Dynamics in large generators due to oval rotor and triangular stator ...

Acta Mech Sin DOI 10.1007/s10409-011-0410-7

RESEARCH PAPER

Dynamics in large generators due to oval rotor and triangular stator shape Niklas L.P. Lundström · Jan-Olov Aidanpäa¨

Received: 24 August 2010 / Revised: 9 December 2010 / Accepted: 9 December 2010 ©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2011

Abstract Earlier measurements in large synchronous generators indicate the existence of complex whirling motion, and also deviations of shape in both the rotor and the stator. These non-symmetric geometries produce an attraction force between the rotor and the stator, called unbalanced magnetic pull (UMP). The target of this paper is to analyse responses due to certain deviations of shape in the rotor and the stator. In particular, the perturbation on the rotor is considered to be of oval character, and the perturbations of the stator are considered triangular. By numerical and analytical methods it is concluded for which generator parameters harmful conditions, such as complicated whirling motion and high amplitudes, will occur. During maintenance of hydro power generators the shapes of the rotor and stator are frequently measured. The results from this paper can be used to evaluate such measurements and to explain the existence of complex whirling motion. Keywords Generator · Rotor · Dynamics · Asymmetric · Hydropower 1 Introduction Hydropower generators have small air-gaps between the rotor and the stator. Usually the gap is about 0.2% of the stator inner radius. Measurements on the shape of hydropower generators are frequently carried out during maintenance. N.L.P. Lundström (¬) · Jan-Olov Aidanpäa¨ Division of Solid Mechanics, Department of Applied Physics and Mechanical Engineering, LuleåUniversity of Technology, SE-97187, Sweden e-mail: [email protected]

Measurements done indicate that all hydropower generators are associated with some degree of asymmetry in the air-gap. These asymmetries distort the air-gap flux density distribution and produce an attraction force between the rotor and the stator which is called unbalanced magnetic pull (UMP). The effect of UMP can damage the machine. There are documented cases, e.g. Talas and Toom [1], where the rotor has been in contact with the stator which has been caused by airgap asymmetries. Measurements also indicate the existence of complicated whirling motions, including both backward and forward whirling, in these machines. A literature survey indicates intensive studies of methods for calculating UMP caused by eccentricity, as well as studies on the vibration characteristics of a rotor system due to UMP. An overview of the research in the area of modeling and calculating UMP in electrical machines was presented by Frosini and Pennacchi, see Ref. [2] and the reference therein. In Ref. [3], Lundström and Aidanpäa¨ derived the UMP for an arbitrary disturbed air-gap through the principle of virtual work applied to a simple generator model. They studied the generator shape and proved the existence of stable equilibrium for the rotor for certain cases of deviations of shape. They evaluated the robustness of the generator by simulations of the basin of attraction to attractors without rotor-stator contact using a Jeffcott rotor model. In Ref. [4], Lundström and Aidanpäa¨ resumed the work on the model in Ref. [3]. They used a power series expansion to prove a theorem about the angular frequency and the amplitude of the UMP for certain deviations of shape. This paper uses the same simple generator model and the same derivation of the UMP as in Refs. [3,4]. The paper analyses responses due to certain deviations of shape in the rotor and the stator. In particular, the perturbation on the rotor is considered to be of oval character, and the perturba-

N.L.P. Lundström, Jan-Olov Aidanpäa¨

2

tions of the stator are considered triangular. The setting of a triangular stator is of interest since old stators often stand on three feet, and when the temperature of the stator changes, the shape may also change to produce triangularity. The case of an oval rotor and a triangular stator will be referred to as a (3,2)-perturbation. Concerning results, simulations of trajectories indicate that harmful conditions can occur due to high amplitudes and also due to alternating whirling of the rotor. In most cases of (3,2)-perturbation, the whirling of the attractor is backward, and the whirling frequency is twice as high as the driving frequency. This was expected from the previous results presented in Ref. [4]. The attractors in the (3,2)-perturbation case will be divided into mainly two types, simple and loops. An attactor will be called simple if the rotor center moves in an orbit which does not intersect itself. Loops are attractors in which the rotor center orbit intersect itself three times, symmetrically as Fig. 1 shows. In case of a loop, the whirling approaches zero several times during one revolution of the rotor, and this occurs when the rotor is located far from the origin. This may produce a harmful wearing of the rotor bearings. It is concluded that these harmful attractors occur for large rotor ovalities. In particular for rotor ovality above approximately 10% of the air-gap if the stator triangularity is less then approximately 20% of the air-gap.

bearings while point Cr is the geometrical center line of the rigid rotor core. The coordinate system has its origin at C s , r is the rotor radius and s is the stator radius.

