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Research Papers

# A Novel Method for the Design of Proximity Sensor Configuration for Rotor Blade Tip TimingPUBLIC ACCESS

[+] Author and Article Information
David H. Diamond

Centre for Asset Integrity Management,
Department of Mechanical and
Aeronautical Engineering,
University of Pretoria,
Pretoria 0002, South Africa
e-mail: dawie.diamond@up.ac.za

P. Stephan Heyns

Centre for Asset Integrity Management,
Department of Mechanical and
Aeronautical Engineering,
University of Pretoria,
Pretoria 0002, South Africa
e-mail: stephan.heyns@up.ac.za

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received November 24, 2017; final manuscript received March 29, 2018; published online May 7, 2018. Assoc. Editor: Costin Untaroiu.

J. Vib. Acoust 140(6), 061003 (May 07, 2018) (8 pages) Paper No: VIB-17-1510; doi: 10.1115/1.4039931 History: Received November 24, 2017; Revised March 29, 2018

## Abstract

Blade tip timing (BTT) is a noncontact method for measuring turbomachinery blade vibration. Proximity sensors are mounted circumferentially around the turbomachine casing and used to measure the tip displacements of blades during operation. Tip deflection data processing is nontrivial due to complications such as aliasing and high levels of noise. Specialized BTT algorithms have been developed to extract the utmost amount of information from the signals. The effectiveness of these algorithms is, however, influenced by the circumferential spacing between the proximity sensors. If the spacing is suboptimal, an algorithm can fail to measure dangerous blade vibration. This paper presents a novel optimization approach that determines the optimal spacing between proximity sensors.

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## Introduction

Blade tip timing (BTT) is a noncontact method for measuring turbomachinery blade vibration during operation [14]. BTT is an alternative to strain gauge systems. Strain gauge systems are considered the conventional technique for measuring rotor blade dynamics [3]. Strain gauge technology, however, has several drawbacks: being expensive or more expensive than BTT systems [1,2,510], being complex and time consuming to install [1,5,814], having a limited lifetime due to harsh operating conditions [5,7,9,10,12,13,1519,], and the fact that only a limited number of blades can be instrumented [1,2,57,913,2023]. These limitations and faster turbomachine development cycles have led to a decrease in use of strain gauge systems in favor of BTT in combination with finite element models [7].

Blade tip timing uses casing mounted proximity sensors spaced circumferentially around a turbomachine rotor stage (Fig. 1(a)). The time of arrival (ToA) of each blade is measured as it passes underneath each sensor. A blade that is not experiencing any vibration has an expected ToA. The expected ToA is calculated using the shaft speed. A vibrating blade will arrive earlier or later than expected due to its tip deflection, x (Fig. 1(b)). The difference between these two ToAs, $Δt$, is used to calculate the blade tip displacement, usually with the following equation: Display Formula

(1)$x=ΔtΩR$

where R is the rotor's outside radius and Ω is the rotor's rotational speed. The BTT system measures one tip deflection each time a blade passes a sensor. One can attempt to infer the blade vibration characteristics by analyzing measurements from several sensors over multiple shaft rotations.

The inference of rotor blade vibration is, however, notoriously difficult to perform, especially if the vibration is an integer multiple, or engine order (EO) of the shaft speed [2,22,24,25]. The frequency of EO excited vibration is calculated with the following equation: Display Formula

(2)$f=EOΩ$

where f is the excitation frequency. This type of vibration is difficult to measure because the vibration cycle of the blade is repeated every revolution. This means that a blade's measured tip deflection remains approximately constant over multiple revolutions; no substantial information is therefore added by measuring for more than one shaft revolution.

Another factor that contributes to the difficulty of processing BTT signals is aliasing [3,6,8,10,22,2631]. In most other measurement activities, aliasing is caused by a limitation in data acquisition sampling rate. The sampling rate of a BTT system is, however, completely determined by the rotational speed of the rotor, the number of sensors, and the circumferential spacing between sensors in the rotor casing.

Consider, for instance, a turbomachine stage rotating at a speed of 50 Hz. The stage is surrounded by eight proximity sensors. The sensors are installed circumferentially equidistant from one another around the casing. This leads to a sampling rate of 400 Hz for each blade (eight measurements taken 50 times per second) and a Nyquist frequency of 200 Hz. Rotor blades often have natural frequencies above 200 Hz, making these signals difficult to process using conventional signal processing techniques. In addition to this, BTT signals are extremely noisy and contain latent signals from shaft and casing vibration. Specialized BTT signal processing techniques have therefore been developed to infer as much meaningful information from the signals as possible [14,13,16,22,2426,28,3137].

All of these methods require tip displacement measurements of the highest possible quality to maximize the accuracy of the inference. One aspect that greatly influences a BTT system's signal quality is the sensor circumferential configuration. Poorly or inadequately spaced sensors can lead to signals that are more sensitive to noise and which contain less information. The machining of sensor mounting holes [1,3,5,6,911,15,16,20,24,28,29,32,36, 3845] is financially expensive and the resulting sensor configuration is permanent. An incorrectly designed sensor configuration can therefore lead to a BTT system that does not deliver on its intended goal. Some methods require sensors spaced in an equidistant manner [1,36] and other methods can use any type of sensor spacing.

This paper presents a novel method to determine the non-equidistant sensor configuration that maximizes the accuracy of BTT algorithms. An approach used by some in industry [46,47] is described and developed further into a constrained optimization problem. The method is then illustrated using a sensor configuration design example.

## Mathematical Model for Vibration Inference

Various BTT algorithms use a single degree-of-freedom sinusoidal model during inference of a blade's vibration characteristics [2,4,5,20,23,26,2932,35]. The blade vibration is expressed as an orthogonal sinusoid. During a single revolution, the tip deflection, xi, measured at sensor i with circumferential position θi is equated to an orthogonal sinusoid with angle $EOθi$. This is shown in the following equation below: Display Formula

(3)$xi(t)=A sin(EOθi)+B cos(EOθi)+C$

If each tip deflection for all N sensors is expressed in this manner, it creates a system of equations. The system is denoted as in the following equation below: Display Formula

(4)$Φw=x$

where $Φ$ is called the design matrix and is given by the following equation: Display Formula

(5)$Φ=( sin(EOθ1) cos(EOθ1)1 sin(EOθ2) cos(EOθ2)1⋮⋮⋮ sin(EOθN) cos(EOθN)1)$

The parameters controlling the vibration amplitude, phase, and offset are $A, B$, and C, respectively. They are contained within the parameter vector $w$Display Formula

(6)$w=(ABC)$

Note that the EO of vibration needs to be known beforehand. This can be accomplished through the use of a Campbell diagram [14,41,48] or through a variety of algorithms that attempt to retrieve the vibration frequency [2,4,13,16,24,25].

