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Proceedings of ASME Turbo Expo 2018: Turbomachinery Technical Conference and ExpositionGT2018

June 11-15, 2018, Oslo, Norway

GT2018-75488

THE DETERMINATION OF STEADY-STATE MOVEMENTS USING BLADE TIP TIMING DATA

Mohamed MohamedSchool of Mechanical, Aerospace and Civil

Engineering, University of Manchester,M13 9PL, Manchester, United Kingdom

Philip BonelloSchool of Mechanical, Aerospace and Civil

Engineering, University of Manchester,M13 9PL, Manchester, United Kingdom

Peter RusshardEMTD Ltd

NG8 2RD, Nottingham, United Kingdom

ABSTRACT

One of the main challenges of the Blade Tip Timing (BTT) measurement method is to be able to determine the sensing position of the probe relative to the blade tip. It is highly important to identify the measurement point of BTT since each point of the blade tip may have a different vibration response. This means that a change in measurement position will affect the amplitude, phase and DC component of the results obtained from BTT data. This increases the uncertainty in the correlation between BTT measurements and Finite Element (FE) modelling. Also, the measurement point should ideally be located to measure as many modes as possible. This means that the probe’s position should not coincide with a node, or a position at which the sensor misses the blade tip. Changes in the sensing position usually arise from the steady state movements of the blades (change in mean displacement). Such movements are caused by changes to the static (thermal and pressure) loading conditions that result from changes in the rotational speed. Such movements usually have a constant direction at normal operating conditions, but the direction may fluctuate if the machine develops a fault. There are three main types of movements of the sensing position that are considered in this paper: (1) axial movement; (2) blade lean; (3) blade untwist. Ideally, the sensing position is known based on the geometries of both the blade and the probe, but due to different types of movements of the blade this position is lost. Very few works have researched the extraction of the sensing position. Such preliminary works have required a pre-knowledge of

mode shapes and additional instrumentation. The aim of this paper is to present a novel method for the identification of the BTT sensing position of the probes relative to a blade tip, which can be used to quantify the above movements. The developed method works by extracting the steady state offset from measurements of blade tip displacements over a number of revolutions as the speed changes from zero to a certain value. Hence, that part of the offset that is due to the angular positioning error of the probes (outside the scope of this work) is cancelled out (since it is independent of speed). The change in steady state offset is then processed to identify the three possible movements. The new method is validated using a novel BTT simulator that is based on the modal model of the FE model of a bladed disk (“blisk”). The simulator generates BTT data for prescribed changes to the sensing position. The validation tests show that the novel algorithm can identify such movements within a 2% margin of error.

NOMENCLATURE

Blisk Bladed diskBTT Blade Tip TimingFE Finite ElementOPR Once per revolutionSFP Sine fitting with preparationTOA Time of arrivalA1 Movement of the blade tip in either axial or lean

directions

1 Copyright © 2018 by ASME

A2 Change in measurement position along the chord line due to either axial or lean movements

An , x Force amplitude at node n in x-directionAP1

, AP2New sensing position of probes (1 and 2) due to untwist

ao , a1 , a2 Coefficients of the integral part of the displacement

b1 , b2 Coefficients of the non-integral part of the displacement

D A diagonal matrix containing the squares of the natural frequencies

d j BTT displacement at a probe jEO∫¿¿

Integral engine order

EOn∫ ¿¿Non-integral engine order

Fn , x Excitation force at noden of the FE model in x-direction

f General force vectorH Matrix of the normalized mode shapesl Distance between the axis of twist and the locus

of probe (1) in untwist methodNb Number of blades

n Node numberP Steady state offsetPi Initial value of BTT offset

Pf Final value of BTT offset

q Modal coordinates vectorR Radius of the blade tipt Timet act Actual arrival time

t exp Expected arrival time

t f Final time

u Vector of absolute positions of the nodesuoffset Vector of prescribed steady-state shift of nodes

urigid Vector of nodal positions due to rotation only

∆ P Change in BTT offset∆ t Change in arrival time∆ x p Axial difference between two probes

θ j Probe angle

φn , x Phase of the force at node n in x-directions∅ Untwist angleψ Angle of the blade’s chord lineΩ Angular velocityΩi Initial angular velocity

Ω f Final angular velocity

[ ]T Transpose of a matrix

[̈ ] ∂2()/∂ t2

1. INTRODUCTION

Turbomachinery applications are usually subjected to high stresses due to harsh operating conditions such as speed, pressure, and temperature. Most of these stresses are dynamic, which raises the attention towards the fatigue limits and dynamic behaviour of such systems. Vibration measurements of the bladed assembly provide the designers of these applications with the required information in order to determine the limitations posed by the above mentioned conditions. Also, they can be used to enhance the design of components to improve their structural integrity.

