Dynamics of a Rotating System With a Nonlinear Eddy-Current Damper

Dynamics of a Rotating System With a Nonlinear Eddy-Current Damper Y. Kligerman 0. Gottlieb Faculty of Mechanical Engineering, Technlon-lsrael Institute of Technology, Technlon City; Haifa, 32000, Israel

We investigate the nonlinear dynamics and stability of a rotating system with an electromagnetic noncontact eddy-current damper. The damper is modeled by a thin nonmagnetic disk that is translating and rotating with a shaft in an air gap of a direct current electromagnet. The damper dissipates energy of the rotating system lateral vibration through induced eddy-currents. The dynamical system also includes a cubic restoring force representing nonlinear behavior of rubber o-rings supporting the shaft. The equilibrium state of the balanced rotating system with an eddy-current damper becomes unstable via a Hopf bifurcation and exact solutions for the limit cycle radius and frequency of the self-excited oscillation are obtained analytically. Forced vibration induced by the rotating system mass imbalance is also investigated analytically and numerically. System response includes periodic and quasiperiodic solutions. Stability of the periodic solutions obtained from the balanced self-excited motion and the imbalance forced response is analyzed by use of Floquet theory. This analysis enables an explanation of the nonlinear dynamics and stability phenomena documented for rotating systems controlled by electromagnetic eddy-current dampers.

1

Introduction An electromagnetic eddy-current damper is an alternative device for reducing lateral vibration of rotating machinery. The lateral vibration energy is dissipated by eddy-currents that are generated by a magnetic flux running perpendicularly through an electrically conductive, but nonferromagnetic, disk which is firmly attached to a rotating shaft. The damping level can be changed by altering the DC current in the exciting electromagnet coils. Several papers (Gunter et al., 1983; Nagaya et al., 1984; Maxfleld et al., 1989) have indicated that electromagnetic eddycurrent dampers are very effective under certain operating conditions. However, conditions (generally at high speed) in which a rotating system with an eddy-current damper becomes unstable have also been documented (Genta et al., 1992, and Frederick and Darlow, 1994). Following a scheme developed by Schieber (1986) an analytical derivation of the forces acting on the rotor in an eddy-current damper was carried out by Kligerman et al. (1995) where the derived forces include the effects of translation and whirl of the diamagnetic disk in the magnetic field. Linearized expressions of these forces include viscous damping terms and cross coupling stiffness terms. The cross coupling terms were found to be the source of the rotor equilibrium state instability. An analytical criterion of stability was obtained and verified experimentally. Morii et al. (1996) also determined the Lorentz's force both as damping and stiffness coefficients to evaluate rotor stability and demonstrate experimentally that their method is sufficiently accurate for practical use. Analysis of the linear model explains the instability mechanism of the rotating system with an eddy-current damper. However, the linear model of the rotor does not enable analysis of rotor dynamics above the threshold of instability. Furthermore the results obtained experimentally (Frederick and Darlow, 1994, and Kligerman et al., 1995) indicate to synchronous (mode-locking) periodic and aperiodic imbalance response. Consequently, Kligerman et al. (1997) included a hardening Contributed by the Technical Committee on Vibration and Sound for publication in the JOURNAL OF VIBRATION AND ACOUSTICS . Manuscript received March 1997. Associate Technical Editor: J. Q. Sun.

848 / Vol. 120, OCTOBER 1998

cubic restoring force representing nonlinear behavior of the rubber o-rings supporting the shaft bearings to complement the linearized magnetic damping mechanism that was considered previously (Kligerman et al, 1995). Analysis of the unforced system revealed that the equilibrium state becomes unstable via Hopf bifurcation resulting with the emergence of a stable self excited limit cycle around the unstable fixed point. Furthermore, numerical analysis of the forced unbalanced system revealed that low speed operation consist of a periodic solution with the system rotational frequency (corresponding to synchronous whirl) whereas high speed operation was found to be governed by coexisting quasiperiodic solutions. A detailed analysis of nonlinear steady-state response of a rotor-support system is given by Choi and Noah (1987). The study of mode-locking and chaotic phenomena were recently shown by Choi and Noah (1994) for a Jeffcott rotor with a geometric nonhnearity describing bearing clearances. Numerous nonlinear phenomena in rotordynamics (subharmonic vibrations, chaotic response) were also reported by Ehrich (1991, 1992 and 1995) for a rotor with a nonlinear bearing model. Shaw and Shaw (1989 and 1991) analyzed an unbalanced nonlinear rotating system with internal damping. They determined that synchronous and nonsynchronous whirl are possible for specific parameter values. Global behavior in terms of the existence and stability of synchronous steady state solutions is examined. This is followed by an investigation of local behavior near instabilities by an application of the central manifold theorem. Ishida et al. (1996) considered forced oscillations of a vertical continuous rotor with geometric stiffening nonlinearity due to the extension of the rotor center line. They showed that among the various kinds of nonlinear forced oscillations, only combination resonances will occur. Vibrations and stability of a nonlinear rotating system with an eddy-current damper and cubic restoring force are considered in the present work. In order to investigate the interactive influence of the damper on the dynamical system, the rotor model derived includes a complete description of the magnetic damper nonlinearities. Exact solutions for the radii of the limit cycle and the frequencies of the self-excited oscillation for the rotating system are obtained analytically. Furthermore, forced vibration system response includes periodic and quasiperiodic solutions.

