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Dynamic Consequences of Shape Deviations in Hydropower Generators

Niklas Lundström

Luleå University of Technology Department of Applied Physics and Mechanical Engineering Division of Computer Aided Design 2006:39|: -1757|: -lic -- 06⁄39 -- 

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Dynamic Consequences of Shape Deviations in Hydropower Generators

Niklas Lundstr¨ om

August 2006, Lule˚ a Division of Computer Aided Design Department of Applied Physics and Mechanical Engineering Lule˚ a University of Technology, SE-971 87, Sweden

Acknowledgement I want to take the opportunity to thank my supervisor Jan-Olov Aidanp¨a¨a, thanks for your supervision, your great interest in this work and for introducing me to the theory of rotordynamics. I would also like to thank Thommy Karlsson, Rolf Gustavsson, Magnus Karlberg, Martin Karlsson, Mattias Lundstr¨om and Anders Angantyr for all interesting discussions regarding rotordynamics and hydropower machines. Thanks to Ludvig Lundstr¨om and Richard Perers for discussions about electro magnetic forces. Finally Elforsk AB and the Swedish Energy Authority by the Elektra research program are acknowledged for the financial support of this project.

Abstract Earlier measurements on hydropower generators have indicated the existence of backward whirling motion, and, also relatively large shape deviations 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 thesis is to analyse the dynamic consequences of shape deviations in hydropower generators. A mathematical model is developed to describe the shapes of the rotor and stator, and the corresponding UMP is obtained through the law of energy conservation. Theorems about the existence of stable equilibria and whirling motion of the rotor response are proved mathematically for certain cases. The results indicate that different whirling motions, including both backward and forward whirling can occur. Generator dynamics are further investigated by simulating the basin of attraction, giving a measure of the robustness of the attractors. From this, the magnitudes are approximately obtained when the shape deviations become dangerous for the generator. It is concluded which shape deviations that are more dangerous than others. In hydropower generator maintenance the shapes of the rotor and stator are frequently measured. The results from this thesis can be used to evaluate such measurements and explain the existence of complicated whirling motion.

Thesis This licentiate thesis includes the following papers;

Paper A N. L. P. Lundstr¨om and J. -O. Aidanp¨a¨a. Dynamic Consequences of Electromagnetic Pull due to Deviations in Generator Shape, Journal of Sound and Vibration. (Submitted)

Paper B N. L. P. Lundstr¨om and J. -O. Aidanp¨a¨a. Whirling Frequencies and Amplitudes of the Response due to Deviations in Generator Shape, International Journal of Non-Linear Mechanics. (Submitted)

Contents 1 Introduction 1.1 Unbalanced Magnetic Pull 1.2 Rotordynamics . . . . . . 1.3 Nonlinear Dynamics . . . 1.4 The Research Problem . .

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3 4 5 6 6

2 Modelling 7 2.1 The Generator Model . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Calculation of the Unbalanced Magnetic Pull . . . . . . . . . . 9 2.3 The Equation of Motion . . . . . . . . . . . . . . . . . . . . . 12 3 Analysis 3.1 Stability . . . . . . . . . 3.1.1 Equilibrium Point 3.1.2 Periodic Motion . 3.1.3 Simulations . . . 3.2 Robustness . . . . . . .

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4 Results 19 4.1 Theorems from symmetry . . . . . . . . . . . . . . . . . . . . 19 4.2 Simulations of the Robustness . . . . . . . . . . . . . . . . . . 23 5 Discussion and Conclusion 5.1 Contribution from the Authors 5.2 Scientific Contribution . . . . . 5.3 Industrial Relevance . . . . . . 5.4 Future Work . . . . . . . . . . .

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25 26 26 27 27

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Chapter 1 Introduction The occurrence and the effect of imperfections in electrical machines have been discussed for more than one hundred years and is still a question of research. Measurements indicates that all hydropower generators, i.e. usually large salient-pole synchronous generators, are associated with some degree of asymmetry in the air-gap. Usually, it is about 0.2 % of the stator radius. These asymmetries distort the air-gap flux density distribution, producing an attraction force between the rotor and the stator, called unbalanced magnetic pull (UMP). The dynamic consequences of the UMP can be dangerous to the machine. There are documented cases, e.g. Talas and Toom [1], where the rotor has been in contact with the stator due to air-gap asymmetry. . The simplest case of asymmetry in the air-gap is referred to as eccentricity. This is when the generator axis does not coincide with the centre of the stator. Two kinds of frequently studied eccentricities are; static eccentricity, which means that the rotor rotates at an equilibrium point which does not coincide with the centre of the stator, and; dynamic eccentricity, the rotor centre follows an arbitrary trajectory. Due to unbalance, the later are often assumed to be simple synchronous whirling motion, i.e. a circular or elliptic limit cycle with a periodicity equal to the driving frequency. In this thesis, dynamic eccentricity due to shape deviations of the generator rotor and stator will be studied.

3

1.1

Unbalanced Magnetic Pull

A literature survey indicates that methods of calculating the UMP caused by eccentricity, as well as the vibration characteristics of a rotor system due to the UMP is still a topic of intensive research. Early papers, such as Behrend [2] and Robinson [3], suggested linear equations to calculate the magnetic pull for an eccentricity up to 10 % of the average air-gap. Covo [4] and Ohishi et al. [5] improved the equations for calculating the magnetic pull by considering the effect of saturation on the magnetization curve. Belmans et al. [6] developed an analytic model for vibrations in induction motors. They showed that the UMP acting on the rotor also consisted of harmonic components. The paper put forward the idea of modulating the magnetic flux by the air-gap permanence, expressed as a Fourier series. Fruchtenicht et al. [7] studied the self-exited transverse vibration of a four-pole machine, Belmans et al. [8] investigated the radial stability of the shaft in a two-pole machine whereas Smith and Dorrell [9] studied the UMP by winding analysis. Finite element analysis technique can today provide solutions to the UMP problem. However, this approach is still very computationally expensive and can often not provide deeper insight into the origin of the found results, see Debortoli et al. [10] and Arkkio [11]. Dorrell [12] calculated UMP for non-uniform rotor eccentricity and tooth saturation, Guo et al. [13] studied the effects of UMP and the vibration level in three-phase generators with any number of pole pairs, Wang et al. [14] derived the UMP due to eccentricity through the law of energy conservation and studied the free and forced vibrations for rotors of electric motors, and Holopainen et al. [15] studied the rotordynamic effects of electromechanical interaction on induction motors. Burakov and Arkkio [16] showed that the damping winding affects the UMP a lot depending on the whirling of the rotor in salient-pole synchronous machines, Karlsson and Aidanp¨a¨a [17] studied the dynamic behaviour in a hydropower rotor system due to the influence of generator shape and fluid dynamics and Williamson and Abdel-Magied [18] calculated the UMP in induction motors with asymmetrical rotor cage. They showed that with an even distribution pattern of bar faults the UMP may be vanishingly small. Tereshonkov [19] determined the angular frequencies of the UMP due to static and dynamic eccentricity, and also due to rotor and stator core ovalities. An overview of the research in the area of modelling and calculating UMP in electrical machines was presented by Frosini and Pennacchi [20].

4

1.2

Rotordynamics

Rotating machines have a wide scope of applications, for example power generating units and manufacturing units. The research on rotating machines started with Rankine’s paper [21] on whirling motion of a rotor in 1869. Incorrectly, he concluded that a rotating machine will never be able to operate above its first critical speed. By an experiment, De Laval showed around 1900 that he’s machine was able to operate above it’s critical speed. Theoretically, Jeffcott [22] derived a theory in 1919 which shows the possibility for rotating machines to operate above their critical speeds. Still in our days, Jeffcott and De Laval are the names associated with the simplest rotor model, a lumped mass in the middle of a rotating shaft. In 1924, the influence of gyroscopic effect was presented by Stodola [23]. Green [24] continued the work with gyroscopic effect in 1948 with his four degree of freedom overhung rotor. Bishop [25] started the research of continuous rotors in 1959. Booker and Ruhl [26] where the first to use finite element methods in the area of rotordynamics 1972. Nelson and McVaugh [27] generalised this model 1976 by also including gyroscopic moment, rotating inertia and axial force. Rotordynamics are normally considered as a separate area within structural dynamics. The major differences from structural dynamics are that the eigenfrequencies will depend on the rotational speed of the rotor because of the change in gyroscopic moments. For theory about rotordynamics, see for example the book by Yamamoto and Ishida [28]. Research in the field of rotordynamics have historically been focused on high-speed turbo machinery. In the area of hydropower rotor systems, there are several specific problems that need to be considered. Development of the electro-mechanical interactions in the generator, the dynamics from the bearings and the influence from the fluid forces are essential in order to predict the dynamics of these systems. Only a few papers are published in this area. In addiction to the work presented in this thesis, see for example Aidanp¨a¨a et al. [29], [30] and [31].

5

1.3

Nonlinear Dynamics

For linear systems the solution is usually composed by superposition of a transient and a steady state solution. The transient part is due to the free oscillations and in the presence of damping it is damped out after some time, leaving the steady state solution with oscillations of the frequency of the forcing. This means that for damped linear systems the steady state solution is independent of initial conditions, and therefore multiple solutions can not exist for such systems. Three kinds of motions are generally predicted by the linear theory. They may be classified as equilibrium point, periodic motions and quasiperiodic motions. These motions are called attractors since the presence of damping causes the solution to be attracted to one of these states after sufficient time. For nonlinear systems the principle of superposition is not valid and a rich variety of motions are possible. These motions includes all of the linear responses mentioned above and also motions like for example, subharmonic motion, superharmonic motion, self exited motion and jump phenomena. Morover, more complex attractors can be found, namely the strange attractor. The motion associated with the strange attractor is referred to as chaotic motions or simply chaos. In nonlinear systems a number of attractors can also exist simultaneously, meaning that small differences in initial condition can lead to different kinds of motions. For an introduction to nonlinear dynamics, see for example the book by Hilborn [32], Devaney [33] or Arrowsmith and Palace [34]. For more advanced bifurcation theory, see Kuznetsov [35], Guckenheimer and Holmes [36], or Wiggins [37].

1.4

The Research Problem

To focus on the research problem one research question has been formulated: What are the dynamic consequences of shape deviations in the rotor and the stator in hydropower generators? The approach will be to use simple models, giving the possibility to do theoretical analysis to obtain general results, and thereby get fundamental understanding of the dynamics.

6

Chapter 2 Modelling 2.1

The Generator Model

Fig. 2.1 shows the geometry of the generator model, having an arbitrary non-circular shaped rotor and stator. The generator is treated as a balanced Jeffcott or Laval rotor having length l0 , mass γ and stiffness k of the generator axis. The rotor rotates at a constant counterclockwise angular speed ω. Point Cs gives the location of the bearings while point Cr is the rotational centre of the rotor. The coordinate system has the origin at Cs , r is the rotor radius and s is the stator radius.

γ y k

r

s

k

Cs

l0

ϕ

Cr

ω x

z

(b) The cross-section of the generator.

(a) The Jeffcott rotor.

Figure 2.1: The generator model.

