Sviluppo di tecniche automatizzate per la caratterizzazione dinamica sperimentale di macchine rotanti.

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Università di Pisa

Scuola di Dottorato “Leonardo da Vinci”

PhD Programme in

Mechanical Engineering

PhD Dissertation

Development of automatic

techniques for experimental

dynamic characterization of

rotating machines.

Paolo Neri

XXVIII CICLO 2012 - 2015

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Università di Pisa

Scuola di Dottorato “Leonardo da Vinci”

PhD Programme in

Mechanical Engineering

PhD Dissertation

Development of automatic

techniques for experimental

dynamic characterization of

rotating machines.

Author:

Paolo Neri . . . .

Tutors:

Prof. Ing. Leonardo Bertini . . . .

Prof. Ing. Ciro Santus . . . .

Ing. Bernardo Disma Monelli . . . .

XXVIII CICLO 2012 - 2015

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Introduction

Bladed wheels are crucial components in many rotating machinery, which are widely used in several industrial fields. Product development often leads the machine to work in really severe conditions. The resulting cyclic load can have catastrophic results on the main components, causing efficiency loss, noise, fatigue failures and expensive plant shut down. A deep knowledge of bladed wheels dynamic behaviour is then needed in order to face all these issues. At design stage, some tools are needed to predict dangerous working conditions or possible resonances. Once the wheels are designed and machined, a validation step is also needed to ensure the specification fulfilment in terms of vibrational response. Finite element analysis are surely useful, but experimental analysis are also essential. Modal analysis can be conducted in order to validate FE results, and harmonic response analysis can help to estimate the component response under specific simulated working conditions.

In this dissertation a study of the commonly adopted resonance prediction methods for bladed wheels is firstly presented. Then, two test bench are described along with experimental results: a robotic station for automatic modal analysis and a test bench for harmonic response analysis. The whole activity was conducted in cooperation with GE Oil&Gas, a worldwide leading company whose core business is represented by rotating machinery design and production. The research was a part of the program “Bando Unico Ricerca e Sviluppo 2012” (POR CReO FESR 2007-2013_REGIONE TOSCANA), set in the framework of the Regione Toscana funded project ATENE: “Advanced Technologies for ENergy Efficiency”.

In this dissertation the whole activity is reported, starting from a description of the two mainly used methods for resonance prediction, Chapter 1. The setup of a robotic station for fully automated experimental modal analysis is then presented in Chapter 2, along with some results obtained testing two different bladed wheels of the shrouded and un-shrouded type respectively. Chapter 3 shows a wide range harmonic response analysis test bench, able to

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simulate operational conditions by controlling the load spatial distribution in the frequency range 1 - 10 kHz. The design of the exciters and of the stinger system involved is also presented, along with some preliminary results. Chapter 4 describes a method for bladed wheels online monitoring, studied and developed during a research activity in cooperation with LMS - Siemens (Leuven, Blegium), which is a leading company in the field of the design of software and hardware for experimental dynamic analysis. Finally, some conclusions are presented: the whole activity is summarised and remarks about the main results are drawn. All the described activity were also presented in several research papers, from which most parts of the dissertation are drawn.

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Chapter 1

Bladed wheels resonances

prediction

1.1 Introduction

Any rotating component can experience different kinds of cyclic load, which could generate dangerous vibrations. Failures of turbo-machinery rotating equipment represent a huge cost to industry. Turbine or compressor impeller failures lead to plants being shut down and fixed, which is both expensive and time consuming. Blade fatigue damage caused by vibrations is the most commonly observed impeller failure mechanism [1, 2]. In order to reduce this fatigue loading, avoiding resonance conditions is one of the most critical issues regarding bladed disk design [3, 4]. The cyclic load acting on the blades is determined by the rotation speed combined with the flow distortion produced by a non-uniform inlet or outlet pressure distribution [5, 6]. Due to the combination between the load and dynamic property of the structure, resonance phenomena may occur [7, 8, 9]. The most common and simple approach to prevent resonance excitations is the Campbell diagram criterion [10], i.e. avoiding working and natural frequency matching. Although this criterion is very useful for simple dynamic structure design, it has severe limitations when studying complex components such as bladed wheels. Impellers have a large number of different modes with very close natural frequencies, and thus the Campbell criterion is not satisfied for (almost) any rotating speed regime, Fig. 1.1. A more effective criterion for bladed wheel design is needed. Since the load has a periodic trend both in time and in space domain, it is possible to claim that resonance happens when two matching conditions are satisfied: working

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Campbell possible resonance n × Working frequency N at ura l fre que nc y

Figure 1.1: Campbell example with very high density of natural frequencies.

and natural frequency matching (the Campbell diagram criterion) and harmonic excitation periodic shape and natural mode shape matching. Shape matching is described here by means of beam examples, Fig. 1.2. Given that the frequency of the applied forces is equal to the excited natural mode frequency, the forces must be in phase with the deformation displacements in order to achieve the maximum mechanical work, Fig. 1.2 (a). However, if the forces are completely out of phase, the mode is not excited since one force produces positive work while the other force produces negative work, Fig. 1.2 (c). There are also many intermediate conditions, Fig. 1.2 (b), where the forces excite the natural mode but the phase combination is not perfectly “constructive” with the excited natural mode. In principle this latter scenario is still a resonance, however it is less dangerous than full resonance and thus may be tolerated. Clearly, a finite element simulation would return unrealistic infinite deformation (and stresses) in any resonance condition, if no damping is introduced in the model. Friction, fluid viscosity, and any other kind of damping, may significantly reduce the amplitude of the resonance vibrations. These effects can even be simulated with finite elements by introducing a specific model input, as shown by Rao and Saldanha [11]. However, the avoidance of resonance is still assumed to be a very reliable approach, and thus a more precise detection of possible resonances is highly desirable at the design stage. The importance of both matching conditions is explained in the book by Bloch and Singh [12] and reported in earlier technical reports by Singh et al. [13, 14, 15, 16, 17, 18]. These works report the basic idea behind Singh’s Advanced Frequency Evaluation (SAFE) diagram, which includes both frequency and shape matching conditions. Useful equations were provided to detect resonance conditions, however, instead of

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1.1. Introduction 5 2 Second natural mode, ω work: max( )+ 2 Harmonic forces Ω ω= work: max( )+ 2 Second natural mode, ω work: ( )+ 2 Harmonic forces Ω ω= work: ( )− 2 Second natural mode, ω work: max( )+ 2 Harmonic forces Ω ω= work: max( )− (a) (b) (c)

Figure 1.2: (a) Force and mode in phase, full resonance. (b) Partial resonance. (c) Out of phase, no resonance.

a rigorous analytical demonstration, a graphical construct was pursued. This explains why in their more recent works the graphical method is commonly used to find dangerous resonance conditions, while analytical expressions are not used or sometimes not even correctly reported.

Using the pioneering investigation by Weaver and Prohl [19] as a starting point, McGuire and Knipe [20] studied resonance in bladed wheels in the same period as Singh. They introduced an interesting formula, which is equivalent to Singh’s formulation, including the excitation order and considering both counter rotating waves (though under slightly different assumptions regarding fluid harmonics). This dual modal wave direction was also evident in the equation reported by Jaiswal [21] for the experimental investigation to find the number of nodal diameters of a single mode. Bidaut and Baumann [22] described the use of the SAFE diagram, referred to as an “interference diagram”, and they experimentally retrieved the mode frequencies and shapes with standstill shaker-test measurements. They also reported an expression relating vane and blade numbers to the engine order and the number of nodal diameters of the excited mode, referred to as the Tyler-Sofrin rule. Tyler and Sofrin’s result [23] was really close to Singh’s approach, but erroneously considered just one of the two counter rotating waves of the modal vibration thus just approximately half the number of full resonance combinations were identified.

