numerical analysis of a rotor dynamics in the magneto

17

Numerical Analysis of a Rotor Dynamics in the Magneto-Hydrodynamic Field

Jan Awrejcewicz1 and Larisa P. Dzyubak2 1Technical University of Łódź

2National Technical University “Kharkov Polytechnic Institute” 1Poland

2Ukraine

1. Introduction

In general, rotating machinery elements are frequently met in mechanical/mechatronical engineering, and in many cases their non-linear dynamics causes many harmful effects, i.e. noise and vibrations. In particular, nonlinear rotordynamics plays a crucial role in understanding various nonlinear phenomena and in spite of its long research history (see for instance (Tondl, 1965; Someya, 1998; Rao, 1991; Gasch et al., 2002; Muszyńska, 2005) and the references therein) it still attracts attention of many researchers and engineers. Since the topics related to nonlinear rotordynamics are broadband and cover many interesting aspects related to both theory and practice, in this chapter we are aimed only on analysis of some problems related to rotor suspended in a magneto-hydrodynamics field in the case of soft and rigid magnetic materials. The magnetic, magneto-hydrodynamic and also piezoelectric bearings are used in many mechanical engineering applications in order to support a high-speed rotor, provide vibration control, to keep lower rotating friction losses and to potentially avoid flutter instability. There are a lot of publications devoted to the dynamics analysis and control of a rotor supported on various bearings systems. The conditions for active close/open-loop control of a rigid rotor supported on hydrodynamic bearings and subjected to harmonic kinematical excitation are presented in (Kurnik, 1995; Dziedzic & Kurnik, 2002). The methodology for modeling lubricated revolute joints in constrained rigid multibody systems is described in (Flores et al., 2009). The hydrodynamic forces, used in the dynamic analysis of journal-bearings, which include both squeeze and wedge effects, are evaluated from the system state variables and included into the equations of motion of the multibody system. To analyze the dynamic behavior of rub-impact rotor supported by turbulent journal bearings and lubricated with couple stress fluid under quadratic damping the authors of (Chang–Jian & Chen, 2009) have used the system state trajectory, Poincaré maps, power spectrum, bifurcation diagrams and Lyapunov exponents. It was detected the dynamic motion as periodic, quasi-periodic and chaotic types. In (Zhang & Zhan, 2005; Li et al., 2006) a rotor–active magnetic bearings (rotor–AMB) systems with time-varying stiffness are considered. Using the method of multiple scales a governing nonlinear equation of motion for the rotor-AMB system with 1-dof is transformed to the averaged equation and then the bifurcation theory and the method of detection function are used to analyze the bifurcations of multiple limit cycles of the averaged equation.

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Numerical Simulations of Physical and Engineering Processes

368

Fig. 1. The cross–section diagram of a rotor symmetrically supported on the magneto–hydrodynamic bearing

In the present chapter 2-dof nonlinear dynamics of the rotor supported on the magneto-hydrodynamic bearing (MHDB) system is analyzed in the cases of soft and rigid magnetic materials. In the case of soft magnetic materials the analytical solutions have been obtained by means of the method of multiple scales (Nayfeh & Mook, 2004). Rigid magnetic materials possess hysteretic properties which are realized in the frames of the present work by means of Bouc-Wen hysteretic model. This model allows simulating hysteretic loops of various forms for systems from very different fields (Awrejcewicz & Dzyubak, 2007). Chaotic regions and the amplitude level contours of the rotor vibrations have been obtained in various control parameter planes.

2. Mathematical model of the rotor suspended in the magneto-hydrodynamic field

Consider a uniform symmetric rigid rotor (Fig. 1) which is supported by a magneto–hydrodynamic bearing system. The four–pole legs are symmetrically placed in the stator. Fk is the electromagnetic force produced by the kth opposed pair of electromagnet coils. This force is controlled by electric currents

0k ki i i

can be expressed in the form

2

0 02*

2

2k k

AN iF i

l

,

y

x

y

Q0

Fy Pr

P

Fx

k

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where i0 denotes bias current in the actuators electric circuits, 0 is the magnetic

permeability of vacuum, A is the core cross–section area, N is the number of windings of the

electromagnet, is the air gap in the central position of the rotor with reference to the

bearing sleeve, l is the total length of the magnetic path, the constant value *0s sB H

denotes the magnetic permeability of the core material; the values of the magnetic induction

Bs and magnetizing force Hs define the magnetic saturation level. k is the angle between

axis x and the kth magnetic actuator. Q0 is the vertical rotor load identified with its weight,

(Pr ,P) are the radial and tangential components of the dynamic oil–film action, respectively.

Equations of motion of the rotor are represented in the following form (Kurnik, 1995;

Dziedzic & Kurnik, 2002; Osinski, 1998)

* * * * * * * * *

1

* * * * * * * * * *0

1

, , cos , sin cos ,

, , sin , cos sin ,

K

r k k xk

K

r k k yk

m x P P F Q t

m y P P F Q Q t

2 * * * *

* * * *2 2 1

, , 21

rP C arctgp q p p

,

* *

* * *2

,P Cq p

.

Here m* denotes the rigid rotor mass, (x*, y*) are the Cartesian coordinates of the rotor center; *xQ t , *

yQ t are the external excitation characterizing bearing housing movements. We

are considering vibrations of the rotor excited by harmonic movements of the bearing

foundation in the vertical direction

* 0xQ t , * * * *sinyQ t Q t ,

where Q* and * are the amplitude and frequency of the external excitation, respectively. Constant C* is defined as

*2

6 s c c

s

R LC

.

Parameters s, s, Rc, Lc denote oil viscosity, relative bearing clearance, journal radius and

total bearing length, respectively. (, ) are the polar coordinates, 21p , 22q are the functions conditional .

