Key words: Rotors, Magnetorheological Dampers, Steady State Response, Collocation Method

Summary. To achieve optimum attenuation of lateral vibration of rotors, the damping effect must be controllable. This is enabled by magnetorheological squeeze film dampers. In this paper, there is presented a computational procedure for calculation of the steady state response of rigid rotors coupled with their casings by flexible elements and short magnetorheological dampers. Application of this approach makes possible to propose the control of the damping effect to achieve optimum performance of the dampers.

1 INTRODUCTION

Lateral vibration of rotors can be significantly reduced by inserting damping elements between the shaft and its casing. To achieve their optimum performance, the damping effect must be controllable. This is enabled by application of magnetorheological squeeze film dampers. Control of the damping effect is carried out by changing intensity of magnetic field in the lubricating layer.

At present time, the magnetorheological dampers are a subject of intensive research. E.g.

Forte et al. [1] presented results of the theoretical and experimental investigations of a long

magnetorheological damper. In [2], Zapomel and Ferfecki introduced the mathematical model

of a short squeeze film damper lubricated with magnetorheological liquid. The equations of

motion of rotors attenuated by magnetorheological damping devices are nonlinear. An

efficient tool for obtaining their steady state solution is a trigonometric collocation method. It

was used e.g. by Zhao et al. [3] to analyze behaviour of a flexible rotor supported by classical

squeeze dampers. Zapoměl and Malenovský [4] extended application of this method for

investigation of rotors mounted with hydrodynamic bearings and excited by baseplate

excitation. In this paper, the emphasis is put on application of the collocation method for

determination of the steady state response of rotors damped by magnetorheological dampers.

2 CALCULATION OF THE DAMPING FORCES

The magnetorheological dampers consist of two rings, between which there is a thin film of magnetorheological liquid. The outer ring is coupled with the casing of the rotating machine directly, the inner one by a flexible element. The shaft is supported by a rolling element bearing, whose outer race is coupled with the inner ring of the damper. The damping effect is produced by squeezing the liquid in the lubricating layer due to lateral vibration of the rotor. In the stationary part of the damper, there are the coils, which are the source of magnetic field. Its intensity in the gap of the damper influences resistance of the magnetorheological liquid against its flow and therefore the change of magnitude of the applied electric current can be used to control the damping effect.

In the developed mathematical model, it is assumed that ratio of the length of the damper to the diametre of its rings is small (short damper) and that the lubricant is Bingham liquid.

On these conditions, the distribution of the pressure gradient in the lubricating layer is described by the differential equation

. (1) 0

4 4

3 ^{2 2 3}

3

3p′ ± (h _{y B}hZ)p′ _y =

h τ m η & m τ

p' is the pressure gradient in the axial direction, τ

y

, η

B

are the yield shear stress and viscosity of the Bingham liquid, Z is the axial coordinate, h is the lubricating film thickness and (

^·

) denotes the first derivative with respect to time. The upper signs are valid for negative, the lower signs for the positive pressure gradient.

In the case of the simplest design of the damper, the inner and outer rings form a core of an electromagnet that is divided by two gaps. Then the relation between the yield shear stress and the applied current in the coil can be expressed

nB

B

y h

k NI⎟

⎠

⎜ ⎞

⎝

= ⎛

τ 2 . (2)

N is the number of the coil turns, I is the electric current and k

B

, n

B

are the material constants.

The pressure profile is calculated by integration of the pressure gradient in the axial direction with the boundary condition expressing that the pressure at the edge of the damper is equal to the atmospheric one. At location, where the thickness of the oil film rises, a cavitation takes place. Pressure of the medium in cavitated areas is assumed to be constant and equal to the pressure in the ambient space. Components of the damping force are then obtained by integration of the pressure distribution.

3 THE EQUATIONS OF MOTION OF THE INVESTIGATED ROTOR

The investigated rotor (Fig.1) consists of a shaft and of one disc. The rotor is mounted

with rolling element bearings, whose outer races are coupled with the casing by flexible

elements. The magnetorheological dampers are inserted between the spring elements and the

casing. The system is symmetric relative to the middle plane of the disc. The rotor turns at

constant angular speed, is loaded by its weight and is excited by a centrifugal force produced

by the disc unbalance. The springs are prestressed in the vertical direction to eliminate their

deflection caused by the rotor weight.

