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The paper presents a mathematical multibody model of a soft mounted induction motor with sleeve bearings regarding forced vibrations caused by dynamic rotor eccentricities considering electromagnetic field damping. The multibody model contains the mass of the stator, rotor, shaft journals and bearing housings, the electromagnetic forces with respect of electromagnetic field damping, stiffness and internal (rotating) damping of the rotor, different kinds of dynamic rotor eccentricity, stiffness and damping of the bearing housings and end shields, stiffness and damping of the oil film of the sleeve bearings and stiffness and damping of the foundation. With this multibody model, the bearing housing vibrations and the relative shaft vibrations in the sleeve bearings can be derived.

Fast running induction motors with high power ratings, ( P N > 1 MW ; n N ≥ 2900 rpm ) are often equipped with sleeve bearings, because of the high circumferential speed of the shaft journals, and are often mounted on soft foundations (

To guarantee a safe operation, the vibrations at the sleeve bearings are often monitored [

Increasing requirements in standards and specifications of electrical machines regarding vibration limits [

The three most important dynamic eccentricities for induction motors―eccen- tricity of rotor mass, bent rotor deflection and magnetic eccentricity―are here considered in the paper (

・ Eccentricity of rotor mass e ^ u which is e.g. caused by residual unbalance, which remains after the balancing process.

・ Bent rotor deflection a ^ , which is e.g. caused by thermal bending of the rotor.

・ Magnetic eccentricity e ^ m , which is e.g. caused by deviation of concentricity between the inner diameter of the rotor core and the outer diameter of the rotor core. The so caused mechanical unbalance is compensated by a placed unbalance, so that the centre of rotor mass U is not displaced from the rotation axis.

If the magnetic centre M of the rotor is displaced from the centre of the stator bore (

These electromagnetic forces act on the rotor but in opposite direction also at the stator. For forced vibration caused by dynamic rotor eccentricity the whirling angular frequency ω F is equal to the rotational angular frequency Ω :

ω F = Ω (1)

Referring to [

p > 1 : { c m d = c m 2 ⋅ ( α p + 1 + α p − 1 ) d m = − 1 ω F ⋅ c m 2 ⋅ ( δ p + 1 − δ p − 1 ) with : c m = π ⋅ R ⋅ l 2 ⋅ μ 0 ⋅ δ ″ ⋅ B ^ p 2 (2)

p = 1 : { c m d = c m ⋅ α p + 1 d m = − 1 ω F ⋅ c m ⋅ δ p + 1 with : c m = 1 2 ⋅ π ⋅ R ⋅ l 2 ⋅ μ 0 ⋅ δ ″ ⋅ B ^ p 2 (3)

The constant c m describes the magnetic spring constant, without electromagnetic field damping, l the length of the core, R the radius of the stator bore, μ 0 the permeability of air, δ ″ the equivalent magnetic air gap width, B ^ p the amplitude of fundamental air gap field, α p + 1 and α p − 1 the real parts and δ p + 1 and δ p − 1 the imaginary parts of the complex field damping value. For 2-pole motors ( p = 1 ) the components α p − 1 and δ p − 1 do not exist, neglecting the homopolar flux. Without electromagnetic field damping, the field damping coefficients become [

α p + 1 = α p − 1 = 1 ; δ p + 1 = δ p − 1 = 0 (4)

With the ordinal number ν = p ± 1 for an eccentricity field wave, the electromagnetic field damping coefficients can be calculated as follows [

α ν = 1 − K ν ⋅ s ν 2 ; δ ν = − K ν ⋅ β ν ⋅ s ν with:

β ν = R 2 , ν ω 1 ( L 2 h , ν + L 2 σ , ν ) ; K ν = 1 β ν 2 + s ν 2 ⋅ ξ Schr , ν 2 ⋅ ζ K , ν 2 1 + L 2 σ , ν L 2 h , ν (5)

R 2 , ν presents the resistance of a rotor bar and ring segment, ω 1 the electrical stator angular frequency, L 2 h , ν the main field inductance of a rotor mesh, L 2 σ , ν the leakage inductance of a bar and ring segment, ξ Schr , ν the screwing factor and ζ K , ν the coupling factor. A very important parameter is here the harmonic slip s ν , which can be described by [

s ν = ω ν ν − Ω ω 1 ν with : Ω = ω 1 p ( 1 − s ) (6)

Here, s presents the fundamental slip of the induction motor, ω 1 the electrical stator angular frequency and ω ν / ν the angular frequencies of the eccentricity fields, depending on the kind of eccentricity:

・ Static eccentricity : ω ν = ω 1 ,

・ Dynamic eccentricity as a circular forward whirl: ω ν = ω 1 ± ω F ,

・ Dynamic eccentricity as a circular backward whirl: ω ν = ω 1 ∓ ω F .