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electromagnet, respectively; B^ is the axial component of the primary magnetic flux density. The primary magnetic flux density Bj,, in the narrow air gap of the cylindrical electromagnet, can be expressed as NI B, = M» •

Fig. 1 Layout of the rotating system with an eddy-current damper

Loss of periodicity is analyzed by use of Floquet theory. Results obtained from the complete nonlinear model are compared with those obtained in previous work (Kligerman et al., 1997) for both self excited and forced vibration. 2

Model of Rotating System The model of an eddy-current damper consists of the thin electrically conductive diamagnetic disk attached to a shaft so as to follow its vibration and rotation (Frederick and Darlow, 1994, and Kligerman et al., 1995). The disk moves in the air gap of the direct current excited axial cylindrical electromagnet, Fig. 1. The axis of the electromagnet coincides with the axis of the nonrotating shaft. The path of the ring cross-section magnetic flux runs perpendicularly the disk. Geometrical and physical relations in the damper allow the following assumptions: (i) (ii) (iii) (iv)

the air gap is narrow with respect to the radii of the electromagnet pole; the relative permeability of the electromagnet core material is suppose to be infinite (/i^ ~* Ky = SK' COS KT (Jt' + y 2 ) ] 2 l

(11)

y" + y + \^ +

Ra

v-v

Eq. (15)—synchronous forward whirl; Eq. (16)—nonsynchronous forward whirl; Eq. (17)—nonsynchronous backward whirl.

However, as the physical rotor amplitudes are small with respect to the diamagnetic disk radius (i.e., A< 1), we consider only the solution denoted by line 2 in Fig. 2:

[l-(x'+y')]' Ra

- r j \ l

[1 -(x'

+ y')y

KX = BK

Sin KT

After expanding the terms in curly brackets into a power series we obtain:

Furthermore, it can be shown that the magnitude for this amplitude obtained from Eqs. (12) with e = 0 is p = 1; A =

+ {0

-

Uo(x^

+ y^)}

Ky = BK^ cos KT

(12)

y" + y + [y - u^ix^ + y ' ) } y ' -

{(3 -

l/o(x^

+ y^)}KX

(19)

1)

The limit cycle solution [Eqs. (14) and (19)] in dimensional form can be written as u = A cos tjj;

w = A" sin ujj

= BK^ s i n KT

The model of the rotating system with nonlinear eddy-current damper, Eq. (12), has two degrees of freedom (x and y) and is characterized by five independent parameters y, I3,v„, B and K. The nondimensional yalues /? and y are defined by Eq. (9) and Ua = 2RaT]. Self-Excited Oscillation The unforced dynamical system (Eq. (7) with e = 0) has a unique fixed point at the origin (x = 0) (the trivial equilibrium

where

A=a-J-^

liy„{K

-

^

1)

(20)

The expression under square root in Eq. (19) is positive if /3K:

- 7 s 0 or

(K:

3

850 / Vol. 120, OCTOBER 1998

0K - y l^o(K -

x" + X + {y - u„(x^ + y^)}x'

Since yl0 following

- 1) s 0

{0K - 7 > 0 \ [(/c - 1) > 0

(21)

= (P + t,)/l3 a 1, Eq. (21) is equivalent to the

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Ala 1

I

I

I

1

1

.

- Stable Limit Cycle - - Unstable Limit Cycle

r '

-1

1

-p-n

T •

-- "^ -- -

2.5

1 •

51.5

2

-

•

Z 0.2

—e—'—o—

-1 — > « —

1 X ' Hopf bifurcalion point

2

-•—--L^

'

— 1 — - ^ — 1 — 1 -

1

4 6 8 Nondimensional Rotational Frequency, K

or

^>I_^ll ^ - / 3 "

^

(22)

4 6 Nondimensional Rotaliontil frequency, K

8

the order of the damper radius (a). Thus, we do not consider it in this work as it is beyond the scope of the model formulated. 3.2 Rotor With Cubic Restoring Force and Nonlinear Eddy-Current Damper. This section deals with the self-excited oscillation corresponding to the solution of the homogeneous differential equations (Eq. (8) with e = 0) above the Hopf bifurcation point. Numerical simulation of Eqs. (7) and (8) shows the solution corresponding to the stable limit cycle of self excited oscillation over a wide range of the operational speeds is given by Eq. (14). After substitution of Eq. (14) into Eq. (8) the following nonlinear algebraic equations for the evaluation of the limit cycle radius and the self excited oscillation frequency are obtained 'A^ = p^-

1

Kp^

+ l V + H _ ^ L = o (25)

