Multi-frequency vibrators have advantages in bulk materials processing but their design is usually complicated. This article presents the synthesis of design parameters of a two-frequency inertial vibrator according to the specified power characteristics. Based on the developed mathematical model, the parameters of variable periodic force is derived for two angular velocities 157, 314 rad/s and their ratios 0.5 and 2. In the case of the 0.5 ratio, the instant angular velocity of the resulting force vector is 2.0–3.5 times greater than for ratio 2. A dynamical model of vibrating screen with the synthesized inertial drive is considered. It was found that at the ratio of angular velocities 0.5, the second harmonic of acceleration prevails at 50 Hz, while at the ratio of 2, the first harmonic has a greater amplitude at 25 Hz. For the first variant, the power does not depend on the initial angle between unbalances, and at the second variant, it can vary. The angle of rotation of unbalances affects the trajectory of the centre of mass and the phases of the harmonics but does not affect their amplitude. Due to such dynamical features, the two-motor inertial drive allows the vibrating machines to operate at a wider range of frequencies and amplitudes.

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Dynamic Analysis of an Enhanced Multi-Frequency Inertial Exciter for Industrial Vibrating Machines

The low energy efficiency and excessive power of electric motors of large-scale vibrating machines for processing bulk materials motivated a new design of the inertial drive. This drive consists of one motor and two coaxial unbalanced masses, whose rotational frequencies are related in the ratio 2:1. This approach allows for a generation of the excitation force with variable amplitude and frequency, which changes depending on the inertial characteristics and shaft rotation frequency and does not relate to the phase difference of the unbalanced masses. Because of this, the symmetry axis of the resulting vector hodograph can be changed. The spectral composition of the exciting force up to 200 Hz contains higher harmonics, the energy share of which is 25.4% from the 2nd harmonic and 14.1% from the 3rd and higher harmonics that correspondingly improves bulk material treatment in comparison to single-frequency vibrators. The finite element model is used for checking the strength capacity of the most loaded units of a dual-frequency drive. Its use allows the realization of complex trajectories of motion that are more technologically efficient for variable parameters of the treated media and energy saving in sieving screens and other vibrating machines.

Energy-Saving Inertial Drive for Dual-Frequency Excitation of Vibrating Machines

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

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Equations of motion of vibration machines with a translational motion of platforms and a vibration exciter in the form of a 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

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

The study has revealed an influence of the parameters of corrective weights (balls and cylindrical rollers) in auto-balancers on the balancing capacity and the duration of the transition processes of auto-balancing in fast-rotating rotors. A compact analytical function has been obtained to determine the balancing capacity of an auto-balancer (for any quantity of corrective weights – balls or rollers), with a subsequent analysis thereof. It is shown that the process of approach of the auto-balancing can be accelerated if the auto-balancer contains at least three corrective weights. It has been proved that at a fixed radius of the corrective weights the highest balancing capacity of an auto-balancer is achieved when the corrective weights occupy nearly half of the racetrack. The study has revealed that it is technically incorrect to formulate a problem of finding a radius of the corrective weights that would maximize the balancing capacity of the auto-balancer. The statement implies that if it is a ball auto-balancer, the racetrack is a sphere, but if it is a roller-type balancer, the racetrack is a cylinder. This leads to a practically useless result, suggesting that the highest balancing capacity is achieved by auto-balancers with one corrective weight. Besides, with n≥5 for balls and n≥8 for rollers, there happens a false optimization, which consists in several corrective weights being “excess”. Their removal increases the balancing capacity of the auto-balancer. It is correct (from the engineering point of view) that the mathematical task is to optimize the balancing capacity of an auto-balancer. Herewith, it is taken into account that the racetrack of the auto-balancer is torus-shaped, which restricts the radius of the corrective weights from the top. It is shown that the balancing capacity of an automatic balancer can be maximized if in a fixed volume the corrective weights have the largest possible radius and occupy almost a half of the racetrack. The research on the duration of the transition processes for the smallest value has produced the following conclusions: – to accelerate the achieving auto-balancing, the corrective weights should occupy nearly half of the racetrack; – the shortest time of the auto-balancing is achieved with three balls or five cylindrical rollers.