If the true vibration frequency is a sinusoid, the error between the true vibration and the inferred vibration characteristics can be calculated. The error is also a vector and is expressed as $Δw$Display Formula

(7)$Δw=wtrue−w$

A scalar value for the error size, ε, is obtained by taking the two-norm, also known as the Euclidean norm [49], of $Δw$ as shown in the following equation: Display Formula

(8)$ε=∥Δw∥$

This error can be used to evaluate the accuracy of the inferred vibration characteristics.

## Sensor Positioning

It is well known that sensor positioning influences the accuracy of a BTT system [18,28,47,50]. The effect of three different sensor configurations on the accuracy of $w$ is now demonstrated.

Consider three different BTT systems, identical in their number of sensors but each with a different sensor configuration. Each BTT system has five sensors with their configurations shown in Table 1.

The BTT systems are installed onto the same rotor. The rotor has a first natural frequency of 300 Hz and is rotating at a rate of 50 Hz. The first natural frequency is excited by an EO 6 disturbance, caused by obstructions in the working fluid flow path [51,52]. The true vibration of the blade can be expressed in terms of the following equation: Display Formula

(9)$w=(141.42141.420)μm$

The blade's vibration amplitude is therefore 200 μm at a phase angle of $(π/4)$. The measurements taken by each BTT system in the absence of noise are shown in Fig. 2.

Figure 2 illustrates the varying quality of tip deflection measurements one can obtain from BTT systems with an identical number of sensors but different configurations. All the sensors in BTT system 1 measure the blade vibration at exactly the same location in the rotor blade's vibration cycle. The value of 50 μm is measured repeatedly. It would be impossible for an algorithm to calculate the vibration characteristics from what is essentially a constant signal. This arrangement is suboptimal and should be avoided.

Blade tip timing system 2 measures two different tip displacements, −67 μm and −158.3 μm. Although some variability among the measurements is present, making it better than system 1, only one direction of vibration is measured (negative direction). This would also make accurate inference difficult if not impossible.

Blade tip timing system 3 measures unique parts of the vibration signal in both directions of motion. An algorithm has a much better chance of inferring the true natural frequency and vibration amplitude from this signal as all its values are unique. This signal therefore contains more unique information than the other two signals and should always be preferred above the other two.

Simulations evaluating the accuracies of the vibration vector inference from the measurements in Fig. 2 are now shown. Normally distributed random noise with a mean of 0 μm and a standard deviation of 10 μm is added to each set of measurements 50 times. The average inference error, $ε¯$, is calculated for each case. The results are shown in Table 2.

Table 2 shows that BTT system 1 results in the poorest performance and system 2 performs better than system 1. System 3 results in the lowest average error by far. These quantitative results support the qualitative assessment given of the sensor configurations earlier.

## Condition Number

The solution to linear systems such as the one shown in Eq. (4) is affected by the design matrix' conditioning [49,53]. The conditioning of the matrix Φ indicates how sensitive the solution of the system is to perturbations in Φ and/or x. The condition number, κ, of the design matrix is an approximate numerical value for the matrix conditioning [49,53]. A condition number of 1, the minimum possible number, indicates an optimally conditioned system. A condition number tending to infinity indicates the design matrix is singular and cannot be used to solve the equation. The condition numbers for the three BTT systems in Table 1 are reported in Table 3.

In Table 3, $θ$ is the shorthand notation for the parameter vector describing the sensor configuration. There is a clear relationship between the condition numbers listed in Table 3 and the average accuracy values listed in Table 2. Higher condition numbers lead to higher average error values. This is in line with the definition of a condition number.

It is possible to perform multiple simulations to further prove the relationship between BTT algorithm accuracy and the design matrix condition number. Five hundred simulations are performed where a BTT system with four sensors located randomly around the casing is generated. For each sensor configuration, the following is performed:

1. (1)Two sets of tip deflections are calculated according to Eq. (3) where EO = 8 and EO = 12, respectively. The vibration parameter is given in Eq. (9).
• (a)The tip deflections are corrupted with normally distributed random noise. The mean of the normally distributed noise is 0 μm and the standard deviation is 10 μm. In statistical notation the error, ε, is distributed according to $ε∼N(0,102)$.
• (b)The vibration parameters for each set of corrupted tip deflections are calculated. The error, ε, is also calculated.
• (c)Repeat steps (1.a) and (1.b) 50 times.
2. (2)The mean error, $ε¯$, of all ε values from steps (1.a) to (1.b) is determined. This mean error indicates the accuracy of the BTT system in inferring the vibration characteristics.

The condition number and mean error for each sensor configuration are shown in Fig. 3.

It is clear from Fig. 3 that a strong correlation between condition number and mean error exists. As the condition number increases, the average error also increases. The condition number can therefore be used as a surrogate value for the mean error and can be minimized to obtain a BTT sensor configuration that has the highest likelihood of resulting in an accurate vibration estimate.

The use of condition numbers to assess the quality of BTT sensor configuration is prevalent in industry. The first published instances of this, to the best knowledge of the authors, can be found in Refs. [46] and [47].

## Optimization Problem

When faced with a new BTT installation, the sensor configuration can be designed by formulating and solving an optimization problem. The three main steps in formulating the problem are presented below.

###### Establish the Number of Sensors to Be Used.

The maximum number of sensors should be preferred; more information is always better. There are, however, limitations to the number of sensors one can use. Sensors that operate for long periods of time in harsh conditions tend to be expensive. The cost of a data acquisition system also increases as more channels need to be measured. Furthermore, there may be limited space for the sensors in the turbomachine casing. The maximum allowable number of sensors subject to the constraints described earlier should be chosen. The number of sensors determines the number of rows in the design matrix.