Let r0 and s0 be the average radius of the rotor and the stator respectively. An arbitrary non-circular shape can be described by adding a Fourier series of cosine terms to the rotor radius, r, and the stator radius, s, 7

∞ 

r = r0 (z) +

δnr (z) cos n (ϕ + αnr (z)) ,

(2.1)

s s δm (z) cos m (ϕ + αm (z)) ,

(2.2)

n=1

s = s0 (z) +

∞  m=1

where δnr

(z) ≥ 0,

s δm

(z) ≥ 0,

∞  n=1

δnr

(z) +

∞ 

s δm (z) < g0 (z) . (2.3)

m=1

s Here, g0 = s0 − r0 is the average air-gap, δnr and δm are referred to as the r s rotor and stator perturbation parameters, while αn and αm are the corresponding phase angles. s = 0 for To simplify notations, it is hereinafter assumed that δnr = δm all m, n ∈ N , if nothing else is mentioned. N is here the set of all natural numbers. Fig. 2.2 shows the rotor shape for δnr = r0 /3, n = 1, 2, 3, 4, and s phase angles αnr = 0. Note that the stator has the same shape for δm = δnr and m = n.

Figure 2.2: The shape of the rotor for δnr = r0 /3, n = 1, 2, 3, 4. The case δ1r > 0 and δ1s > 0 will correspond to rotor eccentricity and stator eccentricity respectively. Since dynamic eccentricity is normally small compared to the dimensions of the generator, it is assumed that the perturbed air-gap (g) is g = s (z, ϕ) − r (z, ϕ) − x cos ϕ − y sin ϕ,

(2.4)

where (x, y) gives the position of Cr . Eqs. (2.1), (2.2) and (2.4) gives, after adding the ω rotation 8

g = g0 (z) +

∞ 

s δm

(z) cos m (ϕ +

s αm )

−

m=1

∞ 

δnr (z) cos n (ϕ + αnr (z) − ωt)

n=1

− x cos ϕ − y sin ϕ.

(2.5)

The parameter values used everywhere in this thesis are taken from an 18 MW hydropower generator. These values are given in Table 2.1. Table 2.1: Numerical values from the 18 MW hydropower generator. s0 l0 g0 γ k ω km μ0

2.2

Average stator radius Length of the generator Average air-gap Mass of the rotor Stiffness of the axis Rotor rotation speed Magnetic stiffness Permeability of air Number of poles

2.775 m 1.18 m 0.0125 m 98165 kg 3.456 · 108 N/m 14.2 rad/s 1.4715 · 108 N/m 4 π · 10−7 Vs/Am 44

Calculation of the Unbalanced Magnetic Pull

In this section, a model based on the principle of virtual work for finding the UMP for an arbitrary disturbed air-gap is presented. Based on the theory of magnetic field [38], the potential energy reserved in the air-gap can be expressed as  E = all space

B (x, y, z, t, ϕ)2 dV, 2 μ0

(2.6)

where B is the magnetic flux density (also called the B-field) in the air-gap and μ0 is the permeability of air. For an approximation, the relations between the B-field and the air-gap widths are assumed as in [14],

B =

B0 (z) g0 (z) . g (x, y, z, t, ϕ) 9

(2.7)

r

dV ϕ Cs

Figure 2.3: The volume element dV . B0 is the uniformly distributed B-field for a perfect circular geometry, i.e. g = g0 . Next, consider a volume element dV as shown in Fig. 2.3. According to Eqs. (2.6) and (2.7), the potential energy, δE, reserved in dV is given as B0 (z)2 g0 (z)2 δE = dV. 2 μ0 g (x, y, z, ϕ, t)2

(2.8)

Eq. (2.8) 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 increments of mechanical energy input to dV and dEelectric increments of the electric energy output from dV . When considering the energy conversion between the magnetic and mechanical fields over an infinitesimal period of time, the law of energy conservation requires, after neglecting losses dEmech = d (δE) + dEelectric .

(2.9)

As in the case of eccentricity [14], it is assumed that the electric energy output from the generator is independent of the air-gap variations, thus assume that 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 df . Then, the virtual work done by this force is df dg = − d(δE), which gives df = −

d (δE) . dg

10

(2.10)

In Eq. (2.8), note that, since dV = r dr dz dϕ, the potential energy δE will increase if r increases when g is constant. This small change in df cannot be considered in Eq. (2.10). 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 dr dz dϕ, where u0 = (r0 + s0 )/2, and dr = g. Eq. (2.8) then yields

δE =

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

(2.11)

According to Eq. (2.10), the force df is given by

df = −

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

(2.12)

Hence, the total forces in the x− and y−directions can be expressed as 1 fx = 2 μ0 1 fy = 2 μ0



2π

2π 0

l0 0

0





 0

l0

B0 (z)2 g0 (z)2 u0 (z) cos ϕ dz dϕ, g (x, y, z, t, ϕ)2

(2.13)

B0 (z)2 g0 (z)2 u0 (z) sin ϕ dz dϕ. g (x, y, z, t, ϕ)2

(2.14)

Eq. (2.13) and (2.14) gives the UMP and completes this section.

11

2.3

The Equation of Motion

From Newton’s laws of motion, the equations of motion for the Jeffcott rotor are easily found. They consist of the two non-autonomous nonlinear second order differential equations γ x¨ + c x˙ + k x = fx (x, y, t) , γ y¨ + c y˙ + k y = fy (x, y, t) .

(2.15)

Here, γ is the mass of the rotor, k is the stiffness of the rotor axis and c being a linear viscous damping. In non-dimensional form, system (2.15) yields X  + 2 ζ X  + X = FX (X, Y, τ ) , Y  + 2 ζ Y  + Y = FY (X, Y, τ ) .

(2.16)

Here, the non-dimensional quantities X =

x , g0

Y =

y , g0

ΔR n =

δnr , g0 

g (x, y, t, ϕ) , G = g0

Ω = ω

ΔSm =

S δm , g0

(2.17)

 γ , k

τ = t

k , γ

(2.18)

have been introduced and τ is a non-dimensional time. The air-gap G, and finally the forces FX and FY yield FX

FY

G = 1+

∞ 

ΔSm

km = 2πk km = 2πk



2π

0



2π 0

cos m (ϕ +

cos ϕ dϕ, G (X, Y, τ, ϕ)2

(2.19)

sin ϕ dϕ, G (X, Y, τ, ϕ)2

(2.20)

s αm )

m=1

−

∞ 

r ΔR n cos n (ϕ + αn − Ω τ )

n=1

− X cos ϕ − Y sin ϕ.

(2.21)

12

Chapter 3 Analysis The governing equations derived in the previous chapter are analyzed as far as possible using analytical methods, and also by using some iterative technics.

3.1 3.1.1

Stability Equilibrium Point

Consider a general autonomous dynamical system on an n-dimensional state space x˙ = f (x)

f, x ∈ Rn .

(3.1)

An equilibrium point to system (3.1) is a point xeq ∈ Rn such that f (xeq ) = 0.

(3.2)

To evaluate the stability of the point xeq , consider an arbitrary point x close to the equilibrium point x = xeq + z.

(3.3)

Substituting Eq. (3.3) into Eq. (3.1) and Taylor expanding about the equilibrium point gives x˙ = x˙ eq + z˙ = f (xeq ) + Df (xeq ) z + O(|z|2 ). 13

(3.4)

Here, Df (xeq ) denotes the Jacobian matrix to the vector field f . From Eq. (3.4) it follows that the system (3.1) can be approximated by a linear system in a neighbourhood of the equilibrium point xeq . The linear system yields z˙ = Df (xeq ) z.

(3.5)

Now, by assuming a solution of the form z = u eλ t , u ∈ Rn , λ ∈ Z and substituting into Eq. (3.5) yields the eigenvalue problem λ u = Df (xeq ) u.

(3.6)

The total solution is of the form z = C1 u1 eλ1 t + C2 u2 eλ2 t + . . . + Cn un eλn t ,

(3.7)

where Ci are constants, λi are the n eigenvalues, called the characteristic values to the equilibrium, and ui are the eigenvectors to the Jacobian matrix Df (xeq ) for 1 ≤ i ≤ n. From Eq. (3.7) it is clear that xeq is an asymptotically stable equilibrium if and only if Re(λ) < 0, for all λ. This linearization gives also more information about the dynamics near the equilibrium. See for example the book by Kuznetsov [35].

3.1.2

Periodic Motion

Consider again system (3.1). The stability of a periodic motion can be studied by defining a map g over a period of oscillation such that xk+1 = g(xk ).

(3.8)

A periodic solution is a fixed point xf ix of the map g. Let xk = xf ix + z k .

(3.9)

Using Taylor expansion as in the case of a stationary point explained in the former section, the linearized map is given by z k+1 = Dg(xf ix ) z k . 14

(3.10)

Now, assume a solution of the form z k = u mk , where u ∈ Rn and m ∈ Z. Inserting this into Eq. (3.10) yields an eigenvalue problem m u = Dg(xf ix ) u.

(3.11)

The total solution is of the form z k = C1 u1 mk1 + C2 u2 mk2 + . . . + Cn un mkn ,

(3.12)

where Ci are constants, mi are the n eigenvalues, called the characteristic multipliers of the periodic solution, and ui are the eigenvectors to the Jacobian matrix Dg(xf ix ) for 1 ≤ i ≤ n. As in the case of an equilibrium point, the stability of the periodic solution is determined from the position of the characteristic multipliers in the complex plane. The periodic solution is asymptotically stable if and only if |m| < 1 for all m. The difficulty with this method to study the stability of periodic motion is usually to find the map g. For most dynamical systems it is impossible. Therefore, in this thesis, the stability of periodic attractors are studied only from numerical simulations.

3.1.3

Simulations

Simulations of trajectories to system (2.16) are carried out using a forth order Runge-Kutta method. The force integrals given by Eqs. (2.19) and (2.20) are solved numerically at each time step by Simpson’s rule. Since the stability of a nonlinear dynamical system can be dependent of the initial conditions, a cell mapping technique is used to estimate the stability for each parameter combination. With this method, an approximation of the basin of attraction to the attractor in question is obtained. Properties of this approximation will be referred to as measures of the robustness of the attractor.

