**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 (*C _{r}*) 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:

*N _{0}* = (

*h*/

_{0}P*C*)/{4.678µ [(

_{r}*L*/

*D*)^1.044] [(

*D*/2

*C*)^2]}

_{r}
where P = bearing radial unit load (P = *W/LD*), 222 psi, and *h _{0}* = lift-off film thickness, defined as 3 R or 60 μin.

This *h _{0}* 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= h _{0}*/[(

*R*^2 +

_{jrnl}*R*^2)^0.5]

_{brg}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:

*d**ø*/*dt* = *α* + [(*T _{0}*

*ω*sin(

*t-t*))/

_{0}*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 – t _{0}*) = –1 at the extreme unwinding of the harmonic vibration. That requires:

*a* > *T _{0}*

*ω*/

*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* > *T _{0}*

*ω*/

*k*

where α_{0} = 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 10^{6} lb-in./rad and ω is 188.5 rad/sec (30 Hz), minimum turning gear rotational speed α0 then becomes:

α_{0} = (0.25 3 in. 8,000 lb) (188.5 rad/sec)/(7.5 10^{6} 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.