###### Find the Engine Orders of Interest.

The condition numbers calculated in the previous examples represented a single EO at a time. It is unlikely that a BTT system is used to measure blade vibration caused by only a single EO. Multiple EOs can be included into the optimization problem; say for EOs 1–1000. This can, however, result in a BTT system optimized to measure vibration modes that are not damaging (such as highly damped high-frequency modes) or modes that are unlikely to be excited.

Practically, one can limit the EOs being taken into account. For instance, suppose an aircraft engine operates between 3000 rpm (50 Hz) and 6000 rpm (100 Hz) for the majority of its life. The first four natural frequencies of the compressor row being monitored have values of 200, 300, 400, and 500 Hz. If one is only interested in measuring the first four natural frequencies, one can calculate the minimum EO to be measured by dividing the smallest natural frequency of interest with the largest operational speed, as done in the following equation: Display Formula

(10)$EOmin=200100=2$

The maximum natural frequency of interest can be obtained by dividing the largest natural frequency with the lowest operational speed, as done in the following equation: Display Formula

(11)$EOmax=50050=10$

All EOs between EOmin and EOmax may be excited and the sensors should be optimized to measure them all as accurately as possible.

In addition to optimizing for all the EOs in this range, some EOs might be of particular interest. There are many possible reasons why a specific EO might be of particular interest. Possible reasons are given below:

1. (1)Turbomachines are often operated at constant shaft speeds for extended periods of time. A certain EO excitation at a set point may coincide or near-coincide with a natural frequency of the machine. Take the example of an aircraft with a shaft speed of 4000 rpm for one of its cruising speeds. If the lowest natural frequency of the first compressor row blades is approximately 540 Hz, this is fairly close to the eighth-order excitation at 4000 rpm of 533.3 Hz. This natural frequency might shift downward to the excitation frequency because of temperature effects or changes in boundary conditions at the root attachment. An EO = 8 excitation is therefore of particular interest.
2. (2)One may know, from simulations or previous experience, that some modes are more damaging than others. Lightly damped modes may vibrate more excessively per unit of input force than highly damped modes. These modes tend to be more damaging and are therefore of particular interest.
3. (3)One may be aware of a particular failure mode that suggests a specific frequency of vibration in a narrow shaft speed range. The EOs that excite this failure mode in that range are therefore of particular interest.

A weighting factor that places specific emphasis on measuring a certain EO of vibration can be assigned during optimization. This weighting factor is denoted by αEO. The size of the weight should be chosen based on engineering intuition, at least until further research is conducted.

###### Find the Smallest Sum of Squared Weighted Condition Numbers.

The optimal sensor spacing is now defined as the sensor spacing leading to the minimum sum of squared weighted condition numbers. This is expressed in the following equation: Display Formula

(12)$θopt=argminθ(∑EO=EOminEOmax(αEOκ(θ,EO))2)$

Most of the weights will be one, except for those that have been identified as particularly important to measure.

###### Particle Swarm Optimization.

This optimization problem does not have a single local solution. Consider the case of a three sensor BTT system being analyzed to determine the sensor configuration. All EOs between 2 and 10 are of equal importance and the first sensor is fixed at 0 rad. A heat map of the minimization surface for the remaining two sensors is shown in Fig. 4.

Lighter shades indicate better sensor configurations. It is clear that there are many local minima and a large amount of local maxima. It is therefore a global optimization problem.

Particle swarm optimization (PSO) is a well-known optimization technique for solving global optimization problems. PSO is the preferred optimization technique used in this paper but could be exchanged for other global optimization techniques.

To determine N sensor positions, the parameter being optimized is $θ$ as shown in Eq. (13) below Display Formula

(13)$θ=(θ1,δθ2,δθ3,…,δθN)$

where θ1 is the absolute circumferential position of the first sensor and Eq. (14) can be used to determine the absolute circumferential positions of the other sensors Display Formula

(14)$θn=θn−1+δθn$

This form of parameter vector ensures that all sensor angles are monotonically increasing. All values of $θ$ are limited to the range 0–0.5π. The maximum distance between two subsequent sensors is therefore 0.5π. It has been found that this value leads to the most rapid convergence and best possible solutions for the PSO algorithm. For all optimization runs conducted in this paper, a particle swarm size of 5000 was chosen and the optimization is run for 30 iterations.

## Constraints

Certain limitations to the positioning of sensors are taken into account. The implementation of the PSO algorithm used [54] allows for the specification of constraint functions. These functions return a positive value if the solution is valid and a negative function if not. Two types of constraints are taken into account below.

###### Minimum Sensor Distance.

Sensors are typically mounted by machining holes into the turbomachine casing [1,3,5,6,911,15,16,20,24,28,29,32,36,3845]. A constraint is therefore necessary that keeps the sensors a minimum circumferential distance apart. It is good design practice to keep hole center distances at least 1.5 hole diameters apart. Additionally, some sensors such as eddy current sensors need to be spaced a minimum distance apart such that the sensors do not interfere with one another. A general recommendation of having three sensor head diameters between eddy current sensors is often used. One can formulate the constraint between any two sensors i and j as done in the following equation: Display Formula

(15)$|Δθi,j|=|θj−θi|≥dminR$

where dmin is the minimum length on the arc of the casing that must separate sensors. Note that the angles here are mapped into the 0–2π range if some are larger than 2π.

###### Casing Fixtures.

Turbomachines are intricate structures with many fixtures and components on the casing. It may not be possible to install sensors on the entire casing circumference. A constraint is therefore added that prevents sensors from being placed in some circumferential ranges. Mathematically, this is given by the following equation: Display Formula

(16)$θi∈[θ1,min∗,θ1,max∗]∪[θ2,min∗,θ2,max∗]…∪[θM,min∗,θM,max∗]$

In Eq. (16), $θm,min∗$ and $θm,max∗$ indicate the minimum and maximum fixture locations for the mth fixture.

## Illustrative Example

The proposed method is now illustrated with an example. The BTT sensor configuration for a rotor row is determined with two different sets of EO weights.

###### Rotor Row.

Consider the characteristics of a compressor blade row in an aircraft engine. The blades have an outside radius of 440 mm. The natural frequencies of the rotor blades increase with increasing rotor speed. The rotor blades' first five natural frequencies are given in Table 4 for rotor speeds at 6800 rpm, 14500 rpm and 15600 rpm, respectively. These frequencies correspond to the frequencies presented in Ref. [55] for an SO-3 engine. Suppose further that this engine operates between 6800 rpm and 15600 rpm for the majority of its life, and that a BTT system should be installed to measure the blade vibration in this operating range.

