Passive Auto-balancer Research Articles

Synthesizing a resonance anti-phase two-mass vibratory machine whose operation is based on the Sommerfeld effect

This paper reports the synthesized two-mass antiphase resonance vibratory machine with a vibration exciter in the form of a passive auto-balancer. In the vibratory machine, platforms 1 and 2 are viscoelastically attached to the stationary bed and are tied together viscoelastically. A passive auto-balancer is mounted on platform 2. It has been established that the vibratory machine has two resonant frequencies and two corresponding forms of platform oscillations. Such values for the supports’ parameters have been analytically selected at which: ‒ there is an antiphase mode of motion at which platforms 1 and 2 oscillate in the opposite phase and the principal vector of forces acting on the bed (when disregarding the forces of gravity) is zero; ‒ the frequency of platform oscillations under an antiphase mode coincides with the second resonance frequency. The antiphase mode occurs when the loads in an auto-balancer get stuck in the vicinity of the second resonance frequency, which is caused by the Sommerfeld effect. The dynamic characteristics of a vibratory machine have been investigated by numerical methods. It has been established that in the case of small internal and external resistance forces: ‒ there are five theoretically possible modes of load jamming; ‒ the antiphase (second) form of platform oscillations is theoretically implemented under jamming modes 3 and 4; ‒ jamming mode 3 is locally asymptotically stable while jamming mode 4 is unstable; ‒ for the loads to get stuck in the vicinity of the second resonance frequency, the vibratory machine must be provided with the initial conditions close to jamming mode 3, or the rotor must be smoothly accelerated to the working frequency; ‒ the dynamic characteristics of the vibratory machine during operation can be controlled in a wide range by changing both the rotor speed and the number of loads in the auto-balancer. The reported results are applicable for the design of resonant antiphase two-mass vibratory machines for general purposes

Research of anti­resonance three­mass vibratory machine with a vibration exciter in the form of a passive auto­balancer

A three-mass anti-resonance vibratory machine with a vibration exciter in the form of a passive auto-balancer has been analytically synthesized. In the vibratory machine, platforms 1 and 2 are visco-elastically attached to platform 3. Platform 3 is visco-elastically attached to the base. The motion of loads relative to the auto-balancer housing is hindered by the forces of viscous resistance. A theoretical study has shown that the vibratory machine possesses three resonance frequencies and three corresponding forms of platforms' oscillations. Values for the parameters of supports that ensure the existence of an anti-resonance form of motion have been analytically selected. Under an anti-resonance form, platform 3 is almost non-oscillating while platforms 1 and 2 oscillate in the opposite phase. In the vibratory machine, platform 1 can be active (working), platform 2 will then be reactive (a dynamic vibration damper), and vice versa. At the same time, the vibratory machine will operate when mounting a vibration exciter both on platform 1 and platform 2. An anti-resonance form would occur when the loads get stuck in the vicinity of the second resonance frequency of the platforms' oscillations. Given the specific parameters of the vibratory machine, numerical methods were used to investigate its dynamic characteristics. Numerical calculations have shown the following for the case of small internal and external resistance forces in the vibratory machine: ‒ theoretically, there are seven possible modes of load jam; ‒ the second (anti-resonance) form of platform oscillations is theoretically implemented at load jamming modes 3 and 4; ‒ jamming mode 3 is locally asymptotically stable while load jamming mode 4 is unstable; ‒ for the loads to get stuck in the vicinity of the second resonance frequency, one needs to provide the vibratory machine with the initial conditions close to the jamming mode 3, or smoothly accelerate the rotor to the working frequency; ‒ the dynamic characteristics of the vibratory machine can be controlled in a wide range by changing both the rotor speed and the external and internal forces of viscous resistance. The results reported here are applicable for the design of anti-resonance three-mass vibratory machines for general purposes

Motion equations of the single­mass vibratory machine with a rotary­oscillatory motion of the platform and a vibration exciter in the form of a passive auto­balancer