BTT is one of the promising techniques in vibration measurements. It is a non-contact, non-intrusive, system which depends on measuring the arrival times of the blade tips at the angular positions of a number of probes during one or more revolutions of the rotor. An additional once per revolution probe (OPR) is used to determine both the time of each revolution and the number of rotations. However, some recent attempts have been made for the generation of OPR signal from the arrival times, without the need to use an additional probe, reducing the cost [1]. The measured arrival times (from the probes) would be exactly the same as the expected ones if there is no vibration and all probes are exactly placed at their prescribed positions relative to the blades. However, they would be different when vibration and/or steady state movements of the blades exist as illustrated in Figure 1.

BTT measurement consists of two main phases. The first one includes the measurement of timing data and the transformation of these times into displacement values. The second phase is the analysis of data in order to obtain the vibration parameters (frequency, amplitude, and phase). Although non-intrusive measurement systems have existed from the nineteen-sixties, the most significant development of the current BTT technology has occurred during the last two decades. It started with the work of Heath [2][3][4]. This was followed by the Auto-regressive methods of [5][6][7] and the sine fitting with preparation (SFP) method by Russhard [8], which is currently used at Rolls-Royce. More recent work has been done by


Figure 1: Change in arrival time due to vibration

Deformed blade

Non-deformed blade

Probe∆ t

Ω

Rigosi et al. [9], who improved the two parameter plot method introduced by Heath [3], and the method of Jun Lin et al. [10] for monitoring multi-mode vibration signals. However, there are many other methods that have developed within industry which have not been published or have insufficient information.

Steady state movements of the blades such as axial movement, lean, and untwist have a huge effect on BTT measurements. These usually occur due to thermal expansion, bearing wear, non-uniform gas loading, and centrifugal loads. Such movements would either increase or decrease the displacement values by many multiples of the vibration amplitude, and are usually considered as a DC error. This causes a problem in the validation of FE models using BTT methods. The change in the measurement position of the probe relative to the blade tip increases the uncertainty of the validation since every node on the blade tip may have different vibration amplitude from the other nodes. Also the movement of the blades may make it difficult to measure all desired modes since the probe may be located at a position of small or zero displacement, as shown in Figure 2a, or it may be unable to detect large displacement if it was located at the far edge of the blade as shown in Figure 2b.

The current BTT algorithms and software are used only for the calculation of the vibration amplitude, and they may also consider the error in probe angular position (probe offset) which is constant. However, the different types of steady movements (axial, lean, and untwist) are neglected, despite

their significant effect on the results. Only few research works have focused on the extraction of the sensing position of the probes relative to the blade tip. Olivier Jousselin [11] has presented a method to extract the measurement position for turbine shrouded blades, due to lean, sweep, or untwist. Some preliminary methods have been developed at Rolls-Royce by using additional probes and a pre-knowledge of mode shapes, in order to find the relative movements [12]. Hatcher Jr et al. [13] have used an image capture device to capture the blade tip at the same time of blade tip detection, and then used either image processing or manual observation to find the measurement position. Kominsky [14] has presented a method for detecting the rotating stall of compressor blades by measuring the untwist angles of the blades. An array of two or more optical fibres were used, and by measuring the intensity of the reflected lights from the blade, the untwist angle can be calculated. All these methods require the use of additional equipment and/or an additional feature on the blade tip, thus reducing reliability and increasing costs and/or restricting the method to particular blade types. A technique using the BTT data is much needed not only for finding the sensing positions but also for measuring the amount of movement.