Fig. 2 The generator model. a The cross-section of the generator; b The Jeffcott rotor

Let r0 and s0 be the average radius of the rotor and the stator, respectively. An arbitrary non-circular shape of the rotor radius r, and the stator radius s, can be described by the Fourier series ∞ δrn (z) cos{n[ϕ + αrn (z)]}, (1) r = r0 (z) + n=1

s = s0 (z) +

∞

δms (z) cos{m[ϕ + αms (z)]},

(2)

m=1

where δrn (z) ≥ 0, ∞ n=1

Fig. 1 The two types of attractors in the (3,2)-perturbation case. a Simple; b Loops

The rest of the paper is structured as follows: Sect. 2 presents the mechanical model used in this paper and Sect. 3 presents the derivation of the UMP for an arbitrary disturbed air-gap through the principle of virtual work. These sections are similar to Sects. 2 and 3 in Refs. [3,4]. Section 4 gives the equation of motion and introduces some numerical values and parameters. Section 5 is devoted to the investigation of (3,2)-perturbation and finally Sect. 6 gives discussion and conclusion. 2 Generator geometry Figure 2 shows the geometry of the generator model. The generator is treated as a balanced Jeffcott or Laval rotor having a rigid core with length l0 , mass γ and stiffness k of the generator shaft. The rotor rotates counter-clockwise, at a constant angular speed ω. Point C s gives the location of the

δrn (z) +

δms (z) ≥ 0, ∞

δms (z) < g0 (z),

(3)

m=1

where g0 = s0 − r0 is the average air-gap, δrn and δms are referred to as the rotor and stator perturbation parameters, while αrn and αms are the corresponding phase angles. To simplify notations, it is hereinafter assumed that δrn = δms = 0, ∀m, n ∈ N, if nothing else is mentioned. N is here the set of all natural numbers. Figure 3 shows the rotor shape for δrn = r0 /3, n = 1, 2, 3, 4, and phase angles αrn = 0. Note that the stator has the same shape for δms = δrn and m = n. The cases δr1 > 0 and δ1s > 0 will correspond to rotor eccentricity and stator eccentricity respectively. Note that, since rotor eccentricity almost surely results in a dynamic eccentricity, it is often referred to as dynamic eccentricity. Similarly, stator eccentricity is often referred to as static eccentricity. Since dynamic eccentricity is normally small compared to the radius of the air-gap, it is assumed that the perturbed air-gap (g) is g = s(z, ϕ) − r(z, ϕ) − x cos ϕ − y sin ϕ,

(4)

where (x, y) gives the position of Cr . After adding the ω rotation, Eqs. (1), (2) and (4) give

Dynamics in large generators due to oval rotor and triangular stator shape

3

Fig. 3 The shape of the rotor for δrn = r0 /3, n = 1, 2, 3, 4

g = g0 (z) +

∞

dEmech = d(δE) + dEelectric .

δms (z) cos{m[ϕ + αms (z)]}

(9)

m=1

−

∞

δrn (z) cos{n[ϕ + αrn (z) − ωt]}

n=1

−x cos ϕ − y sin ϕ.

(5)

The geometric model is now completed. 3 Unbalanced magnetic pull Analytical expressions for the UMP is derived as in Refs. [3–5] by the principle of virtual work. In particular, the two following simplifications are made; the generator will be treated as a continuum and the B-field (also called magnetic flux density) in the air-gap will be assumed as B0 (z)g0 (z) B= , g(x, y, z, t, ϕ)

(6)

where B0 is the uniformly distributed B-field for a perfect circular geometry, i.e. g = g0 . These assumptions mean that the effect of the poles is not taken into account and this is justified since there is often a large number of poles in a hydropower generator. Based on the theory of magnetic fields [6], the potential energy stored in the air-gap can be expressed as B2 (x, y, z, t, ϕ) dV, (7) E= 2μ0 air-gap where μ0 = 4π×10−7 Vs/Am is the permeability of air. Next, consider a volume element dV as shown in Fig. 4. According to Eqs. (6) and (7), the potential energy (δE) reserved in dV is given by δE =