15

3.2

Robustness

A nonlinear dynamical system can have several coexisting attractors, and therefore, a stable solution favourable for the machine can exist for some parameters but only attracting a very small neighbourhood in the state space. Thus, to investigate the robustness of a stable solution, the basin of attraction to this attractor is considered. Both the size and the shape of the basin of attraction will be approximated. It is advantageous if the basin of attraction is large and convex. (A region D is said to be convex if for points A, B ∈ D, the line segment AB lies also in D). Approximating the basin of attraction can be done, for example with a cell mapping method in the following way; For fixed parameters, the state space is covered by a uniform grid of initial conditions. From each initial condition, the trajectory is examined to see if it converges to the attractor in question. The quantity (N) of such initial conditions gives an approximation of the size of the basin of attraction. To estimate the shape, a satisfactory choose of a distance (D) is considered. For example, if the attractor is a stationary point, the shortest distance from this point to an initial condition not converging to the attractor can be a satisfactory choose for D. The largest difficulty with this method is; How to compare distance to velocity? One proposal is to consider the energy added to the system from the initial conditions. However, in this thesis the initial velocities are set to zero as a simplification, reducing the cell mapping to the two-dimensional XY −subspace giving much less simulation time. The procedure is carried out in the following way; The XY −plane is covered by a uniform grid of initial conditions. The simulation of system (2.16) continues until the rotor hits the stator, comes close to an equilibrium, or reaches a time limit. If the time limit or an equilibrium is attained, the corresponding initial condition is said to be converging (otherwise diverging). The condition for reaching an equilibrium is set to (X  )2 + (Y  )2 < 10−9 . The time limit is set to 100 revolutions of the rotor. The converging initial conditions gives an approximation of the two-dimensional XY −subspace of the basin of attraction to attractors favourable for the machine. This approximation is denoted AXY . Let nXY be the number of elements in AXY , and dXY the distance from the origin to the diverging point closest to the origin. This means that dXY gives the radius of the largest circle, centred at the origin that is covered by AXY . Then, to scale these measures, define NXY =

nXY , n0XY

DXY =

16

dXY , 0 dXY

(3.13)

0 where n0XY and dXY correspond to the case of ideal circular generator geometry. One example considering the generator model in this thesis is shown beS low; Fig. 3.1 illustrates AXY for the shape perturbation ΔR 2 = Δ3 = S 0.2, Δ1 = 0.01. The simulation uses a uniform grid of 361201 initial conditions and yields nXY = 17237 and dXY = 0.1244. Origin is marked by a dot. The smaller circle has radius dXY (dashed). The larger circle has 0 radius dXY = 0.6588 (solid), and indicates the boundary of AXY for an ideal circular generator for comparison. The phase angles are chosen to zero and the uniform grid gives n0XY = 122697. According to Eq. (3.13), in this case the two scaled measures of the robustness yield NXY = 0.1405 and DXY = 0.1888.

0.6

0.4

Y

0.2

0

−0.2

−0.4

−0.6 −0.6

−0.4

−0.2

0 X

0.2

0.4

0.6

S S Figure 3.1: AXY for the shape perturbation ΔR 2 = Δ3 = 0.2, Δ1 = 0.01. Origin is marked by a dot. The smaller circle has radius dXY (dashed). The 0 larger circle has radius dXY (solid).

17

18

Chapter 4 Results In this chapter a summary of the most important results obtained in the appended papers are presented.

4.1

Theorems from symmetry

From symmetries in the air-gap, the following theorem about the existence of stable equilibria is proved: Theorem 4.1 Let N S be the set of all natural numbers m such that ΔSm > 0, and N R the set of all natural numbers n, such that ΔR n > 0. Then define   Σ = ΔSm + ΔR (4.1) n. m ∈ NS

n ∈ NR

The UMP is zero when the rotor is rotating at the origin if; 1) 2) 3) 4)

Both N S and N R are empty. N R is empty and m ≥ 2, ∀ m ∈ N S . N S is empty and n ≥ 2, ∀ n ∈ N R . It is possible to find integers pn and pm such that q =

m n = ≥ 2, pn pm

q ∈ N,

∀ n ∈ N R,

∀ m ∈ N S.

(4.2)

When 1,2,3 or 4 holds, system (2.16) has a stable equilibrium at the origin if √ √ 2 π + π2 + 4 π2 + 4 2πk < + . (4.3) 3 km (1 − Σ) (1 + Σ)3 19

A proof of Theorem 4.1 is presented in Paper A. For the generator parameters given in Table 2.1, Eq. (4.3) yields Σ < 0.16385. By using a Taylor series expansion of Eqs. (2.19) and (2.20), the following theorem about angular frequency and amplitude of the UMP is proved: Theorem 4.2 Assume a shape perturbed generator of which one rotor perS turbation parameter, ΔR n , and one stator perturbation parameter, Δm are dominating such that ∞ ∞        R q S X  + Y  + Δi + ΔR 0, Δ2 > 0. Note also the R nonlinear effect near ΔR 2 = 0.03 and Δ2 = 0.25. Figs. 4.3 and 4.4 illustrate that the effect of the shape perturbations on NXY and DXY decreases when m and n increases. Thus, assuming the same amount of perturbation, this S shows that eccentricity (ΔR 1 and Δ1 ), are more severe than perturbations corresponding to higher m and n. Sharp knees are observed near ΔR 1 = 0.05 S and near Δ1 = 0.05 in Fig. 4.4, meaning that perturbations less than 5% of the air-gap of the generator only marginally affect the robustness.

5.1

Contribution from the Authors

In the appended papers one co-writer, my supervisor Jan-Olov Aidanp¨a¨a has been involved in the work. The idea of investigate the consequences of shape deviations was set up by Jan-Olov Aidanp¨a¨a together with Rolf Gustavsson. The generator shape modelling was developed by Me and Jan-Olov Aidanp¨a¨a. The derivation of the UMP, the mathematical analysis and the simulations in the appended papers have been carried out by Me with advice from Jan-Olov Aidanp¨a¨a.

5.2

Scientific Contribution

From an academic point of view, an advice is given how to model shape deviations in a generator using a simple model. Some basic results are shown both analytically and numerically that will help when trying to investigate the dynamics using more detailed models. Also, a method for approximating the robustness to attractors in a nonlinear mechanical system is suggested. 26

5.3

Industrial Relevance

From an industrial point of view, the results indicate which shape deviations are more or less dangerous to the generator. Since the UMP can cause large vibrations in hydropower generators which can destroy the machine, the shapes of the rotor and stator are frequently measured during maintenance. The results from this thesis can be used to evaluate such measurements and estimate the whirling, amplitude, stability and robustness of the rotor response. Thus, better decisions can be made regarding operations, maintenance and redesign. Many results presented here are general and can be applied to different electric motors and generators.

5.4

Future Work

S An interesting work is to analyse the shape perturbation ΔR 2 > 0, Δ3 > 0, including a mass unbalance. This case is of interest since stators often stand on three feats, which can result in a three-angularity shaped stator. With this geometry, the UMP due to shape deviation will force the rotor to backward whirling. However, an unbalance mass will force the rotor to synchronous whirling. Due to these contracting driving forces, some interesting dynamics will occur. Measurements on a real hydropower generator are needed for validation. This can be difficult since real generators have several shape deviations at the same time and also other imperfections. Thus, probably only the strongest effects can be expected to be seen, such as the whirling frequency. Therefore, a measurement of the rotor and stator shape on such generators (whirling with a known frequency) would be interesting in order to compare with the results obtained in this thesis. A more detailed model to calculate the UMP due to shape deviations in the generator is needed. A general expression for the forces due to the generators damping winding, saturation etc. is of interest. Burakov and Arkkio [16] showed that the damping winding affects the UMP a lot depending on the whirling of the rotor in salient-pole synchronous machines. When detailed calculations of the UMP are available, an update of the results obtained in this thesis can be carried out, and a detailed rotor model describing the complete hydropower rotor system can be used in simulations.

27

28

Bibliography [1] P. Talas, P. Toom, Dynamic Measurement and Analysis of Air-Gap Variations in Large Hydroelectric Generators. IEEE, 83 WM 226-8, 1983, 9p. [2] B. A. Behrend, On the Mechanical Force in Dynamos Caused by Magnetic Attraction. AIEE, Vol. 17, Nov. 1990, p. 617. [3] R. C. Robinson, The Calculation of Unbalanced Magnetic Pull in Synchronous and Induction Motors. Electr. Engng. Vol. 62, 1943, p. 620-624. [4] A. Covo, Unbalanced Magnetic Pull in Induction Motors with Eccentric Rotors. Trans. AIEE, Vol. 73. pt III, Dec. 1954, p. 1421-1425. [5] H. Ohishi, S. Sakabe, K. Tsumagari, K. Yamashita, Radial Magnetic Pull in Salient Poles Machines with eccentric rotors. IEEE, 86 SM 4758, 1986, 5p. [6] R. Belmans , W. Geysen, H. Jordan and A. Vandenput, Unbalanced Magnetic Pull in Three Phase Two Pole Induction Motors With Eccentric Rotors. IEE Conference Publication, n 213, 1982, p 65-69. [7] J. Fruchtenicht, H. Jordan and H. O. Seinsch, Running Instability of Cage Induction Motors Caused by Harmonic Fields Due to Eccentricity. Part 1: Electro-Magnetic Spring Constant and Electro-Magnetic Damping Coefficient. Part 2: Self-Exited Transverse Vibrations of the Rotor. Archiv fur Elektrotechnik, v 65, n 4-5, 1982, p 271-292. [8] R. Belmans, A. Vandenput, W. Geysen, Influence of Unbalanced Magnetic Pull on the Radial Stability of Flexible-Shaft Induction Machines. IEE Proceedings, Part B: Electric Power Applications, v 134, n 2, Mar. 1987, p 101-109. [9] A. C. Smith, D. G. Dorrell, Calculation and Measurement of Unbalanced Magnetic Pull in Cage Induction Motors with Eccentric Rotors. Part1: 29

Analytical Model. IEE. Proc. Electr. Power appl., Vol. 143, No 3, p. 193-201, 1996. [10] M. J. Debortoli, S. J. Salon, D. W. Burrow, C. J. Slavik, Effect of Rotor Eccentricity and Parrallel windings on Induction Machine Behaviour: A Study Using Finite Element Analysis. IEEE Transaction on Magnetics 29, p. 1676-1682, 1993. [11] A. Arkkio, Unbalanced Magnetic Pull in Cage Induction Motors with Asymmetry in Rotor Structures. Eighth International Conference on Electrical Machines and Drives, 1-3 Sept., Conf. Publ. No. 444, Cambridge, 1997. [12] D. G. Dorrell, Modelling of Non-Uniform Rotor Eccentricity and Calculation of Unbalanced Magnetic Pull in 3-Phase Cage Induction Motors. ICEM. 28-30. August. Espoo Finland. 2000. [13] D. Guo, F. Chu and D.Chen, The Unbalanced Magnetic Pull and its Effects on Vibration in a Three-Phase Generator With Eccentric Rotor. Journal of Sound and Vibration, v 254, n 2, 4 July 2002, p. 297-312. [14] Y. Wang, G. Sun and L. Huang, Magnetic Field-induced Nonlinear Vibration of an Unbalanced Rotor. Proceedings of the ASME Design Engineering Division - 2003 Volume 2, p 925-930, 2003. [15] T. P. Holopainen, A. Tenhunen, A. Arkkio, Electromechanical Interaction in Rotordynamics of Cage Induction Motors. Journal of Sound and Vibration, v 284, n 3-5, Jun 21, 2005, p. 733-755. [16] A. Burakov, A. Arkkio, Low-order Parametric Force Model for a Salientpole Synchronous Machine with Eccentric Rotor. Electrical Engineering, 2006. [17] M. Karlsson, Jan-Olov Aidanp¨a¨a, Dynamic Behaviour in a Hydro Power Rotor System due to the Influence of Generator Shape and Fluid Dynamics . ASME Power, April 5-7, 2005, p. 905-913, Chicago, Illinois. [18] S. Williamson, M. A. S. Abdel-Magied, Unbalanced Magnetic Pull in Induction Motors With Asymmetrical Rotor Cages. IEE Conference Publication, n 254, p. 218-222, 1985. [19] V. A. Tereshonkov, Magnetic Forces in Electric Machines with Air Gap Eccentricities and Core Ovalities. Elektrotekhnika, Vol. 60, No.9, pp.5053, 1989. 30