The SAFE diagram answers the need for a unique design method, although the optimization of wheel performances led to two different kinds of impellers: open or close, which can both be successfully studied with this method. Ex-amples of open or close impellers are shown in Ref. [24] by Castanier and Pierre and Ref. [25] by Kushner, respectively. Open impellers can have low frequency modes even with a large number of nodal diameters, while close impellers usually have low frequency and large deformation critical modes only with a small number of nodal diameters.

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Clearly, the SAFE diagram approach has some limitations. One drawback is the sector symmetry assumed for both the impeller and the inlet vanes, which introduces exact cyclic periodicity both for the fluid forces and for the dynamic modes. A more comprehensive study for the “mistuned” impeller is required [24, 26, 27], where the cyclic symmetry of the wheel is not perfect. Moreover, shape matching is assessed by the SAFE diagram only in terms of the number of nodal diameters without taking into account harmonic intensity and/or load direction alignment with respect to the displacement of the corresponding mode.

In this paper we provide a complete analytical formulation of the SAFE diagram approach applied to the resonance detection of bladed wheels. The shape matching conditions are analytically derived, and the effectiveness of the graphical construct is proved. In addition, for what we believe is the first time in the literature, a compact and completely general formula is provided. This formula correlates the main parameters of the problem, thus leading to the discovery of all the dangerous combinations in the SAFE diagram, overcoming the graphical method. This analytical expression is very useful for the drawing scripts of the SAFE diagram, and more importantly it allowed to find the symmetry and periodicity properties that were hidden in the graphical approach. Finally, a validation was obtained both with finite element simulations and experimental activity highlighting the advantages of the SAFE diagram compared to the Campbell criterion.

1.2 The SAFE diagram

1.2.1 Derivation of shape matching conditions

Fig. 1.3 shows a schematic bladed wheel interacting with its stator. NBis the

number of blades of the rotor wheel, Ω is the rotating speed of the wheel, and NVis the number of stator vanes. The fluctuating force acting on each blade is a

periodic function. The period is determined by the time needed by the i-th blade to cross a vane length. A blade interacts NVtimes with different vanes after

a complete revolution of the wheel, thus the frequency of the main harmonic component is:

Ω1= NVΩ (1.1)

The blades are equally spaced by the sector angle ∆ ϑ = 2π/NB. The force

acting on each blade is shifted by this phase with respect to the previous blabe. A generic angular position is given by the product of time and angular speed, plus the shift angle, Fig. 1.4 (a). The periodic blade force can be decomposed

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1.2. The SAFE diagram 7 V N B N Ω B -th Blade 1,2,..., i i= N B 2 N ϑ π ∆ = ϑ ∆

Figure 1.3: Bladed wheel and stator.

according to the Fourier series as a steady component plus the main harmonic (period: 2π/NV), plus the second harmonic (period: 2π/(2 × NV)), plus the

third and so on, Fig. 1.4 (b). The i-th blade total force can thus be written as [28]: fi(t) = ∞

∑

n=0 fn,i(t) = F0+ ∞

∑

n=1 Fncos(nNV(Ωt + i∆ ϑ ) + ϕn) (1.2)

where F0, F1, . . . , Fn, . . . are the harmonic force amplitudes and ϕnis a generic

phase for each n-th harmonic. Finally, the n-th harmonic component angular frequency can be defined as:

Ωn= n Ω1= n NVΩ (1.3)

Similarly, the vibration of the blades can be written by following the modal decomposition. The displacement with respect to the undeformed shape of the generic i-th blade is [29]:

xi(t) = ∞

∑

m=0 xm,i(t) = X0+ ∞

∑

m=1 Xmcos(ωmt+ ϕtm) cos(dm∆ ϑ i + ϕϑ m) (1.4)

where Xmis the m-th mode component, ωmis its angular frequency and ϕtm, ϕϑ m

are generic phases for the time and the angle variables, respectively. dmis the

number of nodal diameters of the m-th natural mode, and a nodal diameter is the locus of points which have null displacement. Equation 1.4 is valid for each harmonic shape mode, even when the number of nodal diameters is zero and the angular dependence vanishes, reducing to:

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(a) 0 -th traveling blade i t = + ϑ Ω ϑ B 2 N ϑ π ∆ = Anglecoordinate, ϑ Blade force i f 0 ( 1)-th traveling bladei t ϑ Ω ϑ ϑ + = + ∆ + V 2 / Nπ (b) Blade force i f V 2 N π V 2 2 N π × _V 2 n N π ... 0 F Anglecoordinate, ϑ 2 F F1

Figure 1.4: Blade force: (a) angular shift, (b) Fourier decomposition.

(a) (b) (c)

Figure 1.5: Example of modes: (a) zero nodal diameters, (b) one nodal diameter, (c) two nodal diameters.

where ¯mis a zero nodal diameters mode: dm¯ = 0. Figure 1.5 shows examples

of modes with zero, one and two nodal diameters. Since the effect of mistuning was not considered in this analysis, the bladed wheel was a perfectly cyclic symmetric body and thus all the modes were harmonic [29]. This enabled just one of the NB sectors of the bladed wheel to be studied, by setting suitable

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1.2. The SAFE diagram 9

There is no simple relation between the natural mode order and the number of nodal diameters. Modes with a high natural frequency can have a low number of nodal diameters, and many different modes unavoidably share the same num-ber of nodal diameters. Obviously, all bladed wheels have an infinite numnum-ber of modes, m = 1, 2, . . ., however the maximum number of nodal diameters is limited: 0 ≤ dm≤ dmax, where dmaxis not larger than half the number of blades:

dmax= NB 2 if NBis even dmax= NB− 1 2 if NBis odd (1.6)

When each blade is out of phase with the next, along the hoop coordinate, the maximum number of displacement sign reversals is reached together with the maximum number of nodal diameters (being equal to the zero displacement points divided by 2). This is shown in Fig. 1.6 where the displacement signs are reported near each blade. If NBis even, the maximum number of reversals

is just NBand thus dmax= NB/2. The maximum number of reversals is reduced

to NB− 1 if NBis odd, in fact two consecutive blades by necessity have the

same sign, thus dmax= (NB− 1)/2. As stated in Eqs. 1.2, 1.4, both the force

+ + − ₋ − Nodal diameters B odd N + + + + − Nodal diameters B even N −

Figure 1.6: Maximum number of nodal diameters, even or odd NB.

and the displacement can be decoupled into components with a harmonic time dependency. Each force component is a traveling wave, since the bladed wheel is moving with respect to the stator. On the other hand, each displacement modal component is a stationary wave from a bladed wheel point of view, Fig.1.7. However, any stationary wave can be decoupled as the sum of two

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angle coordinate, ϑ Traveling wave:

cos(ω ϑt+ )

Stationary wave:

1 1 cos( )cos( ) cos( ) cos( )

2 2 t t t ω ϑ = − +ω ϑ + ω ϑ+ Blades: angle coordinate, ϑ 1 2 3 4 ... Force Displacement

Figure 1.7: Relative time dependency of a force harmonic component and a displacement modal component.

traveling waves with opposite directions:

xm,i= Xmcos(ωmt+ ϕtm) cos(dm∆ ϑ i + ϕϑ m)

= Xm

2 [cos(ωmt− dm∆ ϑ i + ϕ1m) + cos(ωmt+ dm∆ ϑ i + ϕ2m)]

(1.7)

where the generic phase angles are: ϕ1m= ϕtm− ϕϑ mand ϕ2m= ϕtm+ ϕϑ m.

