From: Charles R Patton charles.r.patton@........

Date: Sat, 05 Jul 2008 09:23:51 -0700

Hi Brett, I tried to send this yesterday, but think I messed up the addressing, but it gave me a chance to correct a couple of errors. Question: Is a Lehman geometry rolling pivot inherently unstable? Discussion: Assume: 1) That the Lehman is constructed in a typical “garden gate” fashion with a horizontal main beam with rolling pivot and a suspension wire to the pivot bearing. 2) At the point of the rolling pivot, the wire does not bend. I.e., the wire/pivot may be considered rigid in that area. This constraint will hold true if the wire rigidity is greater than the torque required to roll the pivot. Something I believe is a reasonable constraint/assumption. So, if the Lehman is adjusted to a long period, a very small change in the geometry will lead to big changes in period/stability. In particular, with the suspension wire version it seems to me that as the beam moves from a centered position, the effective length of the suspension wire increases due to the movement of the pivot contact point around the diameter of the pivot rod along with the contact point moving sideways along the line of contact. Making it simple to do the mind experiment imagine going through 90 degrees. The wire lengthens by ½ the rod diameter. The contact point moves sideways by ¼ x pi x dia or approx 0.78 dia. So as the pivot rotates, the wire length starts to lengthen by 0.5 dia, which is the condition for stability. Now the lower beam pivot does the same thing but acts in the direction to shorten it by 0.5 dia. The sideways motions do not cancel but lead to increased rotation of the gate. By definition, the wire is at an angle to the beam, so that means that in all real constructions, the lower beam shortening effect is larger than the suspension wire effect (cos(wire/beam angle) x 0.5), but both effects lead to a lowering of the bob as it moves sideways. Energy constraints say that the bob wants to go to the lowest potential energy (flopping). This does not consider the additional effects that the gate is twisting due to the rolling pivots. So is this a possible explanation of the difficulty many people describe in trying to adjust a Lehman? I pose this to you since you’ve always been good with the mechanical simulations to see if I’m way off in this conjecture. An added question is this: What is the effect of the moving effective pivot point in a flexture pivot? Since a standard rectangular shim (or rod/wire) flexure point cannot not have a fixed point due to the stress/strain relationships as the flexure bends and the weight shifts to a side load the point would move back (?) towards the upright on the upper wire/beam and probably towards the support (again shortening the beam and lowering the bob) on a lower tension type flexure. Am I all wet --a welcome condition on a hot, hot 4th of July? (I started this email on the 4^th and did some corrections on the 5th) Comments welcome. Regards, Charles R. Patton __________________________________________________________ Public Seismic Network Mailing List (PSN-L)