[20] L. Frosini, P. Pennacchi, Detection and Modelling of Rotor Eccentricity in Electric Machines - an Overview. C623/060/2004. IMechE, 2004. [21] W. J. M. Rankine, On the Centrifugal Force of Rotating Shafts. Engineer, Vol. 27, pp 249-249, 1869. [22] H. H. Jeffcott, The Lateral Vibration of Loaded Shafts in Neighborhood of a Whirling Speed: the Effect of Want of Balance. Philos. Mag, Vol. 37, pp 304-315, 1919. [23] A. Stodola, Dampf- und Gas-Turbinen. Verlag von Julius, Springer, Berlin, 1924. [24] P. Green, Gyroscopic Effect of the Critical Speeds of Flexible Rotors. Journal of Applied Mechanics, No 15, pp 269-275, 1948. [25] R. E. D. Bishop, Vibration of Rotating Shafts. Journal of Mechanical Engineering Science, Vol. 1, pp 50-65, 1959. [26] R. L. Ruhl and J. F. Booker, A Finite Element Model for Disturbed Parameter Turborotor System. Trans. ASME, J. Eng. Ind. Vol. 94, No. 1, pp. 126-132, 1972. [27] H. D. Nelson and J. M. McVaugh, The Dynamics of Rotor Bearing System, Using Finite Elements. Trans. ASME, J. Eng. Ind. Vol. 98, No. 2, pp. 593-600, 1976. [28] T. Yamamoto and Y. Ishida, Linear and Nonlinear Rotordynamics; A Modern Treatment with Applications. John Wily and Sons, ISBN 0-47118175-7, 2001. [29] R. K. Gustavsson and J.-O. Aidanp¨a¨a, Measurement of Bearing Load Using Strain Gauges at Hydropower Unit. Hydro Review, Volume 11, Number 5, pp 30-36, 2003. [30] R. K. Gustavsson and J.-O. Aidanp¨a¨a, The Influence of Non-linear Magnetic Pull on Hydropower Generator Rotors. Journal of Sound and Vibration. (Submitted). [31] M. Karlsson, J.-O. Aidanp¨a¨a, R. Perers and M. Leijon, Rotor Dynamical Analysis of a Synchronous Generator due to the Influence of Reactive Load. Journal of Applied Mechanics. (Submitted). 31

[32] R. C. Hilborn, Chaos and Nonlinear Dynamics; An Introduction for Scientists and Engineers. Oxford University Press, ISBN 0 19 850723 2, 2000. [33] R. L. Devaney, A First Course in Chaotic Dynamical Systems; Theory and Experiment. Addison-Wesley, ISBN 0-201-55406-2, 1992. [34] D. K. Arrowsmith and C. M. Place, Dynamical Systems; Differential Equations, Maps and Chaotic Behaviour. Chapman and Hall/CRC, ISBN 0-412-39080-9, 1992. [35] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory. New York, Springer-Verlag, ISBN 0-387-94418-4, 1995. [36] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. New York, Springer-Verlag, ISBN 0-387-90819-6, 1983. [37] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York, Springer-Verlag, ISBN 0-387-97003-7, 1990. [38] R. K. Wangsness, Electromagnetic Fields, Hamilton Printing Company, ISBN 0-471-81186-6, 1986.

32

A

DYNAMIC CONSEQUENCES OF ELECTROMAGNETIC PULL DUE TO DEVIATIONS IN GENERATOR SHAPE Niklas L. P. Lundstr¨om and Jan-Olov Aidanp¨a¨a Division of Computer Aided Design, Department of Applied Physics and Mechanical Engineering, Lule˚ a University of Technology, SE-971 87, Sweden. E-mails: [email protected]; [email protected] Abstract Results from earlier measurements on hydropower generators have indicated relatively large eccentricities and shape deviations in the rotor and stator. These nonsymmetric geometries produce an attraction force between the rotor and the stator, called unbalanced magnetic pull (UMP). The UMP force can produce large vibrations which can be dangerous to the machine. A mathematical model is developed to describe the shapes of the rotor and stator, and the corresponding UMP is obtained through the law of energy conservation. The target of the paper is to analyse the dynamics of a generator due to shape deviations in the rotor and stator. As rotor-model, a balanced Jeffcott rotor is used. A linearization of the UMP indicates the importance of considering the nonlinear effects. The stability of some attractors are analysed and the generator dynamics are further investigated by simulating the basin of attraction. The magnitudes are approximately obtained when the shape deviations become dangerous for the generator. It is concluded which shape deviations that are more dangerous than others. In hydropower generator maintenance the shapes of the rotor and stator are frequently measured. The results from this paper can be used to evaluate such measurements and estimate the stability and robustness by simulations. Key words: rotor, dynamics, magnetic pull, generator, shape, hydropower.

1

INTRODUCTION

Hydropower generators have small air-gaps between the rotor and the stator. Usually, it is about 0.2 % of the stator radius. Measurements on hydropower generator shapes are frequently carried out during maintenance, indicating that all hydropower generators are associated with some degree of asymmetry in the air-gap. Talas and Toom [1] have reported on a variation of the airgap deviating more than ± 10%. These asymmetries distort the air-gap flux Preprint submitted to Elsevier Science

10 August 2006

density distribution, producing an attraction force between the rotor and the stator, called unbalanced magnetic pull (UMP). The effect of UMP can be vibrations which are dangerous to the machine. There are documented cases, e.g. Talas and Toom [1], where the rotor has been in contact with the stator due to air-gap asymmetry. A literature survey indicates intensive study on the methods of calculating UMP caused by eccentricity, as well as the vibration characteristics of a rotor system due to UMP. Early papers, such as Behrend [2] and Robinson [3], suggested linear equations to calculate the magnetic pull for an eccentricity up to 10 % of the average air-gap. Covo [4] and Ohishi et al. [5] improved the equations for calculating the magnetic pull by considering the effect of saturation on the magnetization curve. Belmans et al. [6] developed an analytic model for vibrations in induction motors. They showed that the UMP acting on the rotor also consisted of harmonic components. The paper put forward the idea of modulating the magnetic flux by the air-gap permanence, expressed as a Fourier series. Fruchtenicht et al. [7] studied the self-excited transverse vibration of a four-pole machine, Belmans et al. [8] investigated the radial stability of the shaft in a two-pole machine whereas Smith and Dorrell [9] studied the UMP by winding analysis. Finite element analysis technique can now provide solutions of the UMP, though this approach is still very computationally expensive and can often not provide insight into the origins and key factors of its production, Debortoli et al [10], Arkkio [11]. Dorrell [12] calculated UMP for non-uniform rotor eccentricity and tooth saturation, Guo et al. [13] studied the effects of UMP and the vibration level in three-phase generators with any number of pole pairs, Wang et al. [14] derived the UMP due to eccentricity through the law of energy conservation and studied the free and forced vibrations for rotors of electric motors, and Holopainen et al. [15] studied the rotordynamic effects of electromechanical interaction on induction motors. Karlsson and Aidanp¨aa¨ [16] studied the dynamic behaviour in a hydropower rotor system due to the influence of generator shape and fluid dynamics. Williamson and Abdel-Magied [17] calculated the UMP in induction motors with asymmetrical rotor cage. They showed that with an even distribution pattern of bar faults the UMP may be vanishingly small. In this paper, the UMP due to an arbitrary disturbed air-gap is derived through the law of energy conservation. It is also compared to a linear model for calculating UMP. The dynamic consequences of UMP due to shape deviations on the rotor and stator are studied using a balanced Jeffcott rotor as a simplified model of a hydropower generator. A mathematical model describes the shapes of the rotor and stator. The dynamic behaviour of the generator is analyzed through the use of symmetries for certain cases. For different shape deviations the basin of attraction is studied for rotor-stator impact motions. From this, the robustness of the generator is approximated for different rotor and stator shapes. 2

2

GENERATOR GEOMETRY

Fig. 1 shows the geometry of the generator model, having an arbitrary noncircular shaped rotor and stator. The generator is treated as a balanced Jeffcott or Laval rotor having length l0 , mass γ and stiffness k of the generator axis. The rotor rotates at a constant counterclockwise angular speed ω. Point Cs gives the location of the bearings while point Cr is the rotational center of the rotor. The coordinate system has the origin at Cs , r is the rotor radius and s is the stator radius. γ y k s

k

r

ϕ

Cs

l0 (a)

ω

Cr

x

z (b)

Fig. 1. The generator model; a) The Jeffcott rotor. b) The cross-section of the generator.

Let r0 and s0 be the average radius of the rotor and the stator respectively. An arbitrary non-circular shape can be described by adding a Fourier series of cosine terms to the rotor radius, r, and the stator radius, s, r = r0 (z) +

∞ 

δnr (z) cos n (ϕ + αnr (z)) ,

(1)

s s δm (z) cos m (ϕ + αm (z)) ,

(2)

n=1

s = s0 (z) +

∞  m=1

where δnr (z) ≥ 0,

s δm (z) ≥ 0,

∞  n=1

δnr (z) +

∞ 

s δm (z) < g0 (z) .

(3)

m=1

s are referrd to as the rotor Here, g0 = s0 − r0 is the average air-gap, δnr and δm r s are the corresponding and stator perturbation parameters while αn and αm s = phase angles. To simplify notations, it is hereinafter assumed that δnr = δm 0, ∀ m, n ∈ N , if nothing else is mentioned. N is the set of all natural numbers. Fig. 2 shows the rotor geometry for δnr = r0 /3, n = 1, 2, 3, 4, and s = δnr phase angle αnr = 0. Note that the stator has the same geometry for δm and m = n.

3

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

The case δ1r > 0 and δ1s > 0 will correspond to rotor eccentricity and stator eccentricity respectively. Since dynamic eccentricity is normally small compared to the dimensions of the generator, 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 . Eqs. (1), (2) and (4) gives, after adding the ω rotation g = g0 (z) +

∞ 

s s δm (z) cos m (ϕ + αm )−

m=1

∞ 

δnr (z) cos n (ϕ + αnr (z) − ω t)

n=1

− x cos ϕ − y sin ϕ.

(5)

The geometric model is now completed.

3

UNBALANCED MAGNETIC PULL

3.1 CALCULATION OF THE UMP

Based on the theory of magnetic field [18], the potential energy reserved in the air-gap can be expressed as 

E = all space

B (x, y, z, t, ϕ)2 dV, 2 μ0

(6)

where B is the magnetic flux density (also called the B-field) in the air-gap and μ0 is the permeability of air. For an approximation, the relations between the B-field and the air-gap widths are assumed as in [14], B =

B0 (z) g0 (z) . g (x, y, z, t, ϕ)

(7)

B0 is the uniformly distributed B-field for a perfect circular geometry, i.e. g = g0 . Next, consider a volume element dV as shown in Fig. 3. 4

r

dV ϕ Cs Fig. 3. The volume element dV .

According to Eqs. (6) and (7), the potential energy, δE, reserved in dV is given as B0 (z)2 g0 (z)2 dV. δE = 2 μ0 g (x, y, z, ϕ, t)2

(8)

Eq. (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 increments of mechanical energy input to dV and dEelectric increments of the electric energy output from dV . When considering the energy conversion between the magnetic and mechanical fields over an infinitesimal period of time, the law of energy conservation requires, after neglecting losses dEmech = d (δE) + dEelectric.