To represent the equations of motion in a dimensionless form the following changes of

variables and parameters are introduced:

* *t t , *

*

,

*

*

,

*

*

xx

c ,

*

* *

xx

c ,

*

*2 *

xx

c ,

*

*

yy

c ,

*

* *

yy

c ,

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*

*2 *

yy

c ,

*

* * *

CC

m c , *

* ,

*

* *2 *

QQ

m c , *0

0 * *2 *

QQ

m c ,

*

* *2 *k

k

FF

m c , *

* *2 *r

r

PP

m c , *

* *2 *

PP

m c ,

where * is the rotation speed of the rotor; c* is the bearing clearance. Thus the dimensionless equations of motion take the form

0

, , cos , sin ,

, , sin , cos sin ,

r x

r y

x P P F

y P P F Q Q t

(1)

2 1 2 2 1

, , 21

rP C arctgp q p p

,

1 2, .P C

q p

Here

cosx , siny , 2

yx xy ,

xx yy , 2 2x y ,

2 2cos

x

x y ,

2 2sin

y

x y ;

the magnetic control forces are expressed as follows

0xF x x x , 0yF y y y ,

where 0 0,x y are the coordinates of the rotor static equilibrium, and are the control

parameters.

3. Soft magnetic materials

In this section, we consider 2–dof dynamics of the rotor in the MHDB system without taking hysteresis into account.

3.1 The non-resonant case

The right–hand sides of Eqs (1) were expanded in Taylor’s series and the origin was shifted

to the location of the static equilibrium 0 0,x y for the convenience of the investigation. The

linear and quadratic terms were kept. So, the reformed equations of motion are as follows:

2 2

1 1 2 3 4 5 6 7

2 22 1 2 3 4 5 6 7

ˆ2 ,

ˆ2 cos .

x x y x x y xx xy xy xy yy

y y x y x y xx xy xy xy yy F t

(2)

We seek the first–order solution for small but finite amplitudes in the form

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21 0 1 2 0 1

21 0 1 2 0 1

, , ,

, , ,

x x T T x T T

y y T T y T T

(3)

where is the small, dimensionless parameter related to the amplitudes and nnT t

(n=0, 1) are the independent variables. It follows that the derivatives with respect to t

become expansions in terms of the partial derivatives with respect to Tn according to

20 1 20 1 2

0 1 2

,Td T T

D D Ddt T t T t T t

2 22 2 2 20 1 2 0 0 1 1 0 22

2 2d

D D D D D D D D Ddt

, where kk

DT

.

To analyze the non–resonant case the forcing term is ordered so that it appears at order . Thus, we recall in (2) F=f, ˆn n . Substituting (3) into (2) and equating coefficients of

similar powers of we obtain

Order

20 1 1 0 1

20 1 1 0 1 0

0,

cos .

D x x D y

D y y D x f T

(4)

Order 2

2 2 20 2 2 0 2 0 1 1 1 1 1 1 1 1 2 1

3 1 0 1 4 1 1 5 1 0 1 6 1 0 1 7 1 0 1

2 2 20 2 2 0 2 0 1 1 2 1 1 1 1 1 2 1

3 1 0 1 4 1 1 5 1 0 1 6 1 0 1 7 1 0 1

2

,

2

.

D x x D y D D x x D y x y

x D x x y x D y y D x y D y

D y y D x D D y y D x x y


(5)

The solution of (4) is expressed in the form

1 1 1 1 0 2 1 2 0 1 0

1 1 1 1 1 0 2 2 1 2 0 2 0

exp exp exp ,

exp exp exp ,

x A T i T A T i T i T CC

y A T i T A T i T i T CC

(6)

where CC denotes the complex conjugate of the preceding terms, A1 and A2 are the arbitrary functions of T1 at this level of approximation,

2n

nn

i , 1 2

2 2 22

fi

,

2

2 22 2 2

1

2

f

, (n=1, 2).

n are assumed to be distinct and n2 are the roots of the characteristic equation

1 0 0

0det

0 0 1

0

= 4 2 2 22 = 4 2 2 22 0n n , (7)

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1,2 1i , 3,4 2i , 2 21,2

14 2 2 4

2 ,

2 23,4

14 2 2 4

2 .

Substitution of (6) into (5) gives

20 2 2 0 2 1 1 1 1 1 1 1 0

2 2 1 2 2 2 2 0

20 2 2 0 2 1 1 1 2 1 1 1 0

2 2 2 2 2 2 2 0

2 exp

2 exp ,

2 exp

2 exp .

D x x D y i A A A i T

i A A A i T CC

D y y D x i A A A i T

i A A A i T CC

(8)

The terms, which do not influence solvability conditions, are not presented in the last equations and replaced by dots. To determine the solvability conditions of (8), following to the method of undetermined coefficients we seek a particular solution in the form

2 11 1 0 12 2 0exp expx P i T P i T ,

2 21 1 0 22 2 0exp expy P i T P i T (9)

with unknowns P11, P12, P21 and P22. Substitution of expressions (9) into (8) and collection of

coefficients at 1 0exp i T and 2 0exp i T yields

21 2 1n n n n nP i P R ,

21 2 2n n n n ni P P R (n=1,2), (10)

where

11 1 1 1 1 1 1 12 2 2 1 2 2 2

21 1 1 1 2 1 1 22 2 2 2 2 2 2

2 , 2 ,

2 , 2 .

R i A A A R i A A A

R i A A A R i A A A

Taking into account the characteristic equation (7), the determinant of the set of linear algebraic equations relative to P1n, P2n (10) is equal to zero

22

2 2 2

20n n

n n

n n

i

i

.