In order to consider electromagnetic field damping by a simple magnetic spring element c m d and a simple magnetic damper element d m , the determination has to be made, that the calculation of c m d and d m is here only based on circular forward orbits [

s ν = s (7)

The vibration model is on the one side an enhancement of the model in [

Additional masses are the mass of the shaft journal m v and the mass of the bearing housing m b . The rotor, rotating with the rotary angular frequency Ω , presents a concentrated mass and has no inertia moments (no gyroscopic effect is considered).The movement of the shaft journal in the sleeve bearing is described by the shaft journal centre point V. The point B, which is positioned in the axial middle of the sleeve bearing shell, describes the movement of the bearing housing. The rotor mass is linked to the stator mass by the stiffness c and internal (rotating) damping d i of the rotor, the oil film stiffness matrix C_{v} and oil film damping matrix D_{v} of the sleeve bearings, which suppose to be equal for both sides, as well as the bearing house and end shield stiffness and damping matrix C_{b} and D_{b}. The stator structure can be defined to be rigid, compared to the soft foundation. The foundation stiffness matrix C_{f} and the foundation damping matrix D_{f} connect the stator feet, F_{L} (left side) and F_{R} (right side), to the ground. The foundation stiffness and damping on the right side and on the left side is identical and the foundation stiffness values c_{fy} and c_{fz} and the foundation damping values d_{fy} and d_{fz} are the values for each motor side.

The electromagnetism is considered by the electromagnetic spring and damper matrix C m and D m , where also electromagnetic field damping is included. Excitations are all three kinds of dynamic rotor eccentricity―eccentricity of rotor mass, bent rotor deflection and magnetic eccentricity―but are not pictured in

The oil film stiffness and damping coefficients c_{ij} and d i j ( i , j = z , y ) of the sleeve bearing can be calculated by solving the Reynolds differential equation [

c i j = c i j ( Ω ) ; d i j = d i j ( Ω ) (8)

The stiffness of the rotor c is constant. According to [

d i ( Ω ) = c ⋅ tan δ i Ω (9)

The same approach is used for the bearing housing with end shield and the foundation. The stiffness of the bearing housing with end shield ( c b z ; c b y ) and of the foundation ( c f z ; c f y ) is constant. The damping of the bearing housing with end shield ( d b z ; d b y ) and of the foundation ( d f z ; d f y ) can be again described by the mechanical loss factor of the bearing housing with end shield tan δ b and of the foundation tan δ f :

d b z ( Ω ) = c b z ⋅ tan δ b Ω ; d b y ( Ω ) = c b y ⋅ tan δ b Ω (10)

d f z ( Ω ) = c f z ⋅ tan δ f Ω ; d f y ( Ω ) = c f y ⋅ tan δ f Ω (11)

The electromagnetic stiffness coefficient c m d and damping coefficient d m are depending on the harmonic slip s ν , which is here equal to the fundament slip s , and on the whirling angular frequency ω F , which is here equal to the rotary angular frequency Ω . If the motor is converter driven, the angular rotor frequency Ω as well as the fundament slip s may variate arbitrarily. Therefore c m d and d m become:

c m d = c m d ( Ω , s ) ; d m = d m ( Ω , s ) (12)

The forces at the rotor mass, at the shaft journals, at the bearing housings and at the stator mass can be derived in the fixed coordinate systems

( y W , z W ; y V , z V ; y B , z B ; y S , z S ) (

( y r w , z r w ) in

deriving the equilibrium of forces and moments, for each single system.

