## Abstract

Many of the challenges that limited aero-engine operation in the 1950s, 1960s, 1970s, and 1980s were static in nature: hot components exceeding temperature margins, stresses in the high-speed rotating structure approaching safety limits, and turbomachinery aerodynamic efficiencies missing performance goals. Modeling tools have greatly improved since and have helped enhance jet engine design, largely due to better computers and improved simulations of the fluid flow and supporting structure. The situation is thus different today, where important problems encountered past the design and development phases are dynamic in nature. These can jeopardize engine certification and lead to major delays and increased program costs. A real challenge is the characterization of damping and the related dynamic behavior of rotating and stationary components and assemblies, and of the fluid–structure interactions and coupling. The theme of this lecture is instability in the broadest sense. A number of problems of technological interest in aero-engines are discussed with a focus on dynamical system modeling and identification of the underlying mechanisms. Future perspectives on outstanding seminal problems and grand challenges are also given.

## 1 Introduction

Each generation of aero-engine gas turbines has seen advances in technology overcoming major hurdles in the quest for high thrust-to-weight ratio and cycle efficiency. For example, in the 1940s, turbine cooling was used in the German Jumo 004 engine, the world's first mass-produced jet engine, to increase turbine inlet temperature for higher cycle work and efficiency. In the 1950s, multiple spools, variable geometry, and compressor bleed were introduced to enable higher overall pressure ratios while meeting operability requirements. With continued advances in materials, rotordynamics, bearings, and seal technologies, a new engine architecture, the turbofan engine, was introduced in the 1960s achieving step changes in propulsive efficiency and noise. The evolution and maturation of computational fluid dynamics and finite element analysis in the 1970s and 1980s enabled improved high-speed turbomachinery designs and reduced engineering margins for better performance.

With massively parallel computing and better processors, the flow through an entire engine can now be calculated [1]. The progress in three-dimensional simulations of engine flows and structures has improved component designs but has also taken away some of the attention to problems which are often dynamic in nature and where damping is marginal. These are the type of problems that typically surface during engine testing and certification and can have a dramatic impact on an engine program.

The development of an engine for single-aisle aircraft takes about 4–5 years and costs about $1 billion. Ideally, time and cost could roughly be cut by half if everything in a gas turbine jet engine were designed without the inaccuracy of current methods and made to the correct specification and bill of material. Jet engine design aspects that cannot be calculated accurately so that engine certification can proceed without extensive testing are (1) emissions, (2) noise, (3) operability, (4) vibrations, and (5) durability. Prediction tools still rely on a large amount of empirical data, and the assessments are often based on relative changes from previous designs. Such challenging problems encompass non-equilibrium combustion chemistry, nonlinear acoustics (e.g., in fan noise and afterburner screech), characterization of damping to quantify stall and flutter boundaries and to identify emergent vibrations, and the prediction of excessive wear and premature failure of highly stressed hot components.

The scaling laws for fluid flow and heat transfer are well established and the quasi-steady performance of components and sub-systems can be computed over the entire engine operating range and flight envelope.

The hard problems left are related to component dynamic behavior and often lead to system-level instability. One challenge is that instabilities do not scale as simply as performance. This was recognized early in the history of gas turbine jet engines as illustrated by the sketch of a compressor map from 1952 (Fig. 1). The changes with altitude in the steady operating line and stability boundary are primarily governed by Reynolds number effects. As the stability boundary cannot be computed adequately, designers add extra stall margin to account for this, plus other factors such as transient operation, inlet distortion, engine-to-engine variation, and field service deterioration. A similar challenge holds for flutter: the aero-mechanics scale differently from the fluid dynamics such that the flutter boundaries cannot be drawn usefully in a compressor or fan performance map. Another aspect is the impact of geometric variability on component dynamic behavior, such as in an integrally bladed disk (IBR). While full-wheel calculations are possible, not every blade is geometrically identical and the impact of non-uniformities on resonant stress cannot be predicted adequately.

These dynamical issues typically arise when the level of damping is marginal as is in turbomachinery flutter inducing high-cycle fatigue (HCF). A “Class A” mishap occurs when the damage to the aircraft exceeds $2 million, the aircraft is destroyed, or its pilot or crew is killed or permanently disabled. “Engine-related” events exclude mishaps caused by foreign object damage (FOD), bird strike, or failure of support systems external to the engine (e.g., fuel starvation). Figure 2 summarizes the number of engine-related Class A mishaps per thousand engine flight hours (EFH) for single-engine fighter aircraft^{1}. The abscissa indicates aircraft and engine types and approximately time. High-cycle fatigue accounted for 56% of Class A engine-related failures for 15 years. These failures cost about $400 million per year and required 850,000 maintenance person-hours [3]. Table 1 defines the objectives of the HCF Science and Technology Program aimed at eliminating HCF as a major cause of aircraft turbine engine failure. Typically, increasing damping is an effective fire-fighting method to mitigate flutter and resonant stress problems. But, as shown in Table 1, gas turbine jet engines still don't have enough damping.

A historical challenge is the quantification of damping in rotating machinery, best exemplified by the frequency-speed diagram in Fig. 3, Campbell's 1923 study of turbine disk vibrations. In Campbell's diagram, the resonant conditions are identified at the crossings of the natural modes with engine order excitations. This seminal, one-hundred-year-old analytical technique is widely used to determine resonant operating conditions but the amplitude and damping of the vibrations are not directly identifiable. New tools and approaches are required to characterize and evaluate engine dynamic behavior.

The analysis of dynamical problems such as flutter and surge needs different approaches than the partial differential equation (PDE) approach common to fluid flow and structural simulations. One example is that of self-acting hydrodynamic bearings, seals, and dampers. To determine the stability of a fluid film, the Sommerfeld equation [5] needs to be solved. This is a challenging task due to the nonlinear nature and the unsteady properties of the compressible lubrication equations. With the advent of computers, the first steady numerical solutions were reported in the 1950s and the first solutions to the unsteady equations in the 1960s [6,7].

With today's abundant computer power, the problem is therefore to decide *what* to calculate, not *how* to calculate. This inherently makes analysis more, not less, important so as to organize, interpret, and make sense of the very large calculations and data sets. Instead of solving for small perturbations and hunting for the eigenvalues that satisfy the relevant ordinary differential equations (ODEs), some researchers study instability by computing initial-value problems governed by the underlying PDEs. An immediate advantage is that large amplitude effects are taken into account, but a serious limitation is that a complete picture of the dynamic behavior cannot be obtained as only a few cases can be considered. The choice and impact of the initial excitation are always arguable, and it is often difficult to discern numerical instability from the actual phenomenon being investigated. An example is the numerical simulation of the path into instability during the onset of rotating stall in compressors.

### 1.1 Lecture Theme and Scope.

This lecture differs from previous IGTI Scholar Lectures in two major ways (1) the ideas span several disciplines (such as fluid–structure interaction and mechanical design–aero interaction), and (2) the underlying concepts are less covered in the academic literature but are important (e.g., damping, a concept that engineers for all parts of the gas turbine engine need to pay attention to). A central theme of this lecture is instability in the broadest sense and from an overall engine perspective. While the problems addressed are different in nature, they have the following features in common: (i) the level of damping is marginal and is difficult to characterize, (ii) the dynamic behavior leads to instability often with catastrophic consequences, and (iii) the technical approach to solve these problems calls for dynamical system modeling.

For each of the topics considered, there are many publications on relevant methods that lay out *how* to solve the equations governing the underlying fluid flow and structural motion. With an emphasis on *what* to model, this lecture shows that real-world turbomachinery gas turbine engine problems can be solved by pivoting from computing continuum fluids and structures to applying modern computer power for modeling dynamical systems. The value in adopting such dynamical models is direct insight into root cause, characterization of mechanisms, and the quantification of damping, all providing solutions to the problems at hand.

A key concept is forced-response system identification and testing of full engines to quantify engineering and operability margins. Before illustrating the hard aero-engine problems related to surge, flutter, and rotordynamic vibrations that shape this IGTI Scholar Lecture, a short introduction to key concepts relevant to dynamical systems modeling is given. The problems addressed occur in separate parts of the engine and are driven by different mechanisms but a unifying perspective is the dynamic behavior of a second-order mass-spring-damper system. This will serve as a useful analogy for the characterization of the dynamics and stability of the aero-engine systems considered.

## 2 A Primer on Aero-engine Dynamical System Behavior

A central feature of the unsteady phenomena and vibrations of interest is cyclic oscillation. The dynamic behavior of the system is governed by the periodic conversion of potential energy into kinetic energy and back again. In idealized systems, where only elastic and inertial elements are considered, this behavior is clearly manifested as the natural frequencies and natural modes or as group and phase velocities of wave propagation. Real systems exhibit dissipation. A *damping* mechanism is responsible for the removal of energy from the oscillating system which leads to the eventual decay of the vibratory motion. There are situations where the element providing damping is active such that under certain operating conditions instead of dissipating energy, the element is adding energy to the oscillating system. This *negative damping* can lead to dynamic instability if the cyclic addition of energy is outweighed by the total dissipation in the system.