Six microwave sensors, each with a probe diameter of 14 mm, are used. The diameter corresponds to the sensor described in Ref. [56]. These sensors are fixed into the casing and must be placed at least 1.5 times the diameter, or 21 mm, apart. This leads to a minimum distance between sensors as calculated in the following equation: Display Formula

(17)$dminR=21440=0.0477 rad$

Permanent casing fixtures are present around the row to be monitored. The permanent fixtures are such that no sensors can be placed in the circumferential ranges given in Table 5.

###### Possible Engine Orders to Be Excited.

The natural frequencies of the rotor blades increase as a function of shaft speed. To be conservative, the minimum and maximum natural frequencies are calculated using the overall minimum and maximum frequencies and the overall minimum and maximum speeds, i.e., Display Formula

(18)$EOmax=4051.26800/60=35.7≈35$
Display Formula
(19)$EOmin=374.115600/60=1.4≈2$

When calculating the maximum and minimum EOs and ending with a fraction, the maximum EO is always rounded down to the next lowest integer and the minimum EO is always rounded up. All the EO weights of the system are taken as 1, meaning no EO is deemed more important to measure than any other.

The PSO algorithm is now used along with the constraints mentioned in Secs. 6 and 7.1. As can be seen from Fig. 4, there are many local solutions. It might not always be possible to find a global minimum, especially for problems such as this example with many different EOs and many constraints. To demonstrate this point, four PSO optimization runs are performed. The objective function value of the optimal solution per iteration is shown in Fig. 5.

It is seen that all of the runs start out with an objective function value of between 175 and 200 and decrease gradually to a value of between 150 and 155. Because of the random nature of the PSO algorithm and the multitude of local minima, one should not expect to obtain the same sensor spacing repeatedly. All the solutions are, however, good choices for the sensor spacing. It is advisable to perform many different optimization runs and compare different solutions; the solutions will, however, be very close to one another in terms of objective function value. As a rule, the sensor spacing leading to the lowest objective function value over all runs should be used. The values of the parameter vector for each run are shown in Table 6.

It is seen that the first sensor for all four runs is located between 1.67 and 2.56 rad, all starting positions are therefore located in a 90 deg arc. All sensor increments are between 0.1 and 1.03 rad. Recall that the limits of the sensor increments are 0–1.57 (or 0.5π) rad. It is seen that no increment reaches the upper limit of 1.57, which suggests the maximum increment limit is reasonable.

The EO condition numbers resulting from the use of the optimal solution can be visualized. This reveals inherent characteristics of the BTT system. These characteristics can be used to sensitize the designer of some EOs that might be more difficult to measure accurately than others. Figure 6 shows the condition numbers for EOs 2–35 for run 2, the run with the best optimization result.

It is seen from Fig. 6 that the condition number values for different EOs are fairly constant. There are no obvious outliers. This is consistent with the nature of the objective function. The objective function used to find the parameter vector solution, Eq. (12), is referred to as the l-2 norm in compressive sensing literature [57]. The use of the l-2 norm does not result in sparse solutions but rather uniform solutions. The maximum condition number of 7.3 occurs for EO = 2.

###### Weighted Engine Orders.

Suppose it has been decided, through an investigation of the rotor vibration characteristics, that blade excitation below EO = 9 is more dangerous than larger EO vibrations. These EOs can enjoy more importance in the optimization routine by setting the condition number weights, αEO, of all EO values lower than nine to a value higher than 1. The weights for EOs 2–8 are set to 10 here while the other weights are kept at 1.

If the PSO algorithm is repeated with this set of weights, the optimal parameter vector is given below in the following equation: Display Formula

(20)$θ=(1.60,0.46,0.27,0.30,0.36,0.65)$

It is seen from Eq. (20) that the first sensor value, 1.60, is on the lower end of the previous optimization results' first sensor spacing. The spacing between the sensors is between 0.27 and 0.65, a much narrower range than for the previous results.

The condition numbers for the weighted optimization result are shown in Fig. 7 along with the difference between the weighted and unweighted results.

Figure 7 shows that the condition numbers for the EOs between 2 and 8 are generally lower than most of the other condition numbers. They are also lower than the equivalent values of the unweighted solution. It is seen that the weighted solution contains outliers at EOs 18 and 19.

## Future Research and Limitations

The method presented in this paper takes account of only one row of sensors. It is possible that multiple sensor rows requires an amended version of this method and/or may have different constraints. Also, multimode vibration, where more than one natural frequency is excited simultaneously for each blade, requires an amended version of design matrix construction and should be developed where multimodal vibration is expected. The method proposed in this paper should, by definition, not be used where algorithms requiring equidistant sensor spacing will be used. This method can, however, be contracted to accommodate equidistant sensor spacing. The optimization parameter will then only contain two values, the first sensor position and the equidistant spacing.

## Conclusion

This paper presents a novel method that can be used to calculate optimal circumferential positions of BTT sensors. The sensor positions are determined such that several blade vibration frequencies at different EO excitations can be measured. The method uses the sum of weighted design matrix condition numbers as a function that is minimized subject to constraints that may be present in a BTT system installation. It is shown that the optimization problem requires a global optimization algorithm as the problem does not lend itself to local solutions. The method is demonstrated using an example rotor.

## Acknowledgements

The authors gratefully acknowledge support from the Eskom Power Plant Engineering Institute (EPPEI) in the execution of this research.