B20 (z)g20 (z) dV. 2μ0 g2 (x, y, z, t, ϕ)

(8)

Equation (8) shows that if the air-gap is disturbed from the current value g to a new value g + dg, δE will increase if dg < 0 and decrease if dg > 0. Let dEmech be increment of mechanical energy input to dV and dEelectric the electric energy output from dV. When considering the energy conversion law between the magnetic and mechanical fields over an infinitesimal period of time, the law of energy conservation requires, after neglecting losses

Fig. 4 The volume element dV

As in the case of eccentricity [5], it is assumed that the electric energy output from the air-gap is independent of the air-gap variations, thus dEelectric = 0. If dg < 0, then d(δE) = dEmech > 0. Since the mechanical energy input increases when g decreases, a force acting in the radial direction has to be present. Denote this force by d f . Then, the virtual work done by this force is d f dg = −d(δE), which gives df = −

d (δE). dg

(10)

In Eq. (8), note that, since dV = rdrdzdϕ, the potential energy δE will increase if r increases when g is constant. This small change in d f can not be considered in Eq. (10). But, since the change of g and r is of approximately the same size and g r, the change of δE due to r is negligible, and therefore, to simplify the calculations it is assumed that dV = u0 drdzdϕ, where u0 = (r0 +s0 )/2, and dr = g. Equation (8) then yields δE =

B0 (z)2 g0 (z)2 u0 (z) dzdϕ. 2μ0 g(x, y, z, t, ϕ)

(11)

According to Eq. (10), the force d f is given by df = −

d B0 (z)2 g0 (z)2 u0 (z) (δE) = dzdϕ. dg 2μ0 g(x, y, z, t, ϕ)2

(12)


4

Hence, the total forces in the x- and y-directions can be expressed as 2π l0 B0 (z)2 g0 (z)2 u0 (z) fx = cos(ϕ)dzdϕ, (13) 2 0 2μ0 g(x, y, z, t, ϕ) 0 2π l0 B0 (z)2 g0 (z)2 u0 (z) fy = sin(ϕ)dzdϕ. (14) 2 0 2μ0 g(x, y, z, t, ϕ) 0 The generator geometry and the B-field are from now on assumed constant in z-direction through the generator length l0 . This is the case through the rest of the paper. For ideal circular generator geometry and y = 0, the integral in Eq. (13) can be solved analytically to yield l0 g20 B20 u0 2π cos ϕ dϕ fx = 2μ0 (g − x cos ϕ)2 0 0 x = km . (15) 2 (1 − x /g20 )3/2 Here, the magnetic stiffness (km ), is defined as πl0 B20 u0 km = . μ0 g0

Equation (15) is similar to results obtained by Wang et al. [5] and Sadarangani [7]. 4 Equations of motion and parameters The equations of motion for the forced Jeffcott rotor is nonautonomous and nonlinear and consists of two second order differential equations γÿ + c˙y + ky = fy (x, y, t),

(17)

where γ is the mass of the rotor, k is the stiffness of the rotor shaft and c is a linear viscous damping. In this paper, c will always be chosen such that the damping ratio ζ, introduced in Eq. (20) below, satisfies ζ = 0.1. In non-dimensional form, system (17) yields X + 2ζX + X = F X (X, Y, τ), Y + 2ζY + Y = FY (X, Y, τ).

(18)

Here, prime denotes differentiation with respect to nondimensional time τ and δr x y X= , Y= , ΔRn = n , g0 g0 g0 (19) s δm k S Δm = , K= , g0 km c = 0.1, 2 kγ γ Ω=ω , k

ζ=

g(x, y, t, ϕ) , g0 k τ=t , γ

G =1+

∞

ΔSm cos[m(ϕ + αms )]

m=1

−

∞

ΔRn cos[n(ϕ + αrn − Ωτ)]

n=1

−X cos ϕ − Y sin ϕ.