*k*, and mass,

*m*, the undamped natural frequency is $w0=k/m$. Introducing viscous damping,

*c*, via a dashpot, the

*damping ratio, ζ*, is

*c*is the critical damping evident from the time-domain solution of Eq. (1)

_{c}*s*

_{12}=

*σ*±

*jω*

*A*and

*B*depend on the initial conditions. For a dashpot providing critical damping $cc=2km$, the oscillations vanish. The complex frequency pair

*s*

_{12}are the

*eigenvalues*of the system where the real part,

*σ*, is the growth rate and the imaginary part,

*ω*, is the rotation rate or frequency of the damped oscillation. As illustrated in Fig. 5, the eigenvalues or modes of the system are usefully plotted in the complex plane, and the non-dimensional frequency response can be determined via a Laplace transform

There are three basic effects of eigenvalue motion in the complex plane: (1) Eigenvalue *growth rate*, equivalently damping, indicates the amplitude of the mode's resonant peak and stability. (2) Eigenvalue *rotation rate*, equivalently frequency, determines the proximity of the resonance peak to the forcing frequency (red dashed line in Fig. 5 indicates forcing at unity frequency). When the frequency of the eigenvalue is near the forcing frequency (unity), a large response is observed. (3) *Natural frequency*$(\omega 0=\sigma 2+\omega 2)$ reflects the compliance of the eigenvalue, and thus, its susceptibility to forcing. Based on these three factors, eigenvalue locations with imaginary parts near one and real parts near zero will yield the highest response to forcing at unity frequency.

Returning to the mass-spring-damper model in Fig. 4, if the mass is held constant, and *ω*_{0} is allowed to vary while holding *ζ* constant, two of the three effects described earlier are felt and illustrated in Fig. 5 on the bottom. First, the frequency of the resonant peak approaches one. Second, the overall compliance *k* of the system is reduced so the forced-response amplitude increases because, at a steady-state, the amplitude is *F*/*k*.

### 2.1 The Nature of Damping.

*loss factor*,

*η*, is defined as the ratio of the work lost in a cycle relative to the peak potential energy [8]

For simple harmonic motion, *η* is the energy lost per radian divided by the peak energy available. The loss factor in general depends on amplitude and frequency of the oscillation; for linear systems, both the lost work and peak potential energy scale with amplitude squared so the loss factor is independent of amplitude. For linear damping mechanisms, such as for example a linear dashpot, the loss factor yields an important dependency on frequency, *η*(*ω*).

Typically, the key damping mechanisms at play are internal dissipation and radiation of energy away from the system through the surrounding medium (e.g., acoustic radiation or radiation of energy through supporting structures). The radiation damping of a system can be altered by changing the coupling of the system with the surroundings. Changing material properties of or introducing energy-absorbing devices in a system can alter the internal dissipation. Both mechanisms can be expected to be frequency dependent if the radiation occurs in a linear external system, the frequency dependence of the loss factor will be governed by the frequency response of the external system. If the internal dissipation is driven by linear relaxation mechanisms, the loss factor will be enhanced near the relaxation frequency.

*c*= const) undergoing harmonic oscillation

*y*=

*A*sin(

*ωt*), the lost work in a cycle is

The spring can store potential energy *E*_{p} = 1/2*A*^{2}*k* so the loss factor for the ideal dashpot becomes

*ω*, the absolute value of the frequency is taken. In many situations, the damping is marginal

^{3}, and its effect on the oscillations is only appreciable near resonant conditions. With the loss factor at resonance yielding $\eta (\omega 0)=\eta 0=c/km$, the damping ratio,

*ζ*, can be interpreted as half of the lost work at resonance relative to the potential energy stored in the elastic element

*quality factor*,

*Q*= 1/2

*ζ*= 1/

*η*

_{o}, which determines how underdamped the system is, and the

*logarithmic decrement*,

*δ*, defined as the natural logarithm of the amplitude ratio of two successive oscillation peaks

Most linear damping mechanisms can strongly depend on the frequency, and ideal dashpots do not represent the frequency dependence of the actual loss factor. Nevertheless, actual damping can be adequately modeled by ideal, linear dashpots as will be shown in this lecture. Even if the model misses the damping over a range of frequencies, the effect on the dynamic response will not be significant if the damping is light. For multi-modal systems, such as rotating shaft assemblies, a good practice is to assign an ideal dashpot to each natural mode of vibration with the loss factor matched at the mode's resonant frequency. This does not work if coupling of modes due to damping is important.

### 2.2 Dynamic Stability of Non-conservative Systems.

Damping is critical to the stability of non-conservative systems such as flexible rotating shaft assemblies or aero-elastic structures interacting with fluid flow in turbomachinery. It is important to discern static instability or pure divergence from the initial state (e.g., negative spring stiffness, *k*) from dynamic instability or exponentially growing oscillatory motion as evident in Eq. (3) for negative damping, *c*. As will be shown in the following example of a whirling shaft, dynamic instability is often reached before the static instability limit. It is thus crucial to evaluate damping and to interrogate the dynamic behavior of the system.

Another example is classical shaft whirl in Fig. 6, which illustrates the remarkable fact that a stable undamped system can be destabilized by adding damping [8]. The simplified model of a whirling shaft is comprised of a point mass, *m*, suspended by symmetric springs of stiffness, *k*, inside a ring rotating at rotation speed Ω. When the mass is displaced from the center by *ɛ*, there is a restoring force *k–ɛ* pointing to the center, independent of the rotational speed Ω. If the ring is stationary, the mass can move with two degrees-of-freedom, and all natural motions are linear combinations of vertical and horizontal translation modes or, equivalently, of a pair of circular precession modes one whirling forward and the other backward. The modes have the same natural frequency $\omega 0=k/m$.

*Ω*, can introduce an inherent instability. Constraining the vibratory motion of the mass in the rotating frame along the

*ξ*-axis by a guide rail (for small displacements, this is equivalent to infinite stiffness in the perpendicular direction

*η*), Eq. (15) reduces to

As soon as the centrifugal force outweighs the elastic force, the system is *statically* unstable with unbound growth of *ξ*. This occurs for supercritical rotation speeds $\Omega >k/m=\omega 0$.

*ω*in the stationary frame, the whirl will be perceived at frequency

*ω*− Ω in the rotating frame. The whirl stability boundary can be assessed for different loss factor frequency dependencies using Eq. (8)

_{0}when the stationary loss factor equals the rotational loss factor

The instability boundary is set by the intersections of the loss factor curves and thus depends on their frequency distributions as illustrated in Fig. 7. Damping of motion in the stationary frame always stabilizes the system but, similar to diametrally constraining the vibratory motion in the centrifugal field, the damping of motion in the rotating frame provides energy input and destabilizes the system. The analysis also shows that this is only possible at supercritical rotational speeds, Ω > *ω*_{0}, and that the conditions for *dynamic* instability occur before those for *static* instability. Typically, for short hydrodynamic fluid film bearings, the speed of whirl instability onset occurs at twice the natural frequency also referred to as half-frequency whirl. This limits the stable operating range of rotating machinery to shaft speeds below twice the first critical speed of the rotordynamic system [9].

One means by which the stable operating range of the system can be extended is by breaking symmetry. This can be done by arranging the elastic elements with anisotropy so as to provide different levels of stiffness in the two orthogonal directions. The beneficial impact of stiffness anisotropy on the whirl instability boundary will be illustrated in a later section.

The basic ideas and dynamical system concepts laid out here for mechanical systems apply equally to fluid dynamic systems. Modeling the dynamic behavior of turbomachinery flows effectively reduces the same eigenvalue problem, and instability can be addressed by pursuing similar strategies.

### 2.3 Instabilities *Everywhere*.

Instability is a widespread phenomenon that occurs in all fields of engineering and everyday life. For example, from simple force equilibrium the ambient pressure of air in a room would be sufficient to push a layer of liquid against the ceiling. However, by heuristic experimentation, such a layer of liquid collapses immediately. The root cause of the collapse is a gravity-induced instability as illustrated in Fig. 8. Unlike a liquid in a straw, capillary forces cannot oppose gravity in a density stratification layer. More specifically, for any perturbation of the interface, the gravity-induced pressure gradient and the air-to-water density gradient set up a baroclinic torque inducing a destabilizing velocity field. This leads to Rayleigh–Taylor instability for all wavelengths of the perturbations. It follows from the discussions earlier that, if an appropriate stabilizing mechanism were introduced, the inverted water layer could be stably supported. As demonstrated in Fig. 9, vertical shaking at the natural frequency of the air layer supporting the liquid (effectively a mass-spring-damper system) can stabilize the interface. The required damping is provided by shearing due to the relative motion between the liquid layer and the bath walls [10].

More familiar instabilities in mechanical systems are buckling of slender columns under compressive load or wrinkling of plates under shear, stick-slip or chatter instability between two sliding surfaces, and rotor dynamic instability such as shaft whirl.

In fluid systems, hydrodynamic instabilities typically occur in the form of Taylor instability in rotating flows, Kelvin–Helmholtz instability in shear layers, and Rayleigh–Taylor instability in fluid interfaces as illustrated earlier. Common turbomachinery and aero-engine system instabilities include rotating stall and surge, turbomachinery blade flutter, aero-elastic instability in labyrinth seals, hydrodynamic instability in fluid film bearings and seals, and contact instability in rub events between turbomachinery blades and casing abradable.