According to Kronecker–Kapelly’s theorem, the set of linear algebraic equations is compatible if and only if the matrix rank of the linear set is equal to the extended matrix rank. Therefore, the solvability conditions are

1

22

0n n

n n

R i

R

(n=1,2),

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373

otherwise the set of linear algebraic equations (10) has no solutions. So,

21 2

n nn

n

i RR

and the solvability conditions can be written in the form

21

nn

n

RR (n=1,2). (11)

The differential equations to define A1(T1) and A2(T1) are the consequence of solvability conditions (11)

1 1 1 1 21 1 1 1 1 1

1 1

2 2 2 2 22 2 2 2 1 2

2 2

2 22 2 0,

2 22 2 0.

i ii A i A

i ii A i A

(12)

It follows from (3), (6) and (12) that the complex solution of the differential set (2) is

21 1 1 2 2 2 1

21 1 1 1 2 2 2 2 2

exp exp exp exp exp ,

exp exp exp exp exp .

x t a i t t a i t i t CC O

y t a i t t a i t i t CC O

Then the real solution is as follows

2

1 1 1 1 2 2 2 2 1

1 1 1 1 1 2 2

22 2 2 2

exp cos exp cos 2Im sin ,

Im exp sin Im exp

sin 2 cos ,

x t a t t a t t O

y t a t t

a t t O

(13)

where 1 22

14 Im

Im

nn

n nn

, an and n are the real constants.

Figure 2 shows a comparison of the numerical integration of (2) and the perturbation

solutions (13). The following parameters of set (2) were accepted for all cases (a), (b), (c)

=1500, =70, 1=9.985102, 2=2103, 3=7.9588103, 4= 0.002, 5= –4.0794103,

6=4.0002103, 7=8.0005103, 1=29.9975, 2= –0.001, 3= –4.1594103, 4= –1.9997103, 5= –

7.9188103, 6=0.7959, 7= –0.4083; initial conditions are the following x(0)=10-12, y(0)=10-10, 0 0 0x y .

In the case of non–resonant undamped vibrations of the rotor (Fig. 2 (a)) it is accepted for

numerical integration that 1̂ =0, 2̂ =0, F=0. According to (13), the perturbation solution is

presented by the expressions

x=8.2686044·10–6 cos (17.2015t)+1.6313956·10–6 cos (87.2015t),

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y=8.2686044·10–6 sin (17.2015t)–1.6313956·10–6 sin (87.2015t).

Fig. 2 (b) corresponds to the non–resonant damped vibrations of the rotor. For this case

1̂ =0.1, 2̂ =0.15, F=0. The perturbation solution has the form

x=8.2686044·10–6 exp (–0.0412t) cos (17.2015t)+1.6313956 ·10–6 exp (–0.2088t) cos (87.2015t),

y=8.2686044·10–6 exp (–0.0412t) sin (17.2015t)–1.6313956 ·10–6 exp (–0.2088t) sin (87.2015t).

For the non–resonant forced damped vibrations of the rotor (Fig. 2 (c)) it is accepted for numerical integration that 1̂ =0.1, 2̂ =0.15, F=0.005, =10, = –/2. The perturbation solution is

x=5.8241·10–6 exp (–0.0412t) cos (17.2015t)+1.69495·10–6 exp(–0.2088t) cos (87.2015t) –

2.38095·10–6 sin(10t–/2),

y=5.8241·10–6 exp (–0.0412t) sin (17.2015t)–1.69495·10–6 exp (–0.2088t) sin (87.2015t) +

4.7619·10–6 cos (10t–/2).

Fig. 2 demonstrates good agreement of the numerical and analytical solutions.

0.5 1.0 1.5 2.0

-1.0x10-5

-5.0x10-6

0.0

5.0x10-6

1.0x10-5

y

t

numerical integration

analytical solution

2 4 6 8 10

-1.0x10-5

-5.0x10-6

0.0

5.0x10-6

1.0x10-5

y

t


analytical solution

(a) (b)

0 2 4 6 8 10

-1.5x10-5

-1.0x10-5

-5.0x10-6

0.0

5.0x10-6

1.0x10-5

1.5x10-5

y

t


analytical solution

(c)

Fig. 2. Comparison of numerical integration (2) and perturbation solutions (13) in the case of (a) nonresonant undamped vibrations of the rotor, (b) nonresonant damped vibrations of the rotor; (c) nonresonant forced damped vibrations of the rotor

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3.2 Primary resonance: The cases of no internal resonance and an internal resonance

To analyze primary resonances the forcing term is ordered so that it appears at order 2 or in

the same perturbation equation as the non–linear terms and damping. Thus, we recall in (2)

F= 2f, ˆn n . Consider the case in which 2 . The case 1 is analogous. Let us

introduce detuning parameter 1 and put 2 1 .

Substituting (3) into (2) and equating coefficients of similar powers of we obtain

Order

20 1 1 0 1

20 1 1 0 1

0,

0.

D x x D y

D y y D x

(14)

Order 2

2 2 20 2 2 0 2 0 1 1 1 1 1 1 1 1 2 1

3 1 0 1 4 1 1 5 1 0 1 6 1 0 1 7 1 0 1

2 2 20 2 2 0 2 0 1 1 2 1 1 1 1 1 2 1

3 1 0 1 4 1 1 5 1 0 1 6 1 0 1 7 1 0 1 0

2

,

2

cos .

D x x D y D D x x D y x y


D y y D x D D y y D x x y

x D x x y x D y y D x y D y f T

(15)

The solution of (14) is given in the form

1 1 1 1 0 2 1 2 0

1 1 1 1 1 0 2 2 1 2 0

exp exp ,

exp exp ,

x A T i T A T i T CC

y A T i T A T i T CC

(16)

where 2n

nn

i .

