ISSN 1052�6188, Journal of Machinery Manufacture and Reliability, 2013, Vol. 42, No. 3, pp. 192–195. © Allerton Press, Inc., 2013.Original Russian Text © I.I. Blekhman, V.B. Vasil’kov, N.P. Yaroshevich, 2013, published in Problemy Mashinostroeniya i Nadezhnosti Mashin, 2013, No. 3, pp. 18–22.

192

Discovered in the Soviet Union in 1947, self�synchronization of mechanical vibration exciters has, bynow, been studied quite thoroughly. This phenomenon is widely used in vibration equipment, both in Rus�sia and abroad [1–3]. At the same time, the opportunities for improving vibration machines based on thisprinciple are far from being exhausted.

The basic model considered in this study is above�resonance vibration machines with vibroinsulationand straight�line vibrations of the operating part. The machines are driven with two self�synchronizingvibration exciters with a 2nd configuration. That is exactly the scheme on which a large number of modernvibration screens are based [4]. The results, however, allow one to apply the generalization to other typesof machines.

Two opportunities for improving these machines are considered that follow from the studies [1–3, 5–7].The first opportunity is to use a consecutive startup of exciter motors. As a result, the vibration ampli�

tude of the operating part during startup resonance becomes considerably weaker; thus the dynamic pres�sure on machine structures and supports decreases considerably as well. In addition, the electric motorgenerating capacity can be decreased, often determined by the overcoming of a potential machine startupbarrier: if the motor does not have enough power, the velocity of its rotation approaching the free vibrationfrequencies of the machine housing on springs can get stuck (Sommerfeld effect). Ultimately, there is apossibility of decreasing the startup resonance surge current.

The second opportunity is to use vibratory maintenance of unbalanced rotor rotation [1–3]. As aresult, a machine with two unbalanced exciters can run with one electric motor switched off. Hence, thereare practically no electric losses in one of the motors.

Thus the above�described improvements allow one to save energy and the cost of their implementationis minimal.

System Diagram and Equations of Motion

For the dynamic diagram of a vibration machine with two self�synchronizing vibration exciters see thefigure. The two unbalanced vibration exciters (rotors) are placed on a supporting solid body with weightM, that can move in parallel relative to the plane perpendicular to the rotation axes of the exciter rotors.The rotation axes are positioned away from the center of gravity C at equal distances r, and the distancebetween the axes is O1O2 = 2a. The solid body is connected with a fixed base with the help of a system ofresilient and damping elements c and β.

Nominally, the exciter rotors are presumed equal. They are actuated using asynchronous electricmotors and spin in opposite directions. The system in question has five degrees of freedom. The shift ofbody center of gravity C relative to fixed coordinates xOy and body rotation angle ϕ can be considered thegeneralized coordinates of the supporting body. At the same time, x = 0, y = 0, and ϕ = 0 for static equi�librium. The position of the exciter rotors depends on their rotation angles ϕ1 and ϕ2. In the present study,these will be counted from the direction of the x axis in opposite directions (see the figure).

On Some Opportunities for Improving Vibration Machineswith Self�Synchronizing Inert Vibration Exciters

I. I. Blekhman, V. B. Vasil’kov, and N. P. YaroshevichSt. Petersburg, Russia

Received December 10, 2012

Abstract—Two opportunities for improving vibration machines with self�synchronizing unbalanced(inert) vibration exciters with an above�resonance operating vibration frequency are considered. Thefirst opportunity regards machine startup modes, and the second concerns operation in the establishedmode. The proposed improvements allow one to decrease generating motor power, power consump�tion, and the operational dynamic load.

DOI: 10.3103/S1052618813030023

MECHANICSOF MACHINES

JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY Vol. 42 No. 3 2013

ON SOME OPPORTUNITIES FOR IMPROVING VIBRATION MACHINES 193

The system’s motion is described with a system of essentially nonlinear differential equations. Both thesystem’s established motion and its transitional processes have been studied in sufficient detail [4–6].