In this work, a novel method is presented for quantifying the movements of the blade’s chord line from its original position due to axial movement, lean, and untwist. This enables the determination of the new (altered) sensing positions of the probes relative to the blade tip. The method depends only on the probe data and the chord stagger angle. The method is divided into two main parts: (1) averaging the BTT displacements and monitoring the change in this average along a range of rotational speed; (2) using the change in average to calculate the amount of steady movement according to the geometrical relations between the BTT displacement and the expected movement. Similar geometrical relations may already be used in previous studies such as [15] for finding out-of-plane vibration amplitudes. However, the goal of the current study is to link them to the probe data in a novel approach that determines the steady movements of the blade with no extra costs or time. Even the most complicated case, which is the untwist movement, only requires data from two probes, and such information is available in all BTT measurement systems. The method is validated using a blisk simulator’s BTT data with prescribed movements. The dynamic model of the blisk is based on the eigenvectors computed from its FE analysis.

2. BTT DISPLACEMENT

Figure 1 shows the change in arrival time of the blade tip due to vibration, which can be calculated as

∆ t=t exp−t act (1)


Figure 2: Misleading Positions of the probes. (a) The probe is placed at a position of small displacement. (b) The probe is placed away from the deformed blade tip.

Pi Pf ≠ Pib-

(b)

Blade tip

R ΩProbe locus

(a)

Small displacement

Blade tip

RΩ

Probe locus

Deformed blade tip is away from the probe

where t exp and t act are, respectively, the expected and actual arrival times of the blade tip relative to the start of revolution. Also, the sign of ∆ t determines whether the blade tip arrived earlier or later than the expected time. The tip displacement for a given blade at a probe j can be calculated by knowing the angular velocity and the radius of the blade tip R measured from the centre of rotation as

d j=ΩR ∆ t j (2)

t exp can be calculated theoretically using the following formula by knowing the difference in angular position between the blade and the probe (θ¿¿ j)¿ at the start of revolution, in addition to the angular velocity Ω, which is assumed constant during a single revolution [1].

t exp=θ j /Ω (3)

3. BTT DATA ANALYSIS

The analysis of the BTT data mainly involves fitting it to a proposed model. However, there is a preliminary preparation step that has to be carried out before fitting the data when SFP method [8] is applied. It was assumed that the BTT displacement of a given blade at a probe j would have the following form

d j=P+¿

+¿(4)

where P is the steady state offset due to probe position error, ao , a1 , a2 are the coefficients of the integral engine order terms, b1 , b2 are the coefficients of the non-integral engine

order terms, EO∫¿¿ and EOn∫ ¿¿ are the integral and non-

integral engine orders respectively, and θ j is the probe angle.

The four terms of equation (4) should be separated in order to obtain the best results. Both the non-integral EO and the noise parts can be removed by smoothing the data. The offset can then be removed by zeroing the remaining signal outside the resonance regions to leave only the integral EO component of the signal [16]. On the other hand, if the asynchronous vibration signal is to be analysed, the integral and offset parts can be removed by zeroing the original data. P is a result of the probe offset from its original position due to inaccurate probe angle, axial movement, lean, and untwist. The error in probe angle will always keep a constant value ofP, while the different types of movements are results of the gas loading and thermal conditions which are dependent on the rotational speed of the rotor. So, P would have a different value per revolution as the speed is changing as illustrated in Figure 3.

4. BTT SIMULATOR

The testing and evaluation of newly developed methods in BTT using real vibration measurements is very costly and is usually considered time consuming. Thus, data from simulators are nowadays frequently used in place of the actual measurements, at least during the early stages of the evaluation of new methods. BTT simulators presented in the literature have been based on simple mathematical models (discrete multi-degree-of -freedom spring-mass systems) to study the dynamic behaviour of bladed systems under controlled conditions [17][18]. These models had many limitations: blade deflection limited to a single direction; elastic coupling between blades limited to adjacent blades only; the use of fixed supports for the connection between the blades and the disk. Moreover, many types of loadings and/or movements cannot be applied to such systems.

On the other hand, the data generated by most commercially available BTT simulators is not derived from a simulation of the actual mechanics of a blisk. Hence, the dynamics of currently available or published simulated systems are not likely to correspond to those of a real blisk, thus compromising the results of evaluations of BTT algorithms. Indeed, it has been observed [8] that algorithms may work very well with


Figure 3: Steady state offset. (a) Final value of P is equal to the initial one; (b) Final value of P is different from the initial one.

data from simplistic simulators but fail to achieve sufficient accuracy with real data.