Substitution of (16) into (15) yields

20 2 2 0 2 1 1 1 1 1 1 1 0

2 2 1 2 2 2 2 0

2 2 21 1 1 2 1 3 1 4 1 1 5 1 1 6 1 1 7 1 0

2 2 22 1 2 2 2 3 2 4 2 2 5 2 2 6 2 2 7

2 exp

2 exp

exp 2

exp 2

D x x D y i A A A i T

i A A A i T

A i i i i i T

A i i i i i

2 0T

(17)

1 2 1 1 2 2 1 2 3 1 2 4 2 2 1 1 5

2 1 1 2 6 1 2 1 2 7 1 2 0

1 2 1 1 2 2 2 1 3 2 1 4 2 2 1 1 5

2 1 1 2 6 2 1 1 2 7 2 1 0

1 1 1 1 1 2 4 1 5

2 2

exp

2 2

exp

A A i i i i

i i i i i T

A A i i i i

i i i i i T

A A i

6 2 2 1 2 2 2 4 2 5 6 ,A A i CC

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20 2 2 0 2 1 1 1 2 1 1 1 0

2 2 2 2 2 2 2 0

2 2 21 1 1 2 1 3 1 4 1 1 5 1 1 6 1 1 7 1 0

2 2 22 1 2 2 2 3 2 4 2 2 5 2 2 6 2 2 7

2 exp

2 exp

exp 2

exp 2

D y y D x i A A A i T

i A A A i T

A i i i i i T

A i i i i i

2 0T

(18)

1 2 1 1 2 2 1 2 3 1 2 4 2 2 1 1 5

2 1 1 2 6 1 2 1 2 7 1 2 0

1 2 1 1 2 2 2 1 3 2 1 4 2 2 1 1 5

2 1 1 2 6 2 1 1 2 7 2 1 0

1 1 1 1 1 2 4

2 2

exp

2 2

exp

A A i i i i

i i i i i T

A A i i i i

i i i i i T

A A i

1 5 6 2 2 1 2 2 2 4 2 5 6

2 0 1 1

1exp .

2

A A i

f i T T CC

Let 2>1 for definiteness. We need to distinguish between the case of internal resonance

2 12 and the case of no internal resonance, i.e., 2 is away from 21. The case 1>2,

1 22 is analogous. When 2 is away from 21 the solvability conditions (11) are written

in the form

1 1

2 2

1

1 12 2

10,

1 1exp 0,

2

q p

q p f i T

where

1 21 1 1 1 1 1 2 2 1 2 2 22 , 2 ,q i A A A q i A A A

1 1 1 1 2 1 12 ,p i A A A

2 2 2 2 2 2 22p i A A A .

Thus, when there is no internal resonance, the first approximation is not influenced by the non–linear terms; it is essentially a solution of the corresponding linear problem. Actually, the solutions of the differential equations below

1 1 1 1 2

1 1 1 1 1 11 1

2 2 2 2 22 2 2 2 1 2 1 1

2 2 2

2 22 2 0,

2 2 12 2 exp

2

i ii A i A

i ii A i A f i T

are

1 1 1 1 1 1

1exp

2A T a T i ,

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2 12 1 2 2 1 2 1 12 2

2 2 2 1

1exp exp

2 2Im Im

f iA T a T i i T

,

where an and n are the real constants,

1 22

14 Im

Im

nn

n nn

,

2 2 22

14 Im

Imi i .

As t, T1 and

A10, 2 12 1 12 2

2 2 2 1

exp2Im Im

f iA i T

(19)

according to (16), we obtain the following steady–state response:

2 11 2 0 1 12 2

2 2 2 1

exp2Im Im

f ix i T T CC

,

2 11 2 2 0 1 12 2

2 2 2 1

exp2Im Im

f iy i T T CC

.

Therefore, the real solution is

22 12 2

2 2 2 1

1cos sin

Im Im

Fx t t O

,

21 22 2

2 2 1

1cos sin

Im

Fy t t O ,

or it can be rewritten in the form

2

11 22 22 2 2 1

221 2

2 22 2 1

1sin ,

Im Im

1sin ,

Im

Fx t O

Fy t O

(20)

where 1 2 1arctg , 2 1 2arctg . Other situation occurs when the internal resonance 2 12 exists. Let us introduce detuning parameter 2 and put 2 1 22 . Taking into account (11), the solvability conditions for this case become

1 1 2 1 2 1

2 2 1 1

1 2 2 11 1

22 2 1 2 1 1 1

2 2 2

1 1exp 0,

1 1 1exp exp 0.

2

q p q p A A i T

q p q p A i T f i T

(21)

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Here coefficients q1 , q2 , q2-1 , q21 are the expressions in the bracket at the exponents

with the corresponding powers (17) and p1 , p2 , p2-1 , p21 are the expressions in the bracket at the exponents with the corresponding powers (18):

1 21 1 1 1 1 1 2 2 1 2 2 22 , 2 ,q i A A A q i A A A

1

2 22 1 1 2 1 3 1 4 1 1 5 1 1 6 1 1 7 ,q i i i i

2 1 1 1 2 2 2 1 3 2 1 4 2 2 1 1 5

2 1 1 2 6 2 1 1 2 7

2 2

,

q i i i i

i i i i

1 1 1 1 2 1 12 ,p i A A A

2 2 2 2 2 2 22 ,p i A A A

1

2 22 1 1 2 1 3 1 4 1 1 5 1 1 6 1 1 7 ,p i i i i

2 1 1 1 2 2 2 1 3 2 1 4 2 2 1 1 5

2 1 1 2 6 2 1 1 2 7

2 2

.

p i i i i

i i i i

For the convenience let us introduce the polar notation

1exp , 1, 2

2m m mA a i m , (22)

where am and m are the real functions of T1. Substitution of (22) into (21) yields

1 1 1 1 1 1 2 21

22 2 2 2 2 1 2 1

2 2 2

1exp 0,

2

1exp exp 0.

