Limit Cycle Due To Interlocked Shroud Friction
Instrumental To The High Bypass Turbofan

As Presented at ASME Turbo Expo 2000

Nicholas Klompas, Nklompas@localnet.com -- 2000-GT0364

Vibration. Vibration. Vibration was the plague that largely influenced the evolution of today’s turbo-engine. Then, over thirty years ago, experience aided by traditional experimentation and new computer analyses of longstanding science created the marvelous high bypass turbofan, which seemed to paradoxically overcome vibration.

Using test data provided by an engine manufacturer to the British accident investigation board, as they investigated an engine failure, I will illustrate how the high-bypass turbofan overcame the plague of vibration by an natural mechanism that is not yet in the literature, not yet included in engine analysis.

Here are slender blades, retained through flexible disks in the fan and in the LP turbine, supported by a slim, highly torqued, two-bearing shaft. This rotor would seem vulnerable even we ignore the possibility of high frequency excitation by the HP rotor.

The two enabling features, which may seem old hat now, are the squeeze-film damper and the interlocked shroud, at part span in the fan and at tip in the LP turbine. Today, we will examine the role of the interlocked shroud.

 Excerpts from the 1997 IGTI Scholar Paper:

--On bladed disk stiffening:

The introduction of shrouds to provide an elastic support at part-span of fan blades was motivated by flutter prevention.

-On rub at shroud interfaces:

Because the blades bear against each other as they untwist at speed to form a "continuous ring,," they have a tendency to rub at these interfaces during vibration. The extent of this rub and the resulting potential for damping have been subject of scrutiny for over two decades, yet no acceptable analysis has emerged.

These excerpts from the 1997 IGTI Scholar Paper reveal the currently accepted perceptions of the interlocked shroud.

Stiffening of bladed disks by contact between the shrouds is well understood. But the widespread assumption of beneficial damping due rub at shroud interfaces has not yet been proven and we will see that it is really groundless.

Results of tests on whole system:
engine as installed in airframe:

-Blade vibration is nonsynchronous

-Mistuning is disproven;
amplitude increases with sharper tuning

-Anticipated friction damping is negative; higher friction increases amplitude

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Manufacturer’s tests performed in the investigation ot the British Midland crash on January 8, 1989

AAIB 1990 U.K.Air Accidents Investigation Branch 1990
AAIB Aircraft Accident Report No: 4/90 Aircraft G-OMBE (EW/C1095) 1 Farnborough, U.K. 1990

The failure was related to a model change in shroud interface angle intended to strengthen stiffening

My work on design of jet engines began in the early days, when "expensive noises" in the test cell frequently sent us back to the drafting board.

Today, I offer a lesson that was brought home the hard way -- hard indeed for those who lost lives and loved ones in crashes.

Results of manufacturer’s tests triggered by the 1989 British Midlands crash contradict the assumption of damping. Here are the pertinent results of the tests.

In this investigation, the anticipation of damping backfired, without explanation.

The engine was eventually fixed the old fashioned way -- by strategic retreat. The change in shroud angle, intended to strengthen stiffening, was rescinded.

There is no sign that designers would ultimately learn from these hard-earned test results. On the contrary, the Scholar Paper shows us that researchers did not notice this published destabilizing effect of friction -- they are still focussed on calculating friction damping in isolated models, now via micro-mechanics.

These test results demonstrate conclusively the existence of friction induced, self-limited, blade vibration.

I submit that the observed vibration is associated with an inherent limit cycle that protects the fan from excessive vibration in a fully proven engine.

This term "limit cycle" was coined by the mathematician Poincare in 1892. It is often cited by physicists to explain such nonlinear mechanisms as the chattering of a chalk on a blackboard.

To explain the real effect of friction, let's begin with the linear wave that degenerates into the limit cycle.

The mode illustrated here, that of one nodal diameter, was first demonstrated in 1924 by Wilfred Campbell, known for the Campbell Diagram, but it is not considered by today’s researchers. Instead, they focus on modes with two or more nodal diameters. Since these multi-nodal modes do not interact with the shaft, the full significance of the slender shaft is never studied.

This figure represents a backward traveling wave in an overhung fan. This wave is common; it is a component of synchronous elliptical whirl.

How does this wave interact with engine whirl?

The gyroscopic moment, the moment for the bladed disk as conventionally assumed rigid, is magnified by the wave. The total dynamic moment must account for both, the gyroscopic moment and the inertial moment associated with deflection.

The inertial moment is determined by the difference between the wave speed, 2/rev, and the natural frequency of the bladed disk alone. This moment is significant in the typical fan, where the natural frequency with no slippage at shroud interfaces is lower than 3/rev.

The inertial moment would rise precipitously if the bladed disk stiffness were to weaken in operation. We will see that slippage does indeed weaken the stiffness.

A solution of this wave can be secured by conventional linear analysis.

Crawley, E.F. et al.,1986,
"Analytical and Experimental Investigation of the Coupled Bladed Disk/Shaft Whirl of a Cantilevered Turbofan",
ASME, Journal of Engineering for Gas Turbines and Power, Vol. 108 October 1986, pp. 567-575

Klompas, N., 1986b,
"A Frictionally Induced Bladed Disk/Shaft Instability:
Physical Explanation of an Experimental Fan Failure,"
ASME Paper No. 86-GT-131

Klompas, N., 1981,
"Blade Excitation by Elliptical whirling
in Viscous Damped Jet Engines,"

ASME Journal of Engineering for Power, Vol. 103 April 1981, pp. 326-330

The theory of this backward wave and reasons why it should be incorporated in engine dynamics are contained in previous published works, yet magnification of the gyroscopic moment is neglected in current literature.

