)

RJSakowski said:


> I hadn't considered different spring constants in my evaluation.  Yes, if you take the more general case of differing spring constants, it is fairly easy to demonstrate that 1/k  = 1/k1 +1/k2 +....1/kn, where ki is the spring constant for each spring and k is the spring constant for the stack,  which is the same equation as the parallel resistor equation.  When the individual springs are all the same, this reduces to 1/k = n/kn or k = kn/n.



I guess I'm not the only one with nothing better to do, ha-ha.  Noting your post (regarding the sum of the reciprocals) and thinking about it for a moment, I realized that the expression I posted is valid only for the special case n=2 .... bad, bad me


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## brino (May 8, 2015)

Hi Rod,



native34 said:


> Does anyone know how I can calculate the pull force and throw for Bellville washers?



I dug out my copy of the Machinery's Handbook and marveled that there's so many pages on this. They say:

_Forces and stresses generated by compression depend on disc spring thickness much more than on any other dimensions. Standard DIN 2093 divides all disc springs into three groups in accordance with their thickness:
Group 1 includes all disc springs with thickness less than 1.25mm (0.0492 inch)
Group 2 includes all disc springs with thickness between 1.25mm and 6.0mm (0.0492 inch and 0.2362 inch)
Group 3 includes all disc springs with thickness greater than 6.0 mm (0.2362 inch)
There are 87 standard disc spring items, which are manufactured in accordance with Standard DIN 2093 specifications for dimensions and quality requirements._​
So much of what you want would be based on the washer dimensions and materials.
The book also seem to distinguish between those "without contact surface" (simple radius corner on the bottom) and "with contact surface" (load bearing surface or flats).

They do provide a table comparing four different formulas for calculating force versus measured values; but I do not want to reproduce the entire thing here due to copyright.

The book also discussed stacking in series (bottom-to-bottom; and top-to-top), parallel (nested, top-to-bottom) and parallel-series stacks.
Whew, it's a tough read....so many big formulas and graphs.....

I'd suggest either:
i) find a friend or library with a copy of the Machinery's Handbook, or
ii) perhaps email the sales-support contact for a Belleville washer vendor.

-brino


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## jererp (May 9, 2015)

I think that it's interesting to see how different people address the same problem from different viewpoints.  I understand electronic concepts by looking at the equivalent mechanical principles. Others understand mechanical concepts using electronic principles. This is what keeps me reading these posts. Always a fresh perspective on problem solving!


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## RJSakowski (May 9, 2015)

My go-to source for springs is Lee Springs.  If they don't have it, it probably doesn't exist.  The one drawback is the relatively high cost for low quantities but, even if you don't purchase from them, their engineering information is great as well as their CAD models.  They sell Bellville washers, although I have never bought any from them personally.


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## randyc (May 9, 2015)

RJSakowski said:


> My go-to source for springs is Lee Springs.  If they don't have it, it probably doesn't exist...



I emphatically agree.  Although I've been retired for fifteen years, during my working era, a Lee catalog was always in my bookshelf !


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