1

CONCEPT PAPER: A NOVEL PERSPECTIVE ON

BLADE/CASING RUB PROBLEM IN TURBOMACHINERY

Nejat Olgac*, Umut Zalluhoglu and Ayhan S. Kammer

University of Connecticut, Mechanical Engineering Department, 191 Auditorium Rd., Unit 3139,

Storrs, Connecticut, 06269-3139

Abstract

This paper presents a new pathway in studying the ubiquitous blade/casing rub problem

in turbomachinery. The novel concept starts from the claim that the dynamics is inherently akin

to internal machining operation using compliant cutters with relatively rigid workpiece (i.e., the

casing). This impingement dynamics introduce a “regenerative mechanism” which is earmarked

by time delays. The ensuing time-delayed dynamics can be stable which is ideal, or unstable

which results in growing interference amplitudes between the blade and the casing, dynamic

nonlinearities become dominant, which typically imparts a limit-cycle behavior. Authors claim

that this “unstable mode” is the common state of operation in most modern-day turbomachinery.

We introduce a new perspective in looking at these problems, departing from the unique

analytical capabilities of authors’ group. The stability resolution of time-delayed systems (TDS)

is enabled using a recent mathematical paradigm, called the Cluster Treatment of Characteristic

Roots (CTCR). This document evolves further in proposing a procedure to mimic the

blade/casing dynamics in a single experimental abstraction which would dramatically reduce the

cost of commonly-used spin pits.

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I. Problem Statement and Motivation

Tip clearances between the rotor blades and casing are very important feature in turbomachinery

industry. They need to be small for functionality, but the complicated interface between the rotor

(with attached blades) and the casing become critical due to wear and more importantly due to

rub-triggered oscillations and ultimate fatigue. This, so called “blade-casing rub problem” has

been a focal point of investigations for over 25 years [Padova et al., 2007; Choy et al., 1989;

Bataille et al., 2011].

This phenomenon was somewhat handled using a soft rub strip layer as a liner inside the casing.

It has been reported recently that even with such a layer, some disturbing wear features and very

importantly some blade root cracks are observed [Padova et al., 2011; Batailly et al., 2012].

These occurrences invited some renewed interest in the problem. This study reflects a novel

angle of approach in this vein.

Most investigators coalesce on several common points relevant to the rub dynamics: (a) It is very

complex due to the engagement of several compliant objects being in the midst of the dynamics,

(b) Rub characteristics being very different to characterize. Therefore typical scientific

investigation on this problem is bound to be entrapped within either sophisticated FEA codes or

experimentally dominated empirical realm [Turner et al., 2012].

3

Figure 1. Load on the blade tip (a) rotor view (b) blade alone view [Turner et al, 2012]

Typical impingement loading scenario is conceived as in Figure 1, when the rotating

blades radially interfere with the casing. This radial interference originates tangential, radial as

well as axial forcing between the blades and the casing which cause some respective

deformations. These deformations are conceived to have a steady-state part supplemented by a

dynamic oscillatory part. From the stability perspective the latter part plays the most crucial role,

and that is what we focus on here. The amount of steady deformation is certainly important but

not as critical from the angle of fatigue concerns, and thus overlooked for the conceptual

arguments we present next.

4

Figure 2. Dynamic milling system [Altintas et al., 2008]

We present a novel conceptualization of this rub dynamics, next. When rotating blades

interfere on the casing (and the abrasive rub-strip), they behave similarly to the cutting tool flutes

in internal milling removing material from the workpiece (which happens to be the casing in this

case). This dynamic similarity yields a regenerative dynamics which feeds onto itself with some

time delay (see Figure 3).

Figure 3. (a) Blade-casing impingement (b) Regeneration mechanism (c) Abstract dynamics

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In Figure 3b leading creates in its wake tx amount of residual deformation on the casing. As

the trailing blade tries to achieve the same radial deformation as the leading blade did sec

earlier [which is tx ], a net radial disagreement amount txtx between the blade

and casing appears. The time delay is, in fact, the blade passing period. Therefore it is the

direct representation of the bladed-rotor’s angular speed the time delay )60/( n , where

is the rotational speed in RPM and n is the number of blades on the rotor.

