A criterion for mode shape tracking: application to Campbell diagrams

1. Introduction

Predicting the lateral dynamics of industrial rotating machines has been a common task for several decades and many reference books can be cited, see for example Dimentberg (1961), Lee (1993), Lalanne and Ferraris (1998) and Genta (2005). In addition to natural frequencies, the main objective of such predictions is stress field distributions, a decisive step in the design process of both rotating and nonrotating machines. Thus estimating deformed shapes is an efficient way of obtaining qualitative stress field distributions. Therefore information related to deformed shapes, i.e. mode shapes, is useful especially in frequency domain responses. However, when dealing with rotating machines, gyroscopic effects induce the well known dependence of natural frequencies and associated mode shapes with respect to the speed of rotation, a dependence that is, most of the time, enhanced by bearing characteristics. It involves a frequently encountered problem in Campbell diagram plotting because, at each speed of rotation, the eigenvalue problem solver classically provides natural frequencies sorted in ascending order, whereas mode shapes can switch their orders. Consequently being able to know which mode shape is associated with the k^th natural frequency at a speed of rotation does not ensure that this shape corresponds to the k^th natural frequency at another speed. The complexity of gyroscopic systems emanates from veering or crossing phenomena which makes the tracking of rotor shapes versus the speed of rotation difficult, as it does determining which mode shape is excited.

So-called mode pairing may overcome the latter problem by correlating two sets of mode shapes. In this way, several correlation criteria have been developed over the past 30 years. They consist of filled matrices with terms between zero and one, the latter indicates a perfect correlation while zero stands for a poor one. The most popular is certainly the mac (modal assurance criterion), introduced in Allemang and Brown (1982), which was tried and tested in models, the job of which was to update to correlate numerical and experimental mode shapes (Ewins, 1984, Heylen and Janter, 1990, Ting et al., 1993 and Blaschke and Ewins, 1997). Other criteria using stiffness or mass matrices were later established to estimate the orthogonality between compared mode shapes such as the nco (normalized cross orthogonality) (Lieven, 1994 and Morales, 2005). In Brehm et al. (2010), an automatic mode-pairing strategy was suggested by using a criterion based on a combination of the mac and modal strain energies. The ieri criterion used in Bodel (2009) estimates the correlation between mode shape difference. Unlike the mac, these criteria are well adapted to correlating real mode shapes because they ensure collinearity and overall orthogonality with respect to structural matrices, as introduced by modal theory. Furthermore an orthogonality guarantee is a powerful concept which implies small correlation terms even if the mode shapes seem to be close. These terms, reaching zero, greatly facilitate the pairing of shapes whose corresponding terms are usually close to unity. Nevertheless some authors still prefer the mac when dealing with the pairing of real shapes of parametric systems which are subject to veering or crossing phenomena such as in Saito et al. (2009) which regards the mode shapes of a constrained eigenvalue problem of a cracked plate. In the same manner, although the authors of Du Bois et al. (2009) had noticed that the off-diagonal terms of the mac were of the order of 0.4, they decided not to benefit from the orthogonality properties of the eigenvectors of the prestressed truss they studied. In contrast to Perkins and Mote (1986), in which criteria were established to distinguish veerings from crossings, in the present paper we intend to provide an efficient criterion for pairing complex mode shapes and tracking any shape needed, especially for the finite element model of complex damped gyroscopic structures without analytical solutions.

This paper presents a new correlation criterion called nc²o (normalized cross complex orthogonality) based on the orthogonality properties between rotor mode shapes calculated at different speeds of rotation ( φ · ) . Indeed the quadratic and nonsymmetric natures of the related eigenvalue problem involve complex eigenvectors which do not satisfy the same properties given in previous criteria. Therefore, applying them to rotor mode shapes does not take advantage of the orthogonality properties in the mode-pairing process. The motion equations for damped gyroscopic structures are presented as well as their associated quadratic eigenvalue problems. After revising the bi-orthogonality properties of complex mode shapes, the nc²o criterion is developed. The nc²o criterion and the other criteria are tested on a finite element model of an industrial laminated rotor composed of disks and two fluid film bearings whose stiffness and damping are clearly φ · -dependent.