2

a ia a a a i i

fa ia a a i i i

(23)

In the expressions above the following notations were introduced

2 1 2 11

1Re q p

, 2 1 2 1

1

1Im q p

, 1 12 2

2

1Re q p ,

1 12 22

1Im q p ,

14 Im , 1,2

Imn n n

n

i i n ,

1 1 1 2 2 2 1 2 1, 2T T ,

1 and 2 are defined as in Eq (13). Separating Eqs. (23) into real and imaginary parts and taking into account that according to

(6) n (n=1,2) is the imaginary value, we obtain

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1 21 1 1 2 2

1

cos sin2Im

a aa a ,

1 21 1 2 2

1

cos sin2Im

a aa , (24)

21

2 2 2 2 2 12 2 2

cos sin cos2Im Im Im

faa a ,

21

2 2 2 2 12 2 2

cos sin sin2Im Im Im

faa .

For the steady–state response 0n na , therefore 1 1 2

1

2 , 2 1 .

Two possibilities follow from (24). The first one is given by (19). It is the solution of the linear problem. Let us find functions a1 and a2 of T1 according to the second possibility. It follows from the first two Eqs. (24) that

1 1 22 2

2

4cos sin

a

,

11 2 2 2

2

Imcos sin

a

.

So,

1 2

2 22 21 1 2 1 1 2

2 2 2

16 Ima

. (25)

Let us take sin2 and cos2 using, for example, the formulas by Cramer

1 22 2cos , sin , where 2 2

,

1 11 1 1 2 1 1 2

1 1 22 2

2Im1 14 Im

Ima a

,

1 12 1 1 2 1 1 2

1 1 22 2

2Im1 1Im 4

Ima a

.

Then a biquadratic equation relative to a1 follows from the last two Eqs. (24)

4 2 2 21 1 2 1 2 2 2 2 2 1 2 2 24 2 cos sin Im cos sina a a a +

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2

22 2 2 22 1 2 2 1 2 2

2

44 4 Im 0

Im

fa .

Finally, we obtain the expression for a1:

11 2

2 2

12 2

p pa q

, (26)

where

22 1 2 2 2 2 1 2 22 2

42 cos sin Im cos sin

ap ,

2

22 2 2 22 2 1 2 2 12 2 2

2

414 4 Im

Im

fq a

.

Thus, the unknown functions in (16) were defined. It follows from (3), (16) and (22) that

21 1 1 0 2 2 2 0

21 1 1 1 0 2 2 2 2 0

1 1exp exp ,

2 2

1 1exp exp .

2 2

x a i T a i T CC O

y a i T a i T CC O

-1.0 -0.5 0.0 0.5 1.0

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

a2

a2

a2

a2

a1

a1

a1

a1 , a

2

0.000 0.004 0.008 0.012

0.0000

0.0002

0.0004

0.0006

a2

a2

a1

a1

a1 , a

2

f

(a) (b)

Fig. 3. (a) Frequency–response curves; 2=0, 2; (b) amplitudes a1, a2 versus the amplitude

of external excitation f; 2, 1= –0.5, 2= 0

Then, the real solution is as follows

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21 1 2 2 1

21 1 1 2 2 2 1

1cos cos ,

2

1Im sin Im sin .

2

x a t a t O

y a t a t O

(27)

Here a1 and a2 are defined by (25), (26).

Let us consider the expression for a1 (26). When 2{[( 2) 0] ( 0)} [( 2) ]p q p q , there are

no real values of a1 defined by (26) and the response must be given by (20). When 2[( 2) ] ( 0)p q q , there is one real solution defined by (26). Therefore, the response is one

of the two possibilities given by (20) and (27). When 2[( / 2) 0] [( / 2) ] ( 0)p p q q ,

there are two real solutions defined by (26). Therefore, the response is one of the three

possibilities given by (20) and (27). In Fig. 3 (a) the frequency–response curves are depicted. a1 and a2 are plotted as a function

of 1 for 2=0. The dashed line having a peak at 1=0 corresponds to a1=0 and it is a solution of the corresponding linear problem. Arrows indicate the jump phenomenon associated

with varying the frequency of external excitation . Perturbation solution obtained is the

superposition of two submotions with amplitudes a1 and a2 and frequencies 1, 2 correspondingly. To compare the perturbation and numerical solutions we performed an approximate harmonic analysis of solutions x(t), y(t) obtained numerically. These functions are expanded in Fourier series formed of cosines

0

1

cos2

kk

a k tx t a

T

, 0

2cos

T

k

k ta x t dt

T T

, k=0,1,2…,

where T is the period of integration, 0tT. The coefficients of the Fourier series were

calculated approximately. The following parameters of set (2) were accepted: =200, = 10

(parameters =200, =10 correspond to natural frequencies 1=10, 2=20, i.e. 2=21), 1=9.985102, 2=2103, 3=7.9588103, 4= 0.002, 5= –4.0794103, 6=4.0002103, 7=8.0005103, 1=29.9975, 2= –0.001, 3= –4.1594103, 4= –1.9997103, 5= –7.9188103, 6=0.7959, 7= –0.4083. The perturbation and numerical solutions of (2) are in good agreement. In Fig. 3 (b) one can see saturation phenomenon. As f increases from zero, a2 increases too

until it reaches the value a2=3.510-4 while a1 is zero. This agrees with the solution of the

corresponding linear problem. Then a2 saves the constant value and a1 starts to increase.