For the above reasons, a novel BTT simulator that uses the modal data of a FE model has been developed. For given excitation forces, the simulator generates the coordinates of the instantaneous positions of the nodes of the blisk relative to the initial undeformed positions as follows:

u=urigid (t )+Hq ( t ) + uoffset (t ) (5)

where: urigid defines the positions that are purely due to rotation of the blisk;uoffset ( t ) is a prescribed steady-state shift that can change gradually with time; H is the matrix of mass-normalized mode shapes of the non-rotating blisk (each column of which being one of the eigenvectors of the system); q is the vector of modal coordinates, which is governed by the following equations:

q̈+ Dq=HT f (6)

where f is the vector of dynamic excitation forces and D is the diagonal matrix containing the squares of the natural frequencies that correspond to the mode shapes in H . The modal data are extracted by FE analysis.

The second term of eq. (5) therefore describes the oscillatory part ofu. The modal transformation enables efficient solution of eq. (6) regardless of the complexity of the blisk since the effective number of modes (i.e. the number of columns used in H ) required for accurate convergence of eq. (5) is typically much less than the number of physical degrees of freedom used in the FE model. The excitation forces in f are calculated at each time step using the instantaneous values of the sweeping vibration frequencies as per the following formula, which uses as an example the element of f acting node no. n of the FE model in the x-direction [17]:

Fn , x (t )=∑j=1

J

Fn , x( j ) (t )

¿∑j=1

J

An , x( j) sin [EO( j )(Ωi t +( Ωf−Ωi

2 t f)t 2)+φn , x

( j) ](7)

where Fn , x is the excitation force at noden of the FE model in x-direction, An , x

( j) and φn , x( j ) are the force amplitude and phase

respectively corresponding to frequency component with engine order (EO) of j, and Ωi, andΩ f are the initial and final angular velocities respectively, t is the time, and t f is the final time. Similarly to the simpler spring-mass model in [17], [19] the vector containing the amplitudes of the forces in f corresponding to a given engine order can be chosen to be

parallel to one of the mode shapes in order to excite only that selected mode. Also, the arbitrary phases can be selected based on the following formula [19] in order to obtain a travelling wave model of the excitation:

φn , x( j ) =2πi EO( j)/ Nb i=0,1 , ….. ,(N b−1) (8)

where i is the number of the blade to which node no. n belongs.

The simulation neglects gyroscopic effects and centrifugal stiffening due to the rotational speeds. This is justified for the lightweight blisk considered in this study. However, these effects shall be included in future work to ensure accuracy for more generic applications. The modal equations are integrated using the MATLAB function “ode45” via SIMULINK. The use of SIMULINK was essential in order to benefit from the in-built function block “Hit Crossing” which compares the angular positions of the blade tip nodes in u to that of the probe in order to find a time of arrival (the angular position being measured from the fixed horizontal axis of symmetry in the plane of rotation). The Hit Crossing function can force the integrator to


Figure 4: Simulator steps

Equation (5)

+

D , H

uoffset (t ) urigid ( t )

Forces, Equation (7)

TOA

Hit Crossing?

Hq (t )

Integration (ode45)

Equation (6)

FE Modal Analysis

q

u ( t )

No

Yes

change the time steps until it finds the required time. This process is done at each time step for each of the nodes on the tip of a blade. In parallel to this, a separate custom-written function is used to identify which nodes on the blade tip are coincident with the axial location of the probe, within a prescribed tolerance. The saved Hit Crossing time (which is based on angular position) for the node that also fulfils the axial location condition is then accepted as the TOA of the blade at the probe. The flow of the simulator steps is shown in Figure 4.

5. DETERMINATION OF THE MEASUREMENT POSITION

Change in the measurement position affects the BTT result for amplitude and phase of the vibration, and also has an effect on the DC component (the offsetP), which appears in the BTT data (TOA, and displacement) as described in section 3. Ideally, the sensing position is known based on the geometries of both the blade and the probe. However, this is not usually the case because of the tolerances and inaccurate positioning of the probes including the OPR, and the static positions of the blades relative to the probe. This is shown in Figure 3 as the initial value ofP. The change Δ P in the offset will be zero if there are no extra movements due to operating conditions (Pf =Pi). Δ P would not be zero at any operating speed if there is a change in the blade tip static position (Pf ≠ Pi). Starting from this point, a novel method has been developed in order to determine the amount of movement. This method is based on extracting the change inP. This enables the determination of the new (altered) sensing position of a probe with respect to the blade tip.