The ϑ term vanishes, for a zero nodal diameters mode, and Eq. 1.7 simply reduces, again, to Eq. 1.5. As a result of the opposite direction traveling waves, each non zero number of nodal diameters mode always appears as a couple of degenerate modes [29]. On the other hand, any zero nodal diameters mode is always a single mode. Given Eq. 1.7, the speed of the i-th blade can be obtained as the partial derivative of the displacement with respect to time. The modal superimposition is still valid because of the derivative linearity. By following the decoupling of the traveling waves, the modal speed components can be written as:

vm,i=

∂

∂ txm,i= Xmωmcos(ωmt+ ϕtm+ π/2) cos(dm∆ ϑ i + ϕϑ m)

= Xm

2 ωm[cos(ωmt− dm∆ ϑ i + ϕ1m+ π/2) + cos(ωmt+ dm∆ ϑ i + ϕ2m+ π/2)] (1.8)

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and thus the total speed is:

vi(t) = ∞

∑

m=1 Xm 2 ωm[cos(ωmt− dm∆ ϑ i + ϕ 0 1m) + cos(ωmt+ dm∆ ϑ i + ϕ2m0 )] (1.9) where the phase angles are: ϕ_1m0 = ϕ1m+ π/2 and ϕ2m0 = ϕ2m+ π/2.

Resonance conditions are excited if the work done by the forces acting on the structure is greater than zero after one cycle of the main harmonic. This produces a positive energy input on the component, resulting as a vibration amplification. The work done on the wheel, over one main harmonic period, is the sum of the work done on all the blades:

W =

NB

∑

i=1

Wi (1.10)

the work on each blade is the integration of the power:

Wi= Z T₁

0

˙

Widt (1.11)

and the main harmonic period is:

T1=

2π Ω1

(1.12)

The i-th blade power is the harmonic force components, Eq. 1.2, times the vibration speed of the blades, Eq. 1.9:

˙ Wi= fivi fivi= ∞

∑

n=1 ∞

∑

m=1 FnXmωmcos(nNV(Ωt − ∆ ϑ i) + ϕn)× [cos(ωmt+ dm∆ ϑ i + ϕ_1m0 ) + cos(ωmt− dm∆ ϑ i + ϕ_2m0 )] (1.13)

The high order terms of the two series can be neglected, and the indexes limited to nmax, mmaxrespectively, instead of infinite. In fact, the harmonic and the

mode amplitudes quickly decrease as the order increases. The steady term F0

of the force harmonic decomposition does not produce an increasingly work contribution, since the speed is a sum of fluctuating terms, thus F0does not

appear in Eq. 1.13. The combination of the n-th harmonic force and the m-th modal speed component can generate a work contribution greater than zero, over an indefinite period of time, whenever the frequencies match:

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or, more simply:

Ωn= ωm (1.15)

Equation 1.15 is the Campbell criterion. The matching between the shapes of the mode and the harmonic force is also required, as mentioned above. Equation 1.13 proves that there is shape matching if, and only if, one of the two following conditions is verified for all blade indexes i = 1, 2, . . . , NBand for any k integer

(positive, negative, or null): k = . . . , −2, −1, 0, 1, 2, . . .

nN_V∆ ϑ i = dm∆ ϑ i + k(2π)

nNV∆ ϑ i = −dm∆ ϑ i + k(2π)

(1.16)

After Eq. 1.15 and one of the two of Eqs. 1.16 are satisfied, the arguments of the multiplying cosines of Eq. 1.13 are the same (regardless of the angular phases), thus a cos2_{() term results, and finally this has a positive integration}

over a time period.

The conditions in Eq.1.16 can be summarized in one single condition by means of the “plus or minus” operator:

nN_V_N2π

Bi= ±dm

2π

NBi+ k(2π)

for all i = 1, 2, . . . , NBand for any integer k = . . . , −2, −1, 0, 1, 2, . . .

(1.17) where the symbol ± implies that either the equation with the + sign or alter-natively the equation with the − sign has to be satisfied. Equation 1.17 is a complete way to determine resonance combinations. Moreover, when Eq.1.17 is verified for i = 1, it is also verified for any other index i:

Eq.1.17 is solved for i = 1 ⇐⇒ Eq.1.17 is solved for i = 1, 2, . . . , NB (1.18)

If i = 1, k = k1 solves Eq.1.17, i = 2, k = 2 × k1 solves Eq.1.17 too, and so

on for higher i. The opposite implication is obvious: if Eq.1.17 is solved for all the blade indexes i = 1, 2, . . . , NBthen it is also solved for the single i = 1.

Therefore, Eq.1.17 can be formally simplified just by introducing i = 1:

nN_V2π

N_B = ±dm 2π

N_B+ k(2π) for any integer k = . . . , −2, −1, 0, 1, 2, . . . (1.19) Eq. 1.19 shows that the dependency on blade index i is not meaningful, indeed blade indexing is just conventional. Finally, by recalling Eq.1.15 and by re-arranging Eq.1.19, the full resonance conditions can be written in their final

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1.2. The SAFE diagram 13 form: Ωn= ωm nNV± dm NB = integer (1.20)

where obviously n is positive, while the integer that satisfies the second con-dition can be either null or positive, but it can never be negative, because dm≤ NB/2 as shown previously. Again, the first condition of Eqs. 1.20 and

one of two of the second (+ or −) have to be true. Generally, when dm> 0 it is

not possible to have both + and − verified, only one is verified, or alternatively neither are. The only condition when both (+ and −) are verified, provided that dm> 0, is when NB is even, dm= NV= NB/2 and n is even as well, as

discussed below. The zero nodal diameters modes dm¯ = 0 are also implicitly

considered in Eq. 1.20, here the term ±dmvanishes, and the two conditions

reduce to:

Ωn= ωm¯

nNV

N_B = integer

(1.21)

After manipulating the analytical expression of the shape matching condition Eq. 1.20, it is possible to calculate the matching nodal diameters directly. Given a set of numbers NB, NVand for any specific harmonic index n the (unique) dm

value can be calculated. The first step is to find the integer k that satisfies Eq. 1.20. Since dm≤ dmaxit follows that:

nNV+ dmax

NB

≥ k ≥nNV− dmax NB

(1.22)

When NBis odd, there is just one single integer in this range. Since dmax< NB/2,

the difference (nNV+ dmax)/NB− (nNV− dmax)/NBis lower than unity, thus

it is not possible to have two integers in this range. In principle, it would be possible to have no integer, however, this does not happen. After calculating the remainder r of the ratio nNV/NB, if this remainder is higher than the ratio

a= dmax/NB, the left term of Eq. 1.22 is higher than the integer given by the

floor1of the ratio nNV/NB, while the other term is lower, so that floor integer

itself is the k integer. If the remainder r equals ratio a, the right hand term is an integer. Finally, when r is higher than a, it should be at least: r = a + 1/NB,

1_{The floor function of x, usually formalized as bxc, is the highest integer that is lower than or}

equal to x, e.g.: b2.0c = 2, b3.8c = 3. Conversely, the ceiling function of x is the lowest integer that is higher than or equal to x, e.g.: d0.001e = 1.

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as a = dmax/NB= (NB− 1)/(2NB), their sum is at least unity or higher so the

left hand term of Eq. 1.22 is at least the floor integer of the ratio nNV/NBplus

unity, and thus k is just this integer. After demonstrating that there is a unique integer between the range of Eq. 1.22, the integer floor and ceiling of these two limits are exactly the same number:

k0= b(nNV+ dmax)/NBc

k00= d(nNV− dmax)/NBe

k= k0= k00

(1.23)

When NB is even, the range of Eq. 1.22 exactly equals unity, thus there is

usually one single integer in this range. The only possibility of having two different integers, again, is when NV= NB/2 (NBis even) and n is odd, so the

remainder r of the term nNV/NBand the ratio a = dmax/NBare both 0.5. In this

very particular case k0= (n + 1)/2, k00= (n − 1)/2.