2. Background equations

The steady-state dynamic behavior of a discretized damped gyroscopic structure is governed by a set of n_δ equations

M δ ·· ( t ) + C ( φ · ) δ · ( t ) + K ( φ · ) δ ( t ) = F ( t )

(1)

Introducing the general solution δ ( t ) = Ψ k e λ k t in the homogenous part of equation (1) involves the following eigenvalue problem

[ λ k 2 M + λ k C ( φ · ) + K ( φ · ) ] Ψ k = 0

(2)

M δ · ( t ) - M δ · ( t ) = 0

(3)

A y · ( t ) - By ( t ) = 0 with y ( t ) = ( δ · ( t ) , δ ( t ) ) t

(4)

A = [ 0 M M C ( φ · ) ]

(5a)

B = [ M 0 0 - K ( φ · ) ]

(5b)

B r ϒ = Λ A r ϒ

(6a)

B t l ϒ = Λ A t l ϒ

(6b)

r ϒ k = ( λ k Ψ k Ψ k ) t

(7a)

l ϒ k = ( λ k Θ k Θ k ) t

(7b)

ω k = 1 2 j ( λ k * - λ k )

(8a)

ξ k 2 = 1 4 ( λ k + λ k * ) 2 λ k λ k *

(8b)

The well known bi-orthogonality properties of left and right eigenvectors with respect to augmented matrices (Nelson, 1976 and Sui and Zhong, 2006) are written

l ϒ q t A r ϒ k = α qk δ qk

(9a)

l ϒ q t B r ϒ k = β qk δ qk

(9b)

The right and left eigenvectors {Ψ _k , Θ _k } and the bi-orthogonality properties defined in equation (9) permit the uncoupling and solving of equation (1) while reducing cpu-time.

3. A new criterion for rotor mode shape tracking

The homogenous part of equation (1) is solved for each φ · i , i = 1 … n φ · and provides the set of the first m sorted natural frequencies ω ⁱ in ascending order, and the associated right Ψ ⁱ and left Θ ⁱ mode shapes; the upperscript stands for the i^th speed. Note that the Campbell diagram has to illustrate the evolution of the natural frequency ω_{Ψ k} related to its own mode shapes Ψ _k and not the ascending sorted ω _k . Defining a criterion that identifies which shape Ψ k i is related to ω q i is therefore required since the inequality q ≠ k frequently holds true if crossings occur, i.e. when Ψ k i associated with ω k i does not correspond to Ψ k i + 1 related to ω k i + 1 .

Equation (9b) can be expressed as follows

( λ q Θ q , Θ q ) t [ M 0 0 - K ( φ · ) ] ( λ k Ψ k Ψ k ) = δ qk β qk

(10)

Θ q t [ λ q λ k M - K ( φ · ) ] Ψ k = δ qk β qk

(11)

NC 2 O k , q = δ qk | β qk | 2 | β qq | · | β kk | = { 0 if k ≠ q 1 if k = q

(12)

NC 2 O k , q i , i + 1 = | L q i + 1 W k , q i , i + 1 R k i | 2 | L q i + 1 W q , q i + 1 , i + 1 R q i + 1 | · | L k i W k , k i , i R k i |

(13)

L ≡ Θ t

(14a)

R ≡ Ψ

(14b)

W k , q i , i + 1 = λ k i λ q i + 1 M - K ( φ · i )

(15)

4. Industrial application

4.1. Finite element model

The industrial application consists of an induction rotor made of a prestressed laminated stack with tie rods as presented in Mogenier et al. (2010a, 2010b and 2011). The total mass and length were 1026 kg and 2.6 m. By coupling Timoshenko theory (Przemieniecki, 1985) and the Euler angles ψ, θ, φ, i.e. precession, nutation and intrinsic rotation roughly around z → , x → and y → respectively, the rotor is discretized in N _e  = 239 two node elements (n_δ = 956 degrees of freedom, DOFs), all of them containing eight DOFs stored in ^e δ such as

e δ = ( w e , u e , ψ e , θ e , w e + 1 , u e + 1 , ψ e + 1 , θ e + 1 ) t

(16)

where u, w are lateral DOFs along x → , z → respectively while ψ, θ stand for associated rotations around z → and x → . It is assumed that the rotor is supported by two identical fluid film bearings located at nodes #6 and #178, see Figure 1. The bearing characteristics are given in Figures 2 and 3 for the [0, 3.5 × 10⁴] rpm speed range. OPEN IN VIEWER

Figure 1.