## Nomenclature

• BTT =

• OFV =

objective function value

• PSO =

particle swarm optimization

• ToA =

time of arrival

Mathematical Symbols
• A =

sine function coefficient of the vibration model

• B =

cosine function coefficient of the vibration model

• C =

offset coefficient of the vibration model

• dmin =

the minimum arc length between sensor centers

• EO =

engine order

• EOmin =

minimum EO which the BTT system should measure

• EOmax =

maximum EO which the BTT system should measure

• f =

vibration frequency

• M =

the number of casing fixtures

• N =

the number of sensors around the rotor stage

• $N$ =

the Gaussian distribution function

• R =

• S =

symbol indicating a proximity sensor

• t =

the ToA of a blade at a sensor

• w =

the inferred vibration characteristics vector

• wtrue =

the true vibration vector

• x =

tip deflection

• x =

the measured tip deflection vector

• Δw =

the error between the true and inferred vibration vectors

• Δt =

time difference between expected and measured ToAs

Greek Symbols
• αEO =

weighting factor for the EO in the loss function

• $Δθi,j$ =

the minimum sensor circumferential distance between sensors i and j

• $δθi$ =

the circumferential distance between sensor i − 1 and sensor i

• ε =

the error size

• $ε¯$ =

the average error size

• θ =

a sensor's circumferential location

• $θ$ =

the PSO optimization parameter

• $θi,min∗,θj,max∗$ =

the minimum and maximum circumferential positions of the ith casing fixture

• $Φ$ =

the design matrix

• Ω =

rotor's rotational speed

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Liu, C. , and Jiang, D. , 2012, “Improved Blade Tip Timing in Blade Vibration Monitoring With Torsional Vibration of the Rotor,” J. Phys.: Conf. Ser., 364, p. 012136.
Agilis, 2014, “Non-Intrusive Stress Measurement Systems vs. Strain Gauges,” Agilis Measurement Systems, Inc., Palm Beach Gardens, FL, accessed Apr. 18, 2018,
Günther, P. , Dreier, F. , Pfister, T. , Czarske, J. , Haupt, T. , and Hufenbach, W. , 2011, “Measurement of Radial Expansion and Tumbling Motion of a High-Speed Rotor Using an Optical Sensor System,” Mech. Syst. Signal Process., 25(1), pp. 319–330.
García, I. , Beloki, J. , Zubia, J. , Aldabaldetreku, G. , Illarramendi, M. A. , and Jiménez, F. , 2013, “An Optical Fiber Bundle Sensor for Tip Clearance and Tip Timing Measurements in a Turbine Rig,” Sensors, 13(6), pp. 7385–7398. [PubMed]
Di Maio, D. , and Ewins, D. J. , 2012, “Experimental Measurements of Out-of-Plane Vibrations of a Simple Blisk Design Using Blade Tip Timing and Scanning LDV Measurement Methods,” Mech. Syst. Signal Process., 28, pp. 517–527.
Hu, Z. , Lin, J. , Chen, Z.-S. , Yang, Y.-M. , and Li, X.-J. , 2015, “A Non-Uniformly Under-Sampled Blade Tip-Timing Signal Reconstruction Method for Blade Vibration Monitoring,” Sensors, 15(2), pp. 2419–2437. [PubMed]
Von Flotow, A. , 2011, “Overview of Blade Vibration Monitoring Capabilities,” Hood Technology Corporation, Hood River, OR, Report.
Heath, S. , and Imregun, M. , 1996, “An Improved Single-Parameter Tip-Timing Method for Turbomachinery Blade Vibration Measurements Using Optical Laser Probes,” Int. J. Mech. Sci., 38(10), pp. 1047–1058.
Gallego-Garrido, J. , Dimitriadis, G. , and Wright, J. R. , 2007, “A Class of Methods for the Analysis of Blade Tip Timing Data From Bladed Assemblies Undergoing Simultaneous Resonances—Part I: Theoretical Development,” Int. J. Rotating Mach., 2007, p. 27247.
Russhard, P. , 2013, “Timing Analysis,” U.S. Patent No. 8457909B2.
Vercoutter, A. , Berthillier, M. , Talon, A. , Burgardt, B. , and Lardies, J. , 2011, “Tip Timing Spectral Estimation Method for Aeroelastic Vibrations of Turbomachinery Blades,” International Forum on Aeroelasticity and Structural Dynamics (IFASD), Paris, France, June 26–30, pp. 1–9.
Grant, K. , 2004, “Experimental Testing of Tip-Timing Methods Used for Blade Vibration Measurement in the Aero-Engine,” Ph.D. thesis, Cranfield University, Bedford, UK.
Szczepanik, R. , Rokicki, E. , Rzadkowski, R. , and Piechowski, L. , 2014, “Tip-Timing and Tip-Clearance for Measuring Rotor Turbine Blade Vibrations,” J. Vib. Eng. Technol., 2(5), pp. 395–406.
Li, M. , Duan, F. , and Ouyang, T. , 2010, “Analysis of Blade Vibration Frequencies From Blade Tip Timing Data,” Proc. SPIE, 7544, p. 75445F.
Russhard, P. , 2010, “Timing Analysis,” Rolls-Royce PLC, Westhampnett, UK, European Patent No. EP2199764A2.
Przysowa, R. , 2014, “Analysis of Synchronous Blade Vibration With the Use of Linear Sine Fitting,” J. KONBiN, 2(30), p. 17.
Guo, H. , Duan, F. , and Zhang, J. , 2016, “Blade Resonance Parameter Identification Based on Tip-Timing Method Without the Once-per Revolution Sensor,” Mech. Syst. Signal Process., 66, pp. 625–639.
Ivey, P. , Grant, K. , and Lawson, C. , 2002, The 16th Symposium on Measuring Techniques in Transonic and Supersonic Flow in Cascades and Turbomachines, Cambridge, UK, pp. 1–7.
Joung, K. , Kang, S. , Paeng, K. , Park, N. , Choi, H. , You, Y. , and von Flotow, A. , 2006, “Analysis of Vibration of the Turbine Blades Using Non-Intrusive Stress Measurement System,” ASME Paper No. POWER2006-88239.
Gallego-Garrido, J. , Dimitriadis, G. , Carrington, I. B. , and Wright, J. R. , 2007, “A Class of Methods for the Analysis of Blade Tip Timing Data From Bladed Assemblies Undergoing Simultaneous Resonances—Part II: Experimental Validation,” Int. J. Rotating Mach., 2007, p. 73624.
Mansisidor, M. R. , 2002, “Resonant Blade Response in Turbine Rotor Spin Tests Using a Laser-Light Probe Non-Intrusive Measurement System,” Master's thesis, Naval Postgraduate School, Monterey, CA, pp. 1–132.
Zielinski, M. , and Ziller, G. , 2005, “Noncontact Blade Vibration Measurement System for Aero Engine Application,” 17th International Symposium on Airbreathing Engines, Munich, Germany, Sept. 4–9, pp. 1–9.
Procházka, P. , and Vaněk, F. , 2012, “Non-Contact Systems for Monitoring Blade Vibrations of Steam Turbines,” International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium, Sept. 17–19, pp. 3359–3372.
Kwapisz, D. , Hafner, M. , and Rajamani, R. , 2012, “Application of Microwave Sensing to Blade Health Monitoring,” First European Conference of the Prognostics and Health Management Society (ECPHM), Dresden, Germany, July 3–5.
Madhavan, S. , Jain, R. , Sujatha, C. , and Sekhar, A. S. , 2014, “Vibration Based Damage Detection of Rotor Blades in a Gas Turbine Engine,” Eng. Failure Anal., 46, pp. 26–39.
Von Flotow, A. , Mercadal, M. , and Tappert, P. , 2000, “Health Monitoring and Prognostics of Blades and Disks With Blade Tip Sensors,” IEEE Aerospace Conference, Big Sky, MT, Mar. 35, pp. 433–440.
Von Flotow, A. , John, S. , and Gray, B. , 2012, “Field Demonstration of Low-Pressure Turbine Blade Vibration Monitoring,” Electric Power Research Institute, Palo Alto, CA, Technical Report No. 1024665.
Szczepanik, R. , Rokicki, E. , and Rzadkowski, R. , 2012, “Analysis of Middle Bearing Failure in Jet Engine Using Peak Detector Method,” Adv. Vib. Eng., 11(2), pp. 2–6.
Procházka, P. , and Vaněk, F. , 2011, “Contactless Diagnostics of Turbine Blade Vibration and Damage,” J. Phys.: Conf. Ser., 305, p. 012116.
Jousselin, O. , 2013, “Blade Tip Timing Uncertainty,” Rolls-Royce PLC, Westhampnett, UK, European Patent No. EP2631617A2.
Russhard, P. , 2012, “Blade Tip Timing Frequently Asked Questions,” Rolls-Royce PLC, Westhampnett, UK.
Choi, Y.-S. , and Lee, K.-H. , 2010, “Investigation of Blade Failure in a Gas Turbine,” J. Mech. Sci. Technol., 24(10), pp. 1969–1974.
Faddeev, D. K. , and Faddeeva, V. N. , 1981, “Computational Methods of Linear Algebra,” J. Sov. Math., 15(5), pp. 531–650.
Rossi, G. , and Brouckaert, J.-F. , 2012, “Design of Blade Tip Timing Measurement Systems Based on Uncertainty Analysis,” 58th International Instrumentation Symposium, San Diego, CA, June 4–8, pp. 1–19.
Petrov, E. , Di Mare, L. , Hennings, H. , and Elliott, R. , 2010, “Forced Response of Mistuned Bladed Disks in Gas Flow: A Comparative Study of Predictions and Full-Scale Experimental Results,” ASME J. Eng. Gas Turbines Power, 132(5), p. 052504.
Madhavan, S. , Jain, R. , Sujatha, C. , and Sekhar, A. S. , 2013, “Condition Monitoring of Turbine Rotor Blade on a Gas Turbine Engine,” Int. Congr. Sound Vib., 20, pp. 7–11.
van der Sluis, A. , 1970, “Stability of Solutions of Linear Algebraic Systems,” Numer. Math., 14(3), pp. 246–251.
Lee, A. , 2013, “Particle Swarm Optimization (PSO) With Constraint Support,” Python Software Foundation, accessed Apr. 18, 2018,
Szczepanik, R. , Rzadkowski, R. , and Kwapisz, L. , 2010, “Crack Initiation of Rotor Blades in the First Stage of so-3 Compressor,” Adv. Vib. Eng., 9(4), pp. 3–8.
Woike, M. R. , Roeder, J. W. , Hughes, C. E. , and Bencic, T. J. , 2009, “Testing of a Microwave Blade Tip Clearance Sensor at the NASA Glenn Research Center,” AIAA Paper No. 2009-1452.
Baraniuk, R. , 2007, “Compressive Sensing,” IEEE Signal Processing Magazine, 24(4), pp. 118–121.
View article in PDF format.

