Dear Brett Nordgren,

    Thanks for your EMail about the hinge design.

>  As you rotate the cylinders, the total length of flexure material in 
>  contact with the cylinders remains constant.  Whatever unwinds from one, 
>  winds up on the other.  Then, because the force on the hinge is constant, 
>  the "s" curves will be exactly the same shape.....
>  Therefore there is no net change of the strain energy in the flexures as 
>  the cylinders rotate. Therefore there can be no moments resisting the 
>  rotation, or conservation of energy would be violated for the system.

    That was what I had concluded.
 
>  I'm pretty sure the same analysis works when the cylinders are of 
different 
>  sizes, but it requires a slightly more involved argument.

    The track of the intersection point of a fixed length belt around two 
cylinders must always lie on a circle. One cylinder will rotate with regard 
to the other in the ratio 2x their diameters. If you want a very small 
sideways movement, use a large fixed cylinder and a small rotating one. 

    The movement between the cylinder axes under changing lateral forces, can 
be fairly accurately calculated. It is the extension of bent beam with a 
directional load parallel to the tip section, with a correction allow for 
small changes in the position of the contact point of on the cylinder.

    There will be a minimum load which just keeps the flexing belt in contact 
around the cylinder - the natural shape of a crossed belt is a couple of 
'tear shapes'.

    If you have flexures with fixed ends, the curve of the intersection 
approximates to an ellipse - the lengths of the two crossover foils is 
constant, the bend points are 'fixed' and the sum of the distances on one 
side between the crossover and the fixing points must be constant - hence it 
is an ellipse. Then you have to add in the four beam bending corrections at 
the ends. For hinge design for seiesmographs, any pair of hinges should have 
the same geometry and similar, if not identical, loading per unit length. 

    Regards,

    Chris Chapman

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