An Electromagnetic Damper for Vibration Control of a Transmission Shaft

Copyright © IFAC Control Science and Technology (8th Triennial World Congress) Kyoto, Japan, 1981

AN ELECTROMAGNETIC DAMPER FOR VIBRATION CONTROL OF A TRANSMISSION SHAFT V. Gondhalekar*, R. Holmes** and B. V. Jayawant* ·Inter-University Institute of Engineering Control, University of Sussex, Brighton BNl 9QT, England ··Mechanical and Structural Engineering Division, School of Engineenng and Applied Sciences, University of Sussex, Bnghton BNl 9QT, England Abstract. Rotor bearing systems operating at high speeds present severe problems when passing through their critical speeds. This paper describes a method for controlling the vibrations using electromagnets to provide a velocity dependent controlled external forcing to increase the overall damping within the rotor-bearing system. An experimental marine shaft system is described and results presented which confirm the potential of an electromagnetic damper for similar situations. ~eywords.

Magnetic suspension; multivariable control; nonlinear systems

INTRODUCTION

flexible power-transmission shaft and primary pinion shaft was simulated by means of the experimental rotor bearing system shown in Fig. 1. Control of the vibrational behaviour of the supercritical transmission shaft is achieved by the application of an electromagnetic damper (EMo) at a point in the span and the dynamic characteristics of the system have been investigated experimentally and theoretically.

Increased demands on the performance of rotating machinery have led to the use of supercritical shafts and rotors with attendant problems of vibration. With adequate vibration control, however, span lengths can be chosen without being too influenced by critical speed considerations. Additionally misalignments may be accommodated more easily as for example in helicopter tail rotor shafts (operati~g through as many as twenty critical speeds) and in marine transmissions where hull distortions must be allowed for. A novel scheme of exerting control on the vibration behaviour of flexible shafts particularly when passing through critical speeds, is described. The scheme involves the use of controlled d.c. electromagnets as a means of contactless forcing across an airgap with the shaft as the reaction surface. Conventionally such superficial systems are supported on oil filter bearings mainly because of their robustness. They are, however, responsible for a type of instability known as whip and i t has been demonstrated (oostal, Roberts and Holmes, 1974, 1977) that light external damping applied at some point in the span of a shaft can satisfactorily augment the damping already present to the extent of controlling passage through critical speeds and of inhibiting the onset of instability. Such damping may be achieved by a so-called squeeze film bearing but with much more versatility by the electromagnetic damper developed by the authors.

The electromagnetic damper is shown in Fig. 2, in position around the transmission shaft. 80th the rotor (fitted round the shaft) and the stator are laminated to avoid excessive eddy currents which are found to attenuate the gap flux and hence severely restrict the operating rotational speed range. A simple control block schematic is shown in Fig. 3. The transverse shaft deflection signal x due to the external forcing P is differentiated and the velocity signal is used as an input to the power amplifier which drives the current I into the damper coil. The flux thus generated produces a force of attraction F which opposes the transverse shaft velocity

x

x.

A more detailed control schematic for dealing with deflections both in x and y directions is shown in Fig. 4. This diagram shows the two differentiators and four bidirectional amplifiers and as the electromagnets can only exert force of attraction but not repulsion the components of velocity x and y in thepositive and negative senses are separated at 8 and used for energising different magnets. It has been shown by one of the authors (J ayawant. 1976) that the force of magnetic attraction

DESCRIPTION OF A SIMULATED MARINE PROPULSION SYSTEM A marine propulsion system with turbine shaft,

2141

2142

V. Gondhalekar, R. Holrnes and B. V. Jayawant

on the rotor due to a current I in the coil is proportional to I2/G 2 where 'G' is the gap between the poleface and the rotur surface. The open loop system is unstable (Braunbeck. 1939; Earnshaw. 1842). The system stability is improved by the provision of flux feedback loop. The flux density at each pole face is measured by a Hall effect probe mounted on the magnet face (Jayawant. 1974) the output of which is fed back to the amplifier thus effectively reducing it to a flux amplifier rather than a current amplifier i.e. making the force of attraction independent of the airgap reluctance and in the process enhancing the stability characteristics. In spite of having reduced each amplifiercoil configuration to a flux amplifier. the flux branching off into adjacent paths is not indep endent of the path reluctances thereby introducing a considerable degree of cross coupling between the x and y directions. Considering the specific case of CP1 (Fig. 2) being energised the flux splitting into paths A and 0 is a function of the airgaps G and G . 2 4 If ~A and ~D are the fluxes in paths A and 0

EXPERIMENTS ON THE SIMULATED SYSTEM The shaft bending modes were excited by strapping an unbalanced [I,ass U.B. positioned at 1/3 span length (Fig. 1). The shaft was observed to have a first critical speed at 2820 r.p.m. The rotor system characteristics were observed near the first critical speed by recording whirl orbits at positions CT1, CT2, CT3. CT4 and CT5 (Fig. 1). Static measurements of the damper force characteristics showed a square law dependence of force on e~ citation current. The flux blocking action was confirmed by injecting a signal into one amplifier and observing the flux at adjacent poles with the flux blocking circuits on and off. Typical results are shown in Fig. 5. Analytical Modelling The electromagnetic damping force was derived using calibration charts obtained experimentally. These show the amplifier current as a function of the input voltage and the magnetic force on the shaft as a function of the coil current. The differentiator transfer function T was found to be given approximately by T = 0.0396 w volts/volt