Approximate harmonic analysis demonstrates good agreement of the theoretical prediction

presented in Fig. 3 (b) and the corresponding numerical solution of (2).

4. Rigid magnetic materials. Conditions for chaotic vibrations of the rotor in various control parameter planes

In the case of rigid magnetic materials the hysteretic properties of system (1) can be

considered using the Bouc–Wen hysteretic model. It was shown (Awrejcewicz & Dzyubak,

2007) that this modeling mechanism for energy dissipation was sufficiently accurate to

model loops of various shapes in accordance with a real experiment, reflecting the behavior

of hysteretic systems from very different fields. The hysteretic model of the rotor–MHDB

system is as follows

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0 1, , cos , sin 1 ,r m mx P P x x x z

0 2 0, , sin , cos 1 sin ,r m my P P y y y z Q Q t (28)

1 1 1sgn sgnn

zz k x z z x ,

2 2 2sgn sgnn

zz k y z z y .

Here z1 and z2 are the hysteretic forces. The case =0 corresponds to maximal hysteretic

dissipation and =1 corresponds to the absence of hysteretic forces in the system,

parameters ( kz, , n )R+ and R govern the shape of the hysteresis loops.

Conditions for chaotic vibrations of the rotor have been found using the approach based on

the analysis of the wandering trajectories. The description of the approach, its advantages

over standard procedures and a comparison with other approaches can be found, for

example, in (Awrejcewicz & Dzyubak, 2007; Awrejcewicz & Mosdorf, 2003; Awrejcewicz et

al., 2005).

The stability of motion depends on all the parameters of system (28), including initial

conditions. We traced the irregular vibrations of the rotor to sufficient accuracy in the

parametric planes of amplitude of external excitation versus hysteretic dissipation (, Q), the

amplitude versus frequency of external excitation (, Q), the amplitude versus dynamic oil–

film action characteristics (C, Q) and the amplitude versus the magnetic control parameters

(m, Q), (m, Q).

It should be noted, that chaos is not found in absence of hysteresis when =1. Chaotic

vibrations of the rotor are caused by hysteresis and for all chaotic regions presented 1. So,

in system (28) chaos was quantified using the following conditions

*1 ,t t T : * * * *

x yx t x t A y t y t A (29)

chaotic vibrations chaotic vibrations

in the horizontal direction in the vertical direction

Here x t , x t and y t , y t are nearby trajectories respectively, Ax and Ay are the

characteristic vibration amplitudes of the rotor in the horizontal and vertical direction

respectively

11

1max min

2x

t t Tt t TA x t x t ,

11

1max min

2y

t t Tt t TA y t y t .

1 0, ,t T t T and 0 ,t T is the time interval over which the trajectories are considered. The

interval 0 1,t t is the time interval over which all transient processes are damped. The

parameter introduced is an auxiliary parameter such that 0<<1. Ax, Ay are referred to

as the divergence measures of the observable trajectories in the horizontal and vertical

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383

directions and, with the aid of the chosen parameter , are inadmissible for the case of

regularity of the motion.

If the inequality (29) is satisfied in some nodal point of the sampled control parameter space,

then the motion is chaotic (including transient and alternating chaos). The manifold of all

such nodal points of the investigated control parameter space defines the domains of chaotic

behaviour for the considered system.

Figure 4 (a) displays the regions of rotor chaotic vibrations in (, Q) plane. The part of this

plane (10-7<0.0017; 0.00125<Q0.00185) was sampled by means of an uniform rectangular

grid. For this aim two families of straight lines were drawn through dividing points of the

axes

i = i (i=0, 1,…, 120),

Qj = jQ (j=0, 1,…, 120).

Here =1.416510-5, Q=510-6.

The time period for the simulation T is of 200 in nondimentional time units. During the

computations, two thirds of the time period T corresponds to the time interval 0 1,t t ,

where transient processes are damped. The integration step size is 0.02 . Initial conditions

of the nearby trajectories are differed less than 0.5% of characteristic vibration amplitudes,

e.g. the starting points of these trajectories are in the rectangle ( 0 0 0.005 xx t x t A ,

0 0 0.005 yy t y t A ). The parameter is chosen to be equal to 1

3.

All domains have complex structure. There are a number of scattered points, streaks and islets here. Such a structure is characteristic of domains where chaotic vibrations are possible. For each aggregate of control parameters there is some critical value of the hysteretic dissipation (1-cr) that if (1-)<(1-cr), chaos is not observed in the system considered.

In Fig. 4 (b) chaotic regions for the vertical vibrations of the rotor are depicted in the (, Q)

parametric plane (0.25<1.2; 0.0015<Q0.0022). The time period for the simulation T and

other numerical integration characteristics are the same as for (, Q) parametric plane,

=7.9166710-3, 5.8333310-6. Numerical experiments show that for the larger hysteretic

dissipation the chaotic regions areas are increased. Figure 5 shows the phase portrait (a), hysteretic loop (b) and Poincaré map (e) of chaotic motion of the rotor. Parameters of motion correspond to the parameters of chaotic region depicted in Fig. 4 (b). The phase portrait (c), hysteretic loop (d) and Poincaré map (f) of the periodic rotor motion are also agree well with the obtained regions of regular/irregular behaviour of the rotor depicted in Fig. 4 (b). The influence of the

magnetic control parameters m, m on chaos occurring in the rotor vibrations can be

observed in Fig. 6. The (m, Q) (a) and (m, Q) (b) parametric planes were uniformly

sampled by120120 nodal points in the rectangles (0<m0.09; 0.00165<Q0.0019),

m=7.510-4, Q=2.0833310-6; (450<m630; 0.00145<Q0.0025), m=1.5, Q=8.7510-6. The influence of the dynamic oil–film action characteristics on chaos occurring in the rotor motion can be observed in Fig. 7. One can see the restraining of chaotic regions with

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decreasing of hysteretic dissipation (1-). The (C, Q) parametric plane was uniformly

sampled by 120120 nodal points in the rectangles (0<C1.5; 0.0015<Q0.0021),

C=0.0125, Q=510-6 (a) and (0<C1.5; 0.0015<Q0.00225), C=0.0125, Q=6.2510-6 (b). The time period for the simulation T and other numerical integration characteristics are

the same as for (, Q) parametric plane.