(23)

The numerical values used in this paper will be taken from an 18 MW hydropower generator. These values are given in Table 1 below. Table 1 Numerical values from an 18 MW hydropower generator

(16)

γ x¨ + c x˙ + kx = f x (x, y, t),

have been introduced. The air-gap G, and the forces F X and FY yield 2π cos ϕ −1 dϕ, (21) F X = (2πK) G(X, Y, τ, ϕ)2 0 2π sin ϕ −1 dϕ, (22) FY = (2πK) G(X, Y, τ, ϕ)2 0

G=

(20)

Parameters

Values

Average stator radius s0

2.775 m

Length of the generator l0

1.18 m

Average air-gap g0

0.0125 m

Mass of the rotor γ

98 165 kg

Magnetic stiffness km

1.471 5 × 108 N/m

Stiffness of the axis k

3.456 × 108 N/m

Rotor rotation speed ω

14.2 rad/s

Number of poles

44

Concerning perturbations of shape, this paper will, as mentioned in the Introduction, only investigate rotor ovality (ΔR2 > 0) and stator triangularity (ΔS3 > 0). All other perturbations of shape will be set to zero. The values of s0 , l0 and g0 will be fixed according to Table 1 during the rest of the paper, and also the phase angles of the shape perturbations are set to zero, i.e. αr2 = α3s = 0, while K, Ω, H, ΔR2 and ΔS3 will be allowed to vary. To simplify notation, the dimensionless constants Ω0 = 0.239 3 and K0 = 2.349 are introduced. These values are defined according to Table 1 and Eq. (20). The parameters are considered to vary in the following intervals ΔR2 , ΔS3 ∈ (0, 0.5),

Ω ∈ (0, 5Ω0 ),

K ∈ (0.5K0 , 5K0 ). (24)

Note from the equations of motion, i.e. Eq. (18)–(20), that increasing K while the rest of the dimensionless parameters are constant, means that the magnetic stiffness km is increasing, or the stiffnessof the generator axis is increasing but then with the ratio w γ/k constant. Hence, increasing K may correspond to an increase in the rotor axis together with an increase in the rotor mass such that γ/k is constant, which is reasonable for a machine. Increasing the dimensionless driving frequency Ω will correspond to an increase


in the driving frequency w. Further, it should be noted that the damped angular frequency of the system is given by k − km c 2 − , (25) ωd = γ 2γ and in dimensionless form, by Eqs. (19) and (20) and since ζ = 0.1 km (26) Ωd = 1 − ζ 2 − . k When K = K0 Eqs. (25) and (26) give ωd ≈ 44.6 rad/s and Ωd ≈ 0.751. 5 The (3,2)-perturbation To begin √ this section, some terminologies are introduced. Let R = X 2 + Y 2 and Θ = arctan(Y/X) be polar coordinates, then Θ , where prime denotes differentiation with respect to non-dimensional time τ, is the angular frequency of the mo denote the average of Θ during tion of the rotor. Let Θ one revolution of the rotor. An attractor will be said to have is positive and backward whirling if forward whirling if Θ is negative. An attractor is said to have n-whirling if Θ /Ω = n. Moreover, n-whirling is positive n-whirling if Θ Θ does not change sign during one revolution of the rotor, and n-whirling is alternating n-whirling if Θ changes sign during one revolution of the rotor. Finally, as mentioned in the Introduction, an attractor will be called a loop if its projection on the XY-plane intersect itself three times, symmetrically as Fig. 1 shows. An attrator will be called simple, if

5

the rotor center moves in an orbit such that its projection on XY-plane not intersect itself. Often, alternating-2-whirling are loops, and loops are alternating-2-whirling, but there exist cases, proved by simulations below, see Figs. 7e and 7f, where loops are not alternating-2-whirling. This section will focus on finding the parameter values when the machine runs with loops. This is interesting since these attractors may be more harmful for the machine. The reason is that in most cases, R is large when Θ = 0, and such a situation may produce a harmful wearing of the rotor bearings. Note that this reasoning makes sense since there are only (3,2)-perturbation present and therefore the attractors that occur will mostly be of the “same style” as seen by the simulations below. When adding other perturbations of shape or unbalance, the situation becomes much more complicated. Figures 5–8 show the projection of the attractor on the XY-plane for some deviations of shape. Figure 5 shows −2whirling attractors for increasing ΔR , In Figs. 5a–5c the attractor is simple while in Fig. 5d it is a loop. Figure 6 shows attractors for increasing ΔS , in Figs.6a–6d the solution is simple. Figure 7 shows attractors for increasing Ω. Figures 7a, 7b, 7h–7l show simple solutions, while Figs. 7c– 7g show loops. In Fig. 7, note that all attractors are −2whirling except in Figs. 7e and 7f. In these cases the periodic solutions show forward whirling, in particular, positive 2-whirling motion. This proves that loops need not be alternating −2-whirling. Figure 8 shows attractors for increasing K. In Figs. 8a–8c the solutions are simple, while in Figs. 8d–8h the solutions show loops. Note the change in scale in the second row of the matrix of subfigures in Fig. 8.