When k = k0 = k00 (regardless of whether NB is even or odd), dm can be

easily obtained, by introducing this integer into the second of Eqs. 1.20:

dm= |nNV− NBk| (1.24)

and it is worth noting that this number is unique. When k0and k00are not equal, for the aforementioned case, the dmresult is the same either with k0or k00. In

this specific case:

dm= |nNV− NBk0| = | − NB/2| = NB/2

dm= |nNV− NBk00| = |NB/2| = NB/2

(1.25)

For this case too the dmvalue is unique, despite the integer k0, k00duality. In

order to give a general rule, one of the two forms for this integer can be chosen without any ambiguity on the final value of dm. Hereafter, the floor integer is

chosen:

k0= b(nNV+ dmax)/NBc

dm= |nNV− NBk0|

(1.26)

We now give an example by considering the combination NB= 10, NV= 7. The

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be easily found:

k0= b(nNV+ dmax)/NBc = b(3 × 7 + 5)/10c = b2.6c = 2

dm= |nNV− NBk0| = |3 × 7 − 10 × 2| = 1

(1.27)

by taking the other integer formula, the result does not change:

k00= d(nNV− dmax)/NBe = d(3 × 7 − 5)/10e = d1.6e = 2

dm= |nNV− NBk00| = |3 × 7 − 10 × 2| = 1

(1.28)

An other example is provided with k0and k00not equal, by considering the case NV= 5, NB= 10 and n = 3:

k0= b(nNV+ dmax)/NBc = b(3 × 5 + 5)/10c = b2.0c = 2

k00= d(nNV− dmax)/NBe = d(3 × 5 − 5)/10e = d1.0e = 1

(1.29)

as mentioned above, the resulting dmis still a unique value:

dm= |nNV− NBk0| = |3 × 5 − 10 × 2| = | − 5| = 5

dm= |nNV− NBk00| = |3 × 5 − 10 × 1| = |5| = 5

(1.30)

Equations 1.26 can be finally combined to have a unique formula:

dm= |nNV− NBb(nNV+ dmax)/NBc| (1.31)

The unique dmcould also be found with the other form with the ceiling operator:

dm= |nNV− NBd(nNV− dmax)/NBe|, even when the floor and ceiling integers

are different. As mentioned above, just the form of Eq. 1.31 is considered in the following. Finally, the floor operator can also be used to calculate the maximum number of nodal diameters: dmax= bNB/2c, and then an even more compact

form can be issued, where just the symbols n, NV, NBand floor function appear:

dm= |nNV− NBb(nNV+ bNB/2c)/NBc| (1.32)

This equation is a closed-form expression that easily highlights the n to dm

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1.2.2 Examples

In this subsection we provide some examples to explain the second condition of Eq. 1.20 and its implications in specific cases. Figure 1.8 shows a design with no matching between the modal shape and force pattern. The parameters are reported in Eq. 1.33. Although the frequency condition is satisfied, there is no full resonance for the bladed wheel. When the harmonic force and the modal displacement are in phase for one blade, they are not in phase for the other blades, Fig. 1.8 (a). In this situation the resonance is generally “partial”, as in Fig. 1.2 (b), or completely absent, in some special cases, as in Fig. 1.2 (c).

N_B= 7, N_V= 5, n= 2, dm= 2 nNV+ dm N_B ≈ 1.714, nNV− dm N_B ≈ 1.143 (1.33) 2 _i_₁ 3 4 ₅ 6 7 Force harmonic Modal displacement 1 i 2 3 4 5 6 7 1 wheel revolution 0 t t1 0 t t1 Time

Figure 1.8: Example 1, time and angular dependency of force and displacement.

From Eq. 1.32, it follows that: dm= |nNV− NBb(nNV+ bNB/2c)/NBc| = 3,

therefore, this example with dm= 2 obviously is not a case of matching.

When the shape matching condition is fulfilled, there may be two different situations, depending on the sign in the second of Eqs. 1.20. It is possible to define a “positive sign matching” when (nNV+ dm)/NB= integer, and a

“negative sign matching” when (nNV− dm)/NB= integer. An example with

negative sign matching mode is reported in Fig. 1.9 and the parameters are listed in Eq. 1.34. Each blade has force and displacement in phase, though a different number of periods along the angular full circle. A two nodal diameters

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mode is excited by this harmonic when the frequency condition is also met.

NB= 10, NV= 6, n= 2, dm= 2 nNV+ dm NB = 1.4, nNV− dm NB = 1 (1.34)

Figure 1.9 also clarifies that the shape condition can be interpreted as an

2 1 i 3 4 5 6 10 1 i 2 3 4 5 6 7 8 9 10 7 8 9 0 t t1 1 wheel revolution 0 t t1 Time Force harmonic Modal displacement

Figure 1.9: Example 2, time and angular dependency of force and displacement.

aliasing phenomenon. The force may have a high angular frequency (i.e. nNV),

but it is “sampled” by NBblades, so that the highest recognizable frequency

can not be higher than bNB/2c = dmax(Nyquist-Shannon theorem). If nNVis

greater than NB/2, the excitation is misinterpreted as a lower frequency load

(i.e. dm). Another example is reported in Eq. 1.35. As in the previous example,

each blade has force and displacement in phase, but with positive sign matching instead of negative. NB= 10, NV= 9, n= 2, dm= 2 nN_V+ dm NB = 2, nNV− dm NB = 1.6 (1.35)

The difference between these two latter cases is that the negative sign matching implies that the modal shape rotates in the same direction as the harmonic force, while the modal shape and the harmonic force are counter rotating for the positive sign matching. The modal decomposition in Eq. 1.8 highlights that only one of the two counter rotating traveling waves is excited, depending on

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which of the shape matching conditions is satisfied. This sign discrimination can also be found in Eq. 1.32. When the negative sign applies, the argument of the absolute value of Eq. 1.32 is positive, whereas the argument of the absolute value is negative for the opposite case. The very special case with NB

even, NV= NB/2 and n odd, have both positive and negative matching, so both

counter rotating traveling waves fully combine with the harmonic component. Clearly, only one will be excited by running a physical example. However, this specific condition is not of interest since there are many harmonic to mode matching cases at the same number of nodal diameters.

Finally, an example with a zero nodal diameters mode is shown in Fig. 1.10. Here the harmonic forces and blade displacements are all in phase and rotating/counter-rotating mode cannot be defined.

NB= 6, NV= 3, n= 2, dm¯ = 0 nNV N_B = 1 (1.36) 2 i1 3 4 5 6 1 i 2 3 4 5 6 0 t t₁ 0

t t₁ Time 1 wheel revolution Force harmonic Modal displacement

Figure 1.10: Example 3, time and angular dependency of force and displace-ment.

1.2.3 Matching map

Equation 1.32 can be easily exploited in order to find the “matching map”, i.e. a graph reporting the matching number of nodal diameters for each har-monic index, showing a clear view of the matching distribution. The matching

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combinations can also be obtained with a counting procedure, which is the common method for drawing the SAFE diagram. Any rational number, such as (nNV)/NB, can be separated as an integer part and a remainder2whose absolute

value is lower than or equal to 0.5. This remainder can be written as an integer dover NB: nNV NB = integer + d NB (1.37)

This decomposition is always possible and unique, with d limited (again) depending on whether NBis even or odd:

−NB 2 ≤ d ≤ NB 2 if NBis even −NB− 1 2 ≤ d ≤ NB− 1 2 if NBis odd (1.38)

The absolute value of d:

|d| = |nN_V− integer × NB| (1.39)

is the matching number of nodal diameters: dm= |d|. In fact, it follows that the

matching condition is satisfied:

if d > 0; nNV− |d| NB = integer if d < 0; nNV+ |d| N_B = integer (1.40)

Given a combination of NB, NVand an index n, it is possible to find the matching

dmby counting from 0 to nNVon the horizontal axis and applying reflections

at NB/2 if NB is even, or at (NB− 1)/2 if NB is odd. Figure 1.11 shows two

examples with an even and an odd number of blades. This counting search is equivalent to finding d regardless of the sign. In fact, Eq. 1.37 can be rewritten as:

nNV= NB× integer + d (1.41)

Due to the reflections, once a multiple of NB is stepped (NB× integer) the

position dm= 0 is reached. Then, by adding or subtracting a number of steps

|d| to complete the counting, the position dm= |d| is found. Once dmfor n = 1

2_{The remainder used here equals x − bxc if x − bxc < 0.5 or x − dxe if dxe − x < 0.5, and an}

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is obtained, the counting of the next nNVsteps can restart from this position to

find the dmfor n = 2, and so on. From Eq. 1.41 the integer equals the number

of experienced reflections, during the counting, which is the number k0 (or k00) introduced and discussed above.