5.1 Axial Movement

Axial movement is defined as the steady-state movement of the blade’s chord line in the direction perpendicular to the plane of rotation. It usually occurs due to thermal expansion, bearing wear, non-uniform gas loading, centrifugal loads, or any other loads in the loading direction of the system. With reference to Figure 5, the relation between the amount of movement of a node on the blade tip in the axial direction ( A1), the change in measurement node along the chord line of the blade ( A2), and the change in offset which is parallel to the plane of rotation∆ P at one of the probes can be expressed as

A1=∆ P/ tan ψ(9)A2=∆ P/sin ψ

where ψ is the initial inclination angle of the chord line relative to the axis of rotation.

Also, the sign of ∆ P defines the direction of movement relative to the direction of rotation.

5.2 Lean

Lean is defined as the steady-state displacement of the blade in a direction perpendicular to the chord line ( A1) [11]. Thus, the relation between A1, A2 and ∆ P according to Figure 6 will be

A1=∆ P cosψ(10)

A2=∆ P sinψ

5.3 Untwist

Untwist is the steady-state twisting of the blade’s cross section around the longitudinal axis of the blade. As shown in Figure 7, two parameters need to be determined in order to completely know the amount of untwist movement. The first one is the


Figure 5: Axial movement

ψ

A2

A1∆ Pψ

R Ω

Moved blade

Probe locus

Initial position of blade chord

Figure 6: Lean

A2

A1

∆ Pψ

ψ

Moved blade

Probe locus

Initial position of blade chord

R Ω

untwist angle∅ , and the other isl, which is the distance between the axis of twist and the point of intersection between the non-deformed chord line and the locus of probe (1). Thus, at least two probes should be used at different axial positions, where ∆ x p is the axial distance between the two probes, ∆ P1 and ∆ P2 are the changes in offsets of probe (1) and probe (2) respectively. These two probes can be at different angular positions.

The following relation can be obtained using the triangles (1−2−3) and (1−4−5) in Figure 7

tan (ψ+∅ )=∆ P2+(l sin ψ−∆ x p tan ψ )

l cosψ−∆ x p=

∆ P1+lsin ψl cosψ (11)

By rearranging the above equation, we can get

l [∆ P2cos ψ−∆ P1 cosψ ]+∆ P1 ∆ x p=0 (12)

So, l can be calculated as

l=∆ P1 ( ∆ x p/cos ψ ) /( ∆ P1−∆ P2 ) (13)

And then, using l and equation (11), we can get a formula for the untwist angle as follows

∅=tan−1 ¿¿ (14)

By calculating land∅ , the new sensing position of probe (1) measured from the twisting axis along the chord line can be calculated as follows

AP1=l cosψ /cos (ψ+∅ ) (15)

Similarly for probe (2)

AP2=(lcosψ−∆ x p)/cos (ψ+∅ ) (16)

The signs of both ∅ and l define the direction of untwist and the position of the axis of twist with respect to the direction of rotation. Theoretically, if two probes are used with different axial positions, the type of movement can be identified as follows

If ∆ P1=∆ P2 Lean or axial movement

If ∆ P1 ≠ ∆ P2 Untwist

6. VALIDATION OF THE METHOD

The new method of determining the sensing positions of the probes relative to the blade tip has been validated against simulated BTT data of a blisk’s FE model, generated as described in section 4 and Figure 4. The blisk has a 40 flat inclined blades integrated to the disk (Figure 8). Each blade has a chord line angle of 60º relative to axis of rotation. The material of the blisk is Aluminium. The FE model has been created using ANSYS 15.0, it had 1791261 degrees-of-freedom. The first five mode shapes of the blisk in addition to their corresponding natural frequencies are shown in Figure 9.