Fig. 5 Increasing ΔR . ΔS = 0.05, K = K0 and Ω = Ω0 . a ΔR = 0.05; b ΔR = 0.12; c ΔR = 0.18; d ΔR = 0.26. The axis show 100 times real value

Fig. 6 Increasing ΔS . ΔR = 0.18, K = K0 and Ω = Ω0 . a ΔS = 0.01; b ΔS = 0.05; c ΔS = 0.12; d ΔS = 0.18. The axis show 100 times real value


6

Fig. 7 Increasing Ω. K = K0 , ΔR = 0.18 and ΔS = 0.05. Ω given in parts of Ω0 ; a 0.1; b 0.3; c 0.5; d 0.6; e 0.7; f 0.8; g 0.9; h 1.0; i 1.1; j 1.5; k 2.0; l 5.0. The axis show 100 times real value

Fig. 8 Increasing K. ΔR = 0.18, ΔS = 0.05 and Ω = Ω0 . K is given in parts of K0 ; a 0.7; b 1.0; c 1.2; d 1.4; e 1.8; f 3.0; g 5.0; h 10. The axis show 100 times real value

To proceed two Poincare sections are introduced. From the state space dimensions R, Θ, R , Θ and the rotor rotation

Ωτ, the two Poincare sections and are chosen according to Θ

and

Θ

Ωτ

= {R, Θ, R , Θ , Ωτ(mod π)|Θ = 0},

(27)

= {R, Θ, R , Θ , Ωτ(mod π)|Ωτ(mod π) = 0}.

Ωτ

To motivate, note that

Θ

(28)

tells when there are alternating n-

whirling or positive n-whirling. Plotting against the radius R gives the distance from the origin, i.e. the shift of the rotor center, to the location where the whirling is changing sign. This gives information of how dangerous the attractor is for

the machine. The more classical Poincare section gives Ωτ


the periodicity and the location when the rotor has done one half revolution.

Figure 9 shows the Poincare sections , marked with Θ

small dots, and , marked with larger dots for Fig. 9a in-

7

creasing ΔR , 9b increasing ΔS , 9c increasing Ω and 9d increasing K. Note from Fig. 9c the limits at which the attractor changes from −2-whirling to 2-whirling and back again, compared with Fig. 7.

Ωτ

Fig. 9 Poincare section

Θ

, marked with small dots, and

, marked with larger dots. a Increasing ΔR with ΔS = 0.05, Ω = Ω0 and K = K0 ;

Ωτ

b Increasing ΔS with ΔR = 0.18, Ω = Ω0 and K = K0 ; c Increasing Ω with ΔR = 0.18, ΔS = 0.05 and K = K0 ; d Increasing K with ΔR = 0.18, ΔS = 0.05 and Ω = Ω0

In the case of small Ω, ΔR and ΔS , the following theorem gives the border, in the ΔR , ΔS -plane, between simple attractors and loops. Theorem 5.1 Consider a generator with (3,2)-perturbation for which the parameters ΔR , ΔS and also the driving frequency Ω are small enough. Then it follows that if Δ2S >

1 [2K −1 + 2K − 4 22 − 17K −1 − 5K

+ΔR (5K −1 − 1 − 4K) + Δ2R (17K −1 − 13 + 5K)], then the attractor is an alternating −2-whirling loop, otherwise it is simple and positive −2-whirling. The following simulations show the border between simple solutions, loops and unstable points in the ΔR ΔS -

plane together with the analytic approximation in Theorem 1 when it is relevant. In all simulations, the initial condition is set to X = Y = X = Y = 0. In Fig. 10, Ω = 0.1Ω0 and K is increasing. In Fig. 11, K = K0 and Ω is increasing. 5.1 Proof of Theorem 5.1 In the proof of Theorem 5.1 in Ref. [4], the UMP is approximated with a Maclaurin series to yield FY = FY(1) + FY(2) + · · · ,

(29)

with FY(1) = 0,

and

FY(2) = −

3km ΔR ΔS sin(2Ωτ). 2k

(30)


8

The higher order terms FY(3) , FY(4) , · · · includes only higher powers of ΔR and ΔS and can therefore be ignored. Similarly 3km FX ≈ ΔR ΔS cos(2Ωτ). 2k Let Γ be the angular frequency of the UMP, i.e.