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An increase of the balancing capacity of ball or roller-type auto-balancers with reduction of time of achieving auto-balancing

We formulated the empirical criterion for the occurrence of auto­balancing for the rotors balanced by passive auto­balancers. The criterion is applicable for rigid and elastic rotors on ductile supports and for elastic rotors on rigid supports. The criterion is intended to answer the question if it is possible in principle, and under what conditions, to automatically balance a particular rotor n by passive auto­balancers of any type in n planes of correction. In accordance with the criterion, the possibility of applying passive auto­balancers for rotor balancing (in “zero approximation”) is determined not by the type of auto­balancers, but by rotor itself. In this case, reaction of rotor to the elementary imbalances, applied in the required planes of correction, is essential. That is why the criterion makes it possible to obtain universal conditions for the occurrence of auto­balancing, applied for any types of auto­balancers. The criterion is applied in the following sequence. 1. A physical­mechanical model of rotor with elementary imbalances, located in the required planes of correction, is described. 2. Differential equations of motion of the unbalanced rotor are derived. 3. Steady motion of a rotor, which corresponds to the applied elementary imbalances, is searched for. 4. A functional of the criterion for the occurrence of auto­balancing is built. As a rule, this is a quadratic form from elementary imbalances. 5. By analysis of the functional (sign definiteness of the obtained quadratic form), conditions for the occurrence of auto­balancing are determined. The result is conditions of two types. The first ones set limitations to the mass­inertia rotor characteristics. The second ones are a range of angular speeds of rotor rotation, at which auto­balancing will occur provided the first conditions are met. The criterion is used for the axisymmetric rotor with a fixed point and isotropic elastic support. It was found that auto­balancing will occur only in the case of a long rotor, relative to the point O, independent of the number of auto­balancers (planes of correction) at the speeds, which exceed the only resonance speed of rotor rotation.

http://journals.uran.ua/eejet/article/download/79970/77250

Empirical criterion for the occurrence of auto-balancing and its application for axisymmetric rotor with a fixed point and isotropic elastic support

The 3D model of the screen stand with the vibrational-rotational duct motion was developed. The ball-type auto-balancer, which makes it possible to create the two-frequency vibrations, is used as the vibration exciter. The main parameters, which influence the stability of the dual frequency vibrations, were defined after adjusting and testing the model. It was established that the ranges of the dual frequency vibrations are relatively large, which makes it possible to change the characteristics of vibrations with a change in the parameters from these ranges. An increase in the summary mass of the spheres increases the amplitude of slow vibrations of the duct masses in direct proportion. This increases in direct proportion the vibration energy directed toward the execution of the main technical process. An increase in the unbalanced mass on the auto-balancer case increases the amplitude of rapid vibrations of the duct masses center in direct proportion. It was established that an increase in the rotation frequency of the rotor increases the amplitude of the rapid vibration speeds of the duct in direct proportion. This increases the vibration energy directed toward the duct self-cleaning and the change through the vibrations of the mechanical properties of the workable material in proportion to the square of rotation frequency of the rotor. The simulation showed that the auto-balancer works as two separate vibration exciters. In the first one, the spheres rotate practically evenly with the resonance frequency of the duct vibrations, at this, independent of its loads, the spheres automatically adjust to this frequency, by which they excite the slow resonance duct vibrations (12 Hz) with a large amplitude. In the second one, the mass on the AB case excites the rapid duct vibrations with (any) existing non-resonant rotation frequency of the rotor.

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Research into excitation of dual frequency vibrational-rotational vibrations of screen duct by ball-type auto-balancer

A discrete N­mass model of a flexible double­support rotor with two passive automatic balancers of pendulum, ball or roller type was constructed. Automatic balancers are placed near supports. The system of differential equations, which describes the motion of a rotor machine, is obtained. The primary (sustained) motions of a system as the motions, in which automatic balancers eliminated displacements of a rotor in supports, were found. It is shown that on the primary motions, the total imbalances of a rotor and AB, reduced to two correction planes (supports), equal zero. It was proposed to examine the stability (of the family) of sustained motions by generalized coordinates, which set the displacement of a rotor in the supports and by dynamic variables that equal total imbalances of a rotor and AB in two correction planes. We obtained differential equations, which describe the change in these variables that describe the process of self­balancing. By the analysis of differential equations of the motion of a system it was established that: – on the primary motions, AB eliminate rotor deflections and vibrations in elastic viscous supports, but do not remove shaft deflections in non­supporting points; – on the primary motions elastic viscous supports are conditionally converted into hinge supports; – shaft deflections in non­supporting points and the primary motions change with the change in angular speed of rotation of the rotor; – primary motions exist at a certain distance of the speed of rotation of the rotor from the critical speeds of flexible rotor rotation with the hinge supports instead of elastic viscous supports; – at the speeds of rotation of a rotor shaft close to any of these speeds, the conditions of existence of the primary motions are disrupted because shaft deflections theoretically grow to infinity and the balancing capacity of AB is not sufficient for the compensation for the imbalances of the rotor; – in practice these deflections are limited and, therefore, proper selection of the balancing capacity of AB can ensure existence of primary motions at all speeds of rotation of rotor.

http://journals.uran.ua/eejet/article/download/75115/71829

Kostyantyn Dumenko

Papers

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

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

Empirical criterion for the occurrence of auto-balancing and its application for axisymmetric rotor with a fixed point and isotropic elastic support

Research into excitation of dual frequency vibrational-rotational vibrations of screen duct by ball-type auto-balancer

Studying the peculiarities of balancing of flexible double-support rotors by two passive automatic balancers placed near supports