The multi-disciplinary aero-engine problems addressed in this lecture illustrate dynamic instabilities of both mechanical and fluid dynamic types and highlight the role of coupling between the structure and the working fluid. In most cases, the full engine participates in the dynamic behavior involving multiple components and sub-systems.

### 2.4 Two Classes of Stimuli.

A dynamical system will vibrate only if there is a source of excitation or stimulus. There are two major classes of stimuli, forced vibration, and self-excited vibration.

In *forced vibrations*, a sustaining or alternating force exists independent of the motion. The system will vibrate at the frequency of the alternating force or a frequency directly related to it. Typical mechanisms for forced vibrations encountered in aero-engines are as follows:

rotor unbalance

rotor bow

geometric instability (misalignment)

operational instability (thermal growth)

inlet flow distortion

blade wakes

upstream influence via potential fields

A key feature, and useful diagnostic, of forced vibrations, is that the excitation continues even when vibratory motion is stopped.

*Self-excited vibrations* are governed by destabilizing forces that sustain, and the motion is created or controlled by the motion itself and, if damping is marginal, leads to instability. The self-excited system will vibrate at its own natural frequency, independent of the frequency of any external stimulus. Typical mechanisms and related instabilities are as follows:

whirling or whipping

stick-slip or chatter

instabilities in forced vibrations

flutter

Key design strategies for dealing with forced-response and self-induced vibrations typically include (1) increasing component damping to reduce critical response peaks or to cancel the destabilizing forces in the engine operating range, (2) placing the system critical frequencies outside the steady-state operating range (if possible) or raising the critical frequency to minimize supercritical whirling or whipping phenomena, and (3) diminishing or eliminating the destabilizing forces and mechanisms.

### 2.5 Role of Engineering Margins.

Aero-engines are complex dynamical systems with strong coupling between components, both from a mechanical and from a fluid dynamic perspective. This coupling makes it sometimes difficult to diagnose the issues at hand in isolated component or sub-assembly analysis or rig tests. Vibration problems often involve the full-engine system with both rotating and static structures participating in the dynamic behavior, encompassing multiple rotors, bearings, dampers, bearing support structure frames, cases, mount-, nacelle- and pylon structures. The fluid dynamic challenges are unsteady in nature and span a variety of time and length scales often with marginal damping levels. As damping is hard to calculate more engineering margin is built into the designs than ideally necessary. This often comes at the cost of reduced performance.

About half of the total compressor stall margin is devoted to transient operations to ensure stable operation even for large excursions from the steady operating line as shown in Fig. 10 by the black lines. During hot engine re-acceleration transients, so-called Bodie or “hot re-slam” maneuvers,^{4} the compression process is non-adiabatic because, due to thermal inertia, changes in metal temperatures lag those of the gas path flow. The dissimilar characteristic time scales result in unequal temperatures and heat transfer via various mechanisms as illustrated on the right in Fig. 10.

During the first part of a hot Bodie re-acceleration transient, heat is transferred from the hot metal to the cooler working fluid. This reduces the flow capacity due to stage mismatching and increases the stage stagnation pressure loss and flow deviation, resulting in less turning and shallower compressor characteristics. As shown by the speed lines in red, this moves the operating point to lower flow and higher pressure ratio. At the same time, the slope of the speed line is less negative and destabilizing so the stalling pressure ratio reduces. During the latter part of the Bodie transient, the metal has cooled down sufficiently while the gas temperature has risen so that heat is extracted from the working fluid and the opposite effects occur. This is illustrated by the colored lines, and a composite compressor map is defined as the ensemble of speed lines throughout time, each realized with the heat transfer, and thus changes in stage characteristics, matching, and stall point, which occur at the time the operating point is drawn [11].

Detailed non-adiabatic transient calculations in an advanced high-pressure ratio core compressor show an eight-point reduction in stall margin from the adiabatic case, with heat transfer predominantly altering the transient stall line. Heat transfer increases loading in the front stages and destabilizes the front compressor block. The effects of transient heat transfer on compressor stability are exacerbated for compressors with highly loaded front stages and as overall pressure ratios are increased for improved cycle efficiency. This challenge must be addressed early in the design process so that sufficient engineering margin can be guaranteed.

## 3 Hard Aero-engine Field Problems and Technology Challenges

Building on the ideas introduced earlier, a number of different aero-engine field problems and technological challenges are now discussed. The common threads are (1) marginal damping, (2) full-engine issues with the coupling of components, and (3) emphasis on *what* to model rather than *how* to model the problem. A key question in the modeling process is at what level to enter the problem. The following examples aim to elucidate what degree of knowledge is needed to tackle the problems.

The lecture also illustrates that some problems have suffered oversimplification such as for example flutter or stall and surge because of real geometry effects. Very talented engineers in the 1940s, −50s, and −60s worked hard to reduce the problems to analytically manageable terms complemented with empirical data where needed. Much has been learned since but in certain areas the challenge to close these empirical gaps remains.

### 3.1 Compressor Deterioration Leading to Engine Surge.

Engines lose performance over their life due to deterioration manifested, for example, in enlarged compressor blade tip-clearances due to rubs, an ovalized casing, or bearing and seal wear. While reduced engine performance yields increased fuel burn, environmental impact, and operating costs, deterioration can also alter the engine’s dynamic behavior and pose a major safety risk.

Repeated, fleet-wide engine surge events occurred during take-off acceleration transients in a family of large commercial aircraft engines. Experiments in deteriorated engines with enlarged and non-uniform tip-clearances revealed large amplitude pressure fluctuations, measured by a wall static pressure sensor in the core compressor, during acceleration transients. Refurbished engines with tight and uniform clearances did not show this behavior as illustrated in Fig. 11. Engine trials shown in Fig. 12 further demonstrated that the unsteady pressure fluctuations were exacerbated and led to surges during cool snap accelerations while warm Bodie re-accelerations transients did not exhibit this behavior. The engine experiments highlight the important difference between nominal and real geometry which can be affected by enlarged tip-clearances, casing ovalization, and wear an engine made to the bill of material may not exhibit instability, while an engine in real operation does.

#### 3.1.1 Effect of Blade Tip-Clearance Changes on Engine Dynamic Behavior.

Operating tip-clearances are hard to measure, and enlarged clearances can lead to rotating stall and surge. To characterize these real-world effects in compressors effectively, a dynamical system approach can be adopted. In the following, a linearized, compressible, two-dimensional core compressor model in input–output state-space form was used to capture the impact of tip-clearance changes on compressor stability [12,13]. The overall analysis consists of models of the inlet and exit ducts, the blade rows, and the interblade row gaps as depicted in Fig. 13. The hub-to-tip ratio is assumed high enough to neglect radial variations of the flow quantities. Effects of viscosity and heat transfer outside of the blade rows are also neglected. The dynamics for each of the components is governed by potential, vortical, and entropic perturbations obeying the linearized equations of motion. These various models are stacked up axially through the compressor, linked through boundary conditions at each interface, and closed by end conditions at the inlet and exit of the compressor ducts to form an eigenvalue problem. Each complex eigenvalue and corresponding eigenvector represents a pre-stall wave exhibiting growing or decaying oscillatory behavior similar to the mass-spring-damper system introduced in Sec. 2.

To address the effects of deterioration, unsteady pressure fluctuation forcing and disturbances in the gas path from non-uniform tip-clearances, an ovalized casing, and unsteady flow phenomena on the blade scale are introduced [14]. Another important modeling aspect is engine transient operation since the dynamics in general varies during transients. The required model inputs are compressor geometry and blade row performance correlations (viscous losses, deviation and incidence effects, air-bleed, and guide vane schedule). The model outputs are the two-dimensional compressor dynamics in terms of the pre-stall modes or eigenvalues, *s*_{[n,m]} = (*σ* + *jω*)_{[n,m]}.

The pre-stall modes are labeled [*n, m*], *n* denoting the nodal diameter or circumferential spatial harmonic number and *m* indicating essentially the axial wave number (spatial frequency), with the sign reflecting the direction of wave rotation. The first spatial harmonic pre-stall modes (*n* = 1) are plotted in Fig. 14 for a part-power operating point. Mode [1,0] corresponds to an incompressible resonance indicated by *m* = 0 since the mode shape of the mass flow perturbation is approximately uniform in the axial direction. If the compressor is throttled into instability, this mode will grow into a single, finite rotating stall cell (*n* = 1). Other modes are compressible and travel in both directions. Note that the first compressible mode [1, 1], denoted as the 1^{st} acoustic resonance, rotates at a rate very close to rotor noise forcing (1-E). This mode governs overall stability as will be shown in the following analysis.