Branched finite element model of the industrial rotor.

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Figure 2.

Bearing stiffness (10⁹ × N m⁻²).

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Figure 3.

Bearing damping (10⁶ × N s m⁻¹).

Thus both steel disks mounted on shaft-ends as well as bearing stiffnesses and dampings should induce a strongly gyroscopic φ · -dependent dynamic behavior in the rotor.

4.2. Campbell diagram calculation

Natural frequencies and complex mode shapes are obtained by solving equation (6), at each constant speed φ · i . For the sake of matrix reduction, the pseudo-modal method is preferred (Lalanne and Ferraris, 1998) by projecting the homogenous part of equation (1) in a nonrotating modal basis containing m = 12 mode shapes, which reduces cpu-time and enables the introduction of modal damping factors e.g. 1% on each mode.

Moreover the speed of rotation dependence of bearing characteristics involves the calculation of a new modal basis Φ ( φ · i ) at each φ · i , by solving the eigenvalue problem

K ( φ · i ) Φ ( φ · i ) = Λ 0 M Φ ( φ · i )

(17)

with Λ₀ containing undamped eigenvalues, at rest.

By setting the step of speed of rotation at 50 rpm, an initial Campbell diagram can be illustrated in Figure 4 as the evolution of the first m natural frequencies ω sorted in ascending order, i.e. without using a mode-pairing process. On the other hand, Figure 5 presents the Campbell diagram which benefits from a mode-pairing process, especially the one using the nc²o criterion. It should be noted that complex shapes are arbitrarily indexed, at the upper bound φ · n φ · , by associating them with the indicies of the corresponding sorted natural frequencies in ascending order. Actually, the Campbell diagram shows several crossing and veering phenomena and it may be stated that each natural frequency crosses another in the considered speed range, see Figure 6. Indeed one veering occurrence clearly appears between complex shapes Ψ₅ and Ψ₈ around 7.5 × 10³ rpm. Furthermore no fewer than 56 crossing occurrences can be counted between the first m natural frequencies which demonstrates the strongly φ · -dependent behavior of the supported rotor. OPEN IN VIEWER

Figure 4.

A Campbell diagram without mode pairing. Natural frequencies are plotted in Hz. The circles (○) and dashed line (–) indicate critical speeds and 1× excitation respectively.

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Figure 5.

A Campbell diagram using the nc²o mode pairing for different φ · -steps: 5 × 10¹ (−), 2 × 10³ (*) and 5 × 10³ rpm (□).

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Figure 6.

Enlargement of a Campbell diagram emphasizing crossing phenomena between: (a) Ψ₁, Ψ₂ and Ψ₅ and (b) Ψ₇, Ψ₁₁ and Ψ₁₂.

5. Overcoming veering and crossing phenomena

To prove the nc²o criterion, Figure 5 includes star and square symbols which correspond to Campbell diagrams obtained with speed of rotation steps of 2 × 10³ rpm and 5 × 10³ rpm respectively. Although the latter steps are larger than the initial step (continuous lines), all of the Campbell diagrams coincide whatever their step lengths and crossings or veerings, e.g. it can be stated that ω_Ψ9 and ω_Ψ10 cross other natural frequencies 13 and 16 times, respectively. This underlines the efficiency of nc²o in the mode-paring process, which provides Campbell diagrams that allow any mode shape tracking.