This time-delayed forcing argument connects the present blade-casing rub dynamics to a very

well-known problem, which is the regenerative force description in machine tool dynamics

(Olgac and Sipahi, 2005 and references therein). This signature is amplified in zoomed

conceptualization in Figure 3b for a “rigid blades – compliant casing” scenario. The causal

avalanche starts from this radial disagreement (a.k.a. impingement). Tangential and axial loading

terms all originate from this radial disagreement. These regenerative forces act on the compliant

structure very much like the “regenerative chatter” phenomenon in machining. Benefiting from

the machine tool literature we can state the following logical sequence: If the dynamics were

stable, the instantaneous perturbations away from the equilibrium deformation would

exponentially decay, and this behavior is desirable. If, on the other hand, the resulting interface

dynamics were unstable it would induce increasingly large excursions and impingements

between the blade and the casing. This is a detrimental outcome due to the increased likelihood

of initiating fatigue on the structural components.

These simple observations on the two-blade scenario, can be extended to a complete

bladed-rotor and casing interface. The end effect of the multiplicity of blades involving into the

dynamics turns out to be marked by a very similar time-delay effect as the blades are periodically

placed on the rotor. We simply declare an important hypothesis that the resultant interface force

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between the blades and the casing carry the insignia of txtx , where x represents the

radial impingement amount.

Further conceptualization of the rotor (with blades) and the casing as two decoupled

compliant structures is explained next. It is clear from earlier investigations [Turner et al, 2012

and references therein] that the blade loading can be treated in 2-D. Axial forces are ignored as

they are much smaller vis-à-vis radial and tangential forces. For stability assessments the most

critical evolution takes place within the linearity zone (i.e., while the variational excursions away

from the equilibrium are still small). The variational components of both radial and tangential

forces are realistically assumed to be proportional to radial deformations at the tip. In order to

facilitate the crucial analytical formation we impose several critical assumptions to the dynamics:

(i) The blade tip deflections in radial and tangential directions are uniform across the blade

width.

(ii) Both rub (tangential friction) and compression (radial) contact forces across the width

and the axial length of the blade are uniform.

These assumptions, in fact, reduce the problem from 3-D to 2-D conceptualization. This paper

goes one step further by proposing the more rudimentary depiction in 1-D. The simple claim is

that the variational interactions between blade and casing are negotiated primarily in radial

direction. Compliances, impingements, friction and other forces all emanate from these

excursion or disagreements in radial direction. This is a remarkable abstraction to make, which

has never been attempted to date to the best of authors’ knowledge. We wish to make exemption

to a very recent study published by a group from McGill University [Salvat et al., 2013]. The

present work is completely independently developed and substantially different as we later

explain in the text.

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The second very important and often neglected property has to do with the nature of the

radial tip force variations. As mentioned earlier, the radial tip force appears with a signature of

txtxkFr (1)

where k is the consolidated stiffness which includes the radial, tangential and axial stiffnesses in

the radial direction. Further abstraction comes by considering casing and rotor as two

independent dynamics which are connected via the blade tip force (1). Figure 3c depicts this

conceptualization.

This 1-D abstraction of rub problem is the mother lode of the ensuing dynamics although

many critical parameters also play a role, such as temperature of the rub surface, blade geometry

in 3-D (Batailly et al, 2012; ), geometric design clearance, rub strip bonding characteristics to the

casing etc. This claim brings us to another point which has never been explored to date in the

scientific community, again to the best of our knowledge. We wish to build the rest of the paper

on this novel conceptualization, its intriguing mathematical implications and potential practical

ramifications in the industrial realm.

II. Time-Delayed Dynamics and Stability Concerns

The depiction in Figure 3c, when considered within the excursion limits that tolerate

linearity assumptions, can provide a wealthy resource of information. In this section we wish to

elaborate on the mathematically intriguing features of this dynamics.

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Figure 4. Conceptualization of blade-casing interface dynamics

A more explicit representation of Figure 3c is given in Figure 4. bc xx , are displacements

of the casing and blade, respectively. rF is the radial tip force as defined in (1). One should note

that in these representations cb xxx . Lumped masses mmmm ,..., 21 , damping coefficients

mccc ,..., 21 and stiffnesses mkkk ,..., 21 are for casing, whereas nmmm mmm ,..., 21 , nmmm ccc ,..., 21

and nmmm kkk ,..., 21 are for the bladed rotor structure. One can interpret these selections as the

proper selections of lumped parameters to yield m (and n ) dominant modes of casing (and

bladed rotor).

