“Introduction:

In elastic structure definitely vibrating in a mode, maximum stress is proportional to the maximum vibratory velocity. I can prove this clearly for bending stress in uniform beams and axial stress in rods. What Pm finding now is that many believe the stress velocity relationship goes well, beyond a single mode and simple structures. Velocity is a common vibration property to measure and monitor, and with very good reason. Its use is increasing and the topic deserves examination. If you are not already convinced, I want to convince you that stress equals pcv, where p is the mass density, c is the velocity of sound, and v is the maximum modal velocity. Chalmers and I discovered that [1] after several others. It has made a huge difference in my life. I remember Prof Carl Freberg, a consultant to our group saying, “Howie, I think you’ve got something there.” Stress velocity I do almost full time right now; I have to tell you about it. Its a little complicated to prove, but more and more people are becoming aware and incorporating it into their analyses. Most of the monitoring charts are in velocity.

The Monitoring guidelines that I have been taught for low frequency and journal bearings include relative displacement and seismic case or bearing housing measurement. You can’t allow the shaft to touch journal so that displacement of the shaft relative to the bearing must be monitored. In a smaller machinery situation, with rolling element bearings requires seismic acceleration and velocity measurement.

Velocity will have upper limits because it’s caused by stress which can only go so high. However the very high frequency impacting with ball bearings, gears require careful acceleration analysis; very high frequency weak acceleration signals emphasize high frequency. And while the consistent acceleration levels may not be cause for alarm, they carry information that can be interpreted to indicate machinery problems, often by their time history characteristics.

MAXWELL [2] tried to organize the monitoring problem, but mainly treated large journal bearing machinery. He says very little on rolling element bearing in his paper, and still doesn’t come to any firm conclusion. He tries to organize the monitoring decision according to low frequency, mid frequency and high frequency. His thinking being that low frequency would be a displacement situation and mid frequency for velocities. In the velocity region he explains that maximum strain energy occurs at maximum deflection and zero velocity, and that maximum kinetic energy occurs as the system is passing thru zero deflection. Equating the two energies shows that maximum stress is proportional to maximum velocity.”