0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 B 2 N B V 10 7 N N   1 2 3 4 5 6 7 Nodal diameter

Excitation harmonic order

0 1 2 3 4 0 1 2 3 4 5 6 7 8 B 1 2 N  B V 9 7 N N   1 2 3 4 5 6 7 Nodal diameter

Excitation harmonic order

(a) (b)

Figure 1.11: Shape matching map examples: (a) even number of blades, (b) odd number of blades.

1.2.4 Symmetry properties of the matching map

Two symmetry properties of the matching map can be observed and proved thanks to Eq. 1.32, thus contributing to a more complete understanding of the matching combination distribution. The first property concerns the matching map periodicity, with a period of NB, for any set of NB, NV, regardless of

whether NB is even or odd. As highlighted by Fig. 1.12, after the first NB

harmonics, the next NB+ 1 to 2NBshare the same matching dmvalues, and so

on. This property can be formalized as:

dm(n + NB) = dm(n) for any harmonic index n = 1, 2, . . . (1.42)

The second property is that within each period the matching condition distribu-tion is also symmetric, with the reflecdistribu-tion axis at NB/2. This symmetry property

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1.2. The SAFE diagram 21 0 1 2 3 4 0 4 8 12 16 0 1 2 3 0 3.5 7 10.5 14 Periodic Periodic B 2 N B 2 N Symmetric Symmetric B N B 2N B 1 2 N  B 2 N B N B 2N Beven N NBodd

Figure 1.12: Symmetry properties of the matching map.

can be written as:

dm(NB/2 + ∆ n) = dm(NB/2 − ∆ n) (1.43)

where ∆ n is the (positive) value of the difference between NB/2 and the two

symmetric harmonic indexes. In reality, Eq. 1.43 formalizes the symmetry just for the first period NB/2 ± ∆ n = 1, 2, . . . , NB, however, as the distribution

is periodic, this property can be extended to any other period. ∆ n and NB/2

obviously are integer numbers if NBis even, otherwise, Eq. 1.43 still is valid

with ∆ n = 0.5, 1.5, . . . These relations are evident in the two examples (NB

even and odd) in Fig. 1.12. In particular, if NB is odd n = NB/2 − 1/2 and

n= NB/2 + 1/2 have the same matching dmand are consecutive, so the same

mode can be excited by two close harmonics. The proofs of these two properties (Eqs. 1.42, 1.43) are reported in the Appendix.

A further property of the matching combination distribution is that Eqs. 1.42, 1.43 can be interpreted in terms of NVrather than n, which share interchangeable

positions in Eq. 1.32. The same matching distribution can be obtained with two different numbers of vanes NV which are periodic with respect to NB

or symmetric with respect to NB/2. For example, NV= 5 and NV= NB+ 5

generate the same matching maps, and also NV= NB/2 + 1 and NV= NB/2 − 1

(with NBeven), again, produce the same matching maps.

All these properties can help in selecting a safe NB, NVcombination.

How-ever NBand NVare usually constrained by several design requirements and thus

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1.2.5 SAFE diagram graphical derivation

After having found the matching map, the SAFE diagram can be easily obtained by multiplying the harmonic index n by the first harmonic frequency Ω1, and

by superimposing the natural modes following the same coordinates: number of nodal diameters and frequency. Figure 1.13 shows two examples with an even and odd number of blades. As described above, the matching number of

0 1 2 3 4 5 0 1 2 3 4 Shape Frequency 1 ( 1), m d n= Ω 1 2 d = 3 4 d = 2 3 d = 2 ( 2), m d n= Ω 1 Ω × int . 0= int . 2= int . 1= B V 10 7 N N = = 0 1 2 3 4 0 1 2 3 4 Shape Frequency B V 9 7 N N = = 1 2 d = 3 4 d = 2 3 d = 2 ( 2), m d n= Ω 1 Ω × int . 0= int . 1= int . 2= (a) (b)

Figure 1.13: SAFE diagram examples: (a) even number of blades, (b) odd number of blades.

nodal diameters can also be obtained with the counting technique. By counting along the dmcoordinate up to NV, moving on the dotted line, the corresponding

frequency is: NVΩ = Ω1. The scheme of the matching map is followed, so

the spotted point is dm(n = 1), Ω1, where dm(n = 1) is the number of nodal

diameters matching the first harmonic n = 1. Counting and reflections can be applied again for the next NVtimes and the point dm(n = 2), Ω2 is found.

This can be repeated indefinitely, at least up to the highest significant harmonic index. Alternatively, the matching number of nodal diameters can be obtained by means of the above introduced analytical expression Eq. 1.32. Counting is no longer required, so the frequency line (dashed line in Fig. 1.13) could even be removed, as in Fig. 1.16 below. The coordinates of the matching points are

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1.3. SAFE diagram validation 23

easily obtained as:

n= 1, 2, . . .

dm(n) = |nNV− NBb(nNV+ bNB/2c)/NBc|

Ωn= n Ω1

(1.44)

Now the resonance conditions are evident: when a natural mode point dm, ωm

is overlapped with a harmonic point dm(n), Ωn a full resonance is expected.

For example, the third mode d3= 4, Fig. 1.13 (a), has shape and frequency

coincident with the second harmonic, thus a full resonance is excited. The second mode d2= 3, reported in Fig. 1.13 (a), does not match perfectly in terms

of frequency with the first possible resonance point dm(n = 1), Ω1, however,

the dynamic amplification is still expected to be high since the shape matching is verified and the frequency values are close. In addition, the angular speed of the wheel Ω can change during working operations, thus an almost matching frequency can turn into a perfectly matching resonance during transient regimes. In fact, the SAFE diagrams are usually reported as a working frequency range, as shown in the following, in order to detect all dangerous conditions.

1.3 SAFE diagram validation

1.3.1 Harmonic response FE analysis

The SAFE diagram analysis was applied to a compressor impeller and FE verification provided. An experimental analysis was also performed on the same wheel, as shown in the next section. This wheel is a small scale design with NB= 15 blades, which reproduces a larger impeller. The FE modal analysis

was performed by considering the whole bladed wheel structure. To obtain the full model, one of the NBsectors was meshed (ANSYS 3D 10-node tetrahedral

structural solid elements) and an identical mesh on the edge-faces of the sector was imposed. The sector mesh was then copied NB− 1 times, and the nodes

on the sector edge-faces were merged, producing 42120 elements. The hub of the wheel was fixed, imposing a zero displacement boundary condition and the first 24 natural modes were then easily found (frequency values and number of nodal diameters). Harmonic response FE simulations were then performed, by applying four load cases with different configurations. The frequency of the first load case was set as equal to the 12th natural mode of the wheel: ω12. The harmonic forces were applied to the wheel blades and

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the load phase distribution was set in order to have a two nodal diameters shape excited, Fig. 1.14 (a). The 12th natural mode actually had two nodal diameters: d12= 2, thus a full resonance condition was expected. Load cases

2 and 3 were performed by shifting the frequency downwards and upwards respectively: 0.9 × ω12, 1.1 × ω12. Finally, the fourth load case was set with

same frequency ω12but with the load phase distribution according to one nodal

diameter, Fig. 1.14 (b). The load cases are summarized in Fig. 1.14 (c) and the maximum axial displacement of a blade is plotted over time in Fig. 1.14 (d). The displacement amplitude of load case 1 was very large, though limited, no damping was introduced but the imposed frequency of the load was 99.9% of the frequency of the excited mode. The amplitude of case 4 is much smaller, despite having the same load frequency, because of the shape mismatch. Cases 2 and 3 also did not show high amplitudes, though the shape matching, since the frequency of the load was significantly different from the frequency of the mode. These numerical simulations confirmed that both frequency and shape matching conditions are needed in order to excite a mode and obtain a high vibration amplitude resonance.