Because of the small displacements of the stator mass ( z s , y s , φ s ) related to the dimensions of the machine ( h , b , Ψ ) , linearization is possible [

z f L = z s − φ s ⋅ b ; z f R = z s + φ s ⋅ b (13)

y f L = y f R = y s − φ s ⋅ h (14)

To derive the inhomogeneous differential equation system, each single system ―Figures 6(a)-(d)―has to be analyzed. In Figures 6(a)-(c) the equilibrium of forces in vertical direction (z-direction) and in horizontal direction(y-direction) has to be determined for each single system. In

M ⋅ q ¨ + D ⋅ q ˙ + C ⋅ q = f u + f a + f m (15)

Coordinate vector q :

q = [ z s ; z w ; y s ; y w ; φ s ; z v ; z b ; y v ; y b ] T (16)

Mass matrix M:

M = [ m s 0 0 0 0 0 0 0 0 0 m w 0 0 0 0 0 0 0 0 0 m s 0 0 0 0 0 0 0 0 0 m w 0 0 0 0 0 0 0 0 0 Θ s x 0 0 0 0 0 0 0 0 0 2 m v 0 0 0 0 0 0 0 0 0 2 m b 0 0 0 0 0 0 0 0 0 2 m v 0 0 0 0 0 0 0 0 0 2 m b ] (17)

Damping matrix D :

D = [ 2 ( d f z + d b z ) + d m − d m 0 0 − d m d m + d i 0 0 0 0 2 ( d f y + d b y ) + d m − d m 0 0 − d m d m + d i 0 0 − 2 d f y ⋅ h 0 0 − d i 0 0 − 2 d b z 0 0 0 0 0 0 − d i 0 0 − 2 d b y 0 0 0 − 2 d f y ⋅ h 0 2 ( d f y h 2 + d f z b 2 ) 0 0 0 0 0 − 2 d b z 0 0 − d i 0 0 0 0 0 0 − 2 d b y 0 0 − d i 0 0 0 0 0 2 d z z + d i − 2 d z z 2 d z y − 2 d z y − 2 d z z 2 ( d z z + d b z ) − 2 d z y 2 d z y 2 d y z − 2 d y z 2 d y y + d i − 2 d y y − 2 d y z 2 d y z − 2 d y y 2 ( d y y + d b y ) ] (18)

Stiffness matrix C :

C = [ 2 ( c f z + c b z ) − c m d c m d 0 0 c m d c − c m d 0 Ω d i 0 0 2 ( c f y + c b y ) − c m d c m d 0 − Ω d i c m d c − c m d 0 0 − 2 c f y h 0 0 − c 0 − Ω d i − 2 c b z 0 0 0 0 Ω d i 0 − c 0 0 − 2 c b y 0 0 0 − 2 c b z 0 0 0 − c 0 − Ω d i 0 − 2 c f y h 0 0 0 − 2 c b y 0 Ω d i 0 − c 0 2 ( c f y h 2 + c f z b 2 ) 0 0 0 0 0 2 c z z + c − 2 c z z 2 c z y + Ω d i − 2 c z y 0 − 2 c z z 2 ( c z z + c b z ) − 2 c z y 2 c z y 0 2 c y z − Ω d i − 2 c y z 2 c y y + c − 2 c y y 0 − 2 c y z 2 c y z − 2 c y y 2 ( c y y + c b y ) ] (19)

For the calculation of the forced vibrations, the complex form is used. Therefore the excitation vectors can be described as follows:

・ Mass eccentricity:

f u = f ^ u ⋅ e j ⋅ ( Ω ⋅ t + φ u ) (20)

・ Bent rotor deflection:

f a = f ^ a ⋅ e j ⋅ ( Ω ⋅ t + φ a ) (21)

・ Magnetic eccentricity

f m = f ^ m ⋅ e j ⋅ ( Ω ⋅ t + φ m ) (22)

with the amplitude vectors:

f ^ u = [ 0 e ^ u ⋅ m w ⋅ Ω 2 0 − j ⋅ e ^ u ⋅ m w ⋅ Ω 2 0 0 0 0 0 ] ; f ^ a = [ 0 a ^ ⋅ c 0 − j ⋅ a ^ ⋅ c 0 − a ^ ⋅ c 0 j ⋅ a ^ ⋅ c 0 ] ; f ^ m = [ − e ^ m ⋅ c m d e ^ m ⋅ c m d j ⋅ e ^ m ⋅ c m d − j ⋅ e ^ m ⋅ c m d 0 0 0 0 0 ] (23)