This paper describes a mechanical model of the single-mass vibratory machine with a rotary-oscillatory motion of the platform and a vibration exciter in the form of a passive auto-balancer. The platform can oscillate around a fixed axis. The platform holds a multi-ball, a multi-roller, or a multi-pendulum auto-balancer. The auto-balancer's axis of rotation is parallel to the turning axis of the platform. The auto-balancer rotates relative to the platform at a constant angular velocity. The auto-balancer's casing hosts an unbalanced mass in order to excite rapid oscillations of the platform at rotation speed of the auto-balancer. It was assumed that the balls or rollers roll over rolling tracks inside the auto-balancer's casing without detachment or slip. The relative motion of loads is impeded by the Newtonian forces of viscous resistance. Under a normally operating auto-balancer, the loads (pendulums, balls, rollers) cannot catch up with the casing and get stuck at the resonance frequency of the platform's oscillations. This induces the slow resonant oscillations of the platform. Thus, the auto-balancer is applied to excite the dual-frequency vibrations. Employing the Lagrangian equations of the second kind, we have derived differential motion equations of the vibratory machine. It was established that for the case of a ball-type and a roller-type auto-balancer the differential motion equations of the vibratory machine are similar (with accuracy to signs) and for the case of a pendulum-type vibratory machine, they differ in their form. Differential equations of the vibratory machine motion are recorded for the case of identical loads. The models constructed are applicable both in order to study the dynamics of the respective vibratory machines analytically and in order to perform computational experiments. In analytical research, the models are designed to search for the steady-state motion modes of the vibratory machine, to determine the condition for their existence and stability

On stability of the dual-frequency motion modes of a single-mass vibratory machine with a vibration exciter in the form of a passive auto-balancer

By employing computational experiments, we investigated stability of the dual-frequency modes of motion of a single-mass vibratory machine with translational rectilinear motion of the platform and a vibration exciter in the form of a passive auto-balancer. For the vibratory machines that are actually applied, the forces of external and internal resistance are small, with the mass of loads much less than the mass of the platform. Under these conditions, there are three characteristic rotor speeds. In this case, at the rotor speeds: – lower than the first characteristic speed, there is only one possible frequency at which loads get stuck; it is a pre-resonance frequency; – positioned between the first and second characteristic speeds, there are three possible frequencies at which loads get stuck, among which only one is a pre-resonant frequency; – positioned between the second and third characteristic speeds, there are three possible frequencies at which loads get stuck; all of them are the over-resonant frequencies; – exceeding the third characteristic speed, there is only one possible frequency at which loads get stuck; it is the over-resonant frequency and it is close to the rotor speed. Under a stable dual-frequency motion mode, the loads: create the greatest imbalance; rotate synchronously as a whole, at a pre-resonant frequency. The auto-balancer excites almost perfect dual-frequency vibrations. Deviations of the precise solution (derived by integration) from the approximated solution (established previously using the method of the small parameter) are equivalent to the ratio of the mass of loads to the mass of the entire machine. That is why, for actual machines, deviations do not exceed 2 %. There is the critical speed above which a dual-frequency motion mode loses stability. This speed is less than the second characteristic speed and greatly depends on all dimensionless parameters of the system. At a decrease in the ratio of the mass of balls to the mass of the entire system, critical speed tends to the second characteristic speed. However, this characteristic speed cannot be used for the approximate computation of critical speed due to an error, rapidly increasing at an increase in the ratio of the mass of balls to the mass of the system. Based on the results of a computational experiment, we have derived a function of dimensionless parameters, which makes it possible to approximately calculate the critical speed.

Supercritical Coexistence Behavior of Coupled Oscillating Planar Eccentric Rotor/Autobalancer System

The automatic balancing and undesirable nonsynchronous behavior of coupled oscillating configured flexible foundation and planar eccentric rotor equipped with a passive autobalancer (AB) system has been thoroughly investigated here. Specifically, it is described that the unified AB/rotor unit is attached to a foundation via a symmetric support and the foundation is also mounted on the spring‐damper isolator which allows oscillating only vertically. Therefore, the AB/rotor unit dynamically interacts with the flexible foundation, which is quite analogous to well‐known vertically coupled two‐spring and two‐mass oscillator. Although the single unit AB/rotor system is widely explored in the related AB studies, such coupled arrangement with AB discussed here has not been previously investigated and thus needs to be explored for further application of AB into various vibration isolation problems of other complicated machines/settings. Therefore, solutions for the synchronous stable balanced and the nonsynchronous unstable limit cycle response of AB/rotor/foundation system are obtained via a fixed equilibrium condition and a harmonic like balancing approach. Furthermore, the stability of each response is assessed via a perturbation and Floquet analysis and, for the system parameters and operating speeds, the undesirable coexistence of the wanted stable balanced synchronous response and undesirable nonsynchronous limit cycle has been thoroughly studied. Due to coupled oscillating feature, it is newly found that the multiple limit cycles are encountered in the range of supercritical speeds and more complicated coexistence is attracted into the system, as well as the damping parameters of coupled components (i.e., flexible foundation) influences of the undesirable limit cycle of AB on the particular supercritical speeds. The findings in this paper yield important insights for researchers wishing to utilize automatic balancing devices in more practical rotor systems coupled with additional vibrating mechanical subsystem such as a washing machine or a reciprocating air conditioning compressor with a flexible foundation.