1.3.2 Experimental results

An experimental test was performed on a scaled bladed wheel. The experimental setup is shown in Fig. 1.15. The tested impeller was instrumented with strain gauges in order to verify high amplitude vibration of the structure during high rotating speed testing. Twelve telemetry wireless strain gauges were applied on the blades. The wheel had NB= 15 blades and the fluid was distributed

by a stator with NV= 16 vanes. The SAFE diagram of the investigated wheel

is reported in Fig. 1.16. Instead of a having one specific speed, a range of values was used. Accordingly, each load harmonic is denoted by two points on the same dm, connected with a solid vertical line. The dotted lines in

Fig. 1.13 are no longer reported, since graphical construct is overcome by Eq. 1.32. Two matching conditions were obtained along the velocity sweep, regarding the second load harmonic and one of two natural modes: d12= 2, ω12

and d17= 2, ω17= 1.21ω12. Figure 1.17 shows the experimental Campbell

diagram (or waterfall diagram) obtained after performing the speed sweep. The horizontal axis shows the normalized rotational speed of the bladed wheel. The vertical axis reports the normalized fast Fourier transform of the most excited strain gauge. Each frequency is thus associated with its vibration amplitude and represented on the graph with a colored dot. For each rotational speed, a higher amplitude is found (and a dot is drawn) for all natural mode frequencies,

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1.3. SAFE diagram validation 25 (a) (b) 0 1 2 3 0.9 1 1.1 Shape d 12 = 2 Load case 1 Load case 2, 3 Load case 4 Shape 12 Frequency /ω 12 Fr eque nc y / ω 1 1 ax Load case 1 Load cases 2, 3, 4 −0.040.04 Uz/Uz m ax −1 Time (c) (d)

Figure 1.14: FE analysis: (a) load distribution for load cases 1, 2, 3, (b) load distribution for load case 4, (c) partial SAFE diagram showing the investigated load cases, (d) normalized axial maximum displacement vs. time.

Pressure, temperature, flow measurements EM driver (variable speed) Instrumented impeller Flow path Strain gauges

Figure 1.15: Experimental setup, strain gauge positions on the impeller.

horizontal lines in Fig. 1.17, and for excitation frequencies, oblique line in Fig. 1.17. The first matching is evident in Fig. 1.17 (a). Ω0is the rotational speed needed to excite the twelfth mode by means of the second harmonic order: Ω0 = ω12/(2NV). Fig. 1.17(b) focuses on the second matching obtained at

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Ω00= ω17/(2NV). These two experimental resonance conditions are reported in

Fig. 1.16 as square markers. The experimentally excited natural frequencies are very close to those predicted by the SAFE diagram, where the natural modes were determined by means of FE analysis. The SAFE diagram reported in

0 1 2 3 4 5 6 7 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Shape

FEM natural mode Load harmonic Experimental resonance 17, 17 d  12, 12 d  Shape Frequency/ ω12

Figure 1.16: SAFE diagram for the bladed wheel of experimental testing.

Ω′ 12 ω 32× Ω′′ 17 ω 32× (a) (b)

Figure 1.17: Experimental Campbell diagram: (a) first resonance matching, (b) second resonance matching.

Fig. 1.16 shows four different possible resonances along the speed sweep with one, two and three nodal diameters. Only the two nodal diameters modes were actually measured. The reason is that the highest deformation concerned the blades for these two nodal diameters modes (“blade” type mode), thus they are strongly excited by the fluid forces and detected by the strain gauges. On

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1.4. Conclusions 27

the other hand, the highest deformation was on the disc, for the one nodal diameter mode (“disc” type mode), and the strain measurements were applied to the blades which mainly experienced rigid body motions. Finally, the three nodal diameters mode was excited by the third order load harmonic, however, the amplitude decreased with the order of the Fourier series decomposition and thus the higher order resonances were just marginally excited. The SAFE diagram prediction only identifies resonance matching (frequency and shape) but obviously does not give any information on the intensity of the predicted resonance, which in specific cases may not be dangerous.

1.4 Conclusions

In the present paper an in depth explanation of the Singh’s Advanced Frequency Evaluation (SAFE) diagram was provided. The maximum possible number of sign reversals along the hoop coordinate was related to the maximum number of nodal diameters. The analytical formulation, Eq. 1.44, to draw the SAFE diagram without graphical procedure, was newly introduced. Periodicity and symmetry properties of the matching map were also deduced thanks to the analytical expression. These properties may be helpful in selecting a combi-nation of the number of wheel blades and stator vanes that best prevent full resonance. The SAFE diagram analysis was then applied to a compressor impeller to prove that frequency and shape matching are both needed to excite a full resonance condition, thus validating the SAFE diagram. FE analysis showed that a pure harmonic load gives a high amplitude vibration result when both frequency and shape conditions are satisfied, and a much smaller ampli-tude if only the frequency, or only the shape matching condition, is satisfied. Experimental analysis confirmed that in a frequency range characterized by a high density of natural modes, the Campbell diagram predicts a high number of resonance conditions, while the SAFE diagram criterion drastically reduced this number. In addition, experimental tests showed that the SAFE diagram is still conservative. Some predicted resonances could not be measured because disc type modes mainly produce a rigid motion on the blades, so the strain gauge signals remained quite low. The shape is not just given by the number of nodal diameters, also directions and maximum displacement point positions are involved, moreover harmonic force components (especially the high order ones) may be of low intensity, thus producing very limited excitation despite the full resonance condition.

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Appendix. Symmetry properties proofs

The first of the symmetry properties concerns the matching map periodicity with a period of NB (Fig. 1.12). This property can be proved by means of

the introduced Eq. 1.32. The second property regards the symmetry of the matching map distribution, within a single NB period. This property is also

demonstrated here by Eq. 1.32.

I. Proof of Eq. 1.42: dm(n + NB) = dm(n)

dm(n + NB) = |(n + NB)NV− NBb(n+NB)N_NV_B+bNB/2cc| (1.45)

after expanding ((n + NB)NV)/NB, since the additive term NVis an integer, it

can be placed outside the floor operator and thus it is canceled out by the other term. Equation 1.45 can then be rewritten and the expression remaining is dm(n):

= |nNV+NBNV− NB(b

nNV+bNB/2c

NB c +NV)| = dm(n) (1.46)

II. Proof of Eq. 1.43: dm(NB/2 + ∆ n) = dm(NB/2 − ∆ n)

The first term of the equality to be proved can be rewritten as:

dm(NB/2 + ∆ n) = NB|N₂V+∆ nNNBV− b NV 2 + ∆ nNV NB + bNB/2c NB c| (1.47)

the following definitions need to be introduced to simplify the formulation:

x= ∆ nNV NB

y=bNB/2c NB

(1.48)

The two terms in Eq. 1.43 become:

dm(NB/2 + ∆ n) = NB|N₂V + x − bN₂V+ x + yc|

dm(NB/2 − ∆ n) = NB|N₂V − x − bN₂V− x + yc|

(1.49)

which can be written in the following form:

|{N_V/2 + x + y} − y| = |{NV/2 − x + y} − y| (1.50)

where the operator {} is the fractional part of any number defined as: {z} = z− bzc.

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1.4. Conclusions 29

Now the proof is divided into four cases (all the combinations NB, NVeven

or odd) and related subcases. If NBis even, y = 1/2 then:

|{NV/2 + x + 1/2} − 1/2| = |{NV/2 − x + 1/2} − 1/2| (1.51)

If NV is even as well, NV/2 is obviously an integer and an additive integer

number can just be canceled out in the fractional part function argument, thus reducing to:

|{x + 1/2} − 1/2| = |{−x + 1/2} − 1/2| (1.52) Having the fractional part of x, it suffices to show the validity of this equality for 0 ≤ x < 1. After rewriting the two terms:

|x − b1/2 + xc| = | − x − b1/2 − xc| (1.53)

in this range, with the exception of x = 1/2, it can be proved that:

b1/2 + xc = −b1/2 − xc (1.54)

so the absolute value arguments of Eq. 1.53 are exactly opposite to each other. On the other hand, if x = 1/2 both terms in Eq. 1.53 are equal to 1/2, hence the equality is demonstrated for any x.