With the complex form for each particular excitation:

q κ = q ^ κ ⋅ e j ⋅ ( Ω t + φ κ ) ; κ = u , a , m (24)

the complex amplitude vector for each single excitation can be calculated by:

q ^ κ = [ − M ⋅ Ω 2 + D ⋅ j ⋅ Ω + C ] − 1 ⋅ f ^ κ ; with: q ^ κ = [ z ^ s , κ z ^ w , κ y ^ s , κ y ^ w , κ φ ^ s , κ z ^ v , κ z ^ b , κ y ^ v , κ y ^ b , κ ] = [ | z ^ s , κ | ⋅ e j ⋅ α z s , κ | z ^ w , κ | ⋅ e j ⋅ α z w , κ | y ^ s , κ | ⋅ e j ⋅ α y s , κ | y ^ w , κ | ⋅ e j ⋅ α y w , κ | φ ^ s , κ | ⋅ e j ⋅ α φ s , κ | z ^ v , κ | ⋅ e j ⋅ α z v , κ | z ^ b , κ | ⋅ e j ⋅ α z b , κ | y ^ v , κ | ⋅ e j ⋅ α y v , κ | y ^ b , κ | ⋅ e j ⋅ α y b , κ ] ; κ = u , a , m (25)

and each single solution can now be described by:

q u = q ^ u ⋅ e j ( Ω t + φ u ) ; q m = q ^ m ⋅ e j ( Ω t + φ m ) ; q a = q ^ a ⋅ e j ( Ω t + φ a )

Afterwards, all single solutions can be superposed:

q = ∑ κ = u , a , m q ^ κ ⋅ e j ( Ω t + φ κ ) (26)

Now, the vibration velocities of the bearing housings can be calculated for each single excitation [

Verticaldirection : v b , z , κ = Ω ⋅ | z ^ b , κ | (27)

Horizontaldirection: v b , y , κ = Ω ⋅ | y ^ b , κ | (28)

Again, the solutions can be superposed:

Vertical direction:

v b , z = Ω ⋅ | ∑ κ = u , a , m | z ^ b , κ | ⋅ e j ( φ κ + α z b , κ ) | (29)

Horizontal direction:

v b , y = Ω ⋅ | ∑ κ = u , a , m | y ^ b , κ | ⋅ e j ( φ κ + α y b , κ ) | (30)

Referring to [

r _ v − b , κ = r ^ _ v − b , κ + ⋅ e j ⋅ ( Ω t + φ κ ) + r ^ _ v − b , κ − ⋅ e − j ⋅ ( Ω t + φ κ ) with : r ^ _ v − b , κ + = 1 2 ⋅ { | z ^ v , κ | ⋅ e j ⋅ α z v , κ − | z ^ b , κ | ⋅ e j ⋅ α z b , κ + j ⋅ [ | y ^ v , κ | ⋅ e j ⋅ α y v , κ − | y ^ b , κ | ⋅ e j ⋅ α y b , κ ] } = = | r ^ _ v − b , κ + | ⋅ e j ⋅ α v − b , κ + r ^ _ v − b , κ − = 1 2 ⋅ { | z ^ v , κ | ⋅ e − j ⋅ α z v , κ − | z ^ b , κ | ⋅ e − j ⋅ α z b , κ + j ⋅ [ | y ^ v , κ | ⋅ e − j ⋅ α y v , κ − | y ^ b , κ | ⋅ e − j ⋅ α y b , κ ] } = = | r ^ _ v − b , κ − | ⋅ e j ⋅ α v − b , κ − (31)

where r ^ _ v − b , κ + and α v − b , κ + describe the absolute value and the phase shift of the forward rotating complex pointer and | r ^ _ v − b , κ − | and α v − b , κ − the absolute value and the phase shift of the backward rotating complex pointer.