Search for two-frequency motion modes of single-mass vibratory machine with vibration exciter in the form of passive auto-balancer

Dynamics of a single-mass vibratory machine with rectilinear translational motion of the platform and a vibration exciter in the form of a ball, a roller, or a pendulum auto-balancer was analytically explored. The steady-state motion modes, close to dual-frequency modes were found. At these motions, loads in the auto-balancer create constant imbalance, cannot catch up with the rotor and get stuck at a certain frequency. In this way, loads operate as the first vibration exciter, exciting vibrations at frequency of the loads getting stuck. The second vibration exciter is formed by unbalanced mass on the auto-balancer body. The mass rotates at rotor speed and excites more rapid vibrations with this frequency. It was found that despite a strong asymmetry of supports, the auto-balancer excites almost perfect dual-frequency vibrations. Deviations from the dual-frequency law are proportional to the ratio of loads’ mass to the mass of the entire machine and do not exceed 2 %. It was established that at small forces of external and internal resistance, when the loads’ mass is much smaller than the platform’s mass, etc., there are three characteristic rotor speeds. These speeds are larger than the resonance velocity of platform oscillations. At the same time: – at the rotor speeds smaller than the first characteristic speed, there is only frequency when the loads get stuck, in this case it is smaller than the resonance velocity of platform oscillations; – at the above-resonance rotor speeds, located between the first and the second characteristic speeds, there are three frequencies when the loads get stuck, among which only one is below-resonance; – at the above-resonance rotor speeds, located between the second and the third characteristic speeds, there are three frequencies of the loads getting stuck, in this case, they are all above-resonance; – at the above-resonance rotor speeds, exceeding the third characteristic speed, there is only one frequency when the loads get stuck, in addition, it is above-resonant and close to the rotor speed. Only at the rotor speeds smaller than the second characteristic speed, there always exists one, and only one, below-resonance frequency of the loads getting stuck

Search for the conditions for the occurrence of auto-balancing in the framework of a planar model of the rotor mounted on anisotropic viscous-elastic supports

Within the framework of the planar model of the rotor mounted on anisotropic elastic-viscous supports and balanced by a passive auto-balancer, conditions for the occurrence of auto-balancing were analytically determined. An empirical criterion for stability of the main motion was applied. It was found that depending on the forces of viscous resistance in supports, the rotor has one or three critical speeds. These speeds are between two natural frequencies of rotor oscillation in absence of resistance forces in supports. Auto-balancing, respectively, occurs when the single critical speed is exceeded or between the first and the second and above the third critical speeds. At low forces of viscous resistance, the rotor has three critical speeds. The first and the third critical speeds coincide with two natural frequencies of rotor oscillation in absence of resistance forces in supports. The second critical speed is between the first two. An additional (second) critical speed appears when the auto-balancer is mounted on the rotor. In the transition of this speed the behavior of the auto-balancer changes: the auto-balancer reduces the rotor imbalance at slightly lower rotor speeds and increases it at somewhat higher speeds. At finite forces of viscous resistance in supports, depending on the magnitude of these forces, the rotor has one or three critical speeds. At large forces of viscous resistance in supports, the rotor has one critical speed. Depending on the relationship between the coefficients of the forces of viscous resistance, this speed is closer to the smallest or the largest natural frequency of the rotor oscillation. The results obtained were confirmed by computational experiments. It was established that the criterion correctly describes the qualitative behavior of the rotor – auto-balancer system: it determines the number of critical speeds and the region of the auto-balancing onset. Accuracy of determining critical speeds (the boundaries of the regions of auto-balancing onset) increases with: – reduction of the auto-balancer mass with respect to the rotor mass; – an increase in forces of viscous resistance to the motion of correction weights

A new hybrid observer based rotor imbalance vibration control via passive autobalancer and active bearing actuation

Many researchers have explored the use of active bearings, such as non-contact Active Magnetic Bearings (AMB), to control imbalance vibration in rotor systems. Meanwhile, the advantages of a passive Auto-balancer device (ABD) eliminating the imbalance effect of rotor without using other active means have been recently studied. This paper develops a new hybrid imbalance vibration control approach for an ABD-rotor system supported by a normal passive bearing in augmented with an AMB to enhance the balancing and vibration isolation capabilities. Essentially, an ABD consists of several freely moving eccentric balancing masses mounted on the rotor, which, at supercritical operating speeds, act to cancel the rotor's imbalance at steady-state. However, due to the inherent nonlinearity of the ABD, the potential for other, non-synchronous limit-cycle behavior exists resulting in increased rotor vibration. To address this, the algorithm of proposed hybrid control is designed to guarantee globally asymptotic stability of the synchronous balanced condition. This algorithm also incorporates with a “Luenberger-like” observer that continuously estimates the states of a balancer ball circulating around within ABD. In particular, it is shown that the balanced equilibrium can be made globally attractive under the hybrid control strategy, and that the control power levels of AMB are significantly reduced via the addition of the ABD because the control is designed such that it is only switched on for the abnormal operation of ABD and will be disengaged otherwise. Moreover, unlike other imbalance vibration control applications based upon ABD such as rotor speed regulator [21,22], this approach enables the controller to achieve the desirable performance without altering rotor speed once the rotor initially reaches the target speed. These applications are relevant to limited power applications such as in satellite reaction wheels, flywheel energy storage batteries or CD-ROM application.