(9)

As in the case of eccentricity [14], it is assumed that the electric energy output from the generator is independent of the air-gap variations, thus assume that 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 df . Then, the virtual work done by this force is df dg = − d(δE), which gives df = −

d (δE) . dg

(10)

In Eq. (8), note that, since dV = r dr dz dϕ, the potential energy δE will increase if r increases when g is constant. This small change in df cannot 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 dr dz dϕ, where u0 = (r0 + s0 )/2, and dr = g. Eq. (8) then yields δE =

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

According to Eq. (10), the force df is given by 5

(11)

df = −

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

(12)

Hence, the total forces in the x− and y− directions can be expressed as fx =

1 2 μ0

fy =

1 2 μ0



2π

0

l0

B0 (z)2 g0 (z)2 u0 (z) cos ϕ dz dϕ, g (x, y, z, t, ϕ)2

(13)

l0

B0 (z)2 g0 (z)2 u0 (z) sin ϕ dz dϕ. g (x, y, z, t, ϕ)2

(14)

0

0





2π

 0

3.2 PROPERTIES OF THE UMP

The generator geometry and the B-field are now assumed constant in z through the generator length l0 . Numerical values from an 18 MW hydropower generator are used. The values are given in Table 1. These values are used in all simulations presented in this paper. Table 1 Numerical values from the 18 MW hydropower generator. s0

Average stator radius

2.775 m

l0

Length of the generator

1.18 m

g0

Average air-gap

0.0125 m

γ

Mass of the rotor

98165 kg

k

Stiffness of the axis

3.456 · 108 N/m

ω

Rotor rotation speed

14.2 rad/s

km

Magnetic stiffness

1.4715 · 108 N/m

μ0

Permeability of air

4 π · 10−7 Vs/Am

Fig. 4 shows UMP for the static case x = y = 0 with phase angles chosen to zero. Fig. 4(a) illustrates fx and fy for rotor and stator eccentricity, Fig. 4(b) shows fx for stator eccentricity together with δnr = 0.1 g0 , n = 1, 2, 3. Fig. 4(c) illustrates fx and fy for δ2r = δ3s = 0.3 g0, and Fig. 4(d) shows fx s for δ2r = δm = 0.3 g0, m = 1, . . . 7. Note that m even gives no UMP. For ideal circular generator geometry and y = 0, the integral in Eq. (13) can be solved analytically to yield 6

0

UMP (MN)

UMP (MN)

1 0.5 0 −0.5

−0.1

−0.2

−1 −1.5

−0.3

−2 −2.5 0

1

2

−0.4 0

3

1

2

πωt

(a)

(b) 0.5

UMP (MN)

0.6

UMP (MN)

3

πωt

0.4 0.2 0

0

−0.5

−0.2 −1 −0.4 −0.6 0

0.5

1

−1.5 0

1.5

0.5

πωt

1

1.5

πωt

(c)

(d)

Fig. 4. UMP as a function of time t; a) fx (dotted) and fy (solid), δ1s = δ1r = 0.3 g0 . b) fx for δ1s = δnr = 0.1 g0 , n = 1 (dotted), n = 2 (solid), n = 3 (dashdot). c) s = 0.3 g , m = 1 fx (dotted) and fy (solid), δ2r = δ3s = 0.3 g0 . d) fx for δ2r = δm 0 (dotted), m = 2 (solid), m = 3 (dashdot), m = 4 (dashed), m even (solid line at zero UMP).

fx =

cos ϕ x l0 g02 B02 u0  2π

3 . 2 dϕ = km 2 2 μ0 0 (g0 − x cos ϕ) 1 − xg2 2

(15)

0

Here, the magnetic stiffness, km , given in Table 1, is defined as

km =

π l0 B02 u0 . μ0 g0

(16)

Eq. (15) is similar to results obtained by Wang et al. [14] and Sandarangani [19]. For x ≤ 0.1 g0 ,the relative error is less than 2 % when approximating Eq. (15) with the linear function f L = km x. Therefore, a linear model of the UMP is interesting for small shape deviations and will be considered in Section 5.1. 7

4

THE EQUATION OF MOTION

The equation of motion for the forced Jeffcott rotor is non-autonomous and nonlinear and consists of two second order differential equations γ x¨ + c x˙ + k x = fx (x, y, t) , γ y¨ + c y˙ + k y = fy (x, y, t) .

(17)

Here, γ is the mass of the rotor, k is the stiffness of the rotor axis and c being a linear viscous damping. In the case of the linear UMP, discussed in Section 5.1, the damped natural frequency becomes ωd =

 k

− km − γ



c 2γ

2

≈ 44.6 rad/s.

(18)

The numerical values of k, km and γ can be found in Table 1. c is chosen to give the damping ratio ζ = 0.1. The angular velocity, ω, of the rotor is 14.2 rad/s through all simulations. In non-dimensional form, Eqs. (17) yields X  + 2 ζ X  + X = FX (X, Y, τ ) , Y  + 2 ζ Y  + Y = FY (X, Y, τ ) .

(19)

Here, the non-dimensional quantities X =

x , g0

Y =

y , g0

g (x, y, t, ϕ) , G = g0

δnr , g0

ΔR n =



Ω = ω

γ , k

ΔSm =

S δm , g0



τ = t

k , γ

(20)

(21)

have been introduced. The notation implies differentiation with respect to the non-dimensional time τ . The air-gap G, and the forces FX and FY yield

G = 1+

∞ 

FX =

cos ϕ km  2π dϕ, 2 π k 0 G (X, Y, τ, ϕ)2

(22)

FY =

sin ϕ km  2π dϕ, 2 π k 0 G (X, Y, τ, ϕ)2

(23)

s ΔSm cos m (ϕ + αm )−

m=1

∞ 

r ΔR n cos n (ϕ + αn − Ω τ )

n=1

− X cos ϕ − Y sin ϕ.

(24)

8

5

ANALYSIS

5.1 LINEAR MODEL OF THE UMP

Recall from Section 3.2 that the UMP is nearly linear for small shape deviations. A linearization of Eqs. (22) and (23) can be obtained using the Maclaurin series 1 ≈ 1 + 2 , (1 − )2

(25)

with = −

∞ 

s ΔSm cos m (ϕ + αm )+

m=1

∞ 

r ΔR n cos n (ϕ + αn − Ω τ )

n=1

+ X cos ϕ + Y sin ϕ.

(26)

From Eqs. (22) and (25), the linearization of FX (FXL ), can be expressed as FXL =

=

km 2πk



2π

(1 + 2 ) cos ϕ dϕ 0

km r X − ΔS1 cos α1s + ΔR cos (Ω τ − α ) . 1 1 k

(27)

Applying the same procedure in the Y −direction gives FYL =

km r Y + ΔS1 sin α1s + ΔR 1 sin (Ω τ − α1 ) . k

(28)

S From Eqs. (27) and (28) it is clear that only ΔR 1 and Δ1 (rotor and stator eccentricity), affect the UMP.

5.2 ROTOR OR STATOR ECCENTRICITY

For ideal circular generator geometry, a stable equilibrium in the center and a circle of unstable equilibria due to the nonlinear UMP exist in Eqs. (19). If stator eccentricity is added to this, i.e. ΔS1 > 0, these equilibria will move, and for a certain value of ΔS1 stability is lost. If rotor eccentricity is added instead, i.e. ΔR 1 > 0, there exists a circular limit cycle having periodicity equal to the driving frequency (synchronous whirling motion). Note that these kind of machines are always operating at undercritical conditions, recall Eq. (18). Therefore, when writing Eqs. (19) in polar coordinates 9

X = R cos Θ, Y = R sin Θ,

(29)

it is assumed that R = 0,

and

Θ =

⎧ ⎪ ⎨ Ω,

if ΔR 1 > 0

⎪ ⎩ 0,

if ΔS1 > 0.

(30)

With Eqs. (29) and (30), Eqs. (19) takes the form − R Θ 2 cos Θ − 2 ζ R Θ sin Θ + R cos Θ = FX , − R Θ 2 sin Θ + 2 ζ R Θ cos Θ + R sin Θ = FY .

(31)

With FR = FX cos Θ + FY sin Θ, this simplifies to



R 1 − Θ 2 = FR .

(32)

The term R Θ 2 corresponds to the centrifugal force. There is symmetry in the air-gap for both cases of eccentricity, and FR will point towards the shortest air-gap. Therefore, FR can be solved similar to Eq. (15), and the equation to solve yields



R 1 − Θ 2 =

R+Δ km

3 , k 2 2 1 − (R + Δ)

Δ =

⎧ ⎪ ⎨ ΔR ,

if ΔR 1 > 0

1

⎪ ⎩ − ΔS , 1

if ΔS1 > 0. (33)

Thus, to use Eq. (33), Δ should be replaced by − ΔS1 if stator eccentricity is considered, and by ΔR 1 if rotor eccentricity is considered. For zero eccentricity and the values given in Table 1, Eq. (33) has the solutions

R = 0

or

R =

 ± 1



−

km k (1 − Θ 2 )

2 3

≈

⎧ ⎪ ⎨ ± 0.6414, ⎪ ⎩ ± 0.6588,

if ΔR 1 > 0 if ΔS1 > 0. (34)

By varying the eccentricity in Eq. (33), bifurcation diagrams relating to Eqs. (19) can be found numerically. The phase angle Θ0 is introduced to give the direction of R, such that the negative solutions in R can be illustrated as positive with Θ0 = π. The bifurcation diagram for rotor eccentricity is shown in Fig. 5(a) and yields three limit cycles; C1 , C2 and C3 . C1 meets C2 at ΔR 1 ≈ 0.1658, only C1 is stable and is therefore of engineering interest. C1 is the resulting attractor corresponding to synchronous whirling motion. C3 corresponds to the phase Θ0 = π whereas C1 and C2 to Θ0 = 0. This means that C2 is the solution corresponding to the case when the rotor is moved in 10

the same direction as C1 , and C3 is the solution when the rotor is moved in the direction opposite to C1 . Thus, C3 will exist for increasing ΔR 1 until the rotor hits the stator. The bifurcation diagram for stator eccentricity is shown in Fig. 5(b) and yields three equilibria; E1 , E2 and E3 . E1 meets E2 in a foldbifurcation at ΔS1 ≈ 0.1810, only E1 is stable and is therefore of engineering interest. E3 corresponds to phase Θ = 0 whereas E1 and E2 to Θ = π. This means that E2 is the solution corresponding to the case when the rotor is moved in the same direction as E1 , and E3 is the solution when the rotor is moved in the direction opposite to E1 . Thus, E3 will exist for increasing ΔS1 until the rotor hits the stator. Note that it can exist also other attractors in these cases. 1

1

E3

C3 0.8

0.6

C2

0.4

0.4

C1

0.2 0 0

E2

R

R

0.6

0.8

0.1

(a)

E1

0.2

0.2

0 0

0.3

ΔR 1

0.1

(b)

0.2

0.3

ΔS 1

Fig. 5. Bifurcation diagrams to System (19); a) ΔR 1 > 0 (rotor eccentricity). R is the radius of limit cycles. b) ΔS1 > 0 (stator eccentricity). R gives the location of equilibria.