The 1986 paper shows early thinking on the limit cycle, recalling a sixties failure. Now, with the 1990 test results and with an easier design, I am able to prove a rigorous and experimentally validated definition of the limit cycle.

We begin before onset of slippage.

This figure shows the parameters that define the linear backward wave. The wave of tangential force will be used as the primary parameter in analysis of the limit cycle, which is nonlinear. It comprises two components, that due to gyroscopic forces, for zero deflection, and that due to inertial forces associated with deflection. Both components are sinusoidal; mathematically, they are treated as similar vectors that can be combined.

Use of the tangential force as the primary parameter was first introduced in the 1986 paper. Its wave represents the constraint to tangential deflections due to the continuous "ring," which is created by interlocking of the shrouds.

Let me describe the wave at the limit. Here the parameters are repeated for the "ring" broken by slippage into two halves. The nature of this solution follows from the linear analysis:

First, the wave of tangential force due to gyroscopic loading represents the precession of the shaft; it sets the reference frame.

Second, the total tangential force and the step at each slippage are predetermined from the frictional resistance at entry and exit of the passing shroud. For now friction is assumed equal for compression and tension, but we will examine this assumption later.

Third, the two pieces of force wave due to inertial effects are displaced in opposite directions from the zero axis. These displacements represent constant equal and opposite forces in each half of the broken "ring." These constant forces do not interact with the inertial forces.

Fourth, the deflected shape derives from the pieces of force wave; only small discontinuities in the sinusoidal shape arise at the node.

Fifth, the blades are exposed to shocks on passing through the node.

Only two variables must be solved: the phase angle and the magnitude of the pieces of force wave.

CONDITIONS FOR SOLUTION

1) The discontinuous wave of total force must be reducible into a Fourier series of only the first term and much higher order terms.

2) The wave of total force must satisfy the frictional resistance at entry and exit into each slippage.

We secure a rigorous solution of the dynamics of the wave by satisfying the two conditions shown here. The analysis is not complicated, only Fourier coefficients are to be computed.

The solution comprises a stable wave and residual high frequency forces that represent the discontinuities. These residual forces are absorbed as shocks in the blades.

The previous figure shows that a limit cycle exists for a wide range of slope and of frictional resistance. Here we show that it is readily calculable.

We look again to describe the changes in the wave from the linear. The node shifts forward into the second quadrant. The inertial moment now is reducible into a component opposite the gyroscopic moment and another component in the transverse plane. The precession of the shaft, when viewed relative to ground, contains a component of angular velocity in phase with the transverse component.

The combination of inertial moment and angular velocity represents a release of energy, which flows into the shaft. This release is a necessary condition of a limit cycle, which is a self-sustained, self-limited wave.

To prove self-excitation, we must show corresponding absorption, how the wave is driven by the spinning blades

Let me go a bit beyond the written paper to show the backward transmission of power by the shrouds.

This solution can be adjusted to explain how the wave is driven. To produce the driving force, there must be a difference between the steps, with the larger step on the tension side.

The shock on the tension side, spreading adjacent blades apart, tends to delay reattachment. Conversely, on the compression side, the shock jams the blades closer together to advance reattachment. These conditions of slip dictate that inherently on the tension side, the step is larger and on the compression side, the breaking force is larger.

The difference in breaking forces adds a constant compressive force in both halves of the broken "ring," with no effect on the dynamics. The difference in steps changes only the small discontinuities in deflection at the node, with negligible effect on the dynamics.

We now have met a rigorous definition of the limit cycle, which draws energy from blade spin and transmits it into backward whirl of the shaft. While, for convenience, synchronous whirl was assumed, the speed of the self-sustained traveling wave can be nonsynchronous.

Interaction of the resulting self-sustained, self-limited wave with the shaft is inevitable. In a fully proven engine, the limit cycle controls peak vibrations. Indeed, as the title says, it is instrumental to the operation of the high-bypass turbofan.

Results of tests on whole system:
engine as installed in airframe:

-Blade vibration is nonsynchronous

-Mistuning is disproven;
amplitude increases with sharper tuning

-Anticipated friction damping is negative; higher friction increases amplitude

----------------------------------------

Manufacturer’s tests performed in the investigation ot the British Midland crash on January 8, 1989

U.K.Air Accidents Investigation Branch
1990 AAIB Aircraft Accident Report No: 4/90 Aircraft G-OMBE (EW/C1095) 1 Farnborough, U.K. 1990

The failure was related to a model change in shroud interface angle intended to strengthen stiffening

Yet, as shown by the British Midlands crash, unusual conditions that lead to an intolerable limit can be missed in engine certification. The paper discusses other in-flight failures that appear to be related to bladed disk flexing/whirl interaction.

The tests that validate the limit cycle also show that the micro-mechanics of friction do not affect such engine behaviors as would worry designers. The blades would have broken regardless of the exact condition of the rubbing surfaces.

There is no way to benefit from analysis of the limit cycle alone; it is incomplete if not incorporated in analysis of the whole engine. Designers need a comprehensive analysis. The written paper discusses how it can be achieved and why it has not yet been disseminated.

I have rushed through a new subject, but I hope that I have encouraged you to examine the written paper. We should have time for questions and comments, either now or later in private.

Mr. Chairman, I thank you for the opportunity to make this presentation and I thank the audience for their attention.