PSN-L Email List Message
Subject: Stability of a Lehman
Date: Wed, 20 Nov 2002 19:27:11 EST
Hello Charles Patton,
>> There have been numerous discussions about lengthening the period of a
Lehman beyond the approximately 10 sec period and it becomes "unstable."
Let's consider a 'big' 1 metre length boom. The 'natural' period is 2
so to get a period of 10 sec, we have to reduce g to g/25. This corresponds
to about 2.3 degrees off the vertical, lowering the mass by 40 mm below the
horizontal. To get 20 sec, we need 0.56 degrees. To get 30 sec we need 0.255
deg.... x60 = 15.3 minutes of arc.... This corresponds to lowering the mass
4.44 mm below the horizontal position, roughly 175 thou.
Now lets consider a small 25 cm boom, with a natural period of 1 sec. To
period of 10 sec, we now have to reduce g to g/100. This corresponds to 0.573
degrees off the vertical, lowering the mass by 2.5 mm below the horizontal.
To get to 20 sec, we need an angle of 0.143 deg. To increase the period to 30
sec, we need an angle of 0.0637 degrees, 3.8 minutes of arc, lowering the
mass by 0.28 mm below the horizontal, roughly 11 thou. This requires
considerable precision. The best that I can easily get with differential
thread supports is ~10 thou per turn.
>> The question is - unstable in what way? The period changes, the centring
fails, sensitivity goes haywire? In particular, if the answer is, "The
centring fails," does the centre just move around, or does the beam "flop" to
one side or the other?
You name it.... With the small suspension angles, a Lehman is very
tilt effects. The support needs to be very rigid. With very small restoring
forces, like g/900, the response becomes sensitive to any frictional and
springy effects in the suspension. The original Lehmans had a knife edge
suspension. Some even had a point suspension. The forces in these suspensions
are high enough to cause deformation of the contact points / edges. This can
give an erratic response such that no two successive swings are similar. This
can be avoided by using either rolling contacts, either a sphere on a flat,
or a cylinder on a crossed cylinder, or by using
flexing wire or foil suspensions.
>> I'm doing some thought experiments during my long commute, and I'm
thinking about the sources of error. In particular it's been mentioned before
that the centre of rotation changes on the flexible hinge designs. That's
intuitive, but how much and in what direction - toward stability or
instability? Has anybody figured out the actual path of the centre of
rotation for a typical Lehman?
In the single foil / wire hinge of the 'Cardan' type, the material must
not be so highly stressed that it can take on a permanent 'set'. The centre
of rotation will be quite close to the frame clamps. You can get erratic
effects with foils 'crinkling' if the two axes are not quite in line or not
parallel. With the crossed foil hinges clamped onto the edges of a plate, the
flex axis moves in an ellipse. Both of these types have a small spring
effect. In the zero force rolling contact type where the foil is essentially
wrapped in a figure of 8 around two cylinders, the flex axis moves in a
circle. The centres of rotation will be at the centres of these curves.
>> Crossed-X flex hinges were mentioned, but in this application with side
force, I don't think they would be any less susceptible to the same
I would expect crossed foil hinges to be ~completely rigid.
>> Another question is the upright's rigidity. Several pounds of weight at
the end of a boom is a fair torque moment which is resisted by the spring
constant of the 'pipe' vertical. How much does this constant vary with
temperature in standard steel?
From memory, Young's modulus varies by about -2x10^-4 / C Deg.
for steel. While this is not a lot, you don't need a lot to change the period
upset the balance. Professional seismometers of this type use a braced
upright post. Bracing the upright with either more tube or with L angle would
tend to prevent flexing problems.
The orientation of the horizontal arm needs to be rigid and any
resonances in the system should be suppressed. Some quick calculations
suggest that direct thermal expansion effects on the suspension are not very
Hope that this helps.
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