(2)

(1)

where w is the vibration frequency in rad/s. where R2 and R4 are the reluctances of airgaps G2 and G4 and ~ is the flux at CP1. R2 and R4 can be represented by R2 = f (x. y ) and R4 = f 4 (x,y) and consequently 2 gap reluctances as functions of x and y have to be taken into account when considering the flux distribution in the configuration. The major detrimental effect of cross coupling between the flux paths is a reduction in force/ current gains. Referring to Fig. 2, if CP1 and CP2 are energised to generate a force along the + y direction for a displacement of the rotor in the - y direction magnetic flux will be created at all four pole faces with the flux at G and G generating a force in 3 4 the - y direction resulting in a small net force i.e. reduced force/current gain. The gain reduction effect has been overcome by making use of the flux feedback signal to generate a counter m.m.f. to minimise the unwanted or stray flux. Referring again to the above case and Fig. 4, a negative input at y would get through to amplifiers 1 and 2 only with zero demand signals to 3 and 4. The flux appearing at CP3 and CP4 (Fig.2) is an output disturbance for the two closed loop amplifiercoil combinations. The flux feedback loop minimises the disturbance the minimum being an inverse function of the loop gains. This effectively blocks stray flux from appearing in gaps G and G • The added complexity of 3 4 implementing the flux blocking schEme is to use bidirectional amplifiers to drive the coils. A bridge configuration running off a single power supply has been designed for thE purpose.

The relation between the shaft deflection x and the position transducer output voltage was found by linear regression analysis of a set of ten experimental readings. The x and y components of the damper force as functions of the shaft deflection rates in the x and y directions have been derived based on the information above. This electromagnetic damping force is approximately proportional to the coil current squared. It is, therefore. also approximately proportional to the transverse shaft deflection rate squared i.e. F

and

x

F

C(X)2 (3)

C(y)2

Y

where C ~ 1.47 • 10 3 • k 2 and k is the velocity gain coefficient as shown on the extreme left hand in Fig. 4. As the numerical method used for predicting the rotor performance is linear the damping forces need to be expressed as a linear relationship. This is done by considering W. the work done per cycle by F (and F ) as x

y

W =~xdX =fFx.X.dt

(4)

Assuming that the resulting vibratio~ is sinusoidal as an approximation. then x = X sin wt the damping force Fx is given by F x

CX2 sin 2 wt.

o

<

x<

-CX2 sin 2 wt. -X <

x<

X or 0 < wt <

TI

}

0 or TI < wt < 2TI (5 )

An Electromagnetic Damper for Vibration Control Thus W=

•

or W=

for the three settings was respectively. The setting gain coefficient k decides ponse of the damper to the

lit

~fC~2

sin 2wt. X sin wt dwt

III

- ~lcx2

sin 2wt X sin wt dwt

. It 8CX 3 3w

(6)

Assuming that the damping force may be written in terms of an effective linear damping coefb x and the work done by ficient b then F x

x

x

Fx per cycle is b l~

W =fbx'X.X dt = w fX X

2

(7)

o

Equating the two work expressions i.e. equations (6) and (7) gives the effective linear damping coefficient as b

x

8

3TT

Thus b

x

CX

(8)

is a function of the deflection velo-

city amplitude X = wX. Therefore a numerical prediction of shaft vibration must be performed for one rotational speed at a time with band b calculated on the basis of experix

y

mentally measured deflection amplitudes X and Y respectively at that speed. The predicted values of X and Y should then concur with the measured values. Discussion of Results The experimental response of the rotor system in the vicinity of the first critical speed as seen by the position transducer pair CT4 of Fig. 1, is shown in Fig. 6. The four curves correspond to four levels of damping capacity expressed in terms of the settings of the velocity gain coefficient k of 0.0, 0.046, 0.072 and 0.232. The results confirm the successful application of the e.m. damper in the control of the shaft response by positioning the damper at a point in the span. The correspondence between the experimental and theoretically predicted responses is also seen to be very close. Two curves are of particular interest. The one corresponding to zero damping is a check on the accuracy of the theoretical predictions and the one for maximum damping which shows good agreement between the two despite the non linearity of the damping force. The experimental results were obtained by using two different unbalanced masses to eliminate the effect of residual unbalance. Traces of four experimental whirl orbits which include the effect of residual unbalance are shown in Fig. 7 (a). The y were recorded photographicall y from position transducers CTA at the crit ical speed of 2820 r.p.m. for the four values of damping capacity k. The damping force exerted on the shaft is zero at zero setting and a constant at about 9N at settings of 0.046, 0 . 0 72 an d 0 .2 32 whilst the power consumption

2143

53W, 70W and 107W of the velocity the level of resdemand.