The stall line is determined at operating points for which the least stable mode, typically the rotating stall mode, is neutrally stable and has zero growth rate (*σ*_{[1,0]} = 0). The model results for a cool snap acceleration transient are shown in Fig. 15 for a refurbished engine. The red circles are the computed operating points matched to the measured transient operating line. The dynamic model-based stall line is marked in red. To obtain the response of the compressor system to forcing, the state-space model is integrated in time. The random forcing was chosen as white noise, augmented with rotor frequency noise in each blade row, thus modeling geometric non-uniformities. Noise levels were set at the level observed in the engine experiments. The spectrogram of the simulated mid-compressor sensor signal, depicted in Fig. 16 on the top, compares well with the engine experiment on the bottom. A strong response is observed near rotor frequency during the initial phase of the acceleration. A more useful view of the data is provided by integrating the power spectral density between normalized frequencies of 0.9 and 1.1. The resulting signal is denoted as the “first acoustic” or “[1,1] signal” since the response is chiefly governed by the [1,1] mode.

Engine deterioration was modeled as an increase in tip-clearance (implemented as local blockage changes) which was assumed to change as a function of rotor speed due to centrifugal and thermal effects. The [1,1] signals of a deteriorated and a refurbished engine are depicted in Fig. 17 in red and in black, respectively. The simulations capture the measured trends well with the ratio of the peak values between deteriorated and refurbished engines roughly a factor of two in both data and computation. The connection between engine transient operation in Fig. 15 and the observed pressure signals in Fig. 16 is established next.

#### 3.1.2 Lack of Damping.

The mechanism for the observed response can be characterized in a modified Campbell diagram where, instead of the structural modes, the compressor pre-stall modes are plotted. This is illustrated for modes [1,0] and [1,1] in Fig. 18 by the circles together with the time history of the [1,1] signal for the refurbished engine on the bottom. The operating line excursion during the acceleration transient yields less damped compressor dynamics as the operating point moves closer to the stall line in Fig. 15. This is also reflected in the motion of the modes (eigenvalues) in the complex plane; the location of the [1,1] acoustic mode coincides with rotor noise forcing during the transient, and both frequency and damping vary during the acceleration. To capture this change in damping, the size of the circles in Fig. 18 is plotted inversely proportional to the mode's growth rate. As the [1,1] mode moves closer to the first engine order rotor forcing frequency (1-E), point A, damping is reduced and more energy is fed into the [1,1] mode. This results in large amplitude oscillations near rotor frequency reflected in time-domain response. As the engine accelerates further, the [1,1] mode moves away from 1-E forcing and damping increases in point B, dropping the frequency response. Approaching take-off power conditions, point C, the mode [1,1] frequency wanders toward the excitation frequency but, since damping is increased, the response remains small.

These effects can be further elucidated by considering the mass-spring-damper system from Fig. 4 as a mechanical analogy for the resonant [1,1] mode. Substituting the eigenvalue of mode [1,1] into Eq. (1) yields a model that ignores all dynamics except those near the synchronous forcing frequency. The frequency response of the mechanical analogue is depicted in Fig. 19 for idle and points A, B, and C during the cool snap acceleration transient. The response is qualitatively similar to that in Fig. 18 but, because the [1,1] signal depends on the square of the frequency response, the variations are more pronounced than in Fig. 19. The motion of the eigenvalue in the complex plane follows the three basic effects described in Sec. 2 and Fig. 5 depending on the eigenvalue locations; more or less energy originating from the 1-E forcing is fed into the [1,1] mode. This is reflected in the amplification of the [1,1] signal.

#### 3.1.3 Engine Monitoring for Fleet Management Purposes.

The instantaneous stall margin can be determined along the acceleration transient using the dynamic model and is plotted in Fig. 20 together with the measured [1,1] signal. The steady-state stall margin^{5} at idle is 30% and drops to about 12% during the acceleration transient. The [1,1] signal from the refurbished engine experiment follows a similar variation and can be used as a direct measure of transient damping and stall margin and therefore as a diagnostic for engine health monitoring.

Based on these ideas, extensive test cell and on-wing engine ground test campaigns were conducted involving close to 140 revenue service engines as depicted in Fig. 21. Core compressor unsteady pressure measurements were acquired during acceleration transients to determine the amplitude of the [1,1] signal, a large amplitude being liable to engine surge. As these engine tests statistically proved themselves strong indicators of the likelihood of an engine to surge in subsequent service, the FAA Airworthiness Directive [15] summarized in Table 2 was issued as a mandated service tool for engine safety.

### 3.2 Impact of Engine Architecture on Bearing, Damper and Seal Technologies, and Vice Versa.

All else equal, the power density of jet engines scales with tip velocity cubed and inversely with the length scale (cube square law). The incentive is thus to spin the turbomachinery as fast as the steady-state stresses allow, mandating lightweight structures and making the rotordynamic system flexible by design. Several bearings typically support multiple long and slender shafts that carry compressor and turbine disk assemblies which retain the turbomachinery blading. The bearing arrangement is strongly dictated by the engine architecture as illustrated in Fig. 22 for a legacy and an advanced turbofan engine for single-aisle aircraft applications. Low fan pressure ratio for improved propulsive efficiency leads to a large fan powered by a small core engine and a rotordynamic layout that differs from the legacy engine in several aspects. To support the lower speed fan plus its fan drive gear system, the no. 1 bearing is split into two tapered roller bearings, bearings no. 1 and no. 1.5. The core engine, comprised of high-pressure compressor, combustor, and high-pressure turbine, is shrunk to a size where the no. 4 bearing typically located beneath the combustor is pushed outward to allow the low spool to pass through the bore. The low spool of the advanced engine is therefore longer and thinner so that an additional bearing, bearing no. 6, is placed outboard of the low-pressure turbine to provide the required stiffness.

No matter what the architecture, aero-engine rotors are softly supported to enable a wide range of supercritical operations relative to the rigid-body rotor modes. In Fig. 23, the flexibility of the bearings is gradually increased from hard mount on the left to soft mount on the right. This reduces the bearing support stiffness and thus decreases the rotor critical frequencies. Further, the rotor mode shapes gradually transform, and for soft-mounted rotors, the mode shapes for the first two critical speeds become the first and second rigid-body mode shapes, with cylindrical and conical deflection distributions thus commonly called the translational and pitching modes. Comparing a hard-mounted rotor with a soft-mounted rotor, the critical speed ratio of the first flexural modes is 2.27 as indicated by the red vertical arrows in Fig. 23 [16]. This more than doubles the viable engine operating speed range which is set by the frequency of the first flexural mode. Safe transient operation through the rigid-body modes is accommodated by careful rigid-body mode balancing and the incorporation of squeeze-film dampers to control the amplitude of the vibrations. The response to unbalance is illustrated in Fig. 24 for a small aircraft engine.

Since rolling contact bearings are relatively stiff, an additional, softer support stiffness is required. A flexible bearing support structure typically used to soft mount rotors in aero-engines is the so-called squirrel cage, formed by a retainer ring with cut-outs as shown in Fig. 25. The squirrel cage acts like a circular array of leaf springs that can be tailored to achieve the required system natural frequencies. Care must be taken so that the soft supports do not subject the engine to excessive blade tip-clearance closure during aircraft maneuvers.

To eliminate rotordynamic instability problems and to decrease rotor peak response when crossing critical speed, additional external damping can be provided by a variety of conventional and advanced rotor damper concepts as sketched in Fig. 25. Aero-engines typically use squeeze-film dampers, which provide viscous damping through an oil film. Sometimes simpler solid state dampers comprised of a solid compound or elastomers are sufficient to provide the required damping. For example, an annular solid damping insert can be used in parallel with the squirrel cage. More advanced, active damper concepts include squeeze-film dampers combined with magnetic bearings. An overview can be found in Ref. [17].

Dampers, bearings, and seals, critical to the stable and safe operation of gas turbine engines, are strongly coupled with the internal flow system and thus prone to system instability. A key challenge is the high-speed operating requirements of the rotating machinery. The achievable surface speed of bearings, dampers, and seals is limited by centrifugal stress. This is commonly characterized by the *DN* number, defined as the shaft diameter, *D*, in mm times the shaft speed, *N*, in rpm. Conventional ball bearings have a limiting *DN* number of 1–2 million mm-rpm. Figure 26 depicts the achievable *DN* number for various bearing and seal technologies as a function of bearing length-to-diameter ratio, *L*/*D*.

This *DN* number limitation of bearings and seals dictates the aero-engine architecture since the turbomachinery is operating at much higher speeds than the bearing surface speed, the bearings must be located close to the shaft centerline as in Fig. 22. If bearings and seals were available with an order of magnitude higher *DN* number capability, engines could look dramatically different such as in Fig. 27.

Aimed at a step change in thrust-to-weight ratio, the exo-skeletal engine concept is a radically different engine architecture based on all-composite drum rotors. This eliminates the heavy shaft and disc assemblies of conventional engines. The drums act as rotating casings that support the turbomachinery blading under compression. For high flight Mach number applications, the resulting open duct at the engine centerline could also provide additional thrust via a ramjet [18]. Apart from advanced materials, a required technology to make this engine concept practical is ultra-short high-speed bearings. Such bearings are one order of magnitude smaller in *L*/*D* and one order of magnitude larger in *DN* number compared to conventional technology and could be achieved by short gas journal bearings as marked in the top left corner of Fig. 26.