(31)

Γ=

FY F X − F X FY F X2 + FY2

,

(32)

where again prime represent differentiation with respect to non-dimensional time τ. From Eqs. (29)–(31) it is concluded that Γ = O(Ω). Hence, if Ω is small enough the problem can be treated as a static problem.

Fig. 10 Ω = 0.1Ω0 . a K = 0.7K0 ; b K = K0 ; c K = 1.2K0 . The analytic approximation is marked with the solid curve. Simple orbits are marked with dots, loops are marked with asterixes and unstable points are marked with crosses

Fig. 11 K = K0 , a Ω = 0.3Ω0 ; b Ω = 0.5Ω0 ; c Ω = Ω0 ; d Ω = 1.5Ω0 ; e Ω = 2Ω0 ; f Ω = 5Ω0 . Simple orbits are marked with dots, loops are marked with asterixes and unstable points are marked with crosses. The analytic approximation is marked with the solid curve


Next, let E denote the equilibrium which occurs when Ω = 0. Recall that it is assumed that αr2 = α3s = 0, therefore E will be located on the negative X-axis. Let (XE , 0) be the coordinates of equilibrium E. From the geometry of the (3,2)-perturbation and the nonlinearity of the UMP it follows that F X (XE , 0, τ) has a strict global minimum at τ = 0. (For a proof, differentiate on Eq. (21)). Hence, the rotor will move in the positive X-direction away from E when it starts to rotate. From Theorem 5.1 in Ref. [4] it is concluded that if ΔR and ΔS are small enough, then the mean value of Γ is −2Ω. Hence, to determine if the orbit will become a loop or not, it suffices to determine if the rotor moves in the positive or negative Y-direction when it starts to rotate. If Ω, ΔR and ΔS are small enough, the orbit will intersect itself (become a loop ) if and only if the rotor moves in the negative Y-direction. Moreover, the orbit becomes alternating -2 whirling if and only if the rotor moves in the negative Y-direction. Hence, to complete the proof it suffices to determine when FY (XE , 0, 0) < 0. To do so the location of the equilibrium E is first approximated. Consider the air-gap given by Eq. (23). This simplifies to G = 1 + ΔS cos(3ϕ) − ΔR cos(2ϕ) − X cos ϕ.

9

Thus, the force F X given by Eq. (21) can be analysed using the Maclaurin series 1 = 1 + 2 + 3 2 + · · · + (q + 1) q + · · · , (1 − )2

(33)

with

= −ΔS cos(3ϕ) + ΔR cos(2ϕ) + X cos ϕ.

(34)

From Eqs. (22) and (33) and by keeping up to third order term F X approximates as 2π 1 FX ≈ (1 + 2 + 3 2 + 4 3 ) cos(ϕ)dϕ 2Kπ 0 3X 2 2 = 2ΔS + 2Δ2R + ΔR + 2K 3 +

3 (−ΔS X 2 + X 3 − ΔS ΔR − ΔS Δ2R ). 2K

(35)

Due to the nature of the problem, if the terms including X 3 and X 2 are ignored, the approximation is still accurate, see Fig. 12. Since such a simplification simplifies the algebra a lot, these terms are ignored.

Fig. 12 Third order approximation compared to first order approximation. K = 0.5K0 is marked with “squares”, K = K0 with “circles” and K = 5K0 with “triangles”. a The steady state solution, the result when including only first order terms is marked with a dot; b Relative error

By deleting terms including X 2 and X 3 , Eqs. (35) and (18) yield XE ≈

3ΔS ΔR (ΔR + 1) . −2K + 3ΔR + 2 + 6Δ2S + 6Δ2R

Finally ries

FY (XE , 0, 0)

(36)

is approximated with the Maclaurin se-

1 (q + 1)(q + 2) q

+ · · · , (37) = 1 + 3 + 6 2 + · · · + 2 (1 − )3

with as in Eq. (34). By taking the derivative of Eq. (22), using Eq. (37) and keeping up to third order terms yields FY (XE , 0, 0)

2ΔR Ω =− Kπ ≈−

2ΔR Ω Kπ

2π

0

2π 0

sin(ϕ) sin(2ϕ) dϕ G3 (1 + 3 + 6 2 + 10 3 )

× sin(ϕ) sin(2ϕ)dϕ


10

=

ΔR Ω [3ΔS (5Δ2R + 5Δ2S − 4ΔR + 2) 4K +15X(2ΔS + Δ2R ) + 15X 2 + 10X 3 ].