Similar proofs can be provided for all the other NB, NVeven/odd

combina-tions, but they are not reported here for brevity, in this way equation 1.43 was proved without any exceptions.

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Chapter 2

Robotic station for automatic

modal analysis

2.1 Introduction

Industrial product development leads to design optimization and continuous per-formance enhancement. This trend forces the designer to carefully consider all the possible failure mechanisms in order to improve the in-service performance. A great effort is spent in preventing catastrophic failures during operational conditions, as proved by the damage monitoring techniques described in litera-ture [31, 32, 33, 34, 35, 2, 36]. Anyway, an accurate dynamic characterization of the mechanical components experiencing cyclic loads is needed to estimate its behaviour under operational conditions. Analytical and numerical analysis are surely useful at the design stage, but experimental tests are always recom-mended for very complex component geometries or large assemblies. Highly three-dimensional structures need a large number of measurement points to properly obtain the modal shapes. By following the usual accelerometer testing approach, many sensors should be placed on the structure, however, this is both time consuming and influences the mechanical response of the component. The cumbersome work of accelerometer monitoring can be circumvented by measur-ing the vibrational speed, rather than the acceleration, with a contactless Laser Doppler Vibrometer (LDV) [22]. Manual positioning and orientation of the laser device can still cause long and defective tests. As preliminarily introduced by the authors [37], in order to solve the handling issues, a robotic station can be used allowing a positioning with good accuracy and high resolution by keeping a stable configuration during the measurements. An ABB anthropomorphic

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arm was used to position an LDV sensor head and, after the initial alignment operation, a precise relative positioning was achieved in a reasonable time. The whole testing duration was further reduced by implementing an automatic pro-cedure for the entire modal analysis. The LMS hardware and software allowed an interface with an external programming code, such as Visual Basic, to drive all the test sequence: positioning, excitation, measurements and data storing. After the test, the saved data was finally elaborated to extract the modal results: eigenvalues, eigenvectors and dumping factors.

In the present study the robotic station was used to test two different cen-trifugal compressor bladed wheels of the shrouded and unshrouded types. The unshrouded wheel had the splitter blade peculiarity, that has some ad-vantages such as efficiency, flow distribution regularization, and noise reduction [38, 39, 40, 41, 42] but impairs the sector cyclic symmetry (as discussed be-low). The literature offers a great number of test procedures for bladed wheels. Kammerer and Abhari [5] presented an operational conditions test (Operational Modal Analysis, OMA) by applying the load through different fluid distortion to excite the various modes. A simpler test can be performed by applying a cyclic load to a stationary bladed wheel using one or more shakers (Experimental Modal Analysis, EMA) as Bidaut and Baumann presented [22]. Apparently this approach neglects the centrifugal effects, however, it can still be considered reliable since for this kind of structure FE analyses proved a stress stiffening lower than 1% in terms of natural frequencies up to 10 000 rpm. A compromise between OMA and EMA can be achieved just by testing a stationary wheel under a complex load field. Indeed, the well known SAFE diagram (Singh’s Ad-vanced Frequency Evaluation) as reported by Bloch and Singh [12], and recently analytically reconsidered by Bertini et al. [43], proved that a particular spatial force distribution is needed to properly energize each blade mode. This can be achieved by applying an excitation source, such as a shaker or an electromag-netic device, on each blade and controlling both amplitude and phase of each applied force, as the experiment by Berruti et al. shows [27]. The complexity of these tests is justified when higher frequency modes are studied and operational conditions need to be simulated. When just the lower frequency modes are studied, which usually are the most dangerous, a single-input/multiple-output test is accurate enough. The present paper shows this latter kind of test: one shaker is placed on the structure and the vibration response is measured on different blade positions for each blade of the wheel. The use of many measure-ment points in each sector of the wheel emphasized the method effectiveness and the results accuracy was confirmed after comparing the mode frequencies and the mode shapes to Finite Element (FE) analyses. FE modelling usually is

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2.2. Experimental setup 33

recommended to be performed along with the experimental activity for this kind of structure for comparison purposes. Due to the investigated component cyclic symmetry, only one sector can be modelled, in order to reduce the number of elements, neglecting the mistuning effect. Different method are available in literature to model mistuning through FE analysis [35, 44, 45, 46, 24], but some experimental data were needed to calibrate the models. Anyway, experimental results confirmed that for the tested bladed wheels the mistuning effect slightly influences a few natural modes, mainly introducing a separation between any couples of degenerate modes.

2.2 Experimental setup

The component to be tested was placed in the robot working volume and ex-cited with a vibration source. The test article was held simulating the “free-free” boundary condition. This allows to analyse the component itself without any effect of the clamping equipment. More accurate comparisons between ex-perimental and numerical results can be obtained, since uncertain boundary conditions are usually unlikely to be properly simulated with a FE model. The free-free constraint condition is usually experimentally achieved by using low stiffness elastic supports, such as inflated inner tubes or elastic bands, just to bear the weight of the component and minimize any action against the vibration [47]. The high mass of the structure, coupled with the low stiffness of the elastic support, produced very low frequency rigid body modes, thus leaving the higher frequency modes of the structure merely unchanged. The cyclic load was applied with a TiraVib electrodynamic shaker (18 N peak force, 20 kHz maximum frequency) with a load cell mounted on the shaker, and a stinger used to connect it to the structure. The component vibrational response was measured with a Laser Doppler Vibrometer (LDV) by Polytec offering a high sensitivity in a wide frequency range (up to 50 kHz). The sensor head produces a laser beam which can measure the velocity component of the vibration speed aligned with the beam direction. Thus the measurement location has to be spotted with the desired orientation and also respecting the sensor stand-off distance requirements. An anthropomorphic robotic arm by ABB was used to ensure proper laser positioning and orientation. The manual placing could take several minutes to be accurately accomplished for each measurement point, moreover giving no feedback in terms of location precision. The robotic station instead allows to rapidly place the sensor with a known maximum error which is less than 1 mm in the whole working volume, and even less if the robot

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working volume is partitioned, with a repeatability uncertainty less than 0.2 mm. By having this positioning reliability, the measurements can be repeated at different times offering high testing procedure versatility.

Any LDV sensor has several discrete stand-off distances which give the maxi-mum measurement signal power, thus the laser must be placed with an offset with respect to the analysed component. For this reason, laser head mounting on robot wrist is a crucial issue to maximize the actual measurement volume of the robotic station. Two different sensor head orientations were considered in the present work: the laser beam parallel to the wrist Z axis and the laser beam orthogonal to the wrist Z axis. This latter configuration (inspired to a robotic welding torch mounting) has proved to be more suitable for the aim of the present work. Fig. 2.1 schematically compares the difference between the two configurations, showing that having the laser beam orthogonal to the wrist Zaxis guarantees a greater number of reachable measurement locations.

Bladed wheel Shaker

Inner tube Robot arm

Figure 2.1: Test set-up scheme and different laser mounting solutions.

Test equipment also involved LMS hardware (SCADAS) and software (Test.LAB). The hardware allows to acquire up to 8 measurements channel, providing 2 output channels for shaker control. The software includes data acquisition and post-processing procedures. It also allows the user to program a specific procedure which can interact with Test.LAB, and then automatize the test. The described equipment is represented in Fig. 2.2 (a), showing an example of laser positioning on a measurement location, while the measurement point pattern on each blade of the wheel is shown in Fig. 2.2 (b).