5.3 STABLE EQUILIBRIUM AT THE ORIGIN

When the rotor rotates at the origin, the UMP is zero for certain shape deviations due to their geometry, recall Fig. 2. The UMP is zero if G is periodic in ϕ with a period of 2 π/q, q ∈ N and q ≥ 2. To see this, consider the UMP from each period of G. Since G is periodic and UMP is a function of only G, all q UMPs are identical. Therefore, the sum of the forces becomes zero if G passes two or more periods during one revolution. In these cases, only a moment in the z−direction can be induced by the electro-magnetic field. Fig. S 6 shows the generator for the two simple cases ΔR 2 > 0, Δ2 > 0 (to the left) R S R and Δ2 > 0, Δ3 > 0 (to the right). In the case Δ2 > 0, ΔS2 > 0, G is periodic with period π and, therefore, the UMP is zero. This can be clarified by drawing a line through the origin with an arbitrary slope (dashed line in S the figure). In the case ΔR 2 > 0, Δ3 > 0 the period of G is 2 π and there will be a resulting UMP.

11

S Fig. 6. The geometry of the generator for the two cases; ΔR 2 > 0, Δ2 > 0 (to the R S left) and Δ2 > 0, Δ3 > 0 (to the right).

Thus, if it is possible to find an integer q ≥ 2 such that 

2π G (0, 0, τ, ϕ) = G 0, 0, τ, ϕ + q



(35)

holds, the UMP is zero. Here, G is given by Eq. (24) with X = Y = 0. Since S Eq. (35) needs to hold ∀ ΔR n ≥ 0, ∀ Δm ≥ 0, n, m ∈ N , and ∀ τ ≥ 0, it reduces to 

cos n (ϕ +

αnr



− Ω τ ) = cos n ϕ +

αnr



cos m (ϕ +

s αm )

2π − Ωτ + , q

∀ n ∈ N R, (36)



= cos m ϕ +

s αm

2π + , q

∀m ∈ N . S

(37)

S is Here, N R is the set of all natural numbers such that ΔR n > 0, and N S the set of all natural numbers such that Δm > 0. From Eqs. (36) and (37); if N R is empty, Eq. (35) holds for m ≥ 2; if N S is empty, Eq. (35) holds for n ≥ 2. If both N S and N R are empty, Eq. (35) holds (ideal circular geometry). Otherwise, Eq. (35) holds if it is possible to find integers pn and pm such that

q =

n m = ≥ 2, pn pm

q ∈ N,

∀ n ∈ N R,

∀ m ∈ N S.

(38)

From Fig. 4(d), recall that fx is zero for m even, since n = 2. If Eq. (38) holds, the UMP is zero, and therefore, Eqs. (19) have an equilibrium at the origin. For small shape deviations this equilibrium is stable.

12

Noting that Σ is introduced to simplify the notation according to Σ =

∞ 

ΔSm +

m=1

∞ 

ΔR n < 1.

(39)

n=1

It is proved (see appendix for the proof) that the equilibrium is stable if √ √ 2 π + π2 + 4 π2 + 4 2πk < + . km (1 − Σ)3 (1 + Σ)3

(40)

Thus, if Eq. (38) is satisfied, Eqs. (19) have an equilibrium at the origin. This equilibrium is stable if Eq. (40) holds. For the specific machine considered in Table 1, that is, if Σ < 0.16385.

6

SIMULATED RESULTS

Note that the numerical values used in all simulations presented in this paper are taken from an existing 18 MW hydropower generator. These values are given in Table 1. Eqs. (19) is simulated using a 4th order Runge-Kutta method. The force integrals given by Eqs. (22) and (23) are solved numerically at each step by Simpsons integrating method. For fixed parameters and different initial positions in the XY −plane, the trajectory is examined to see if it converges to an attractor without any impacts between the rotor and stator. The initial velocities are set to zero. With this method, an approximation is obtained of the two-dimensional XY −subspace of the basin of attraction to attractors without impacts between the rotor and stator. This approximation is denoted AXY . The XY −plane is covered by a uniform grid of points. The simulation continues until the rotor hits the stator, comes close to an equilibrium, or reaches a time limit. The condition for reaching an equilibrium was set to (X  )2 + (Y  )2 < 10−9 . The time limit was set to 100 revolutions of the rotor. If the time limit or an equilibrium is attained, the corresponding initial position is said to be converging (otherwise diverging), and is added to AXY . Since the shape of AXY can be complicated, both the size and the shape of AXY are considered. To measure the size, let nXY be the number of elements in AXY , and to measure the shape, let dXY be the distance from the origin to the diverging point closest to the origin. This means that dXY gives the radius of the largest circle, centred at the origin that is covered by AXY . Then, to scale these measures, define

13

NXY =

nXY , n0XY

DXY =

dXY , 0 dXY

(41)

0 where n0XY and dXY correspond to the case of ideal circular generator geom0 = 0.6588. etry. From Eq. (34), dXY

Figs. 7 - 9 show NXY and DXY as a function of different shape deviations. s The uniform grid is chosen to give n0XY = 1225. The phase angles αnr and αm are chosen due to symmetry. One simulation is needed for each phase angle combination. The values of nXY and dXY is taken as the worst case for each S point in the ΔR n Δm −plane. Six phase angle combinations are considered in Figs. 7 - 9. 1 0.15

ΔS 1

ΔS 1

0.15

0.1

0.8 0.6

0.1

0.4 0.05

0.05

0.2

0 0

0.05

(a)

0.1 ΔR 1

0 0

0.15

0.05

0.1

ΔR 1

0.15

0

(b)

1 0.8

0.3

ΔS 2

ΔS 2

0.3

0.2

0.6 0.2 0.4 0.1

0.1

0

0

0.1

(c)

0.2 ΔR 2

0 0

0.3

0.2

0.1

0.2 ΔR 2

0.3

0

(d)

S Fig. 7. a) NXY and b) DXY as a function of ΔR 1 , Δ1 (rotor and stator eccentricity). S R c) NXY and d) DXY as a function of Δ2 , Δ2 .

14

0.5

1

0.4

0.4

0.8

0.3

0.3

0.6

0.2

0.2

0.4

0.1

0.1

0.2

ΔS 3

ΔS 3

0.5

0 0

0.1

(a)

0.2 ΔR 2

0 0

0.3

0.1

0.2

0.3

0

ΔR 2

(b) 1 0.3

ΔS 2

ΔS 2

0.3

0.2

0.8 0.6

0.2

0.4 0.1

0.1

0 0

0.1

0.2

(c)

0.3 ΔR 3

0.4

0 0

0.5

0.2

0.1

0.2

0.3

0.4

0

ΔR 3

(d)

0.5

0.5

1

0.4

0.4

0.8

0.3

0.6

0.2

0.2

0.4

0.1

0.1

0.2

ΔR 3

ΔR 3

S Fig. 8. a) NXY and b) DXY as a function of ΔR 2 , Δ3 . c) NXY and d) DXY as a S. function of ΔR , Δ 3 2

0.3

0 0

0.05

0.15

0.05

0.1

0.15

0

ΔR 1

(b)

0.5

0.5

1

0.4

0.4

0.8

0.3

0.3

0.6

0.2

0.2

0.4

0.1

0.1

0.2

0 0

ΔS 3

ΔS 3

(a)

0.1 ΔR 1

0 0

0.05

0.1 ΔS 1

0 0

0.15

(c)

0.05

0.1 ΔS 1

0.15

0

(d)

R Fig. 9. a) NXY and b) DXY as a function of ΔR 1 , Δ3 . c) NXY and d) DXY as a S S function of and Δ1 , Δ3 .

15

1

1

0.8

0.8

DXY

NXY

Fig. 10 also shows NXY and DXY as a function of different shape deviations, though as a function of one parameter at a time. The symmetry therefore allows the phase angles to be chosen to zero. The uniform grid is refined to give n0XY = 13637. Deviations of the rotor are considered in Fig. 10(a,b), and deviations of the stator in Fig. 10(c,d).

0.6 0.4

0.4

0.2

0.2

0 0

0.1

0.2

(a)

0.3

0.4

0 0

0.5

ΔR n

1

1

0.8

0.8

0.6

0.4

0.2

0.2 0.1

0.2

0.3

ΔS m

0.2

0.3

0.4

0.5

0.3

0.4

0.5

ΔR n

0.6

0.4

0 0

0.1

(b)

DXY

NXY

0.6

0.4

0 0

0.5

(c)

0.1

0.2

ΔS m

(d)

Fig. 10. NXY and DXY as a function of; a) ΔR n , n = 1 (dotted), n = 2 (dashdot), n = 3 (solid) and n = 4 (dashed). b) ΔSm , m = 1 (dotted), m = 2 (dashdot), m = 3 (solid) and m = 4 (dashed).

Figs. 11 and 12 shows AXY and the corresponding attractor(s) for different 0 = shapes of the rotor and stator. In the figures the circle with radius dXY 0.6588 is included to indicate the boundary of AXY for the ideal circular generator. The phase angles are chosen to zero and the uniform grid is chosen to give n0XY = 13637. In Figs. 7 - 10, the location of some of these illustrations S can be found by using the values of ΔR n and Δm for each case. Deviations of the rotor are considered in Fig. 11(a-d), and deviations of the stator in Fig. S 11(e-h). Fig. 12 shows some cases of ΔSm and ΔR n with and without Δ1 (stator eccentricity), and also two cases of very small AXY .

16

0.5

Y

Y

0.5

0

−0.5

0

−0.5

−0.5

0

0.5

−0.5

0

(a)

(b) 0.5

Y

Y

0.5

0

−0.5

0

−0.5

−0.5

0

0.5

−0.5

0

X

(d) 0.5

Y

0.5

Y

0.5

X

(c)

0

−0.5

0

−0.5

−0.5

0

0.5

−0.5

0

X

0.5

X

(e)

(f) 0.5

Y

0.5

Y

0.5

X

X

0

−0.5

0

−0.5

−0.5

0

0.5

−0.5

0

X

0.5

X

(g)

(h)

Fig. 11. AXY and the corresponding attractor for some cases of ΔR n > 0 , and ΔSm > 0 together with the boundary of AXY for the ideal circular generator. a) R R R S ΔR 1 = 0.15. b) Δ2 = 0.30. c) Δ3 = 0.40. d) Δ4 = 0.40. e) Δ1 = 0.16. f) ΔS2 = 0.35. g) ΔS3 = 0.40. h) ΔS4 = 0.40.

17

0.5

Y

Y

0.5

0

0

−0.5

−0.5

−0.5

0

0.5

−0.5

0

(a)

(b) 0.5

Y

Y

0.5

0

−0.5

0

−0.5

−0.5

0

0.5

−0.5

0

X

(d) 0.5

Y

0.5

Y

0.5

X

(c)

0

−0.5

0

−0.5

−0.5

0

0.5

−0.5

0

X

0.5

X

(e)

(f) 0.5

Y

0.5

Y

0.5

X

X

0

−0.5

0

−0.5

−0.5

0

0.5

−0.5

0

0.5

X

X

(h)

(g)

Fig. 12. AXY and the corresponding attractor for some cases of shape deviations together with the boundary of AXY for the ideal circular generator. a) S R S S R S ΔR 2 = Δ2 = 0.19. b) Δ2 = Δ2 = 0.19, Δ1 = 0.01. c) Δ2 = Δ3 = 0.20. S = 0.20, ΔS = 0.01. e) ΔR = ΔS = 0.24. f) ΔR = ΔS = 0.24, d) ΔR = Δ 2 3 1 3 2 3 2 R S S ΔS1 = 0.03. g) ΔR 2 = 0.33, Δ3 = 0.08. h) Δ2 = 0.32, Δ2 = 0.12.