#### 3.2.1 Scaling Laws for Ultra-short Seals and Gas Journal Bearings.

To explore this uncharted territory, scaling laws are required but unfortunately the bearing and seal dynamic behavior does not scale simply. For example, in ultra-short gas bearings the bearing flow-through time, governed by the hydrostatic differential pressure, is much shorter than the time scale of the viscous effects. The diffusion of vorticity away from the bearing surfaces is therefore limited, and the flow in the bearing land does not fully develop. These unsteady compressible flow effects impact the bearing and seal dynamic behavior and the conditions for whirl instability.

A short externally pressurized (hydrostatic) gas bearing with *L*/*D* < 0.1 is considered to illustrate the scaling challenges. These bearings essentially act like short Lomakin seals due to the very low *L*/*D* where the restoring force in the fluid film is generated through the pressure differential supplied by an external source. For a static shaft offset with no rotation, the restoring force acts in the opposite direction of the shaft displacement. Lomakin [19] first explained this large direct-coupled stiffness that can be developed by annular seals.

In the simplified model of a plain seal with an off-centered shaft (Fig. 28), the same axial pressure difference is imposed on both the top and the bottom half of the seal. In the upper half of the seal, where the clearance is smaller, the resistance is higher, and thus, the flowrate lower than in the bottom half with larger clearance. As the inlet pressure loss scales with dynamic pressure, the inlet pressure drop is lower, and thus, the pressure gradient in the bearing land is higher in the upper half. This yields a potentially large hydrostatic direct-coupled restoring force as indicated by *F ^{hs}*.

*K*, becomes

^{hs}The non-dimensional hydrostatic stiffness scales linearly with both length-to-diameter ratio and static pressure drop but inversely with the square of clearance-to-radius ratio, potentially providing large stiffness as the seal diameter is scaled up and the seal length kept constant.

Introducing shaft rotation Ω, the flow in the short bearing land is unsteady and inherently three-dimensional. At low shaft speed, appropriate simplifications can be made. Under the assumption of incompressible, fully developed viscous flow, the combination of a Couette flow and a Poiseuille flow is locally set up as illustrated in Fig. 29. The motion induces damping and a hydrodynamic force, *F ^{hd}*, as the fluid is pumped into the bearing or seal gap.

At low speed, the characteristic bearing flow-through time is much shorter than the characteristic flow-change time around the circumference (reduced frequency ≪ 1) so the axial flow induced by the external static pressure difference Δ*p*/*p _{o}* can be decoupled from the flow induced by the shaft rotation. The induced pressure field together with the viscous stress field can be integrated to determine the hydrodynamic force. This force acts perpendicular to the shaft deflection and is thus a cross-coupled force potentially contributing to whirl instability. The damping force can be calculated from the pressure field that is set up as the flow is squeezed in the gap due to the radial shaft motion toward the wall [20].

*based on the bearing or seal clearance*

_{C}*C*and Mach number

*M*based on bearing or seal surface velocity Ω

*R*, the so-called bearing number, Λ, can be defined as

The first term is governed by pure hydrodynamic pumping and is denoted *k ^{p}*, and the second term is due to viscous drag and denoted

*k*. The hydrodynamic pumping and viscous drag effects both scale with bearing number but are of opposite sign. By inspection, the cross-coupled hydrostatic stiffness vanishes for (

^{v}*C*/

*R*) = 2(

*L*/

*D*)

^{2}when the hydrodynamic pumping balances the viscous drag.

#### 3.2.2 Whirl Instability Criteria.

To determine the whirling motion of the shaft,^{6} a rotordynamic analysis with the hydrostatic and hydrodynamic stiffness and the damping coefficients was carried out. Solving an eigenvalue problem, the whirl instability limit is determined by the rotor frequency Ω = Ω* _{w}* at which the least stable system mode becomes unstable. The whirl ratio Ω

*/Ω*

_{w}*in Fig. 30 characterizes the limit of stable operation [20].*

_{N}Ω* _{N}* is the system's natural frequency which is the frequency of the whirling motion as seen in Sec. 2.2. The whirl instability limit was computed for three levels of shaft unbalance as a function of

*C*/

*R*. The stable design space for high-speed operation is narrow but there is a range of bearing clearance-to-radius ratio

*C*/

*R*for which the whirl ratio is very large. For a small rotor unbalance of

*a*/

*C*= 0.07, an infinite whirl ratio is obtained at

_{0}*C*/

*R*of about 0.01. It is important to note that the trend of whirl ratio with

*C*/

*R*resembles singular behavior and that the condition (

*C*/

*R*) = 2(

*L*/

*D*)

^{2}, where the stiffness due to hydrodynamic pumping balances that due to viscous drag is roughly aligned with the singularity as marked by the dashed line.

Infinite whirl ratio occurs when *k _{v}* =

*k*, equivalent to the clearance-to-radius ratio taking on the singular value, (

_{p}*C*/

*R*) = 2(

*L*/

*D*)

^{2}. Inspection of the system modes and mode shapes shows that for

*C*/

*R*less than the singular value (left branch of curves in Fig. 30), the hydrodynamic pumping effect dominates and the whirling motion is in the direction of rotor rotation (forward whirl). The opposite occurs for

*C*/

*R*greater than the singular value where viscous drag effects dominate and the whirling motion is against the direction of rotor rotation (backward whirl). For larger

*L*/

*D*, the hydrodynamic pumping force is dominant and the whirl ratio approaches the value of 2, consistent with hydrodynamic short bearing theory [9].

Note that the whirl ratio in Eq. (22) is independent of bearing number Λ. But at high speed, the low reduced frequency assumption does not hold and the decoupling of the hydrostatic and hydrodynamic flow fields is no longer valid. It is expected that the unsteady development of the viscous gap flow has an impact on the whirl instability limit. A higher fidelity computation of the unsteady flow needs to be carried out to quantify its impact on the stability of short seals and gas bearings.

There is an indication that in view of the short axial flow-through times governed by the hydrostatic differential pressure, time is insufficient for axial vorticity to diffuse radially outward from the rotating surface [21]. Consequently, the circumferential flow does not fully develop affecting the magnitudes of the hydrodynamic pumping and viscous forces acting on the rotor.

To gain insight into the mechanisms that govern the evolution of the unsteady bearing flow, the impulsive starting of a Couette flow between two infinite parallel plates as shown in the lower inset of Fig. 31 serves as a useful analogy.

The lower plate is suddenly brought to a steady velocity while the upper plate is held stationary. The unsteady velocity distribution obeys the diffusion equation^{7} and takes on a Fourier series solution with characteristic time $\tau d\u223cC2/\upsilon $. This is the time scale for the vorticity to diffuse across the channel.

*t*, of the impulsively started Couette flow is equivalent to the time it takes a fluid particle to reach an axial location

*z*along the journal bearing such that $t=z/U\xaf$, where $U\xaf$ is the mean axial velocity through the bearing set by the external static pressure difference Δ

*p*/

*p*

_{0}. Comparing the flow-through time scale of the axial hydrostatic flow, $\tau f\u223cL/U\xaf$, with the viscous diffusion time scale, $\tau d\u223cC2/\upsilon $, a reduced frequency parameter can be defined that determines the development of the circumferential flow field in the journal bearing:

For *β _{FD}* ≪ 1, the viscous diffusion time is significantly longer than the axial flow-through time and the circumferential flow cannot fully develop leading to appreciable departures from the linear flow profile, impacting the hydrodynamic forces and whirl stability.

Based on the impulsively started Couette flow solution, the characteristic time required for the flow to develop within 1% of the fully developed linear profile, *τ*_{linear}, is half the viscous diffusion time $\tau linear\u224812\tau d$. As such, the circumferential flow in the bearing gap is fully developed if the axial flow-through time is half of the viscous diffusion time, $\beta FD\u22481/2$.

Three-dimensional numerical calculations of the unsteady bearing flow were carried out for a wide range of geometries and external static pressure differences marked by the different symbols as shown in Fig. 31. The rotor was subjected to radial eccentricity perturbations, and the viscous pumping and drag components of the hydrodynamic forces were extracted from the computations via integration of the pressure and viscous stress fields acting on the spinning rotor. The whirl ratio was then determined as before and is plotted as a function of reduced frequency *β _{FD}* in Fig. 31. The data collapse remarkably well and exhibit singular behavior at a reduced frequency of

*β*

_{FD}= 0.43. The whirl instability criterion can therefore be stated more generally as singular behavior of whirl ratio in short seals or gas bearings occurs when the flow-through time is approximately equal to the characteristic time required for the circumferential flow to fully establish [21].

#### 3.2.3 Breaking Symmetry for Stability.

Following the ideas described in Sec. 2.2, the dynamic performance and stability range of plain cylindrical bearings can be improved by breaking symmetry in support stiffness. This can be facilitated, for example, by introducing multiple lobes or pads to create a pre-load (the dimensional difference in clearance between the lobe and the bearing). This pre-load generates direct stiffness at the bearing-centered operating condition. For higher speeds and light loads, multiple-lobe bearings are known to be superior in stability compared with plain cylindrical journal bearings [22,23]. There is evidence that, with a considerable lack of symmetry in bearing support stiffness, the transition speed from stable to unstable operation can be increased dramatically relative to the critical speeds even in the absence of stationary damping [24]. This asymmetry in support stiffness is commonly referred to as anisotropic support stiffness.

