Ryzhik B.; Duckstein H.; Sperling L.

Automatic Balancing of the Unsymmetrical Rigid Rotor

The paper is devoted to the investigation of an automatic balancing device for rigid rotors. It presents some new re-sults concerning automatic balancing, namely the effect of partial unbalance compensation and reduction of vibrationsin the region after the first critical speed.

1. Introduction

The autobalancing device under consideration consists of two circular tracks with moving elements (balls, rollers).Under certain conditions elements seek positions such as to compensate, partially or completely, unbalanced inertiaforces. The first device of this type, comprising two balls in one track, was introduced by Thearle and Schenec-tady [1] in 1932. In 1977 the device was generalized by Hedaya and Sharp [2] proposing a dynamic balancerwith two circular tracks and four balls to compensate both the unbalanced force and the unbalanced moment. Sincethen, autobalancing of rigid rotors by employing devices of this type has become a well-known phenomenon (seebibliography in Blekhman 2000 [3], Sperling et al. 1998 [4]). Nevertheless, some important results, sufficientlyexpanding the field of possible applications, were obtained only recently.

2. Partial Compensation of the Unbalance

The presented results were obtained during investigation of rotor systems with an essential unsymmetry of supporting(sufficient non-diagonal member of the support elasticity coefficient matrix) and a large polar moment of inertia.Earlier [5,6], authors had analyzed the effect of complete unbalance compensation and established that completestable compensation of both the unbalanced force and the unbalanced moment is only possible in systems with atransverse moment of inertia greater than the polar one in the speed range after the second critical speed. Presentpaper reveals that under certain conditions an autobalancing device can partially compensate unbalance and decreasevibrations in the region after the first critical speed. Such compensation can be achieved even if the rotor polarmoment of inertia exceeds the transverse moment.Fig. 1 shows a rotor system model. The equations of motion, linearized in the vibrational co-ordinates, are presentedin [6]. For investigating possible solutions the method of direct separation of motion was employed, which proved tobe an efficient tool for analyzing synchronization problems. The existence conditions for synchronous motions yieldthe equations for determining angular positions of compensating elements. For a rotor with two primary unbalancesand n compensating elements these equations are [6]

V 0i = V 0

i (α∗1, . . . , α

∗n) = fi

n∑

k=1

fkAik sin (α∗i − α∗

k) + fi

2∑

k=1

f0kA0

ik sin(α∗

i − α0k

)= 0, i = 1, . . . , n , (1)

where α∗1, . . . , α

∗n; α0

1, α02 are the angular co-ordinates of the compensating elements and the primary unbalances;

fk = mkεkΩ2, k = 1, . . . , n; f0k = m0

kε0kΩ2, k = 1, 2 are the corresponding centrifugal forces (mk, m0

k are themasses, εk, ε0

k - the eccentricities and Ω - the rotational speed); and Aik, A0ik are the so-called harmonic influence

coefficients, defined as the vibration amplitude at the path center of element i caused by a harmonic force excitationwith an amplitude of value 1 at the path center of element k or of the corresponding primary unbalance.The analysis shows that the equations (1) yield several solutions of different types. To establish the actual motionsthat may occur, a stability investigation is required.The publications [5,6] proved that the equations (1) yield a solution provided complete compensation of unbalancedforce and moment. This solution is stable in the frequency range after the second critical speed. The possibility ofcomplete compensation is very important for symmetrical or nearly symmetrical rotor systems with a comparativelysmall polar moment of inertia.For systems with an essential unsymmetry or a large rotational inertia moment nominal speeds typically lay in thefrequency range after the first, however before the second critical speed, and hence complete compensation by meansof an autobalancing device is not possible. Fortunately, an analysis shows that the equations (1) yield a solution ofa different type, which may be stable and provide a ”partial compensation” of unbalanced forces and a reduction

PAMM · Proc. Appl. Math. Mech. 2, 70–71 (2003) / DOI 10.1002/pamm.200310021

of vibrations in certain frequency range after the first critical speed. This analytical conclusion was confirmed byresults of computer simulation performed employing Advanced Continuous Simulation Language (ACSL).Computations show that after passing the first critical speed balls synchronize with the rotor. Under specificconditions they seek positions such as to partially compensate unbalanced forces. The ”level” of compensationdepends on the axial co-ordinate of the device planes, but for a reasonable choice of planes the decrease of vibrationslooks quite substantial. The process of partial compensation is illustrated by Fig. 2 demonstrating the amplitude-frequency characteristics of vibration of the rotor top point.

Figure 1: Rotor system model.

Vib

rati

on

amp

litu

de,

mm

Rotor speed, rad/s

Figure 2: Amplitude of vibration of rotor top.

3. Conclusion

Partial compensation of unbalance seems to be a very important feature of the autobalancing device. Due to thepartial compensation autobalancing devices can be used in unsymmetrical rotor systems with a large polar momentof inertia, like centrifugal rotors. Among the other applications of device the most promising are hand power toolsand washing machines. Recently an information about the use of autobalancing devices in optical disk drives hasappeared in the press.

Acknowledgements

The authors would like to express gratitude to the Deutsche Forschungsgemeinschaft for providing financial support(No.Sp 462/7-1).

4. References

1 Thearle E.; Schenectady N.: A New Type of Dynamic-Balancing Machine. Trans. of ASME, N54(12) (1932), 131–141.2 Hedaya M.; Sharp R.: An Analysis of a New Type of Automatic Balancer. Journal Mechanical Engineering Science,

N19(5), (1977), 221–226.3 Blekhman I.: Vibrational Mechanics, World Scientific, 509, (2000).4 Sperling L.; Merten F.; Duckstein H.: Analytical and Numerical Investigations of Rotation-Vibration-Phenomena.

Nonlinear Oscillations in Mechanical Systems, Proceedings of the XXV-XXVI Summer Schools, St. Petersburg, Russia,1998, Volume 1, (1997,1998), 145–159.

5 Sperling L.; Duckstein H.: (2001) Zum selbstttigen Auswuchten des starren Rotors in zwei Ebenen. Proceedings of theSIRM 2001, Wien, (2001), 161–168.

6 Sperling L.; Ryzhik B.; Linz Ch.; Duckstein H.: Simulation of two-plane automatic balancing of a rigid rotor.Mathematics and Computers in Simulation, N58, (2002), 351–365.

Dr.-Ing. B. Ryzhik; Dr.-Ing. H. Duckstein; Prof. Dr.-Ing. habil. L. Sperling, Otto-von-Guericke-Universitat Magdeburg, Institut fur Mechanik, Universitatsplatz 2, D-39106 Magdeburg, Germany

Section 1: Linear and Nonlinear Oscillations 71