M. M. Khonsari
Professor
Dept. of Mechanical
Engineering
Louisiana State Univ.
Baton Rouge, La.
E. R. Booser
Engineering Consultant
Vero Beach, Fla.
Stick-slip chatter commonly shows up on
initial start of a machine when static friction
in the bearings temporarily restrains
shaft rotation. As the shaft then winds
up, torsional force rises until it exceeds
the restraining static friction. At this point, friction
drops to its kinetic value as rotation initiates and the
rotor “jumps ahead” as it dissipates its stored elastic
energy.
Repetition of this sequence during slow-speed
rotation often produces a stop-start jerking pattern,
as illustrated in the accompanying figure. A system
with a long, limber shaft is particularly susceptible to
this behavior.
Many rotating machines such as turbines, compressors,
oil-well drill shafts, and water-lubricated
ship stern-tube bearings can be reduced to the concept
of a disk driven by a turning gear through a flexible
shaft. Similar effects are also encountered with
slow-speed sliding contacts in machine tools, metal
cutting, squealing automobile brakes, household appliances,
and shaft seals.
The main culprits of stick-slip are low oil-film
thickness and torsional vibration, both of which are
described next.
Low oil-film thickness
Friction characteristics with oil-film bearings are
shown in the accompanying graph. The coefficient of
friction reaches a minimum at the low oil-film thickness
limit corresponding to the “lift-off ” speed at which a full oil film is first formed. Above this lift-off
speed, the influence of surface asperities on bearing
performance is practically nil.
Once journal surface speed again drops below this
lift-off, asperity peaks on the mating shaft and bearing
surfaces will begin again to interlace, restrain shaft rotation
with increased friction, and build up torsional
elastic energy.
When this rising torsional force overcomes the
static bearing friction with the slowed or halted shaft,
the shaft will unwind and jump ahead. As the wind-up
torque is then relieved, the shaft may then stick again
in a repeating stick-slip vibration.
The negative slope associated with the drop of the
friction coefficient from boundary to mixed lubrication
sometimes gives rise to a related type of self-excited vibration.
As the surface of a lubricated machine element
moves from boundary lubrication to mixed lubrication
where the friction coefficient drops with increasing
speed (∂f/∂V <0), this negative equivalent damping
gives rise to self-excited oscillatory system response.
A similar phenomenon takes place in bearings of tracking devices when the mechanism must swing
back and forth to zero in on a desired target. This is a
major problem in satellite systems in which ball bearings
operate under ultralow speeds. Feedback control
loops are often necessary to compensate for friction.
The accompanying table gives lift-off speeds
needed to generate a full oil film in a journal bearing,
speeds below which stick-slip vibration becomes a
possibility.
The following example can be modified to perform
lift-off calculations for specific machines. Consider
an industrial steam turbine with a bearing radial load
(W) of 8,000 lb. The bearing has a journal diameter
(D) of 6.0 in., a bearing length (L) of 6.0 in., an internal
bearing radial clearance (Cr) of D/1000 or 0.006 in.
and a journal surface finish (R) of 0.000020-in. rms.
The viscosity of the oil at a bearing temperature of
40°C (μ) is 4 10-6 lb-sec/in.2
Applying Reynolds hydrodynamic equation as
modified for low speeds and small oil-film thicknesses
to obtain lift-off speed N0 revolutions/sec:
N0 = (h0P/Cr)/{4.678µ [(L/D)^1.044] [(D/2Cr)^2]}
where P = bearing radial unit load (P = W/LD), 222 psi,
and h0 = lift-off film thickness, defined as 3 R or 60 μin.