Non-uniform bearing geometry, such as in lobed bearings, can affect both direct and cross-coupled stiffness. A simple modification in externally pressurized ultra-short gas bearings and seals can be made to eliminate the singular behavior in whirl instability. The concept is to break symmetry in hydrostatic direct stiffness (without much affecting the hydrodynamic cross-coupled stiffness) by blocking the axial through flow over parts of the circumference. This can dramatically extend the stable design space and increase the whirl instability limit. For ultra-short bearings and seals at large diameters, this also yields a major relief in fabrication-tolerance requirements, allowing some margin in clearance while maintaining stability. The impact of bearing stiffness anisotropy on the stable design space is depicted in Fig. 32 for the ultra-short gas bearing configuration discussed earlier with an *L*/*D* of 0.07.

### 3.3 Aerodynamically Induced Rotor Whirl Instability in Compressors and Turbines.

Non-uniform engine tip-clearance distributions as seen earlier due to an ovalized casing, a compressor shaft offset from its casing centerline or whirling in its bearing journal not only affect compressor stability but can also induce destabilizing rotordynamic forces. The so-called Thomas [26] or Alford forces [27] stem from the strong influence of the blade tip-clearance on the local performance of the turbomachinery. The coupling between the turbomachinery aerodynamics with the structural dynamics can lead to destructive whirl instabilities. Despite a large number of investigations of this topic over the past 60 years, there has been a disparity in the findings on the magnitude and direction of the whirl-inducing force [28,29]. To resolve this issue, extensive experiments were carried out in a multi-stage compressor, and a dynamical model served useful to explain the observed aerodynamic–rotordynamic interaction mechanisms [30].

The coupled aerodynamic–rotordynamic model consists of two parts (1) the flow field through the compressor is represented by an unsteady, two-dimensional, compressor tip-clearance model for stall inception, and (2) a rotordynamic force model based on the unsteady momentum equation applied locally to one blade passage in the circumferential direction. The inputs to the model are the compressor geometry, an axisymmetric compressor characteristic, and the sensitivity of the compressor characteristic to changes in axisymmetric rotor tip clearance, which are matched to the data. The family of axisymmetric compressor characteristics is shown by the dashed line and the two solid lines in Fig. 33, the latter corresponding to the local compressor performance at the smallest and largest tip-clearance, respectively. The circles and stars are the modeled and measured mean compressor operating points for a steady shaft offset 0.7% of blade span.

An unsteady control volume analysis is conducted in a frame locked to the rotor and applied locally to each blade passage to obtain the local tangential blade force. The local tangential blade force is governed by the local blade loading and thus depends on the local tip-clearance. The resulting force in the rotor frame is obtained by integrating the distributed tangential force around the circumference. This force is then transformed back to the absolute frame to obtain the cross-coupled Alford force and stiffness coefficient acting on the rotor shaft.

This modeling approach can evaluate the aerodynamically induced rotordynamic forces for any unsteady flow regime such as a whirling shaft or rotating stall. The assessment can also be extended to deal with the effects of unshrouded stator tip-clearances and stator shroud seal leakage flows [30].

The destabilizing Alford force parameter, defined as the cross-coupled stiffness coefficient normalized by the stage torque over the mean wheel diameter, was experimentally measured in the third stage of a four-stage axial compressor with a steady shaft offset [28]. The measurements were taken over a range of compressor flow coefficients from low loading to high loading near stall and are plotted as the stars and red line in Fig. 34. The model results (black line and circles) capture the behavior well and backward shaft whirl tendency is observed for low flow coefficients, while forward shaft whirl can be induced at high flow coefficients.

*α*and

*β*are the absolute inlet and relative exit swirl angles, respectively, $\varphi \xaf$ is the mean flow coefficient, and

*δϕ*is the perturbation in flow coefficient due to the tip-clearance asymmetry. If the tip clearance is large, the blockage increases and the flow coefficient decreases $(\delta \epsilon \u223c\u2212\delta \varphi )$. The bracketed expression is denoted by the

*blade loading indicator*, Λ

^{b}, so that

For a purely sinusoidal variation in flow coefficient, if Λ^{b} is positive, $\delta f\theta $ is in phase with the tip-clearance distribution. Integration of the tangential force distribution around the annulus thus yields a positive net tangential force *F _{y}^{b}*, inducing forward whirl. The opposite holds for a negative blade loading indicator, inducing a backward whirl. The blade loading indicator therefore determines the whirl direction tendency, and since it is based on mean flow coefficient and flow turning, it is applicable for both compressors and turbines.

This simple model is applied to the four-stage test compressor in Fig. 35. The inlet and exit swirl angles at mid-span are assumed constant over the operating range and yield tan *α* + tan *β* = 1.1. The condition at which the whirl tendency vanishes is set by Λ^{b} = 0, which is satisfied at a flow coefficient of $\varphi \xaf=0.42$. For flow coefficients less than 0.42, the blade loading indicator is negative and a backward whirl is induced. The opposite occurs for flow coefficients greater than 0.42. This is in good agreement with the trend in Fig. 34 and the flow coefficient for which the Alford force parameter vanishes.

If whirl instability occurs, the rotor shaft whirls at an off-center whirl radius at the natural frequency of the system. To investigate the effect of the whirl on the Alford force parameter, dynamic model calculations can be carried out for a range of forced shaft whirl frequencies. The tip-clearance compressor model can also be used to determine the dynamic stability of the compressor [30]. Since the shaft whirl induces flow distortion, the rotating stall frequency varies slightly with the forced whirl frequency. For this four-stage compressor, the rotating stall frequency ranges between 22 and 46 percent of rotor speed as depicted in Fig. 36. Also plotted is the Alford force parameter, computed over a range of flow coefficients for negative and positive forced whirl frequencies. At high flow coefficients and blade low loading, forced shaft whirl does not alter the Alford force parameter. However, at highly loaded, low flow conditions, the Alford force parameter is amplified over the frequency range of the rotating stall. This is due to the enhanced flow field distortion near the rotating stall frequency. The frequency collocation between enhanced whirl tendency and rotating stall can lead to frequency locking and coupling between the structural and fluid dynamic modes of the system. This resonant interaction between shaft whirl and rotating stall can lead to large amplitude flow oscillations and structural vibrations which can have dramatic consequences for the engine. A nonlinear, coupled system analysis, such as reported in Refs. [31,32], can determine the overall dynamic stability of the compressor-rotor system.

## 4 Forced -Response Testing for Diagnostics and Prognostics of Aero-Engine Operability Margins

A familiar example of more effective use of information in propulsion and power systems is the change over the past two decades in automobiles, which include an ever-increasing amount of electronics and software. Cars and trucks not only run for hundreds of thousands of miles and operate in a variety of environments with imperceptible changes in performance, but they diagnose their failures and remind operators when maintenance is needed for safety and convenience. A similar trend has occurred with aero-engines, even more prominently than with ground transportation.

To improve the life cycle operation of commercial and military engines, new intelligent aero-engine technologies in diagnostics and prognostics are being developed by engine manufacturers. For example, advanced data analytics, artificial intelligence (AI), and machine learning techniques are deployed to provide the customers with advanced engine health, performance, and life management information on demand, ensuring predictable asset availability. By embedding models of engine operation and optimization based on condition and mission, maintenance and fleet management costs can potentially be reduced.

In engine development and test programs, novel distributed and embedded fluid sensors and actuators can be used to characterize component performance and diagnose potential component faults. A key idea is *forced-response testing* to explicitly measure the level of damping and the boundaries and engineering margins for instabilities. A system with explicit dynamics is excited with known forcing and the dynamical properties and modes of the system are estimated by measuring the response. Forced-response system identification requires the integration of actuators and sensors tailored to system properties of interest.

Many of the ideas discussed here look toward advanced generations of engines embodying arrays of embedded actuators and sensors, which could be used for the active identification of system behavior. The basic concept is that the unsteady response of a system is dependent on the system's properties. As such, sensing the unsteady signals offers new opportunities, through real-time identification, for the characterization of the condition of the system.

A basic example is to consider the types of circumferentially propagating disturbances seen in Sec. 3.1. If the “steady-state” conditions around the annulus are circumferentially uniform, the waves will be sinusoidal in nature and have a given characteristic speed. If the conditions are non-uniform or asymmetric, for example, due to circumferential flow distortion or tip-clearance asymmetry as it occurs during the life of an engine, the waveforms are no longer monochromatic. The waves can have a rich spatial harmonic content, with the individual harmonics all moving at different phase velocities, and the group velocity of the wave varying around the circumference. The main diagnostic implications are that (1) the dispersiveness of the unsteady wave relates directly to the non-uniformity of the mean flow, (2) one can infer the nature of the latter from sensing and processing of the former, and (3) the sensing and processing, with the advances in instrumentation, can be done rapidly and with high accuracy.