Figure 7: Untwist

Figure 8: Blisk with inclined blades

ZX

Y

The position of the force

A sweeping rotational speed (0 to 200 rev/s) was used, and the total simulation time was 2.1s, although the prescribed movements started at 0.1s. A travelling waving force of constant amplitude of 100 N was applied with a single engine order (EO=1), and in a direction perpendicular to the plane of rotation. The position of the force at one of the blades is shown in Figure 8. Based on the excitation frequency (0 to 200 Hz), the simulator used the first 5 modes of the blisk, which ranged from 340 Hz to 393 Hz. Three different tests have been carried out, one for each type of movement. It is important to note that the following user-prescribed movements refer only to the term uoffset (t ) in equation (5) and Figure 4:

Test (1): The axial position (z-coordinate) of each node was changed gradually with constant rate (0.5 mm /s) in order to validate the axial movement method. According to the simulation time, the total axial movement would be 1 mm.

Test (2): Each node was assumed to be moved in a direc-tion perpendicular to the chord line with rate (0.5 mm /s) in order to validate the lean method. The total movement would be 1 mm.

Test (3): The untwist method was validated by forcing the blade tip to be twisted 5° around its longitudinal axis in 2 seconds. Two probes were used at different axial positions (10 mm apart). The axis of twist was selected to be about 12 mm away from the position of the first probe along the chord line.

7. RESULTS AND DISCUSSIONS

The BTT displacements (calculated using equation (2)) corresponding to the three tests listed above are shown in Figure 10, 11, and 12. The BTT displacement due to the vibration response during the specified range of frequencies was only about 6×10−6m, which is too small relative to the magnitudes of the movements in the specified directions. So, the total amount of BTT displacement was used without averaging as it was almost equal to the steady state offset.

The movements were started at t=0.1 s. Also, it should be noted that the output data had regions with constant or small rate of displacements. These regions are due to the finite distances between the different FE nodes on the blade tip (since the mesh is finite). During a period of time, the simulator can only find the same node within the range of a probe, until the next node enters the probe’s range. At the instant this happens, the simulated BTT displacement would change suddenly and then remain constant again until a different node. To avoid this effect, the FE model of the blade tip should be refined as much as possible to reduce distances between nodes.

The results and percentage error of each of the movement types are listed in Table 1. The validity of the new method for determining the amount of movement and specifying the new measurement position is seen to be proven, since the error in all cases was less than 2%. Part of that error occurred due to the constant or small rate displacement regions of the BTT displacements that resulted from the finite distance between the FE nodes, as explained above. Such a source of error would not appear with real measurements. Moreover, the averaging technique may have an effect on the results once it is used. Hence, the next crucial step will be to validate the methods experimentally in order to enhance their effectiveness.


Figure 9: Mode shapes of the blisk

Table 1: Validation results of the measurement position method

8. CONCLUSIONS

A novel method has been developed for determining the amount of steady-state transition of a blade tip from its original position due to one of three different types of movement: axial shift, lean, or untwist. This enables the determination of the instantaneous measurement position of the probes relative to the blade tip. The method uses BTT displacement data and is based on the extraction of the change in its steady state offset. This method has been validated using a FE model of a blisk using a novel BTT simulator which has the capability of adding steady-state movements to the FE model in any direction. The results were promising since the errors were always below 2%. Most of these errors occurred due to the limitations of the FE model, such as the mesh size. Hence, the accuracy of the method would be better with real measurements since this source of error will not exist. The method should be validated on real data in order to enhance its effectiveness, particularly with regard to the effect of noise, probe size, and number of averaging points for extracting the offset from BTT displacement data. Further studies, including the effect of both bending and torsional modes of the airfoil, are currently being carried out. Also, a method for the determination of all types blade movement (axial, lean, and untwist) simultaneously is under development.

ACKNOWLEDGMENTS

We would like to thank the Egyptian Ministry of Higher Education, and the British Council in Cairo for funding the project.

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Figure 10: Effect of axial movement on BTT displacement

Figure 11: Effect of lean on BTT displacements

Figure 12: Effect of untwist on BTT displacements

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% error

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Lean 1 mm 0.998 mm 0.2

Untwist 5º 5.01 0.2

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[19] Carrington, I. B., Wright, J. R., Cooper, J. E., and Di-mitriadis, G., “A comparison of blade tip timing data analysis methods,” Proceedings of the Institution of Mechanical Engineers. Part G Journal of Aerospace Engineering, vol. 215, no. 5, pp. 301–312, 2001.


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