## References

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Dimitriadis, G. , and Carrington, I. , 2002, “Blade-Tip Timing Measurement of Synchronous Vibrations of Rotating Bladed Assemblies,” Mech. Syst. Signal Process., 16(4), pp. 599–622.
Beauseroy, P. , and Lengellé, R. , 2007, “Nonintrusive Turbomachine Blade Vibration Measurement System,” Mech. Syst. Signal Process., 21(4), pp. 1717–1738.
Diamond, D. H. , Heyns, P. S. , and Oberholster, A. J. , 2015, “A Comparison Between Three Blade Tip Timing Algorithms for Estimating Synchronous Turbomachine Blade Vibration,” 9th WCEAM Research Papers, Springer, Cham, Switzerland, pp. 215–225.
Lawson, C. P. , and Ivey, P. C. , 2005, “Turbomachinery Blade Vibration Amplitude Measurement Through Tip Timing With Capacitance Tip Clearance Probes,” Sens. Actuators A, 118(1), pp. 14–24.
Sabbatini, D. , Peeters, B. , Martens, T. , and Janssens, K. , 2012, “Data Acquisition and Processing for Tip Timing and Operational Modal Analysis of Turbomachinery Blades,” AIP Conf. Proc., 1457, pp. 52–60.
Russhard, P. , 2015, “The Rise and Fall of the Rotor Blade Strain Gauge,” Vibration Engineering and Technology of Machinery (Mechanisms and Machine Science, Vol. 23), pp. 27–38.
Loftus, P. , 2013, “Determination of Blade Vibration Frequencies and/or Amplitudes,” Rolls-Royce PLC, Westhampnett, UK, U.S. Patent No. 8380450B2.
Heath, S. , 1999,“A New Technique For Identifying Synchronous Resonances Using Tip-Timing,” ASME Paper No. 99-GT-402.
Salhi, B. , Lardies, J. , and Berthillier, M. , 2009, “Identification of Modal Parameters and Aeroelastic Coefficients in Bladed Disk Assemblies,” Mech. Syst. Signal Process., 23(6), pp. 1894–1908.
Hesler, S. , 2004, “Infrared Probe for Application to Steam Turbine Blade Vibration Detection,” Electric Power Research Institute, Palo Alto, CA, Technical Report No. 1004961.
Kumar, S. , Roy, N. , and Ganguli, R. , 2007, “Monitoring Low Cycle Fatigue Damage in Turbine Blade Using Vibration Characteristics,” Mech. Syst. Signal Process., 21(1), pp. 480–501.
Carrington, I. B. , Wright, J. R. , Cooper, J. E. , and Dimitriadis, G. , 2001, “A Comparison of Blade Tip Timing Data Analysis Methods,” Proc. Inst. Mech. Eng., Part G, 215(5), pp. 301–312.
Amoo, L. M. , 2013, “On the Design and Structural Analysis of Jet Engine Fan Blade Structures,” Prog. Aerosp. Sci., 60, pp. 1–11.
Pfister, T. , Günther, P. , Dreier, F. , and Czarske, J. , 2012, “Novel Dynamic Rotor and Blade Deformation and Vibration Monitoring Technique,” ASME J. Eng. Gas Turbines Power, 134(1), p. 012504.
Chen, Z. , Yang, Y. , Xie, Y. , Guo, B. , and Hu, Z. , 2013, “Non-Contact Crack Detection of High-Speed Blades Based on Principal Component Analysis and Euclidian Angles Using Optical-Fiber Sensors,” Sens. Actuators A, 201, pp. 66–72.
Liu, C. , and Jiang, D. , 2012, “Improved Blade Tip Timing in Blade Vibration Monitoring With Torsional Vibration of the Rotor,” J. Phys.: Conf. Ser., 364, p. 012136.
Agilis, 2014, “Non-Intrusive Stress Measurement Systems vs. Strain Gauges,” Agilis Measurement Systems, Inc., Palm Beach Gardens, FL, accessed Apr. 18, 2018,
Günther, P. , Dreier, F. , Pfister, T. , Czarske, J. , Haupt, T. , and Hufenbach, W. , 2011, “Measurement of Radial Expansion and Tumbling Motion of a High-Speed Rotor Using an Optical Sensor System,” Mech. Syst. Signal Process., 25(1), pp. 319–330.
García, I. , Beloki, J. , Zubia, J. , Aldabaldetreku, G. , Illarramendi, M. A. , and Jiménez, F. , 2013, “An Optical Fiber Bundle Sensor for Tip Clearance and Tip Timing Measurements in a Turbine Rig,” Sensors, 13(6), pp. 7385–7398. [PubMed]
Di Maio, D. , and Ewins, D. J. , 2012, “Experimental Measurements of Out-of-Plane Vibrations of a Simple Blisk Design Using Blade Tip Timing and Scanning LDV Measurement Methods,” Mech. Syst. Signal Process., 28, pp. 517–527.
Hu, Z. , Lin, J. , Chen, Z.-S. , Yang, Y.-M. , and Li, X.-J. , 2015, “A Non-Uniformly Under-Sampled Blade Tip-Timing Signal Reconstruction Method for Blade Vibration Monitoring,” Sensors, 15(2), pp. 2419–2437. [PubMed]
Von Flotow, A. , 2011, “Overview of Blade Vibration Monitoring Capabilities,” Hood Technology Corporation, Hood River, OR, Report.
Heath, S. , and Imregun, M. , 1996, “An Improved Single-Parameter Tip-Timing Method for Turbomachinery Blade Vibration Measurements Using Optical Laser Probes,” Int. J. Mech. Sci., 38(10), pp. 1047–1058.
Gallego-Garrido, J. , Dimitriadis, G. , and Wright, J. R. , 2007, “A Class of Methods for the Analysis of Blade Tip Timing Data From Bladed Assemblies Undergoing Simultaneous Resonances—Part I: Theoretical Development,” Int. J. Rotating Mach., 2007, p. 27247.
Russhard, P. , 2013, “Timing Analysis,” U.S. Patent No. 8457909B2.
Vercoutter, A. , Berthillier, M. , Talon, A. , Burgardt, B. , and Lardies, J. , 2011, “Tip Timing Spectral Estimation Method for Aeroelastic Vibrations of Turbomachinery Blades,” International Forum on Aeroelasticity and Structural Dynamics (IFASD), Paris, France, June 26–30, pp. 1–9.