18

7

DISCUSSION AND CONCLUSIONS

In this paper, a mathematical model consisting of a Fourier series representation is developed to describe an arbitrary non-circular shape of the rotor and the stator. Since the length of these generators is relatively small, all parameters are considered constant in the z-direction for simplicity. In Section 3.1, the UMP is derived through the law of energy conservation. The generator is treated as a continuum. This approximation can be done since the number of poles in hydropower generators are high (the generator considered in this paper has 44 poles). This method is used since it works for an arbitrary disturbed air-gap. Fig. 4 illustrates the complexity of the UMP even for simple shape deviations. A linear model of the UMP is proposed in Section 5.1, since the nonlinear UMP derived in Section 3.1 has an almost linear behaviour for small eccenS tricities. When considering the linear model only ΔR 1 and Δ1 (rotor and stator eccentricity), affect the UMP and, therefore, the dynamics in the electric machine. Recall Eqs. (27) and (28). This result differs strongly from the nonlinear UMP, see Fig. 4(b), indicating the importance of considering the nonlinear effects. The results from Eqs. (27) and (28) can be understood by observing that the rotor/stator eccentricity is the only case where the deviation moves the geometrical center of the rotor/stator, recall Fig. 2. Thus, there will be a constant force with stator eccentricity, and an oscillating force with rotor eccentricity. In Section 5.2, the stability due to eccentricity is analysed. In the case of small stator eccentricity, one stable equilibrium exists. Following this equilibrium for increasing eccentricity, it is shown that the stability is lost in a fold bifurcation at ΔS1 ≈ 0.1810. In the case of rotor eccentricity, a stable limit cycle exists. Following this cycle for increasing eccentricity, it is shown that the stability is lost at ΔR 1 ≈ 0.1658, See Fig. 5. This shows, when assuming that no other stable attractors exist, that the generator can not operate without rotor-stator contact if rotor or stator eccentricity exceeds these values. The assumption above is strengthen by the simulations in Section 6. Fig. 7(a) agrees with these maximum values of eccentricity. In Section 5.3, the UMP is proven to be zero for shape deviations according to Eq. (38), i.e. for these cases there exists an equilibrium at the origin. This result is general for electric machines if the UMP is assumed to be a function of only the air-gap width. It is proven that this equilibrium is stable if Σ < 0.16385 for the generator considered. Note that if all shape perturbation parameters are known for a machine, Σ can easily be calculated by adding these parameters according to Eq. (39). By assuming that this equilibrium is 19

reached, this indicates that shape deviations satisfying Eq. (38) are preferable compared to other deviations for a generator. This assumption is strengthen by the simulations in Section 6, showing that in all tested cases, the trajectory reaches the origin. Simulations of the basins of attraction are carried out in Section 6. From these simulations the importance of the shape deviations can be studied for the generator in question. To approximate the robustness of the safe solutions of a shape perturbed generator, both the size and the shape of AXY , (recall that this is the approximation of the two-dimensional XY −subspace of the basin of attraction) will be approximated. It is advantageous if AXY is large and convex. (Recall that the measure of the size is NXY , and the measure of the shape is DXY , see Eq. (41) for the definitions). From Fig. 7(a) it is concluded that the effect of rotor and stator eccentricity, is nearly similar on NXY . From Fig. 7(b) it can be seen that the same holds for DXY . Hence, rotor and stator eccentricity affects the robustness nearly similar. Fig. 7(c,d) shows some differences between the rotor and stator for the case ΔR 2 > 0, S ΔS2 > 0, note some strange nonlinear effects near ΔR = 0.25, Δ = 0.15. 2 2 S > 0, Δ > 0 and Fig. 8 illustrates some differences between the two cases ΔR 2 3 S > 0, Δ > 0. Note from Fig. 8(a,b) and Fig. 7(c,d) the strange behaviour ΔR 3 2 for large stator perturbations and 0 < ΔR 2 < 0.05. Similarities between the rotor and stator are shown in Fig. 9, where the same shape deviations are considered for the rotor in Fig. 9(a,b) and for the stator in Fig. 9(c,d). Fig. 9(a,b) shows rotor eccentricity combined with ΔR 3 > 0 whereas Fig. 9(c,d) S shows stator eccentricity combined with Δ3 > 0. Figs. 7 - 10 all illustrate that the effect of the shape deviations to NXY and DXY decreases when m and n increases. Thus, assuming the same amount of deviation, this shows that S eccentricity, i.e. ΔR 1 and Δ1 , are more severe than deviations corresponding to higher m and n. Sharp knees are observed near ΔR 1 = 0.05 and near ΔS1 = 0.05 in Fig. 10, meaning that deviations less than 5% of the air-gap of the generator only marginally affect the robustness. Moreover, it is of interest to discuss AXY . Complicated AXY occurs in some cases shown in Fig. 12. This illustrates the complexity of the dynamics; therefore, both NXY and DXY has to be considered for investigating how robust different shape deviations are in these cases. Simple periodic attractors occur in some cases, but more complicated attractors also exists. See Fig. 12(b-h). In Fig. 12(b) there are at least two attractors present, with very long period or chaotic behaviour, and in Fig. 12(h), there are two attractors present. This also indicates the existence of multiple solutions. Note that ω = 14.2 rad/s was used in all simulations. Since ω is not close to ωd , recall Eq. (18), and since rotor eccentricity will give synchronous whirling motion, the rotor is not close to resonance in the case of rotor eccentricity. Thus, the results presented in Section 6, showing that the cases of rotor eccentricity being the worst in the case of robustness, cannot be trivially explained 20

by excitations of the damped natural frequency. This can also be understood by observing that stator eccentricity affects the robustness nearly similar to rotor eccentricity, and stator eccentricity gives only a stationary point. Since UMP can cause large vibrations in hydropower generators which can destroy the machine, the shape of the rotor and stator is frequently measured during maintenance. The results from this paper can be used to evaluate such measurements and estimate the stability and robustness through simulations. When using the mathematical methods presented in this paper on a real machine, the unbalance and the dominating shape perturbation parameters has to be included. Then a simulation of the robustness can be carried out, where NXY and DXY will be found for the generator considered. Many results presented in this paper are general and can be applied to different electric motors and generators. This paper indicates which tolerances are more important than others when constructing new machines.

8

ACKNOWLEDGEMENT

Elforsk AB and the Swedish Energy Authority by the Elektra research program are acknowledged for the financial support of this project.

APPENDIX

A ; PROOF OF EQUATION (38)

The eigenvalues of the Jacobian matrix to Eqs. (19) at the origin yield, λ1,2 = − ζ + λ3,4 = − ζ −



ζ 2 + α ± β,



ζ 2 + α ± β,

(A.1)

where 1 α = 2



∂FY ∂FX + ∂X ∂Y

 



− 1,

1 ∂FX ∂FY β = − 2 ∂X ∂Y

2

+4

∂FX ∂FY . ∂Y ∂X (A.2)

All derivatives are hereinafter evaluated at the origin. Clearly Re λ3,4 < 0. From Eqs. (22) and (23), it is concluded that 21

∂FX ∂FY = . ∂Y ∂X

(A.3)

Therefore, β > 0 and real, and it follows that Re λ1,2 < 0 if and only if α + β < 0.

(A.4)

The derivatives of the force integrals have the following bounds, L
0 and δ1s > 0 will correspond to rotor eccentricity and stator eccentricity respectively. Since dynamic eccentricity is normally small compared to the dimensions of the generator, 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 . Eqs. (1), (2) and (4) gives, after adding the ω rotation, g = g0 (z) +

∞ 

s s δm (z) cos{m (ϕ + αm (z))} −

m=1

∞ 

δnr (z) cos{n (ϕ + αnr (z) − ω t)}

n=1

− x cos ϕ − y sin ϕ.

(5)

The geometric model is now completed. III.

UNBALANCED MAGNETIC PULL

Based on the theory of magnetic field [9], the potential energy reserved in the air-gap can be expressed as  B (x, y, z, t, ϕ)2 E = dV, (6) 2 μ0 all space where B is the magnetic flux density (also called the B-field) in the air-gap and μ0 is the permeability of air. For an approximation, the relations between the B-field and the air-gap widths are assumed as in [3], B =

B0 (z) g0 (z) . g (x, y, z, t, ϕ)

(7)

B0 is the uniformly distributed B-field for a perfect circular geometry, i.e. g = g0 . Next, consider a volume element dV as shown in Fig. 3. According to Eqs. (6) and (7), the potential energy (δE), reserved in dV is given by δE =

B0 (z)2 g0 (z)2 dV. 2 μ0 g (x, y, z, ϕ, t)2 3

(8)

r

dV ϕ Cs

FIG. 3: The volume element dV .

Eq. (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 increments of mechanical energy input to dV and dEelectric the electric energy output from dV . When considering the energy conversion between the magnetic and mechanical fields over an infinitesimal period of time, the law of energy conservation requires, after neglecting losses dEmech = d (δE) + dEelectric .

(9)

As in the case of eccentricity [3], it is assumed that the electric energy output from the generator 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 df . Then, the virtual work done by this force is df dg = − d(δE), which gives df = −

d (δE) . dg

(10)

In Eq. (8), note that, since dV = r dr dz dϕ, the potential energy δE will increase if r increases when g is constant. This small change in df cannot 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 dr dz dϕ, where u0 = (r0 + s0 )/2, and dr = g. Eq. (8) then yields B0 (z)2 g0 (z)2 u0 (z) dz dϕ. 2 μ0 g (x, y, z, t, ϕ)

δE =

(11)

According to Eq. (10), the force df is given by d B0 (z)2 g0 (z)2 u0 (z) dz dϕ. (δE) = dg 2 μ0 g (x, y, z, t, ϕ)2

df = −

Hence, the total forces in the x− and y−direction can be expressed as  2π  l0 B0 (z)2 g0 (z)2 u0 (z) 1 fx = cos{ϕ} dz dϕ, 2 μ0 0 g (x, y, z, t, ϕ)2 0

fy =

1 2 μ0

 0

2π

 0

l0

B0 (z)2 g0 (z)2 u0 (z) sin{ϕ} dz dϕ. g (x, y, z, t, ϕ)2 4

(12)

(13)

(14)

The generator geometry and the B-field are from now on assumed constant in z 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 g02 B02 u0 2π cos ϕ x (15) fx =

32 . 2 dϕ = km 2 μ0 2 (g − x cos ϕ) 0 0 1 − xg2 0

Here, the magnetic stiffness (km ), is defined as km = π l0 B02 u0 / (μ0 g0 ). Eq. (15) is similar to results obtained by Wang et al. [3] and Sandarangani [10]. IV.

EQUATION OF MOTION

The equation of motion for the forced Jeffcott rotor is non-autonomous and nonlinear and consists of two second order differential equations γ x¨ + c x˙ + k x = fx (x, y, t) , γ y¨ + c y˙ + k y = fy (x, y, t) .

(16)

Here, γ is the mass of the rotor, k is the stiffness of the rotor axis and c being a linear viscous damping. In non-dimensional form, System (16) yields X  + 2 ζ X  + X = FX (X, Y, τ ) , Y  + 2 ζ Y  + Y = FY (X, Y, τ ) .