The relative orbit between the shaft journal point V and of the bearing housing B can also be described by the ellipse parameters―semi-major axis a v − b , κ , the semi-minor axis b v − b , κ and the angle of the relative major axis ψ v − b , κ related to the vertical axis (z-direction)―can be calculated:

a v − b , κ = | r ^ _ v − b , κ + | + | r ^ _ v − b , κ − | (32)

b v − b , κ = | | r ^ _ v − b , κ + | − | r ^ _ v − b , κ − | | (33)

ψ v − b , κ = ( α v − b , κ + + α v − b , κ − ) / 2 (34)

Again the single solutions can be superposed:

r _ v − b = [ ∑ κ = u , a , m r ^ _ v − b , κ + ⋅ e j ⋅ φ κ ] ︸ r ^ _ v − b + ⋅ e j ⋅ Ω t + [ ∑ κ = u , a , m r ^ _ v − b , κ − ⋅ e − j ⋅ φ κ ] ︸ r ^ _ v − b − ⋅ e − j ⋅ Ω t (35)

In this section the bearing housing vibrations and the relative shaft displacements in the sleeve bearings of a 2-pole converter driven induction motor, mounted on a soft steel frame foundation, is analyzed. First, the boundary conditions have to be described.

The data of the 2-pole induction motor and the sleeve bearings are listed in

The calculated oil film stiffness and damping coefficients of the sleeve bearings are shown in

The magnetic spring constant c m d and magnetic damper constant d m are calculated, depending on the rotary angular frequency Ω and on the fundamental slip s. The fundament slip s is varying between 0 (no load operation) and 0.01 (operation near below the breaking torque) (

Motor data | Description | Value |
---|---|---|

Rated power | ||

Rated speed | ||

Rated torque | ||

Rated slip | ||

Pole-pair number | ||

Undamped magnetic spring constant | ||

Mass of the stator | ||

Mass inertia of the stator at x-axis | ||

Mass of the rotor | ||

Mass of the rotor shaft journal | ||

Mass of the bearing housing | ||

Height of the centre of gravity | ||

Distance between motor feet | ||

Stiffness of the rotor | ||

Horizontal stiffness of bearing housing and end shield | ||

Vertical stiffness of bearing housing and end shield | ||

Mechanical loss factor of bearing housing and end shield | ||

Mechanical loss factor of the rotor | ||

Sleeve bearing data | Description | Value |

Bearing shell | Cylindrical | |

Lubricant viscosity grade | ISO VG 32 | |

Nominal bore diameter | ||

Bearing width | ||

Ambient temperature | ||

Supply oil temperature | ||

Mean relative bearing clearance (DIN 31698) | Y_{m} = 1.6‰ | |

Foundation data | Description | Value |

Vertical stiffness of the foundation at each motor side | ||

Horizontal stiffness of the foundation at each motor side | ||

Mechanical loss factor of the foundation |

First, the vibrations of the induction motor, mounted on a rigid foundation ( c f z = c f y → ∞ ) , are analyzed. The bearing housing vibration velocities and the semi-major axis of of the relative orbit between bearing housing point B and shaft journal point V, are calculated for the different kinds of rotor eccentricity. The vibration velocities and the semi-major axes are related to the corresponding rotor eccentricity. The related bearing housing vibration velocities are shown in

The related semi-major axis of the relative orbit is shown in

Now the vibrations are analyzed for the soft foundation ( c f z = 1.5 × 10 8 kg / s 2 ; c f y = 1.0 × 10 8 kg / s 2 ) . The related bearing housing vibration velocities are shown in

The related semi-major axis of the relative orbit is shown in

The influence of electromagnetic field damping can be shown, by variating the fundamental slip s . For s = 0 no electromagnetic field damping occurs and for s = 0.01 , the electromagnetic field damping is maximal.

First, the vibrations for a rigid foundation (

from s = 0 ―is caused by the change of the electromagnetic damper constant d m (

For soft foundation (

The paper presents a mathematical multibody model of a soft mounted induction motor with sleeve bearings regarding forced vibrations caused by dynamic rotor eccentricities considering electromagnetic field damping. After the mathematical coherences have been shown, a numerical example was presented, where the bearing housing vibration velocities and the semi-major axes of the relative orbits between shaft journals and bearing housings have been analyzed for the different kinds of rotor eccentricity. By analyzing the vibrations for different fundamental slip s , the influence of electromagnetic field damping could be clearly shown. The aim of the paper is to present a method―based on a multibody model―for considering electromagnetic field damping for vibration analysis of a soft mounted induction motor, which can also be adopted in FE-Analysis.

Werner, U. (2017) Mathematical Multibody Model of a Soft Mounted Induction Motor Regarding Forced Vibrations Due to Dynamic Rotor Eccentricities Considering Electromagnetic Field Damping. Journal of Applied Mathematics and Physics, 5, 346-364. https://doi.org/10.4236/jamp.2017.52032