Grant, K. , 2004, “Experimental Testing of Tip-Timing Methods Used for Blade Vibration Measurement in the Aero-Engine,” Ph.D. thesis, Cranfield University, Bedford, UK.
Szczepanik, R. , Rokicki, E. , Rzadkowski, R. , and Piechowski, L. , 2014, “Tip-Timing and Tip-Clearance for Measuring Rotor Turbine Blade Vibrations,” J. Vib. Eng. Technol., 2(5), pp. 395–406.
Li, M. , Duan, F. , and Ouyang, T. , 2010, “Analysis of Blade Vibration Frequencies From Blade Tip Timing Data,” Proc. SPIE, 7544, p. 75445F.
Russhard, P. , 2010, “Timing Analysis,” Rolls-Royce PLC, Westhampnett, UK, European Patent No. EP2199764A2.
Przysowa, R. , 2014, “Analysis of Synchronous Blade Vibration With the Use of Linear Sine Fitting,” J. KONBiN, 2(30), p. 17.
Guo, H. , Duan, F. , and Zhang, J. , 2016, “Blade Resonance Parameter Identification Based on Tip-Timing Method Without the Once-per Revolution Sensor,” Mech. Syst. Signal Process., 66, pp. 625–639.
Ivey, P. , Grant, K. , and Lawson, C. , 2002, The 16th Symposium on Measuring Techniques in Transonic and Supersonic Flow in Cascades and Turbomachines, Cambridge, UK, pp. 1–7.
Joung, K. , Kang, S. , Paeng, K. , Park, N. , Choi, H. , You, Y. , and von Flotow, A. , 2006, “Analysis of Vibration of the Turbine Blades Using Non-Intrusive Stress Measurement System,” ASME Paper No. POWER2006-88239.
Gallego-Garrido, J. , Dimitriadis, G. , Carrington, I. B. , and Wright, J. R. , 2007, “A Class of Methods for the Analysis of Blade Tip Timing Data From Bladed Assemblies Undergoing Simultaneous Resonances—Part II: Experimental Validation,” Int. J. Rotating Mach., 2007, p. 73624.
Mansisidor, M. R. , 2002, “Resonant Blade Response in Turbine Rotor Spin Tests Using a Laser-Light Probe Non-Intrusive Measurement System,” Master's thesis, Naval Postgraduate School, Monterey, CA, pp. 1–132.
Zielinski, M. , and Ziller, G. , 2005, “Noncontact Blade Vibration Measurement System for Aero Engine Application,” 17th International Symposium on Airbreathing Engines, Munich, Germany, Sept. 4–9, pp. 1–9.
Procházka, P. , and Vaněk, F. , 2012, “Non-Contact Systems for Monitoring Blade Vibrations of Steam Turbines,” International Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium, Sept. 17–19, pp. 3359–3372.
Kwapisz, D. , Hafner, M. , and Rajamani, R. , 2012, “Application of Microwave Sensing to Blade Health Monitoring,” First European Conference of the Prognostics and Health Management Society (ECPHM), Dresden, Germany, July 3–5.
Madhavan, S. , Jain, R. , Sujatha, C. , and Sekhar, A. S. , 2014, “Vibration Based Damage Detection of Rotor Blades in a Gas Turbine Engine,” Eng. Failure Anal., 46, pp. 26–39.
Von Flotow, A. , Mercadal, M. , and Tappert, P. , 2000, “Health Monitoring and Prognostics of Blades and Disks With Blade Tip Sensors,” IEEE Aerospace Conference, Big Sky, MT, Mar. 35, pp. 433–440.
Von Flotow, A. , John, S. , and Gray, B. , 2012, “Field Demonstration of Low-Pressure Turbine Blade Vibration Monitoring,” Electric Power Research Institute, Palo Alto, CA, Technical Report No. 1024665.
Szczepanik, R. , Rokicki, E. , and Rzadkowski, R. , 2012, “Analysis of Middle Bearing Failure in Jet Engine Using Peak Detector Method,” Adv. Vib. Eng., 11(2), pp. 2–6.
Procházka, P. , and Vaněk, F. , 2011, “Contactless Diagnostics of Turbine Blade Vibration and Damage,” J. Phys.: Conf. Ser., 305, p. 012116.
Jousselin, O. , 2013, “Blade Tip Timing Uncertainty,” Rolls-Royce PLC, Westhampnett, UK, European Patent No. EP2631617A2.
Russhard, P. , 2012, “Blade Tip Timing Frequently Asked Questions,” Rolls-Royce PLC, Westhampnett, UK.
Choi, Y.-S. , and Lee, K.-H. , 2010, “Investigation of Blade Failure in a Gas Turbine,” J. Mech. Sci. Technol., 24(10), pp. 1969–1974.
Faddeev, D. K. , and Faddeeva, V. N. , 1981, “Computational Methods of Linear Algebra,” J. Sov. Math., 15(5), pp. 531–650.
Rossi, G. , and Brouckaert, J.-F. , 2012, “Design of Blade Tip Timing Measurement Systems Based on Uncertainty Analysis,” 58th International Instrumentation Symposium, San Diego, CA, June 4–8, pp. 1–19.
Petrov, E. , Di Mare, L. , Hennings, H. , and Elliott, R. , 2010, “Forced Response of Mistuned Bladed Disks in Gas Flow: A Comparative Study of Predictions and Full-Scale Experimental Results,” ASME J. Eng. Gas Turbines Power, 132(5), p. 052504.
Madhavan, S. , Jain, R. , Sujatha, C. , and Sekhar, A. S. , 2013, “Condition Monitoring of Turbine Rotor Blade on a Gas Turbine Engine,” Int. Congr. Sound Vib., 20, pp. 7–11.
van der Sluis, A. , 1970, “Stability of Solutions of Linear Algebraic Systems,” Numer. Math., 14(3), pp. 246–251.
Lee, A. , 2013, “Particle Swarm Optimization (PSO) With Constraint Support,” Python Software Foundation, accessed Apr. 18, 2018,
Szczepanik, R. , Rzadkowski, R. , and Kwapisz, L. , 2010, “Crack Initiation of Rotor Blades in the First Stage of so-3 Compressor,” Adv. Vib. Eng., 9(4), pp. 3–8.
Woike, M. R. , Roeder, J. W. , Hughes, C. E. , and Bencic, T. J. , 2009, “Testing of a Microwave Blade Tip Clearance Sensor at the NASA Glenn Research Center,” AIAA Paper No. 2009-1452.
Baraniuk, R. , 2007, “Compressive Sensing,” IEEE Signal Processing Magazine, 24(4), pp. 118–121.