(38)

By substituting Eq. (36) into Eq. (38), keeping zero to second order terms and solving for FY (XE , 0, 0) < 0 gives Δ2S >

1 [2K −1 + 2K − 4 22 − 17K −1 − 5K

+ΔR (5K −1 − 1 − 4K) + Δ2R (17K −1 − 13 + 5K)],

are small, this is due to the fact that if K is small, then the problem becomes more linear and therefore the low-order approximation becomes better. When Ω increases, the assumption that the problem can be threated as static becomes irrelevant and the approximation fails.

The Poincare maps and illustrated in Fig. 9 show Θ

(39)

which completes the proof of Theorem 1. 6 Discussion and conclusions The generator model, together with the derivation of the UMP in Sect. 3, are rather simplified. The generator is modelled as a Jeffcott rotor, the rotor is treated as a continuum (reasonable due to the high number of poles, the generator considered in Table 1 has 44 poles) and the B-field is assumed according to Eq. (6). Even if this way of finding the UMP is a strong simplification, it is justified since more detailed models including arbitrary disturbed air-gaps and giving analytically the UMP are, to our knowledge, not available today. Due to this simple generator model, this work gives a clear guide for understanding the relation between asymmetries in the rotor and complicated whirling motions. From Sect. 5 it is concluded that with (3,2)perturbation, the attractor is most truly either positive -2whirling, alternating −2-whirling or in some rare cases positive 2-whirling. It is relevant to determine when the rotor moves in a loop attractor, espessially with high amplitude. The reason is that in most cases, R is large when Θ = 0, and such a situation may produce a harmful wearing of the rotor bearings, see Figs. 5–8. By simulations and analysis it is in Sect. 5 concluded that these harmful attractors occur for large rotor ovality. In all cases tested in Figs. 5–11, if the stator triangularity satisfies ΔS3 < 0.2, then the attractor is a loop only if ΔR2 > 0.1. In case of small Ω, which can be produced by making either the driving frequency w or γ/k small, the simulated results are strengthen with analytic results, see Theorem 5.1. From Fig. 10 it can be seen that if the dimensionless stiffness K is decreased, then the number of loops, and as expected also the instability comes earlier when increasing the shape perturbation parameters. However, the “shape” of the result is quite similar for different values of K. From Fig. 11 it can be noted that the number of loops increases from Ω = 0.1 to Ω = 0.5, then it is decreasing with increasing Ω. See also Fig. 7 for the attractors in some cases with different Ω. In Fig. 10 and 11, note that the analytic approximation shows good agreement in Fig. 10a, where both Ω and K

Ωτ

the following: Figure 9a shows where the alternating −2whirling loop begins, see also Fig. 5. Figure 9b shows no loops, see also Fig. 5. The loops behavior is more dependent of ΔR2 then ΔS3 , this is better illustrated in Figs. 10 and 11. Figure 9c illustrates that the alternating −2-whirling loop changes to a positive 2-whirling loop at approximately Ω/Ω0 ∈ (0.65, 0.85). This behavior can also be seen from Fig. 7. In Fig. 7d the attractor is alternating −2-whirling, in Fig. 7e and Fig. 7f it is positive 2-whirling and finally in Fig. 7g it is alternating −2-whirling. This proves, in contradiction to what the authors suspected from paper [4], that (3,2)-Perturbation can give forward whirling. Moreover, the

section, note from Fig. phase shift is illustrated by the Ωτ

7 that the amplitude of the solution is rather constant for Ω/Ω0 ∈ (0.1, 1.1). The peak in the amplitude of the motion at Ω/Ω0 ≈ 1.5 can be explained by the damped natural frquency of the system, see Eq. (26). Figure 9d shows where the alternating −2-whirling loop begins, see also Fig. 8. During maintenance of hydro power generators the shapes of the rotor and stator are frequently measured. The results from this paper can be used to evaluate such measurements and to explain the existence of complex whirling motion.

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