(17)

Here, the non-dimensional quantities X =

x , g0

Y =

y , g0

δnr , g0

ΔR n =



g (x, y, t, ϕ) G = , g0

Ω = ω

ΔSm =

S δm , g0

(18)

 γ , k

τ = t

k , γ

(19)

have been introduced and τ is a non-dimensional time. The air-gap G, and the forces FX and FY yield  2π cos ϕ km dϕ, (20) FX = 2 π k 0 G (X, Y, τ, ϕ)2

FY =

G = 1+

∞ 

km 2πk

 0

2π

sin ϕ dϕ, G (X, Y, τ, ϕ)2

s ΔSm cos{m (ϕ + αm )} −

m=1

∞ 

(21)

r ΔR n cos{n (ϕ + αn − Ω τ )}

n=1

− X cos ϕ − Y sin ϕ.

(22)

5

V.

PROPERTIES OF THE UMP

To get understanding about the responses from the UMP due to different shape deviations, properties of the UMP will be investigated in this section. A.

SIMULATIONS

The angular frequency of the UMP is given by Θ =

FY FX − FX FY . FX2 + FY2

(23)

Here, the prime represents differentiation with respect to time τ , FX and FY are given by Eqs. (20) and (21). The MATLAB function quad was used to evaluate the integrals. Numerical values used are from an 18 MW hydropower generator and are given in Table I. TABLE I: Numerical values from the 18 MW hydropower generator. s0 l0 g0 γ k ω km μ0

Average stator radius Length of the generator Average air-gap Mass of the rotor Stiffness of the axis Rotor rotation speed Magnetic stiffness Permeability of air Number of poles

2.775 m 1.18 m 0.0125 m 98165 kg 3.456 · 108 N/m 14.2 rad/s 1.4715 · 108 N/m 4 π · 10−7 Vs/Am 44

Fig. 4 shows Θ / Ω for a centred rotor (X = Y = 0) for the shape perturbations; ΔSm = R S 0.05, ΔR n = 0.1 (dashed) and Δm = 0.1, Δn = 0.05 (solid). Three mn-combinations are considered; [m, n] = [2, 3] giving a positive result, [3, 2] a negative and [1, 1] giving an alternating result. 4 3

Θ/Ω

2 1 0 −1 −2 −3 0

1

2

3

Ωτ

4

5

6

R S R FIG. 4: The angular frequency ratio Θ / Ω for ΔS m = 0.05, Δn = 0.1 (dashed) and Δm = 0.1, Δn = 0.05 (solid). Three mn-combinations are considered, [m, n] = [2, 3] (positive), [3, 2] (negative) and [1, 1] (alternating).

6

Fig. 5(a) shows the result from a simulation of the largest value of the UMP during one revolution of the rotor when X = Y = 0, i.e.   2 2 max FX + FY . (24) Ω τ ∈ [0, 2π]

m

The shape perturbations are ΔSm = ΔR n = 0.3. Fig. 5(b) shows the generator geometry for the case ΔS10 > 0, ΔR 9 > 0. From this figure, it can be seen that if m − n = ± 1, several minimum of the air-gap will occur near the absolute minimum, and therefore, larger amplitudes of the UMP will occur in these cases. Compare to the result presented in Fig 5(a). The same argument can explain the larger amplitudes on the lines 2 m − n = ± 1 and m − 2 n = ± 1, which also can be observed in Fig 5(a). By studying Fig. 5(b) it can be realised that there exists mn-combinations giving no UMP, for example the cases m = n. All such combinations was derived by Lundstr¨om and Aidanp¨a¨a in [8]. 10 9 8 7 6 5 4 3 2 1

0.4 0.3 0.2 0.1

1

2

3

4

5

6

7

8

9 10

n (a) The maximum amplitude for m, n ∈ [1, 10].

FIG. 5:

(b) The generator geometry for R ΔS 10 > 0, Δ9 > 0.

The amplitude of the UMP.

B.

ANALYSIS

In this section, a theorem describing how the average angular frequency and amplitude of the UMP depends on different shape deviations of the generator will be proved. Consider the air-gap given by Eq. (22) including the N first rotor perturbation parameters ΔR n and the M first stator perturbation parameters ΔSm , i.e. G = 1+

M 

s ΔSm cos{m (ϕ + αm )} −

m=1

N 

r ΔR n cos{n (ϕ + αn − Ω τ )}

n=1

− X cos ϕ − Y sin ϕ.

(25)

The forces FX and FY given by Eqs. (20) and (21) can be analysed using the Maclaurin series 1 = 1 + 2 + 3 2 + . . . + (q + 1) q + . . . , (1 − )2

(26)

with = −

M 

s ΔSm cos{m (ϕ + αm )} +

m=1

N 

r ΔR n cos{n (ϕ + αn − Ω τ )}

n=1

+ X cos ϕ + Y sin ϕ.

(27) 7

From Eqs. (21) and (26), FY can be expressed as  2π km FY = (2 + 3 2 + . . . + (q + 1) q + . . . ) sin{ϕ} dϕ 2πk 0 = FY1 + FY2 + . . . + FYq + . . .

(28)

From Eq. (27) and the multinomial theorem, FYq , q ∈ N yields, with κ = km (q + 1)/(2 π k),  2π FYq = κ q sin{ϕ} dϕ 0   2π   q! j1 jA (29) = κ a . . . aA sin{ϕ} dϕ, j ! . . . jA ! 1 0 j +···+j =q 1 1

A

where A = N + M + 2 are the total number of terms in and r an = ΔR n cos{n (ϕ + αn − Ω τ )}, s )}, aN +m = ΔSm cos{m (ϕ + αm aA−1 = X cos ϕ, aA = Y sin ϕ.

n = 1 . . . N,

m = 1 . . . M, (30)

To proceed, the following assumptions has to be made; One rotor perturbation parameter and S one stator perturbation parameter is dominating. Call these perturbation parameters ΔR n and Δm respectively. All other perturbation parameters together with X and Y are assumed to be sufficiently S small compared to ΔR n and Δm such that     X  + Y  +

M  m = 1 m = m

ΔSm +

N 

ΔR ΔR 1 . This can be seen from Theorem V.1 when considering FX and FY . The response will in this case be a synchronous whirling orbit, independent whether Θ is alternating or not. Fig. 7 in Section VI illustrates periodic attractors for 60 cases of shape perturbations. In all cases, a comparison to Theorem V.1 gives a satisfactory result. To mention a few; All whirling frequencies 13

satisfies Table II. Attractors corresponding to n = 1 has a large amplitude, see Figs. 7 (a), (b), (c), (d), (g), (j), (m). Attractors to geometries having an mn-combination on the lines m − n = ± 1, i.e. in Figs. 7 (e), (h), (i), (k), (l) shows larger amplitudes than solutions in Figs. 7 (f), (n), (o). In Figs. 7 (e), (h), (i), (k), (l) the amplitudes are nearly independent of whether ΔSn or ΔR n are larger, but in Fig. 7 (f), the case of larger ΔS2 gives the largest amplitude, and in Figs. 7 (n), (o), the case of a larger ΔR n give the largest amplitudes. These simulations show that attractors behave according to the properties of the UMP proved in Theorem V.1, and also point out that the results from the theorem hold for some cases of shape deviations even if Eq. (31) is not satisfied. From Fig. 8, Section VI, it is seen that resonance occurs at different driving frequencies due to the shape perturbation considered. This is because for a certain mn-combination of perturbation, the UMP has a corresponding angular frequency according to Theorem V.1. Compare the resuls from the theorem to Fig. 8 and note from Eq. (44) that the damped natural frequency, ωd ≈ 44.6 rad/s. Observe that the curve corresponding to [m, n] = [3, 2] has two peaks. The larger peak is explained by considering FX2 and FY2 , while the smaller one can be explained by considering FX3 and FY3 in Eqs. (40) and (41). In this paper shape deviations are analysed of which one rotor perturbation parameter and one stator perturbation parameter are dominating. Other perturbation parameters together with X and Y are assumed sufficiently small. Otherwise the problem becomes complicated and a more general approach is difficult. Therefore, combined shape deviations should be considered by studying the behaviour of a real generator with a known generator shape. VIII.

CONCLUSIONS

The amplitude and the average angular frequency of the UMP due to generator shape are found analytically for certain cases of shape deviations. The results are proved mathematically and are presented in Theorem V.1. These results explain that different whirling frequencies, both backward and forward whirling, can occur in large synchronous generators due to deviations in the generator shape. Simulations of trajectories using a Jeffcott rotor model show good agreement between the whirling and amplitude of the response and the properties of the UMP given in Theorem V.1. The resulting periodic solutions can be complicated for some geometries. See Fig. 7 (h). The results in this paper clarifies which whirling motion and amplitude that can be expected in a machine with a given shape deviation. Since the angular frequency of the UMP is dependent of the shape deviations, different drive frequencies can give resonance. See Fig. 8. Since UMP can cause large vibrations in hydropower generators which can destroy the machine, the shape of the rotor and stator is frequently measured during maintenance. The results from this paper can be used to evaluate such measurements and indicate which tolerances are more important than others when constructing new machines. IX.

ACKNOWLEDGEMENT

Elforsk AB and the Swedish Energy Authority by the Elektra research program are acknowledged for the financial support of this project.

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[1] P. Talas, P. Toom, Dynamic Measurement and Analysis of Air-Gap Variations in Large Hydroelectric Generators. IEEE, 83 WM 226-8, 1983, 9p [2] D. Guo, F. Chu and D.Chen, The Unbalanced Magnetic Pull and its Effects on Vibration in a Three-Phase Generator With Eccentric Rotor. Journal of Sound and Vibration, v 254, n 2, 4 July 2002, p. 297-312. [3] Y. Wang, G. Sun and L. Huang, Magnetic Field-induced Nonlinear Vibration of an Unbalanced Rotor. ASME, Design Engineering Division (Publication) DE, v 116, n 2, Proceedings of the ASME Design Engineering Division - 2003 Volume 2, 2003, p 925-930. [4] T. P. Holopainen, A. Tenhunen, A. Arkkio, Electromechanical Interaction in Rotordynamics of Cage Induction Motors. Journal of Sound and Vibration, v 284, n 3-5, Jun 21, 2005, p. 733-755. [5] S. Williamson, M. A. S. Abdel-Magied, Unbalanced Magnetic Pull in Induction Motors With Asymmetrical Rotor Cages. IEE Conference Publication, n 254, p. 218-222, 1985. [6] V. A. Tereshonkov, Magnetic Forces in Electric Machines with Air Gap Eccentricities and Core Ovalities. Elektrotekhnika, Vol. 60, No.9, pp.50-53, 1989. [7] L. Frosini, P. Pennacchi, Detection and Modelling of Rotor Eccentricity in Electric Machines - an Overview. C623/060/2004. IMechE, 2004. [8] N. L. P. Lundstr¨ om, J. -O. Aidanp¨ a¨ a Dynamic Consequences of Electromagnetic Pull due to Deviations in Generator Shape, Journal of Sound and Vibration. (Submitted) [9] R. K. Wangsness, Electromagnetic Fields, Hamilton Printing Company, ISBN 0-471-81186-6, 1986. [10] C. Sandarangani, Electrical Machines Design and Analysis of Induction and Permanent Magnet Motors. Royal Institute of Technology, Stockholm, 2000-08-07.

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