## Figures

Fig. 1

Principle behind BTT: (a) a compressor fan row with a broken out casing is shown with five proximity sensors (numbered S1 through S5) above the row and (b) the deflected blade tip arrives earlier than the undeflected tip due to the tip's deflection

Fig. 2

Different measurements taken by three different sensor arrangements for an EO 6 vibration of size 200 μm

Fig. 3

The relationship between increasing condition number and increasing BTT algorithm error is illustrated. Two EOs are used: 8 and 12. Each marker represents a single simulation with a randomly generated sensor configuration.

Fig. 4

The error function as a heat map for a three sensor BTT optimization where θ1 is fixed to 0 rad and the EOs between 2 and 10 are taken into account. Lighter shades indicate better solutions. The sensor spacings are constrained to be monotonically increasing.

Fig. 5

Progress of PSO algorithm for four different runs of the same problem

Fig. 6

Condition numbers for different EOs for the run 2 optimal result

Fig. 7

Condition numbers for different EOs when weighting the EO values lower than eight more than the remaining EO values. The difference between the weighted and unweighted EOs are also shown: (a) Weighted EO condition numbers and (b) condition number difference.

## Tables

Table 1 Sensor positions for the three different BTT systems
Table 2 Average simulated error in vibration parameter recovery for each of the three different BTT systems
Table 3 Condition numbers for the design matrices of three different BTT systems
Table 4 Natural frequencies for blades at different rotor speeds
Note: Frequencies in Hertz.
Table 5 Minimum and maximum locations of the permanent fixtures around the compressor casing
Note: Values in degrees.
Table 6 Parameter vector and objective function optimal values for each PSO run
Note: The objective function value has been abbreviated as OFV.

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