### 4.1 Unified Forced-Response System Identification of Full-Engine Dynamics.

Synthesizing the concepts from the earlier Sections, a unified, dynamical engine system modeling framework can be established. The aim of the framework is to characterize the coupled aerodynamic, aeromechanical, and rotordynamic behavior of an engine system at a level of fidelity that is useful to identify the sources and mechanisms responsible for vibrations and instability. The methodology also lends itself to guide the setup of forced-response simulation and experimentation. What follows is a brief overview of the different subcomponents and their interlinkages which constitute the overall methodology depicted in Fig. 37. The detailed descriptions are omitted here as the previous sections lay out the ideas and the level of modeling required. Known forcing can be introduced via commanding a set or an array of actuators such as shakers, air injectors, fast-acting guide vanes, or bleed valves. The actuator dynamics, captured by transfer function *A*(*s*), provides the actual forcing input to the dynamic engine system, *G*(*s*), comprised of the rotating stall, surge, and flutter dynamics. The dynamic response of the engine system, in terms of flow field oscillations and structural motion such as blade and rotor deflections, is recorded via a set of sensors. The sensors exhibit their own dynamics captured by transfer function *S*(*s*). There can be significant coupling between the turbomachinery aerodynamics and the rotating structure (e.g., via Alford forces). As such a feedback loop via the rotordynamic transfer function, *R*(*s*), is introduced to characterize the rotor deflection under aerodynamic loads.

The unified dynamic system model can guide the design of forced-response experiments in full-engine systems and help determine the placement of the required actuators and sensors to ensure controllability and observability of the dynamic behavior of interest. The measured response of the dynamic engine system allows the direct characterization of the system eigenvalues and quantification of (i) stall margin and the aerodynamic stability boundary, (ii) flutter damping and the aeromechanical stability boundary, and (iii) rotordynamic damping and the whirl boundary. These engineering margins and damping levels can be assessed for a range of engine operating and health conditions. Furthermore, faults can be introduced to investigate the consequences of dynamic behavior and potential failure modes. This can help establish a “signal catalogue” for typical vibration signatures, instabilities, and pre-cursors of catastrophic failure.

Given the importance of these engine problems, experiments should be targeted at full engines that are instrumented to capture the information needed. An experimental facility was recently developed at the MIT Gas Turbine Laboratory and a 1,500 lbf thrust Pratt & Whitney PWC 615F two-spool turbofan engine was set up for such industry-class experiments. As illustrated in Fig. 38, the engine is mounted on a floating test stand with dual load cell thrust measurement capability. Calibration of the load cells is facilitated via an engine centerline static fan-pull test. Shakers can be installed on the support frame for forced-response testing. A large number of pressures and temperature sensors plus an on-engine turbine-type flowmeter for burning fuel allow the detailed characterization and industry-standard measurement of engine performance. The engine is electronically controlled via a Full Digital Engine Control (FADEC) system, and a pressurized fuel system allows for rapid engine transients. Vibration monitoring is provided in two planes for both the low-pressure and the high-pressure spool. Current research efforts are focused on (1) full-engine rotordynamic system identification, and (2) on-engine forced-response fan aero-damping characterization [33]. The type of new information that can be generated is described next.

### 4.2 Forced-Response Identification of Full-Engine Rotordynamics.

Identifying the source of a rotor vibration-driven response is challenging as the sensed response can depend on the frequency and operating conditions. Furthermore, distributed internal thrust, aircraft altitude, and Mach number, and rotor speed can significantly influence the boundary conditions of the rotordynamic system and thus the modal characteristics of the multitude of vibrational modes of the propulsion system. New methods of identifying vibration sources must be developed to meet the goal of reducing sustainment costs.

The challenges are further complicated by the continuing trend in engine architecture of longer, more flexible core and fan/low-pressure rotor systems as described in Sec. 3.2. This has resulted in increased sensitivity to assembly error-related unbalance as well as component and module residual imbalance. Relative to legacy engines, modern rotor systems and static frames are more flexible and the external components constitute an increasingly greater proportion of the system mass. The propulsion system dynamics has transformed from isolated rotordynamic problems into large engine system dynamic problems. Thus, families of rotor critical speeds exist, where there can be strong coupling between the rotor modes and the static frame response, rendering these modes suitable for external detection.

Damping is incorporated in multiple forms, all of which play a key role in system response. Squeeze-film dampers are placed at multiple locations to reduce the response at critical speeds. Their dynamic behavior is complex in nature, as it changes across a wide range of engine operating conditions. The hysteretic structural and material damping is not well understood. Furthermore, the speed of sound effects, bubbly damper oil due to air entrainment, and time-varying stiffness of the support structure can all affect the rotor mode shapes and dynamic response. In summary, early knowledge of the problematic critical frequencies and mode shapes can yield dramatic cost and time savings in an engine development program.

Motivated by this, the PW615-F engine serves as a test bed to explore forced-response system identification in a full engine. One objective is to infer the rotor mode shapes and least stable modes of the low-pressure and high-pressure spools, and to determine the level of damping across the entire operating range. To guide the design of the experiments, a dynamic engine model was set up including actuation via shakers attached to the front engine mount on the support frame and multiple displacement sensors to measure the motion of the two spools as illustrated in Fig. 39. In the planned experiments, two air-cooled electrodynamic shakers will be used to induce forced whirl over the full-engine operating speed range and the rotordynamic response will be measured by four displacement sensors for each of the spools. The first few critical frequencies of the low-pressure (LP) and high-pressure (HP) spools, estimated by the dynamic engine model, are summarized in the Campbell diagram in Fig. 40. The low-pressure and high-pressure spool modes are marked in red and blue respectively; solid lines refer to forward whirling modes while dashed lines refer to backward whirling modes. The modes are labeled based on their mode shapes which are illustrated in Fig. 41. The first engine order forcing frequencies for each of the spools, 1-E N1 and 1-E N2, are plotted as the solid black lines in Fig. 40. For clarity, higher order rotor modes and static casing modes are omitted in the Campbell diagram.

Virtual forced-response tests were carried out using the model dynamics as a representation of the actual engine system to demonstrate the feasibility of the actuator and sensor setup. Following the methodology and sensor locations in Fig. 39, frequency-by-frequency system identification using sinusoidal command signals was carried out. The rotor transfer functions were then inferred from spectral analysis of the shaker command and displacement sensor response signals at each of the discrete frequencies. The estimated frequency response of the low-pressure and high-pressure spools is plotted in Fig. 42. Multiple, lightly damped modes can be identified in the frequency response. Forward and backward whirling modes can be discerned by inspecting the phase response (not shown here). Coupling between the low-pressure and high-pressure spools is observed for the fan and HPT modes as the response is visible in both measurements. Note that the mode near 80% corrected low-spool speed is identified as a casing mode as it does not change the frequency with rotor spool speeds (not shown here) and is visible in all traces. Also, analysis shows that the HPC forward whirling mode is so highly damped that it is not detectable at the levels of forcing implemented.

The damping ratio of the dominant forward whirling modes (Fan, HPT, and LPT), inferred from the frequency response curves at various engine operating conditions, is depicted in the inset in Fig. 43. The forward whirling HPT mode has marginal damping at low corrected fan speed. To illustrate the change in damping ratio along the fan operating line, the size of the operating point is plotted inversely proportional to the damping ratio of the forward whirling HPT mode. While the corrected fan speed is low when damping is marginal, the 1-E N2 crossing of the HPT forward whirling mode occurs at about 50% corrected core speed (N2) as shown in the Campbell diagram in Fig. 40. At higher levels of rotor unbalance, increased damping is required to avoid excessive vibration levels. This can for example be provided by installing an additional squeeze-film damper near the location of the HPT bearing support.

### 4.3 Characterization of Fan Flutter Damping.

Self-excited aero-elastic vibration of fan blades, or fan flutter, has historically been a major problem in aero-engines. For a neutrally stable system, the unsteady work done by the air on the fan blades balances the work dissipated by friction and material damping. To compute this unsteady work, knowledge of the vibrational deflections and unsteady aerodynamic forces is required. Great progress has been made in finite element analysis (FEA) and unsteady CFD but, as conditions change across the fan map, damping and the flutter boundaries are still hard to predict. The latter is predictable if damping can be characterized but plotting the flutter boundaries in a compressor map as in Fig. 44 is an oversimplification.

The challenge is that the flutter is lightly damped over a wide range of operating conditions,^{8} and since flutter damping depends on stagnation pressure and temperature (the elastic modulus is temperature dependent), the flutter boundaries are very sensitive to flight conditions and altitude. As such, Fig. 44 should capture the flutter boundaries in a three-dimensional representation with density as the third dimension.

A series of commercial aircraft accidents occurred in the late 1990s attributable to fan flutter [35]. The stretched version of an aircraft required approximately 7% higher engine thrust. This could be achieved by the same engine operating at 2–3% increased fan rotational speed. Since this was a minor increase in fan speed, a ground test was considered appropriate not requiring flight testing the engine at the higher thrust rating. To increase the fan speed at sea level without overheating the turbine, the bypass nozzle area was increased to down-match the fan operating line. This is sketched in the fan map in Fig. 45. The engine passed the ground test operating at 103% speed with the fan working line between the flutter regions (shaded area in Fig. 45). With the flight nozzle, the flutter boundary would have been “kissed” at the max climb operating condition but was not detected in the ground test. After certification, engine failures occurred on several flights. The probable cause was determined to be flutter-induced HCF leading to the failure of the outer part of the fan blade. Subsequent flight testing demonstrated that flutter indeed occurred at maximum climb thrust at altitudes of 25,000 ft and greater [35].