Equations of motion of vibration machines with a translational motion of platforms and a vibration exciter in the form of a passive auto-balancer

Generalized models have been built of one-, two-, and three-mass vibration machines with a rectilinear translational motion of platforms and a vibration exciter in the form of a ball, a roller, or a pendulum auto-balancer. In the generalized model of a single-mass vibration machine, the platform relies on an elastic-viscous support with the guides enabling the platform’s rectilinear translational motion. A passive auto-balancer is installed on the platform. In the generalized models of two- and three-mass vibration machines, each platform relies on a fixed external elastic-viscous support with the platforms coupled in pairs by elastic-viscous inner supports. The guides allow the platforms to move rectilinearly translationally. A passive auto-balancer is installed on one of the platforms. We have derived differential equations of the motion of vibration machines. The equations are reduced to the form that is independent of the type of an auto-balancer. The models of particular one-, two- and three-mass vibration machines can be obtained from the generalized models by selecting a specific type of the auto-balancer. The models of particular two-mass vibration machines can also be obtained from the corresponding generalized model by rejecting one of the external elastic-viscous supports. The models of particular three-mass vibration machines can also be derived from the corresponding generalized model by rejecting: – one or two external elastic-viscous supports; – one of the three inner elastic-viscous supports; – one or two external elastic-viscous supports and one of the three inner elastic-viscous supports. The constructed models are applicable both for analytical studies into dynamics of the relevant vibration machines and for performing computational experiments. When employed in analytical studies, the models are designed to search for the established modes of a vibration machine motion, to determine conditions for their existence and stability

Parameters optimization of centrifugal juicer with auto-balancer by minimization of time of autobalancing occurred

A method for optimizing the parameters of passive auto-balancer in rotary machines in terms of minimizing the functional quality as a decay time of transition. It is based on the theory of planning multifactorial experiment, analysis of the experimental data using programs STATISTICA_6, MathCad. Technique is aimed at:– construction of the approximating model correlation between functional quality and controlling factors of a rotary machine with auto-balancer;– search the factor values for constructed model at which the functional quality takes the smallest and largest value.Correlation is constructed as the equality of two functional expressions. Left-hand side is the expression relative to the functional quality. Right-hand side is the expression on the factors. The specifics of the problem is that the predicted minimum value of the functional quality should not be less than the acceleration time of the rotor.As the right-hand side offered two varieties of functions: theTaylorseries expansion in powers of the factors of the first and second order; with hyperbolic components. As the left side proposed three types of functions, suitable for finding the values of the functional quality, respectively the largest, smallest, as the smallest and largest.Was conducted testing the proposed method for 3D model of juicer with auto-balancer.The proposed method may be standard in the optimization of the parameters of passive auto-balancers in rotary machines by time decay of transients.

Automatic Balancing of Bladed-Disk/Shaft System via Passive Autobalancer Devices

Autobalancers for rotor/bearing systems are passive devices consisting of eccentrically mounted balance masses that freely revolve around the rotor's axis of rotation. At certain supercritical speeds, the balancer mass positions naturally adjust to cancel the rotor imbalance. This automatic-balancing phenomena occurs as a result of nonlinear dynamic interaction between the balancer masses and the rotor's lateral vibration. Previous studies have found that autobalancers can effectively compensate for mass imbalance in planar rigid rotors such as hard-disk drives and flywheels, however, their use in bladed-disk and turbomachinery applications has not been previously considered. This study explores the dynamics and stability of a flexible bladed-disk/rotor-bearing system equipped with a dual-ball automatic-balancing device. It is found that the autobalancer effectively compensates for both mass and aerodynamic imbalances produced by a bladed-loss condition over a wide revolutions/minute range at speeds above the first lateral natural frequency. It is also shown that for stable automatic balancing to occur, the ratio of automatic balancer damping to the blade aerodynamic drag coefficient must be above some critical value. The analysis demonstrates that the automatic balancer is able to simultaneously reduce both bearing loads and blade deflections for a simulated blade-loss event.