Figure 1: The investigated rotor system

In the computational model, the rotor and the stationary part are considered to be absolutely rigid. The magnetorheological liquid is modelled by Bingham material.

Lateral vibration of the rotor is then described (taking into account the system symmetry)

( )

R T

(

o

)

dy D P

Ry b y k y F y z y z m e t

m +0.5 + = , , , +0.5 ω cosω +ψ 5

.

0 && & & & ² , (3)

(

y z y z

)

F m e

(

t

)

m g

F z k z b z

m_R 0.5 _{P D dz} , , , _PS 0.5 _{R T} sin _o 0.5 _R 5

.

0 ^&&+ ^&+ = ^{& &} + + ω² ω +ψ − . (4)

m

R

is the mass of the rotor, k

D

is the stiffness of the rotor flexible support, b

P

is the coefficient of external damping, e

T

is eccentricity of the rotor centre of gravity, ω is angular speed of the rotor turning, ψ

o

is the phase leg, g is the gravity acceleration, F

dy

, F

dz

are components of the damping force, F

PS

denotes the prestress force, t is time, y, z are displacements of the rotor centre and (

^·

) and (

^··

) denote the first and second derivatives with respect to time.

Because of the prestress, trajectory of the rotor centre has a circular shape. This enables to assume the steady state solution of the equations of motion (3) and (4) in the form

t r t r

y= _Ccosω − _Ssinω , z=r_Csinωt+r_Scosωt. (5)

The unknown values of coefficients r

C

and r

S

can be calculated by application of a collocation method. This requires to substitute (5) and their first and second derivatives with respect to time into (3) and (4) and to express the resulting equations at the collocation points of time. As the number of unknown parameters and the number of equations is 2, only one collocation point is needed. After carrying out the mentioned manipulations related to the collocation time equal to 0 s, the resulting set of nonlinear algebraic equations takes the form

(

k_D−0.5m_Rω²

)

r_C −0.5ωb_P r_S−0.5m_Re_Tω²−F_dy

(

r_C,r_S

)

=0, (6)

(

0.5

) (

,

)

0

5 .

0 ωb_Pr_C+ k_D− m_Rω² r_S−F_dz r_C r_S = . (7)

4 THE COMPUTATIONAL SIMULATIONS

The task of the computational simulations was to analyze the steady state vibration of the

rotor system, whose principal parameters are: mass of the rotor 425.9 kg, stiffness of the

supporting springs 2.10

⁶

N/m, length, diameter and width of the gap of the damper 50 mm,

150 mm, 1.0 mm, number of the coil turns 400, Bingham viscosity 0.003 Pa·s and the yield

stress material coefficients of the magnetorheological liquid 2.5 10

^-8

N/A

²

and 2 (exponent).

In Fig.2 and 3, there are drawn orbits of the rotor centre and time history of the force transmitted into the stationary part in the horizontal direction for two magnitudes of the applied current. It is evident that increasing current leads to reduction of the rotor oscillations but on the other hand also to rising magnitude of the force transmitted into the rotor casing.

0 0.02 0.04 0.06 0.08 0.1

-1000 -500 0 500 1000

Time [ s ]

Horizontal force [ N ]

ROTOR

2 A

1 A

-1 -0.5 0 0.5 1

-1 -0.5 0 0.5 1

Displacement y [ mm ]

Displacement z [ mm ]

ROTOR

1 A

2 A

Figure 2: Orbits of the rotor centre Figure 3: Force transmitted into the casing

5 CONCLUSIONS

The described approach represents a computational procedure for determination of the

steady state response of rigid symmetric rotors supported by flexible couplings combined with

short magnetorheological squeeze film dampers. It is intended for proposing control of the

damping effect to achieve the optimum performance of the dampers.

In document Stress Intensity Factors for Bolt Fixed Laminated Glass Fröling, Maria; Persson, Kent (Page 112-115)