0.0005 0.0010 0.0015

0.0014

0.0016

0.0018A

mpli

tude

Q

(a)

0.4 0.6 0.8 1.0 1.2

0.0016

0.0018

0.0020

0.0022

Am

pli

tude

Q

Frequency

(b)

Fig. 4. (a) The influence of hysteretic dissipation parameter on chaos occurring in vertical vibrations of the rotor (28) in the case of rigid magnetic materials. The following parameters are fixed: C=0.03, m=0.001, m=450, kz=0.000055, =15, =0.25, n=1.0, =0.87, Q0=0, x0=0, y0=0, 80 0 10x y , 0 0 0x y , z1(0)=z2(0)=0; (b) chaotic regions for the vertical vibrations of the rotor in the (, Q) parametric plane with other parameters of the system fixed: =0.0001, C=0.2, m=0, m=500, kz=0.000055, =15, =0.25, n=1.0, Q0=0, x0=0, y0=0, 80 0 10x y , 0 0 0x y , z1(0)=z2(0)=0.

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385

-0.002 -0.001 0.000 0.001

-0.001

0.000

0.001

y

y-0.002 -0.001 0.000 0.001

-4.0x10-6

-2.0x10-6

0.0

2.0x10-6

4.0x10-6

z2

y

(a) (b)

-0.002 0.000 0.002

-0.002

0.000

0.002

y

y-0.002 0.000 0.002

-4.0x10-6

-2.0x10-6

0.0

2.0x10-6

4.0x10-6

z2

y

(c) (d)

-0.002 -0.001 0.000 0.001

-0.001

0.000

0.001

y

y-0 .0 0 2 0 .0 0 0 0 .0 0 2

-0 .0 0 2

0 .0 0 0

0 .0 0 2

y

y

(e) (f)

Fig. 5. Phase portraits (a), (c), hysteresis loops (b), (d) and Poincaré maps (e), (f) of the chaotic (a), (b), (e) and periodic (c), (d), (f) rotor motion that agree well with the chaotic/regular regions in Fig. 4 (b). The parameters =0.0001, C=0.2, m=0, m=500, kz=0.000055, =15, =0.25, n=1.0, Q0=0, x0=0, y0=0, 80 0 10x y , 0 0 0x y , z1(0)=z2(0)=0 are fixed; (a), (b), (e) =0.87, Q=0.00177; (c), (d), (f) =1.2, Q=0.0017

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386

0.00 0.02 0.04 0.06 0.08

0.0017

0.0018

0.0019

Am

pli

tude

Q

m

450 500 550 600

0.0015

0.0020

0.0025

Am

pli

tude

Q

m

(a) (b)

Fig. 6. The influence of the magnetic control parameters m (a) and m (b) on chaos occurring

in vertical vibrations of the rotor (28) in the case of rigid magnetic materials. The parametric

planes are depicted at (a) m=500 and (b) m=0 with other parameters of the system fixed: =0.000001, C=0.2, kz=0.000055, =15, =0.25, n=1.0, =0.87, Q0=0, x0=0, y0=0, 80 0 10x y , 0 0 0x y , z1(0)=z2(0)=0

0.0 0.5 1.0 1.5

0.0016

0.0018

0.0020

Am

pli

tude

Q

C0.0 0.5 1.0 1.5

0.0016

0.0018

0.0020

0.0022

Am

pli

tude

Q

C

(a) (b)

Fig. 7. The influence of the dynamic oil–film action characteristics on chaos occurring in vertical vibrations of the rotor (28) in the case of rigid magnetic materials. The parametric planes (C, Q) are depicted at (a) =0.000001, m=0 and (b) =0.001, m=0.03 with other parameters of the system fixed: m=500, kz=0.000055, =15, =0.25, n=1.0, =0.87, Q0=0, x0=0, y0=0, 80 0 10x y , 0 0 0x y , z1(0)=z2(0)=0

In order to see if the rotor chaotic motion is accompanied by increasing of the amplitude of vibration, the amplitude level contours of the horizontal and vertical vibrations of the rotor

have been obtained. In Fig. 8 (a) the amplitude level contours are presented in (m, Q)

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parametric plane with the same parameters as in Fig. 6 (a). Some “consonance” between the chaotic vibrations regions and the amplitude level contours is observed. At that the amplitudes of chaotic rotor vibrations are greater in comparison to the periodic vibrations. In Fig. 8 (b) the amplitude level contours are presented in (C, Q) parametric plane with the same parameters as in Fig. 7 (a). Although some “consonance” between the chaotic regions of vibrations and the amplitude level contours is observed, it can not be concluded that chaos leads to essential increasing of the rotor vibrations amplitude.