Collocated force-displacement transfer functions of the two substructures are given by

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sF

sXs

sF

sXs bc

2

2

1

1 (2)

Substituting the blade-casing interaction force defined earlier in (1) in the transfer functions in

(2) and using their Laplace domain representation, the governing equation of the unforced

dynamics becomes

cc

s

b

s XXekXek 11 11 (3)

bc

s

b

s XXekXek 11 22 (4)

Rewriting these equations in matrix form

0111

111

22

11

b

c

b

c

ss

ss

X

X

X

X

ekek

ekek

(5)

The determinant of leads to the characteristic equation of the combined system, which is

11)det( 21 seksCE (6)

The stability of this particular dynamics is handled uniquely using the Cluster Treatment of

Characteristic Roots (CTCR) paradigm of UCONN researchers [Olgac et al., 2002, 2005,

2006]. CTCR can create exhaustive and non-conservative declaration of stable and unstable

operating conditions in the domain of ),( k or equivalently ),( nk where n (RPM) is the rotor

speed. Notice that /160/ nN , where N is the number of blades on the rotor.

The point of interest to the industrial community is that we can create a parametric

declaration of stability zones. For instance, one can obtain a stability outlook in the domain of k-

which has been missing in the literature. Such a stability tableau can bring a considerable

analytical capability in the hands of the turbo-machinery designer of rub strips. A proper

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stiffness can be selected so that a given bladed rotor RPM (which determines the delay ) will

render stable interference mechanism, thus avoiding the growth of the blade-casing chatter.

Such a capability could potentially ease the starting steps of the design work. If rotor-casing

interface was operating in stable mode the ensuing chatter excursions would decay, and there

would be no fatigue concerns. As the industry is continuously alarmed about the onset of fatigue,

we are making a qualified generalization that “in today’s turbo-machinery industry the typical

operation is in unstable mode”.

The natural consequence of this remark is an invitation to design the operating conditions

such that the incursions will remain “stable”. This will assure disturbance rejection capabilities.

We can provide the design parametric boundaries for such an analytical as exemplified in the

next section.

CTCR can provide a further capability, beyond stability. It can also declare disturbance

rejection levels at various design points, again in the ,k parametric space. A designer can

depart from the baseline provided, with this information. These are the key contribution of the

preset paper, differently from [Salvat et al., 2013]. The latter presents an unclear path of

modeling the blade-casing dynamics. They deploy a methodology called the “semi-discretization

technique” for stability assessment of the dynamics which needs to be periodic-time-variant

linear-time-delayed in nature. We believe this assumption is unnecessary and even incorrect, as

there are more than a few blades in engagement during the rub event, the rub forces should not

be considered time variant, let alone periodic.

Furthermore if there is an external excitation force on the casing as shown in Figure 4, the casing

and blade excursions can be determined as

11

0

11 ext

b

c F

X

X (7)

Then transfer functions for the combined system can also be found. For example the casing

displacement with respect to external force Fext acting on it becomes:

1))(1(

)1(

21

121

s

s

ext

c

ek

ek

F

X (8)

III. Case Study

We consider a realistic example in this section, with 3-mode representation for both the casing

and the bladed rotor side. Obviously both structures exhibit infinite number of modalities with

some dominant select group of them being dominant modes. In practice, these dominant modal

characteristics are expected to be obtained from a series of dynamic response experiments (for

instance, impact hammer tests). Typically, this information is summarized as undamped natural

frequencies and damping ratios for each mode. These features are, indeed, sufficient to construct

the factors in the denominators of the respective transfer functions of ss 21 , . For our

example case study, these system parameters and corresponding transfer functions for the

isolated blade and casing dynamics are taken as in Table 1. For these selections we used the

available data from two recent theses [Arzina, 2011; Salvat, 2011] as departure values, assumed

25-bladed disk.

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Table 1. System parameters for 3-mode casing and bladed rotor model

Casing Blade

#1 #2 #3 #4 #5 #6

m [kg] 3 5 2 3 6 8

c [Ns/m] 60 40 80 30 50 25

k [N/m]

22172163114956

1592934

1104.81093.11069.71015.11032.96890240

101.61025.91048.1109048

ssssss

sssss (9)

21172153104956

15102834

21091045.11026.71071023.1292030

1098.11012.11068.586015

ssssss

sssss (10)

The respective modal natural frequencies and damping ratios to casing and the bladed rotor are

given in Table 2. Using these transfer functions one obtains the corresponding system

characteristic equation including the interface forcing, similar to (6). The deployment of our

paradigm CTCR follows next. Mathematical details are left to [Olgac et al., 2002] and the

references therein in order to focus this presentation to the practicality of the findings. The

CTCR routine starts with an exhaustive determination of the parameter set (k, ), where this

system exhibits marginal stability (i.e., the characteristic equation has imaginary roots). It

further evolves into declaring those design conditions (i.e. the selections of rub-strip stiffness k

and blade passing period ) which will render stable and unstable rub interface behavior.

Remember the direct connection of )*60/( n which facilitates the declaration of the safe

(stable) operating conditions in (k, ) space. We will return to this connection later in this

section.