This h0 assumes lift-off is completed when the oil
film thickness rises to three times the rms journal
finish R as mated with a soft bearing surface polished
smooth by initial operation. This corresponds in the graph to a composite film parameter at breakaway of:
Λ= h0/[(Rjrnl^2 + Rbrg^2)^0.5]
or 60 μin./20 μin. = 3
Applying the Reynolds equation:
N0 =[(6E-5 in. x 222 lb/sq. in.)/6E-3 in.]/
{4.678 x 4E-6 lb-sec/sq. in. x (6.0 in./6.0 in.)^1.044 x [6.0 in./(2 x 6E-3 in.)]^2}
Torsional vibration
On start-up of a machine, initial rotor acceleration
at breakaway may induce a cycle of torsional vibration:
torsional windup of the shaft to overcome static
bearing friction, followed by unwinding as shaft speed
jumps ahead of the turning gear drive speed. At the
completion of such a cycle, journal surface velocity
in its bearing drops and a new “stick” cycle may be
initiated if the bearing is devoid of its full film. This
repetition of sticking can usually be expected only following
completion of at least to of a full torsionalvibration
cycle.
Torsional-shaft displacement, ø radians, after the
shaft breaks free involves harmonic vibration at ω radians/
sec superimposed on turning gear displacement
α radians/sec to give a response pattern as illustrated
in the first graph. Neglecting damping effects:
dø/dt = α + [(T0ω sin(t-t0))/k]
To avoid a repetition of sticking from loss of torsional-
shaft velocity within the bearing, dø/dt must be
kept above zero when sin (t – t0) = –1 at the extreme
unwinding of the harmonic vibration. That requires:
a > T0ω/k
That is, the drive gear rotational
velocity α must remain significantly
above the oscillating journal
velocity. This torsional low-speed
limit is calculated in the following
example for the previously mentioned
industrial steam turbine.
As the second criterion for avoiding low-speed stick-slip behavior,
the gear drive should be
set above the oscillating journal
velocity:
a > T0ω/k
where α0 = minimum turning
gear rotational speed in radians/
sec, k = the shaft’s torsional stiffness,
and ω = the torsional natural
frequency. T0 is the breakaway
torque, f (D/2) W, lb-in., where
the coefficient of breakaway friction,
f, is 0.25.
If k is 7.5 106 lb-in./rad and ω
is 188.5 rad/sec (30 Hz), minimum
turning gear rotational speed α0
then becomes:
α0 = (0.25 3 in. 8,000 lb)
(188.5 rad/sec)/(7.5 106 lb-in./rad)
= 0.151 radians/sec = 1.4 rpm.
To avoid stick-slip vibration,
both this 1.4-rpm torsional requirement
and the earlier 28-rpm
film lift-off requirement must be
exceeded. The designer should
calculate both to ensure that minimum
requirements are satisfied.
Significant secondary effects
may also be introduced in some
cases either by rotor system damping
or by partial preservation of
an oil film as the journal velocity
passes through zero.
Once established, this calculated
minimum speed will also assist in
sizing system components. With
large turbines, for instance, turning
gear drives have been typically set at
five to 10 times their required minimum
speed with corresponding excess
size and power requirements.
On the other hand, increasingly
large turbines were once supplied
with the same size turning-gear
motor used on existing turbines,
but with a higher gear ratio to overcome
the greater breakaway torque
with the heavier bearing loads. As
turning-gear speed dropped below
2 rpm, chatter developed along
with fatigue damage.
Means for avoiding stickslip
and chatter
Any of the following steps can
be considered as possibilities for
eliminating stick-slip vibration in
low-speed bearings:
- 1. Boost oil viscosity, either by
using a more viscous grade
of oil or lowering the oil-feed
temperature.
- 2. Idle faster as with a turning-gear
device.
- 3. Use externally pressurized oil
lifts to generate a full oil film for
starting and at low speeds.
- 4. Use a bearing material that gives
a lower static coefficient of
friction such as filled PTFE or
filled nylon.
- 5. Apply ball bearings with their
low friction at breakaway and at
very low speeds.
Further Reading
Khonsari, M.M. and E.R. Booser. Applied Tribology — Bearing Design
and Lubrication, Wiley Book Co., 2001.
Lu, X. and M.M. Khonsari. “On the lift-off speed in journal bearings,” Tribology Letters, Vol. 20, Nos. 3-4, pp. 299-305, 2005.
Michalopoulos, D. and A. Dimarogonas. “Stick-Slip of Rotors in
Fluid Bearings at Very Low Speeds,” Wear, Vol. 70, No. 3, pp. 303-309,
1981.