2.2.1 Reference frame definition

Laser spot positioning and orientation with respect to the component geometry is a crucial issue in the presented application. The considered ABB robot itself is programmed to move with respect to a factory-defined reference frame, which is centred in the middle of its basement, having a vertical Z axis. Another

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(a)

(b)

Figure 2.2: (a) Robotic station and sensor positioning. (b) Measurement points on each blade of the wheel.

factory-defined reference frame is the wrist Tool Centre Point (TCP0), which is placed on the centre of the wrist flange. Obviously, it is possible to create several user-defined reference frames, which can be useful for the programming usage. Those coordinate systems can be TCPs placed on relevant tools location or Work Objects (Wobj), which are usually centred on a significant location of the target object. Any movement instruction defines the final position and orientation of the chosen TCP with respect to the desired Wobj. For the present application, a Wobj properly placed on the studied wheel and a TCP centred and aligned with the laser beam need to be defined. In this way it is possible to precisely place the laser spot by knowing the measurement locations and orientation from a CAD file of the analysed component. The robot embedded Wobj definition procedure requires a TCP to touch three points of the object, to define X , Y and Z axes. A sharp tip was mounted on the sensor-head protection cage and its vertex was used as a pointer. To define the TCP1, which is the one centred at the sharp tip, a fixed location P0 in the workspace is chosen,

and the TCP1 origin is manually positioned on that location with four different orientations. This procedure is very efficient when dealing with tangible tools, like the sharp tip used to define TCP1. However, the laser beam is not tangible and the laser spot location on a surface is not easy to be detected since the focused spot is really smaller than the naked eye reflected pattern. To overcome this issue, a webcam sensor was used to precisely locate the laser spot on a

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plane, acquiring the image on a PC. This allowed to define the TCP2, centred and aligned with the laser beam, by choosing a location P1on the sensor. The

laser spot is then pointed to P1with four angular orientations. The distance

between the laser head and P1was kept constant for each positioning. Finally,

the TCP2 Z axis was easily defined by translating (and not rotating) the robot wrist, pointing the location P1at any distance. The Z axis definition precision

increases if the chosen distance increases.

2.2.2 Automatic testing procedure

After having defined the reference frames, the robot can automatically move the sensor head to reach any desired target location. In order to have a fully automated testing procedure an integration between all hardware and software involved in the measurement is needed. A control software was developed to coordinate all the test components. Visual Basic (VB) has been chosen as programming language for this application, since it is compatible with all the equipment involved and technical support is provided for the LMS products. The data acquisition and the elaboration was performed through LMS Test.Lab software and a Test.Lab project was initially prepared, by implementing the geometry and the channel setup (i.e. the sensor types and the sensitivity). Mea-surement points, and their orientation, were stored in the project by providing a target list available for the robot. Other test setup information were needed, such as acquisition sampling frequency, bandwidth range and excitation intensity. The developed VB program was then used to import geometry information from the Test.Lab project and to communicate the first target location to the robotic arm. After the robot completed its positioning task, the VB program received a feedback signal and the laser autofocus was started by sending a command string to the laser controller. When the laser was properly positioned and fo-cused, the measurement was started. The Test.Lab switched on the excitation source and the acquisition channels, recording the force applied by the shaker and the vibrational velocity. Once the measurement process was completed and data stored, a feedback was sent to the VB program, so that the procedure could be repeated for the following measurement points. The described procedure is summarized in Fig. 2.3 flow chart.

Two different strategies were possible for the robot positioning. The first and simpler one was to subsequently move the robot through the different measure-ment locations. This is the fastest solution, as the movemeasure-ment are limited to a minimum, anyway, this is neither safe nor robust. This strategy does not allow to study each location independently, but the whole measurement path must

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Test.Lab project setup

Target reached? VB: import point list

Start robot positioning

Check robot position

no

yes

Start laser autofocus

Check focusing Laser focused? no yes Start excitation Start measurement Check measurements Meas. completed? yes no Last point? yes Finish no Start: i=1 Set target to Pi i=i+1

Figure 2.3: VB program flow chart.

be considered. Two different locations (P1and P2) may be both reachable for

the robot if starting from the rest position but it can happen that the location P2is not reachable anymore if the robot starts from the location P1. Moreover,

the movement from the location P1to the location P2could cause the robot to

hit the testing structure with the sensor head. These issues can be solved by carefully planning the measurement path, anyway, it would be time consuming and also unreliable since any time a new target is added, the whole path needs to be redefined and checked. A more robust strategy is to define a rest position (P0) and to move the robot to that location before starting the actual positioning.

When point Pi is the target location (i = 1, 2, . . .) the subroutine will contain

the instruction for the following positioning sequence: P0→ Pi and then all the

measurement location can be reached by just updating the definition of each measurement point. Several tests were performed implementing both these positioning strategies, showing up that the overall testing time is mainly influ-enced by focusing and measuring time, while positioning requires the shortest time. For this reason, the latter more robust strategy has been usually preferred. The small testing time increase was largely compensated by the very shorter time needed to plan the test. The overall testing process required from 60 to 120 seconds to be completed for each measurement location, depending on the test set up and the component geometry.

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2.3 Experimental modal analysis

The described robotic station is suitable for different applications in experimen-tal structural dynamics. Any component with a complex shape, requiring a large number of measurement points, can be tested in a relatively short time whit the described automatic procedure. Possible example of mechanical components suitable for this kind of analysis are vehicle chassis, small hulls and brake disks. In this paper, modal analysis based on the robotic station is presented for bladed wheels designed for centrifugal compressors. Bladed wheels are indeed complex three-dimensional components, which typically show many different modal shapes with a narrow frequency range, thus requiring several measurement points to fully describe structural vibration.

2.3.1 Bladed wheels modal shapes

Bladed wheels have a cyclic symmetric geometry, thus the natural modes are not only periodic with respect to time but also with the angular coordinate. The displacements δmk(t) associated to corresponding points on each k-th sector for

the m-th mode can be represented by means of the product of two harmonic terms. These harmonic components can be expressed by using the complex exponential notation and then computing the real part of the obtained complex number, Eq. 2.1. The real derivation can be actually done either by calculating the real part of the complex product, or alternatively extract the real parts before calculating the product. The first is referenced to the complex form while the second is the real form.

δmk(t) = AmRe(ei(dmk∆ ϑ +ϑm)ei(ωmt+φm))

δmk(t) = AmRe(ei(dmk∆ ϑ +ϑm)) Re(ei(ωmt+φm))

(2.1)

In Eq. 2.1 Re(z) is the real part of the complex number z, Amrepresents the

oscillation amplitude of the m-th mode, dmis the Number of Nodal Diameters

(NND) of the m-th mode, ∆ ϑ = 2π/NBis the sector angle (being NBthe number

of blades), ωmis the mode angular natural frequency, and finally ϑmand φmare

generic angular and time phases. A nodal diameter is the locus of points which do not move in the mode shape, and it is equal to the half of displacements sign reversals along the hoop coordinate. Even if a continuous body has an infinite number of natural modes, it can be proved that the NND is limited by dmax= bNB/2c (where bxc is the “floor” function). Since the component only

Sviluppo di tecniche automatizzate per la caratterizzazione dinamica sperimentale di macchine rotanti.

Università di Pisa

Scuola di Dottorato “Leonardo da Vinci”

PhD Programme in

Mechanical Engineering

Development of automatic

techniques for experimental

dynamic characterization of

rotating machines.

Paolo Neri

Università di Pisa

Scuola di Dottorato “Leonardo da Vinci”

PhD Programme in

Mechanical Engineering

Development of automatic

techniques for experimental

dynamic characterization of

rotating machines.

Author:

Paolo Neri . . . .

Tutors:

Prof. Ing. Leonardo Bertini . . . .

Prof. Ing. Ciro Santus . . . .

Ing. Bernardo Disma Monelli . . . .

Contents

Introduction

Chapter 1

Bladed wheels resonances

prediction

1.1

Introduction

1.2

The SAFE diagram

∑

∑

∑

∑

∑

∑

∑

∑

1.3

SAFE diagram validation

1.4

Conclusions

Appendix. Symmetry properties proofs

Chapter 2

Robotic station for automatic

modal analysis

2.1

Introduction

2.2

Experimental setup

2.3

Experimental modal analysis