Period of oscillation of a ring on a cylinder in gravity field
Period of oscillation of a ring on a cylinder in gravity field
(OP)
I'm having trouble calculating the oscillation period of a ring on a shaft. If I use the compound pendulum calculation it gives reasonable values if the shaft is very small compared to the ring inner diameter. Increasing the shaft size causes a shorter oscillation period experimentally (observing parts here at my desk) but my simple mathematical model doesn't correlate. I think the missing piece is related to slippage between the shaft and inner diameter of the ring. My math model assumes slippage as the ring moves about it's center of oscillation but the real parts do not slip. It appears that when the shaft and inner diameter of the ring are very close the period approaches that of a simple pendulum. Any help in calculating the period of this pendulum will be much appreciated.
Thank you to all in advance for your time and advice.
Thank you to all in advance for your time and advice.





RE: Period of oscillation of a ring on a cylinder in gravity field
RE: Period of oscillation of a ring on a cylinder in gravity field
RE: Period of oscillation of a ring on a cylinder in gravity field
I*omega^2/2 - m*g*(D-d)/2*cos(theta)= constant
where I is the moment of inertia of the ring about an axis at the center of the shaft, m is the mass of the ring, and theta is the angle of swing of the "pendulum". For a normal compound pendulum, omega is d(theta)/dt.
However, for this pendulum, omega is (1-d/D)*d(theta)/dt, so the usual solution must be modified slightly unless d is very small compared with D.
Hope my math is right.
RE: Period of oscillation of a ring on a cylinder in gravity field
m*((D-d)/2)^2*(d(theta)/dt)^2/2+I*(d(theta)/dt)^2*(1-d/D)^2/2 - m*g*(D-d)/2*cos(theta)= constant
where I is now the moment of inertia about the center of the ring.
The first term is the instantaneous translational KE of the ring as it pivots about the center of the shaft. The second term is the rotational KE of the ring about its mass center - which is less than it would be for a normal compound pendulum because of the rolling action - and a constant minus the third term is the gravitational potential energy.
Something like that anyway - I'm in too much of a hurry really. I expect someone else will correct me if I'm wrong
RE: Period of oscillation of a ring on a cylinder in gravity field
I'll study this for a bit and see if I can wrap my brain around it.
This is not a homework problem. I'm trying to construct an absorber to combat a torsional oscillation problem using a concept similar to pendulous absorbers in engine crankshafts. One difference here is the potential energy is stored in the height of the ring (against gravity) instead of the radius relative the the crank axis (against centrifugal force).
Any other thoughts? Thank you!
RE: Period of oscillation of a ring on a cylinder in gravity field
Cheers
Greg Locock
RE: Period of oscillation of a ring on a cylinder in gravity field
RE: Period of oscillation of a ring on a cylinder in gravity field
RE: Period of oscillation of a ring on a cylinder in gravity field
RE: Period of oscillation of a ring on a cylinder in gravity field
Simply increasing the machine mass only increased the oscillation period.
What I like so much about the pendulous ring is it's low cost and easy retrofit on the existing machinery. I have some geometry constraints that the ring concept fits well within.
Tuned mass torsional absorber using spring elements is also on the agenda. I think I'll revisit the friction damper as well if I can get it to be more sensitive. A viscous fluid damper sounds very attractive. I think I'll give it a try also but at this point I don't see how it could be done cheaply enough. This will have to be done in the 1-2 dollar range to be economically feasible. However, experimentally trying other concepts is certainly valid to learn what does and does not work.
RE: Period of oscillation of a ring on a cylinder in gravity field
RE: Period of oscillation of a ring on a cylinder in gravity field
An important consideration is how exactly you think the energy is going to be absorbed. I suspect it is air damping and a bit of friction at the interface - not very much power with the operating speed and size you give. Oil is a much better damping fluid than air.
Cheers
Greg Locock
RE: Period of oscillation of a ring on a cylinder in gravity field
RE: Period of oscillation of a ring on a cylinder in gravity field
another possibility - use a collar-mounted assembly of springs and weights arranged circumferentially about the shaft (think of a crankshaft TV damper with springs instead of rubber - more flexible, in other words)
RE: Period of oscillation of a ring on a cylinder in gravity field
T=2pi*sqrt[((R-r)/g)(1+(R2^2+R^2)/2R)]
r = hub or shaft diameter
R = inner ring diamter
R2 = outer ring diamter
I derived this based on a textbook example of a disk rolling inside a circular surface. The result is off by about 20 percent but the trend is correct. There is either an error in my derivation or in my desktop measurement. If I find the error I'll post. Also when I try the concept in reality I'll post the results.
Thank you again to all who took the time to help.