In addition, Figure 7 presents the following quantities

NC 2 O ⊥ = ∑ k , q = 1 q ≠ k m NC 2 O k , q i , i + 1 m ( m - 1 )

(18a)

MAC ⊥ = ∑ k , q = 1 q ≠ k m MAC k , q i , i + 1 m ( m - 1 )

(18b)

NCO K ⊥ = ∑ k , q = 1 q ≠ k m NCO K k , q i , i + 1 m ( m - 1 )

(19a)

NCO M ⊥ = ∑ k , q = 1 q ≠ k m NCO M k , q i , i + 1 m ( m - 1 )

(19b)

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Figure 7.

Off-diagonal term averages of mac_⊥ (×), nco _{k ⊥} (.), nco _{m ⊥} (*) and nc²o_⊥ (+) correlation criteria for different step sizes.

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Table 1.

Correspondences between correlation criteria

Criterion	L q i + 1	W k , q i , i + 1	R k i
NCO 2 k , q i , i + 1	Θ q t i + 1	λ k i λ q i + 1 M - K ( φ · i )	Ψ k i
MAC k , q i , i + 1	Ψ q * t i + 1	I	Ψ k i
NCO K k , q i , i + 1	Ψ q * t i + 1	K ( φ · i )	Ψ k i
NCO M k , q i , i + 1	Ψ q * t i + 1	M	Ψ k i

5.1. Veering

A veering obviously appears between complex shapes Ψ₅ and Ψ₈ around 7.5 × 10³ rpm as shown in the zoom-in plotted between speeds φ · i = 7 . 4 × 10 3 rpm and φ · i + 1 = 7 . 65 × 10 3 rpm in Figure 8. Indeed, the natural frequencies ω_Ψ5 and ω_Ψ8 tend to get closer in the vicinity of 7.5 × 10³ rpm while being crossed by ω_Ψ7 and ω_Ψ6 respectively. Actually, Figure 9(a) shows that this phenomenon is distinctly noticed by the nc²o criterion by correlating shapes Ψ 5 i to Ψ 8 i to shapes Ψ 6 i + 1 , Ψ 5 i + 1 , Ψ 8 i + 1 and Ψ 7 i + 1 respectively. OPEN IN VIEWER

Figure 8.

Veering phenomenon between 7.4 × 10³ and 7.65 × 10³ rpm.

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Figure 9.

Criteria calculated between 7.4 × 10³ and 7.65 × 10³ rpm.

On the other hand, whatever the correlation criteria illustrated in Figure 9(b)–(d), artefacts occur, e.g. correlation terms higher than 0.9, which correlate artificially the shape Ψ 6 i to shapes Ψ 5 i + 1 and Ψ 8 i + 1 . Moreover, in the same way the shape Ψ 7 i seems to also correspond to shapes Ψ 5 i + 1 and Ψ 8 i + 1 whereas Figure 8 proves the contrary.

The notion of difference of order of magnitude between nc²o terms is significantly highlighted by Figure 9(b), which illustrates mac matrix calculated with right mode shapes Ψ, and Figure 9(a) that shows the nc²o matrix emanated from right and left eigenelements {Λ, Θ, Ψ} at the same speeds. It can be seen that the mac terms associated with a pair of noncoincident complex mode shapes can be higher than 10⁻¹ (0.5) whereas those obtained with the nc²o reach 10⁻⁴, see Figure 7.

5.2. Crossing

As previously mentioned, several crossing occurrences appear between the first m natural frequencies ω_Ψ in the speed range. It can be stated that ω_Ψ9 and ω_Ψ10 cross other natural frequencies 13 and 16 times, respectively. Figure 10 presents a zoom-in of Figure 5 between φ · i = 9 × 10 3 rpm and φ · i + 1 = 1 . 2 × 10 4 rpm while Figures 11(a)–(d) represent the nc²o, mac, nco _k and nco _m correlation matrices respectively. OPEN IN VIEWER

Figure 10.

Crossing phenomena between 9 × 10³ and 1.2 × 10⁴ rpm.

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Figure 11.

Criteria calculated between 9 × 10³ and 1.2 × 10⁴ rpm.