Transition through Resonance during Routine Startup

Considering soft above�resonance vibration machines (with vibroinsulation), when the free oscillationfrequencies of the supporting body with stopped rotors on resilient elements are far lower (usually at leastthree times as low) than the operating frequency of machine vibrations ω0, i.e., px, py, pϕ < ω0/3. Then theself�synchronization of exciters with equal phases of rotation in different directions appears ϕ1 = ω0t, ϕ2 =ω0t to be steady (see the figure).

In this mode the body executes progressive straight�line vibrations parallel to straight line BC, i.e., per�pendicularly to line O1O2. At the same time x = 2mε sin ωt/M, y ≈ 0, ϕ ≈ 0.

Let us assume that a rotor spinning during startup remains synchronous and in phase. Then the bodyvibrates along axis Ox and the vibration amplitude is calculated according to equations [3, 5]:

(1)

where Bx = , λx = , px = , nx = , and current frequency ω is derivedfrom the following equation:

where L(ω) and R(ω) are, respectively, the rotational torque and the rotation resistance torque of onemotor; I1 is the rotor’s inertial torque;

(2)

is the vibration torque that generates additional pressure on the motor due to vibrations of the body withthe rotor.

The values of λx ≈ 1, λy ≈ 1, and λϕ ≈ 1. Ax and V sharply increase close to resonances at values of nx,ny, and nϕ common for applications. The second and third values indicate relations of frequencies of freebody vibrations along y and rotation vibrations at frequency ω, and ny and nϕ indicate the respectivedimensionless damping coefficients. The resilient and damping elements are supposed to be placed sym�metrically relative to x and y (see the figure). At the same time, the sharp increase in V at the resonanceapproximation is an exact explanation of potential startup sticking of ω0 (Sommerfeld effect). For theexciter to achieve its operating frequency ω0, the motor torque must exceed the vibration torque V(ω).According to equation (2), the weaker the damping nx, the greater is the peak value of this torque. That iswhy it is more difficult to start the machine without a workload than with it.

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β

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m

Figure.

194

JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY Vol. 42 No. 3 2013

BLEKHMAN et al.

Separate Startup of Vibration Exciters

Instead of equations (1) and (2) during separate startup we will have [6]

(3)

where By = , Bϕ = .

Let us consider that frequencies px, py, and pϕ, i.e., λx, λy, and λϕ are sufficiently different from eachother. This can always be achieved by choosing vibration�insulating resilient elements. It will also be reck�

oned that < 1 and nx, ny, and nϕ are approximately the same. Then, with ω changing, each summand

in equation (3), except for the one that corresponds either to ω ≈ py or to ω ≈ px, or ω ≈ pϕ, will be negligiblysmall and the following equation will result:

(4)

To put it differently, if the given conditions are followed the vibration torque affecting each rotor canbe expected to be twice as low during separate startup of vibration exciters. In addition, separate startupallows one to decrease vibration amplitude twofold when resonance areas are passed. With these supposi�tions, rotation body vibrations occur along Oy unlike in the case of combined startup; the amplitude ofthese vibrations, however, is also about twice as low as the combined startup vibration amplitude. It shouldbe noted that, because of some phase difference in the process, certain extraneous vibrations during com�bined startup occur as well. It should also be taken into account that, if px, py, and pϕ are different from

each other the respective resonance vibrations will not be simultaneous.1

Using Vibratory Rotation Maintenance

It has been theoretically and experimentally observed that in many instances self�synchronization isnot disturbed when one or several of the motors are switched off. This effect essentially gave the impetusto the discovery of self�synchronization [2, 3], which is also true for machines constructed according tothe diagram in the Figure. If one or several motors are switched off in these cases only a minor shift in rotorspinning phases will appear. As a result, in addition to main body vibrations along x, minor harmonicvibrations along y and ϕ occur as well, which leads, respectively, to a certain ellipticity in vibrations ofpoints of the body. Usually, it does not significantly disturb the engineering procedure and can evenimprove it on condition that the system’s parameters are set on purpose (in vibratory screens, forinstance). For the conditions of maintaining the synchronous operation of vibroexciters with one of themotors off see [1–3]. In the very same works you will also find the equations for evaluating the degree ofrespective phase shift of the spinning rotor Δα. Because of zero electricity losses in the running motor, theuse of the considered effect allows one to save more power.