The effect of operating the damper on only two channels is also shown in Fig. 7(b). This indicates that the damper can operate quite satisfactorily on two channels so long as they are associated with mutually perpendicular magnets. There is thus a built in safety margin which will enable the damper to tolerate the malfunctioning of at least one channel without a serious build up of vibrations. Another experimental confirmation of the effectiveness of the EMO was obtained at speeds of nearly twice the first critical speed. With zero damping the threshold instability speed was found to be 5510 r.p.m. (predicted 5637 r.p.m.). For damping coefficient settings of 0.046 and 0.072 the threshold speed increased to 5700 r.p.m. and 5795 r.p.m. respectively. For the setting of 0.232 a small steady double loop occupying a small portion of the clearance and indicating a nonsynchronous vibration component of frequency equal to half the rotational speed developed at 5780 r.p.m. This small subharmonic resonance remained constant in size and orientation at all speeds up to the maximum of 6000 r.p.m. For safety reasons in the laboratory this was the limiting speed attainable on the experimental rig. CONCLUSIONS The practicability of controlling the vibration characteristics of a supercriticaltransmission shaft by using controlled d.c. electromagnets has been demonstrated successfully. It has been shown that linearisation leads to good response predictions of the overall dynamics of the system. The damper described in this paper is a composite unit providing damping in both x and y directions. There is the possibility of using two independent orthogonally placed actuator~ Whilst this leads to possible simplification in the control and drive electronics, the electromagnets will be heavier and bulkier. Reference has already been made to the ability of the damper to operate in the case of a malfunction of one channel. Howe ver, the experience of two of the authors (B.V. Jayawant, V.M. Gondhalekar) in connection with their other investigations of vehicles and high speed rotary bearings is such that exceptionally high reliability can be guaranteed for the EMO operation. The extension of the simulated system to a practical marine propulsion or aeronautical transmission system would require larger damping forces but the experience of the two authors with much larger controlled d.c. electromagnets would indicate no serious problems of maintaining adequate bandwidths on the larger electromagnets necessary by increased forcing voltages of the amplifiers. It must

V. Gondhalekar, R. Holmes and B. V. Jayawant

2144

however be borne in mind that the damper characteristics are related to the shaft dynamics and not directly to the transmitted torques. This should. therefore, cover a wide spectrum of possibilities of transmitted torques for similar shaft dynamics. ACKNOWLEDGEMENTS The work described in this paper on shaft dynamics has been funded over the past five years by the Science Research Council (U. K.) by grants to one of the authors (R . Holmes). REFERENCES Braunbeck. W. (1939). Free suspension of bodies in electric and magnetic fields. Zeit Phys .• 112, 11, 753-763. Dostal, M., J. B~oberts, and R. Holmes

(1974). Stability control of flexible shafts supported in oil film bearings. J. Sound and Vibrations, 35, 361-377. Dostal. M., J. B. Roberts, andlR. Holmes (1977). The effects of external damping on the vibration of flexible shafts supported on oil film bearings. J. Sound and Vibrations, 51, 69-87. Earnshaw, S. (1842)-:- On the nature of molecular forces which regulate the constitution of luminiferous ether. Trans. Camb. Phil. Soc .• 7, 97-112. Jayawant. B.V., R.L. Hodkinson, A.R. Wheeler. and R.J. Whorlow (1974). Proc. lEE Conf. on Control Aspects of New Forms of Guided Land Transport, Publ. No. 117. 200-206. Jayawant. B. V., P. K. Sinha. A. R. Wheeler, R. J. Whorlow. and J. Willsher (1976). The development of 1-ton magnetically suspended vehicle using controlled d.c. electromagnets. Proc. lEE,~. 9, 941-948.

eT'

o

I

UB Q

Sl

'\"'l:- "'mm_~'f- m. 'U'

ra

nO'mm

FIG. I

Experimentol Stainless

shaft

FIG.2

Damper

cross-section

steel

shaft

bearing

system

An Electromagnetic Damper for Vibration Control

x

Shaft

F Electro magnets Power amplifi'er FIG, 3

Simplified

loop

control

diagram

signal

differentiator

IspIl'tt ing '

mixing

FI G ,4

flux

t

flux density

time--? Without Flux Block ing - - ma in

t ime--? With Flux Blocking flux

---- 'stray ' flux

Fig

5

Typ ical

Results

of

Flux

B locking

2 145

2146

V. Gondha lekar, R. Holmes and B. V. Jayawa nt ~------r-----~'-'-----r------

12

10 ~

52 E

"~'"

0. E
8

.I

2800 Fig . 6

3200 Speed, rev/min

Synchron oua , . . pon •• - - experime nt.1

- - - - - theoretic . I

o HI (1.2.3.41 11 .31 ---\-Hr-+-j....l\--~~-__

•

Fig .

7

x

Experime nt.1 r•• pon •• orbit.

b