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Table 2. Modal characteristics of the blade and the casing models

ωn [Hz]

Casi

ng

208.50 0.024

360.81 0.065

928.17 0.022

Bla

de 180.50 0.003

498.52 0.008

838.19 0.012

The results of the CTCR paradigm are presented in Figure 5. It contains the key strength and the

novelty of this paper. From the practical perspective the stable operating conditions are marked

in grey shading. The aero-engine designer, now, has a tool to depart from: an appropriate

selection of the rub-strip stiffness and operating speeds (or the corresponding ) in order to

guarantee “stable” rub dynamics. This is not, by any means, the solution elixir to all misdeeds

that can happen in this very complex dynamics, but a departure point for finding a remedial

pathway.

Once again, the uniquely obtained CTCR-based stability picture in Figure 5 is exhaustive and

non-conservative. That is, there cannot be any other stability region(s) than those that are

declared by the CTCR and the boundaries of these regions are exact (not approximate). There is

no other routine in mathematics today, that can claim these capabilities, including those

methodologies which are utilized in [Arzina, 2011; Salvat, 2011]. We offer this unique strength

to address an industry-wide practical problem in the turbo-machinery field.

Short of experimental validations, we can also verify the above claims on Figure 5 using

simulations. They are presented in Figure 6 through 8 for various case selections (marked by

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A,B and C in Figure 5). These simulations are showing the system responses to an impulsive

disturbance (a.k.a. impact response). As expected A and B are showing stable reactions while

the setting in C is unstable. Frequency response of the system in case A is also displayed in

Figure 9 just to provide some additional information beyond the stability of the dynamics. It

shows that there are two critical frequencies for this case which need to be avoided, if possible.

This is a by-product of the CTCR-based analysis, which can also be used as an additional design

tool.

Figure 5. CTCR-generated stability regions for the blade-casing rub dynamics. Grey shading

indicates the stable region.

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Figure 6. Impulse response of the system for A (k = 106 N/m, τ = 0.0008 s)

Figure 7. Impulse response of the system for B (k = 105 N/m, τ = 0.002 s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5x 10

-10

Time [sec]

Dis

pla

cem

ent

[m]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1

0

1

2

3

4

5x 10

-10

Time [sec]

Dis

pla

cem

ent

[m]

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Figure 8. Impulse response of the system for C (k = 105 N/m, τ = 0.004 s)

Figure 9. Frequency response (FR) of casing to external disturbance for parameter configuration

A (k = 106 N/m, τ = 0.0008 s)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-8

Time [sec]

Dis

pla

cem

ent

[m]

103

104

-200

-150

-100

-50

Magnitude [

dB

]

Combined FR

Initial FR

103

104

-200

-100

0

100

200

Frequency [rad/s]

Phase A

ngle

[deg]

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Figure 10. Duplicate of Figure 5 in (k, ) domain.

Figure 10 presents the same stability picture in Figure 5 but this time in the space of (k, ), for

ease of conveyance. The abscissa is now changed in direction due to inverse relation between (τ

and ). As expected the larger brings larger stable region. Desired property is larger

stiffness k. at a desired RPM. The key question arises from this analytical observation: “If

one cannot play with this ideally designed , is there a particular stiffness bound which can be

implemented which will assure stable operating conditions in the event of unavoidable rub

interactions?”. This is an interesting question that the turbo-machinery industry has been

thriving to answer over 25 years [Choy et al., 1989, Padova et al., 2011, Turner et al., 2012].

The perspective which is presented in this paper, with the help of our unique mathematical

paradigm (CTCR), may crack open a novel pathway, never attempted before. With this spirit,

one of our group’s next foreseeable research directions is set.

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IV. Further Practical Ramifications and Conclusions

These observations and the proposed perspective have far reaching ramifications. If the

backbone of the complex blade-casing dynamics is reduced to the abstract construct as depicted

in Figure 4, one can simply create a test platform where the interface forces can be artificially

introduced between the blade and the casing. In fact these forces can be created in the form of

time-delayed feedback forces. Such a structure could replace a very expensive proposition of

building a spin pit with real turbomachinery, rub strip and casing elements. The tests can be

conducted with much less cost, and the analytically obtained rub-strip characteristics can be put

to test. This is a game-changer just from the design point-of-view.

Note that in Figure 5 at point C we can use much harder rub strip which is desired,

without inducing instability. Such a critical information cannot be obtained without the help of

CTCR.

The key novelty in this work, once more, is on the critical proposition that the blade-

casing rub dynamics should be looked at as a time-delayed system structure. Short of that will

simply ignore the most important physical mechanism which imparts instability.

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