#### 4.3.1 Full-Engine Fan Flutter Damping Assessment.

Following the ideas for forced-response system identification of rotordynamic modes and vibrations, subsequent experiments are planned to measure flutter damping over a broad range of operating conditions in the PW 615-F turbofan engine. Actuation will be provided by an array of high-power zero-net mass flux actuators capable of exciting the flutter modes [33]. The methodology is summarized in Fig. 46.

A detailed investigation and bench testing of the actuator array were carried out to demonstrate the required forcing capability. Further, the actuator design and authority apply to large engines as well since the engine mass flow scales with the square of the fan radius whereas the actuator mass flow scales with the square of the speaker radius. As such, the mass flow ratio is independent of engine size, and the actuator authority is constrained only by the availability of sufficiently large speakers as the fan diameter increases [33]. Inlet duct acoustics are known to be an additional source of flutter damping (both positive and negative) for fan speeds where the upstream acoustic field is cut-on and the downstream field is cut-off [37]. While the aim is to measure the blade-only damping for all nodal diameters and fan speeds, there is no single inlet length for which the inlet acoustics have no influence. As such, the inlet length is chosen to minimize this influence on the flutter modes of interest.

The fan blade deflections due to forcing will be measured using a fiber optic tip-timing system which consists of a small probe installed into the fan outer casing. The goal of system identification is to estimate the flutter eigenvalues and thus the mode damping, from the measured blade bending response to a given excitation signal. Virtual forced-response system identification tests were carried out using the dynamic engine model to demonstrate the feasibility of the actuator and sensor setup and to assess the robustness of the methodology in light of a wide range of uncontrollable factors such as sensor and process noise sources. Figure 47 depicts the estimated flutter frequency response for engine operation at take-off conditions. The results illustrate the impact of noise on the measured transfer function for first forward nodal diameter forcing (*n* = 1). The variance in the transfer function magnitude and phase is exacerbated over frequency ranges where the coherence is near zero (e.g., between frequencies of 1 and 2 times rotor frequency), indicating that the response in this frequency range is strongly governed by noise. Near the frequency of the first forward nodal diameter flutter mode, the coherence is close to unity with greatly reduced variance in the transfer function. The useful frequency band for system identification of the first nodal diameter flutter mode is marked by the red dashed lines at 2.5 and 3.1 times rotor frequency. The flutter damping ratio can be inferred from the transfer function as discussed earlier.

The same procedure was carried out for fan operating points near choke, design, and stall at design speed, and a statistical analysis was applied to quantify the error bars in terms of a 95% single-sample confidence interval [33]. Since each vibratory mode of a fan blade yields a family of flutter modes, and in each mode, any individual fan blade is vibrating at the same frequency and with the same mode shape but the phasing between adjacent blades varies from mode to mode, it is useful to characterize the flutter modes by the so-called *interblade phase angle*. Circular symmetry restricts the possible values of the interblade phase angle to a finite set of values, and the flutter modes are often labeled by the nodal diameter *n* set by the number of blades oscillating in phase around the circumference.

The mean damping ratio and its 95% single-sample confidence interval are given in Fig. 48 for fan operating points near choke, design and near stall as a function of interblade phase angle. The error in the damping ratio depends on the amount of modal response for a given level of forcing. Lighter damped modes are more susceptible to forcing and yield stronger responses and thus higher signal-to-noise ratio and lower error. The forcing level is also affected by the fan operating conditions. As the actuator-to-engine mass flow ratio decreases for higher fan flow coefficients, the effective forcing level is reduced. The level of inlet turbulence also increases with the flow coefficient so that both effects can reduce the signal-to-noise ratio.^{9} The opposite occurs at lower flow coefficients with improved signal-to-noise ratios. For the operating points considered, the mean error in damping ratio is generally within ±3% and the 95% single-sample confidence interval within ±15%. The largest error of nearly 25% occurs for the second forward nodal diameter since, at fan design speed, the mode is near the actuator bandwidth limit where actuation power is dramatically reduced.

Using this forced-response system identification methodology, the damping ratio of the relevant flutter modes can be assessed across the entire fan operating map so as to create contours of constant flutter damping ratio, much like it is done for compressor efficiency. Figure 49 illustrates this idea schematically for a selected flutter mode. Such a map can be created for each nodal diameter so as to identify the regions of low damping and the most critical flutter modes. The flutter boundary is set by the contour of zero damping ratio of the least stable flutter mode. The map also allows us to determine how much the damping ratio varies across a speed line or along an operating line for each of the flutter modes. Furthermore, the impact of fan blade deterioration due to rubs with the casing abradable and FOD can be assessed in a controlled setting.

Although these ideas do not yet fully address the dependence of the flutter boundaries on altitude and flight conditions mentioned in the introduction, force response system identification enables the characterization and quantification of engine operability margins under real engine operating conditions. This new knowledge can be invaluable when strategic design choices and trades, with potentially dramatic consequences for engine certification, must be made during an engine development program.

## 5 Concluding Remarks

Many aero-engine problems encountered past the design and development phases are dynamic in nature. The viewpoint taken in this lecture was from an overall engine perspective and, while the problems addressed were different; they all had the following features in common: (1) the level of damping was marginal and difficult to characterize, (2) the dynamic behavior led to instability often with catastrophic consequences, and (3) dynamical system modeling proved useful in addressing the challenges and in solving the technical problems.

With an emphasis on *what* to model, the aim of the lecture was to show that real-world turbomachinery gas turbine engine problems can effectively be solved by pivoting from computing continuum fluids and structures to applying modern computer power for modeling dynamical systems. By deploying mathematical tools to exploit these new models in combination with experimental verification of the concepts in component rigs and full engines, the added value is direct insight into root cause, characterization of mechanisms, and the quantification of damping. All of these are necessary to address the hard problems at hand.

A key idea was forced-response system identification of the dynamic behavior of turbomachinery components and full aero-engines. The methodology provides new approaches for quantifying engineering and operability margins in support of future engine diagnostics and prognostic, critical for engine safety and health monitoring.

## Footnotes

For example, near the fan flutter boundary, the unsteady work done by the air on the blades balances the work dissipated by friction and material damping.

A Bodie maneuver consists of a hot engine throttle chop from full power to idle power, holding the idle power setting for several seconds, and then a throttle slam back to full power. Bodie maneuvers are used during engine and aircraft certification and are typically the worst handling tests an engine may experience.

Stall margin is defined here as the distance between the operating point and the corresponding point on the stall line at the same inlet corrected flow.

A thin turbomachinery disk with *L*/*D* of 0.07 with the bearing on the outer periphery was considered.

The behavior is analogous to the diffusion of heat in a fluid due to a sudden increase in temperature of a plate. The analogy is that vorticity spreads like heat.

In contrast, rotating stall and surge are only marginally damped near the stall boundaries and incipient instability is evident.

The inlet turbulence intensity as % of the mean flow velocity may stay the same but the ratio of turbulent velocity fluctuations to velocity perturbations from the actuator will decrease.

See Note ^{1}.

## Acknowledgment

Much of the work described in this lecture is the result of collaborations with various engine companies and the contributions of a number of individuals and many of whom appear as co-authors of the referenced papers. In particular, the author would like to express his gratitude to (in alphabetical order) Dr. N. Cumpsty, Dr. F. Ehrich, I.-Y. Hur, Dr. S. Jacobson, Dr. A. Kiss, Dr. L. Liu, Dr. J. Paduano, Dr. J. Sabnis, Dr. O. Sharma, Dr. A. Strazisar, Dr. C.J. Teo, and Dr. D. Wisler. The author is also grateful to the anonymous reviewers of the draft manuscript for their useful and constructive comments. Finally, the author is most greatly indebted to Drs. A. Epstein and E. Greitzer for 25 great years as mentors, colleagues, and friends.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

Data provided by a third party are listed in Acknowledgments.

## Nomenclature

*n*=nodal diameter, spatial harmonic

*s*=complex frequency

*t*=time

*p*=static pressure

*A*=area, integration constant

*B*=integration constant

*C*=damping coefficient

*K*=stiffness

*Q*=quality factor

*W*_{lo}_{st}=lost work

*m*, $m\u02d9$ =mass, axial mode no., mass flow

*x*,*y*=coordinates

*F*,*E*=force, energy

*G*(*jω*) =transfer function

*α*=absolute flow angle

*β*=relative flow angle

*γ*=phase angle

*δ*=logarithmic decrement

*Δ*=difference

*ζ*=damping ratio

*ɛ*=displacement

*η*=loss factor, coordinate

*τ*=non-dimensional time

*ξ*=coordinate

*σ*=growth rate

*ω*, Ω =rotation rate, frequency

*ω*, Ω_{0}=_{N}natural frequency

- Ω
=_{W} whirl frequency limit

## References

*Multi-Wafer Rotating*MEMS Machines, MEMS Reference Shelf