Evaluation of Generated Motor Capacity and Power Consumption

Let us use the following designations: Nw = 2Nw1 is the power spent to maintain the established (oper�ating) vibration mode (Nw1 indicates the same but for a single motor); N = 2N1 = qNw = 2qNw1 is the gen�erated motor power necessary to pass the domain of resonance frequencies during synchronous startup(q > 1 is the coefficient that indicates how much N exceeds Nw; N1 is the generated power of one motor;

is the power of one motor that is required to exceed the resonance frequencies during separate startup.

1 According to work [6], from the standpoint of decreasing the amplitude of rundown resonance vibrations, it is reasonable thatthese frequencies be similar (kinetic energy of rotor revolutions has three degrees of freedom). Being more important, however,the requirements facilitating the startup should be followed in the first place.

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MBx

���������, Ay'mε

MBy

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nx

Bx2

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nϕ

Bϕ

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N1'

JOURNAL OF MACHINERY MANUFACTURE AND RELIABILITY Vol. 42 No. 3 2013

ON SOME OPPORTUNITIES FOR IMPROVING VIBRATION MACHINES 195

Let us evaluate the relation among the introduced values. Let us assume that = , i.e., in the

case of separate startup the capacities also relate to one another as vibratory torques (4). However, the sep�arate startup capacity of one motor cannot be lower than Nw1 spent by the motor running. That is why thenecessary separate startup generating capacity of one motor is derived from the following equation:

It follows that generated separate startup capacity of one motor is always less than necessary gen�erated synchronous startup capacity N1 (twice as low at q > 2 and q times as low at q < 2).

The decrease in power consumption during separate startup will result from the elimination of electric�ity losses in the switched�off motor and the increase in the motor efficiency factor resulting from the elim�ination of motor underload in operating mode.

CONCLUSION

The information given above shows that opportunities for improving vibration machines by using self�synchronization are far from being exhausted. It is noteworthy that both proposed measures involve essen�tially no expenses.

ACKNOWLEDGMENTS

This study was supported by the Russian Foundation for Basic Research (project no. 12�08�01009).

REFERENCES

1. Blekhman, I.I., Sinkhronizatsiya dinamicheskikh sistem (Dynamical Systems Synchronisation), Moscow:Nauka, 1971.

2. Blekhman, I.I., Sinkhronizatsiya v prirode i tekhnike (Synchronisation in Nature and Technical Engineering),Moscow: Nauka, 1981.

3. Blekhman, I.I., Vibratsionnaya mekhanika (Vibration Mechanics), Moscow: Fizmatlit, 1994.

4. Vaisberg, L.A., Proektirovanie i raschet vibratsionnykh grokhotov (Design and Calculation of Vibration Screens),Moscow: Nedra, 1986.

5. Blekhman, I.I., Indeitsev, D.A., and Fradkov, A.L., Slow motions in systems with inertial excitation of vibra�tions, J. Mach. Manuf. Reliab., 2008, vol. 37, no. 1, pp. 21–27.

6. Blekhman, I.I. and Yaroshevich, N.P., Transition modes in inertionally excited after�resonance vibrationdevices with a bearing system of several degrees of freedom, in Nelineinye problemy teorii kolebanii i teorii uprav�leniya. Vibratsionnaya mekhanika. Sb. nauchnykh trudov In�ta problem mashinovedeniya RAN (Nonlinear Prob�lems in Oscillation and Control Theories. Vibratory Mechanics. Collection of Scientific Works of the Instituteof Problems of Mechanical Engineering of the Russian Academy of Sciences), Beletskii, V.V., Indeitsev, D.A.,and Fradkov, A.L., Eds., St. Petersburg: Nauka, 2009, pp. 215–238.

7. Rumyantsev, S.A., Dinamika perekhodnykh protsessov i samosinkhronizatsiya dvizhenii vibratsionnykh mashin(Transition Processes Dynamic and Self�Synchronization of Motion for Vibration Machines), Yekaterinburg:Ural Brunch of Russian Acad. Sci., 2003.

Translated by S. Kusnetsov

N1'12��qNw1

N1*

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