(23) Numerical solution of Eq. (25) shows that the discriminant of

where

0

0 A =

2

Fig. 3 Stability analysis of ttie self-excited oscillation limit cycle

The first condition of Eq. (22) defines the range 0 =s w s w,, of limit cycle Eq. (20) existence whereas the second condition defines limit cycle existence beyond the Hopf bifurcation point. Note that the limit cycle between a; = 0 and w = w,, grows from A = a-iylv„. Consequently, we consider the limiting amplitude, Eq. (20), for K > KH only (fine 1 in Fig. 2). Stability of the limit cycle can be demonstrated by Floquet theory for autonomous systems. We formulate the variational equations for the dynamical system by substitution of Zi = x + 6zi and Zi = y + Sz^ instead of jc and y into Eqs. (12) with e = 0, respectively, and the following linearization relative to small perturbation (5z = [Szi Szi &z^ 6z^]'• 6i, = A6z

HopI bifurcalion point

0

10

Fig. 2 Limit cycle radius of self-excited oscillation (0, stable, x, unstable equilibrium state)

1

I.

IvMx

+ Ky) - ) v„ix^ + y^) - y 0 0 _v„[2xy - K(y^ + 3x^)] + KJS 0

i^A^yx + K{X^ + 3y^)] - Kp 0 2i'„y{y - KX) - \

and X and y are defined by Eq. (14). Calculation of the monodromy matrix (Nayfeh and Balachandran, 1995) yields the system Floquet multipliers presented in Fig. 3 versus the nondimensional rotor angular velocity. One of the multipliers is equal to unity and corresponds to the periodic solution of the linearized differential equation for small perturbation. Two other multipliers coincide, are less than unity and correspond to complex conjugate eigenvalues of the monodromy matrix. The fourth multiplier corresponds to a real eigenvalue of the monodromy matrix. This multiplier is less than one only in the range between the Hopf bifurcation point and specific supercritical angular velocity (K, = 5 in Fig. 3) and it becomes greater than one at higher operational speeds. Thus the limit cycle, given by Eqs. (20), is stable in the limited range between KH < K < K^ as, is depicted in Fig. 2. Furthermore, numerical simulation of Eqs. (11) and (12) with e = 0 reveals that the stable periodic limit cycle loses its stability («, = 5) to an additional large amplitude periodic solution. The amplitude of the additional solution is of

Journal of Vibration and Acoustics

0 0 1 v„{x^ + y^)

(24)

cubic polynomial in p is negative for various parameter values. Thus, coexisting limit cycles are found. However, it can be shown that one root is always negative, and that the radius of the second root is great than the damper radius. Thus we consider only the small positive root of Eq. (25). Application of Floquet theory to this solution shows that it is always stable. Dependence of the nondimensional limit cycle radius on the nondimensional rotational frequency for the rotating system with cubic restoring force and nonlinear eddy-current damper, obtained from the Eq. (25), is presented by line 4 in Fig. 2. We also consider the limit cycle obtained for the rotor system with a linear eddy-current damper and a cubic restoring force (Kligerman et al., 1997) and presented by line 3 in Fig. 2. The results for the rotating system with a cubic restoring force and a nonlinear eddy-current damper are close to those obtained for the case of the linear eddy-current damper, particularly in the range of the relatively low operational speeds immediately after Hopf bifurcation point. Consequently, nonlinear terms in the expression of the force acting on the rotor in the eddy-current

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•UF7¥ la r

' ''

' '' ' -

0.025

•s

lU

a o

a 1

n -10

References Choi, Y-S., and Noah, S.T., 1987, "Nonlinear Steady-State Response of a

-20 0 Nondimensional Displacement a)

25 20 15 10 5 0 -5 -10 -15 0 Nondimensional Displacement b)

Fig. 6 Poincare maps (a) rotating system with cubic restoring force and linear eddy-current damper (b) rotating system with cubic restoring force and nonlinear eddy-current damper

and a cubic restoring force. Solutions for the stable limit cycle radius and frequency of self excited oscillation were found analytically. Synchronous imbalance response of the rotating system with a nonlinear eddy current damper and a cubic restoring force was investigated analytically. Stability of the periodic solutions was investigated by use of Floquet theory. Numerical simulation verified the analytical solution and demonstrated that the eddy-current damper losses its stability to quasiperiodic response. The stability threshold was found to be close to the Hopf bifurcation point obtained for the balanced analysis. A comparison of results to those obtained in previous work with a linear damper showed that the influence of nonlinear damping terms is negligible near the stability threshold. It was also shown in the present paper that an eddy-current damper does not dissipate energy of synchronous forced vibration but is an efficient method for reducing nonsynchronous lateral vibration. Note that modification of the eddy-current damper design can lead to stable operation at supercritical conditions above the Hopf bifurcation point and help to limit the amplitude of synchronous imbalance response. This modifica-

Journal of Vibration and Acoustics

Rotor-Support System," ASME JOURNAL OF VIBRATION, ACOUSTICS, STRESS, AND RELIABILITY IN DESIGN, Vol. 109, No. 3, pp. 255-261.

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