0.00 0.02 0.04 0.06 0.08

0.0017

0.0018

0.0019

m

Q

2E-4

4.5E-4

7E-4

9.5E-4

0.001200

0.001450

0.001700

0.001950

0.002200

0.0 0.5 1.0 1.5

0.0016

0.0018

0.0020

Q

C

0

5E-4

1E-3

0.001500

0.002000

0.002500

0.003000

0.003500

0.004000

(a) (b)

Fig. 8. The amplitude level contours of vertical vibrations of the rotor (28): (a) in the

parametric plane (m, Q) that corresponds to Fig. 6 (a); (b) in the parametric plane (C, Q)

that corresponds to Fig. 7 (a)

5. Conclusions

2–dof non–linear dynamics of the rotor suspended in a magneto–hydrodynamic field is studied. In the case of soft magnetic materials the analytical solutions were obtained by means of the method of multiple scales. In the non–resonant case the system exhibits linear properties. The perturbation solutions are in good agreement with the numerical solutions. The cases of primary resonances with and without an internal resonance were investigated. The frequency–response curves were obtained. The saturation phenomenon was demonstrated. When the amplitude of the external excitation increases (or decreases), above some critical value the energy pumping between various submotions of the rotor occurs. A comparison of the analytical and numerical solutions based on the approximate harmonic analysis was made. In the case of rigid magnetic materials, hysteresis was considered using the Bouc–Wen hysteretic model. Using the approach based on the analysis of the wandering trajectories the regions of chaotic vibrations of the rotor were found in various control parameter planes: amplitude of external harmonic excitation versus hysteretic dissipation, versus frequency of external harmonic excitation, dynamic oil–film action characteristics as well as versus the magnetic control parameters. The amplitude level contours of the horizontal and vertical vibrations of the rotor were obtained. Phase portraits and hysteretic loops are in good agreement with the chaotic regions obtained. Chaos was generated by hysteretic properties of the system considered.

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6. References

Awrejcewicz, J. & Dzyubak, L. (2007). Hysteresis modelling and chaos prediction in one– and two–dof hysteretic models. Archive of Applied Mechanics, No.77, pp. 261–279

Awrejcewicz, J., Dzyubak, L. & Grebogi, C. (2005). Estimation of chaotic and regular (stick–slip and slip–slip) oscillations exhibited by coupled oscillators with dry friction. Non–linear Dynamics, No.42(2), pp. 383–394

Awrejcewicz, J. & Mosdorf, R. (2003). Numerical Analysis of Some Problems of Chaotic Dynamics, WNT, Warsaw

Chang–Jian, C. W. & Chen, C. K. (2009). Non–linear analysis of a rub–impact rotor supported by turbulent couple stress fluid film journal bearings under quadratic damping. Non–linear Dynamics, No.56, pp. 297–314

Dziedzic, K. & Kurnik, W. (2002). Stability of a rotor with hybrid magneto–hydrodynamic support. Machine Dynamics Problems, No.26(4), pp. 33–43

Flores, P., Ambrosio, J., Claro, J. C. P., Lancarani, H. M. & Koshy, C. S. (2009). Lubricated revolute joints in rigid multibody systems. Non–linear Dynamics, No.56, pp.277–295

Gasch, R., Nordmann, R. & Pfützner, H. (2002). Rotordynamik, Springer, Berlin Kurnik, W. (1995). Active magnetic antiwhirl control of a rigid rotor supported on

hydrodynamic bearings. Machine Dynamics Problems, No.10, pp. 21–36 Li, J., Tian, Y., Zhang, W. & Miao, S. F. (2006). Bifurcation of multiple limit cycles for a rotor–

active magnetic bearings system with time–varying stiffness. International Journal of Bifurcation and Chaos

Muszyńska, A. (2005). Rotordynamics, CRC Press, Boca Raton Nayfeh, A. H. & Mook, D. T. (2004). Non–linear oscillations, Wiley, New York Osinski, Z. (Ed.) (1998). Damping of Vibrations, A.A. Balkema, Rotterdam, Brookfield Rao, J. S. (1991). Rotor Dynamics, Wiley, New York Someya, T. (1998). Journal–Bearing Databook, Springer, Berlin Tondl, A. (1965). Some Problems of Rotor Dynamics, Chapman & Hall, London Zhang, W. & Zhan, X. P. (2005). Periodic and chaotic motions of a rotor–active magnetic

bearing with quadratic and cubic terms and time–varying stiffness. Non–linear Dynamics, No.41, pp. 331–359

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Numerical Simulations of Physical and Engineering ProcessesEdited by Prof. Jan Awrejcewicz

ISBN 978-953-307-620-1Hard cover, 594 pagesPublisher InTechPublished online 26, September, 2011Published in print edition September, 2011

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Numerical Simulations of Physical and Engineering Process is an edited book divided into two parts. Part Idevoted to Physical Processes contains 14 chapters, whereas Part II titled Engineering Processes has 13contributions. The book handles the recent research devoted to numerical simulations of physical andengineering systems. It can be treated as a bridge linking various numerical approaches of two closely inter-related branches of science, i.e. physics and engineering. Since the numerical simulations play a key role inboth theoretical and application oriented research, professional reference books are highly needed by pureresearch scientists, applied mathematicians, engineers as well post-graduate students. In other words, it isexpected that the book will serve as an effective tool in training the mentioned groups of researchers andbeyond.

How to referenceIn order to correctly reference this scholarly work, feel free to copy and paste the following:

Jan Awrejcewicz and Larisa P. Dzyubak (2011). Numerical Analysis of a Rotor Dynamics in the Magneto-Hydrodynamic Field, Numerical Simulations of Physical and Engineering Processes, Prof. Jan Awrejcewicz(Ed.), ISBN: 978-953-307-620-1, InTech, Available from: http://www.intechopen.com/books/numerical-simulations-of-physical-and-engineering-processes/numerical-analysis-of-a-rotor-dynamics-in-the-magneto-hydrodynamic-field

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numerical analysis of a rotor dynamics in the magneto

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