Even if correlation criteria were calculated between speeds far from the one at which veering occurs, the mac suggests a virtual correlation between the shapes subjected to veering and associated to ω_Ψ5 and ω_Ψ8, i.e. the 7^th and 10^th mode shapes Ψ 7 i , Ψ 10 i calculated at φ · i and the 10^th and 5^th shapes Ψ 10 i + 1 , Ψ 5 i + 1 calculated at φ · i + 1 . Indeed, the noncorrespondence terms of the mac are higher than 10⁻¹ (0.7) whereas both nco _k and nco _m provide terms of order of 0.5.

In addition to the fact that nc²o is not disturbed by high-value terms, and despite being evaluated between speeds spaced out over 3 × 10³ rpm, Figure 11(a) shows a good correspondence between complex shapes of the φ · -interval bounds, and is therefore in good agreement with Figure 10.

6. Mode shape-pairing process

By correlating two bases of complex shapes Ψ ⁱ and Ψⁱ⁺¹ at speeds φ · i and φ · i + 1 , the mode shape-pairing process provides correspondence between the k^th natural frequency ω k i and the q^th one ω q i + 1 . It makes identification of the natural frequencies ω_{Ψ k} associated with their own shape possible. Then let ς be the index matrix which satisfies

If Ψ k i ≡ Ψ q i + 1 then ω k i ≡ ω q i + 1 ∀ k , q = 1 … m i.e. ω Ψ k i = ω ς k i i

(20)

Qualitatively, mode-pairing process could be envisaged, between φ · i , φ · i + 1 , as a double-entry table whose row and column correspond to mode shapes Ψ ⁱ , Ψⁱ⁺¹ respectively, the latter ones being related to sorted natural frequencies ω ⁱ , ωⁱ⁺¹ in ascending order. By subsequently comparing row and column shapes, it becomes pertinent to associate coincident mode shapes of both sets Ψ ⁱ , Ψⁱ⁺¹. Quantitatively, digital translation of this table should consist of assigning to cells of coordinates (k, q) distinct values to pair coincident shapes Ψ k i , Ψ q i + 1 and noncoincident ones.

Thus, choosing any correlation criterion might be a solution to incrementing this table, but a more efficient solution consists of using the nc²o correlation matrix which benefits from bi-orthogonality properties according to the coincidence degree of the mode shapes Ψ ⁱ and Ψⁱ⁺¹, and then assigning values with a high difference in order of magnitude. Thereby the ς matrix may be incremented as follows

ς k i = q * with max q * N C 2 O k , q i , i + 1 ∀ q = 1 … m

(21)

ς k n φ · = k ∀ k = 1 … m

(22)

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7. Conclusion

A novel correlation criterion was proposed by taking advantage of the bi-orthogonality properties of the right and left eigenvectors of damped gyroscopic structures. The nc²o criterion greatly facilitates rotor mode shape pairing even if its associated natural frequencies are highly dependent on the speed of rotation. As a numerical test, the proposed criterion was tested on an industrial application by taking into account hydrodynamic bearings characteristics, the latter being strongly φ · -dependent. Moreover, it should be observed that the tested rotor is not an academic case since its gyroscopic effect and bearings properties give it highly speed-of-rotation varying behavior as illustrated by the Campbell diagram bringing out significant and unusual veerings and crossings phenomena.

The robustness of the nc²o was shown firstly by using different step sizes of speed of rotation from 5 × 10¹ rpm to one hundred times more and secondly by comparing the low values of the off-diagonal terms of the nc²o with those of classical correlation criteria. This showed numerically the efficiency of the criterion in the mode-pairing process by illustrating the evolution of each natural frequency associated with its own complex mode shape. Furthermore, note that a 5 × 10³ rpm step size is very large since the Campbell diagram, Figure 5, contains only seven different speeds of rotation over the considered speed range from 0 to 3.5 × 10⁴. It should be noted that the key here is to present a criterion which provides an efficient tracking of natural frequencies that are highly φ · -dependent. This tracking makes sense only if the Campbell diagram is plotted with a relatively fine step of rotation speed in such a way that all the changes in natural frequency can be tracked and observed.

Finally, a way to improve the robustness of the nc²o might be to consider other mechanical systems which are complex, implying important eigenelements changes with respect to mechanical parameters such as modes of coalescence in rubbing systems.