Research of Stability and Transition Processes of the Flexible Double-support Rotor with Auto-balancers Near Support

Within the discrete model the stability of main motions and transition processes of the flexible unbalanced two-support rotor at its balancing by two passive auto-balancers located in close proximity to supports is investigated.The simplified system of differential equations describing the process of auto-balancing of the flexible rotor with respect to four Lagrange coordinates – displacements of the shaft in supports and the given total rotor unbalances is received.It is shown that the received system of equations accurate within designations matches the equations describing the process of dynamic auto-balancing of the rigid rotor on pliable supports with two auto-balancers. Therefore, main motions of the flexible rotor on condition of their existence are always steady on above resonance velocities of rotation.At velocities close to any critical velocity the conditions of existence of main motions can be violated. For expansion of the area of stability of main motions it is necessary to increase the balancing capacity of auto-balancers.Analytically (using the roots of the characteristic equation) the assessment of duration of passing of transition processes when balancing the flexible rotor is carried out. At the same time, it is established that:– transition processes are divided into: Fast at which Fast relative motions of corrective weights stop and the motion of rotor corresponding to the current given total rotor unbalances of the flexible rotor is established; slow at which corrective weights come to the auto-balancing positions;– at the increase in forces of resistance to relative motion of corrective weights duration of exit of corrective weights to the cruiser velocity of rotor decreases and duration of arrival of corrective weights to the auto-balancing position increases;– duration of passing of transition processes does not decrease at the reduction of the mass of corrective weights, rigidity of supports, remoteness of supports from the center of mass of the flexible rotor;– duration of passing of transition processes does not increase at the increase in the cruiser velocity of the rotor at velocities higher than the first critical (if at the same time the conditions of existence of main motions are not violated)

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

Experimental Research of Rectilinear Translational Vibrations of a Screen Box Excited by a Ball Balancer

The vibrations of the screen box excited by the ball balancer are experimentally investigated. The stand of the screen and two-frequency vibration exciter in the form of the ball balancer has been created for this purpose. The elastic supports of the box allow it to perform three principal vibrational motions corresponding to three resonant speeds of the shaft rotation: vertical rectilinear, angular oscillatory around one of two transverse central axes of the box.During the adjustment of the stand, it has been defined that the ball balancer excites almost ideal two-frequency vibrations of the screen box. The slow frequency corresponds to the rotational speed of the balls centers around the longitudinal axle of the shaft, and the Fast one corresponds to the rotational speed of the shaft with the unbalanced mass attached to it. The two-frequency mode arises in a broad range of parameters and its general characteristics can be changed by changing: the mass of the balls and unbalanced mass; angular velocity of the shaft rotation.It is experimentally defined that the balls get stuck at the lowest resonance frequency, thus exciting the first form of the box vibrations. As a result, the box makes almost pure translational rectilinear vibrations and there are no angular components. In this regard, there is no need to impose additional kinematic constraints on the box motions.On the assumption that the box makes two-frequency vibrations, in the software package for statistical analysis Statistica, the coefficients in the relevant law have been picked up. At the same time, it has been defined that:– the process of determination of the coefficients values is stable (robust), the coefficients practically do not vary with the time interval of measurement of the law of the box motion;– the amplitude of the slow vibrations is directly proportional to the total mass of the balls;– the amplitude of the Fast vibrations is directly proportional to the unbalanced mass.The discrepancy between the laws of motion obtained experimentally and with the statistical analysis methods is less than 1 %

Experimental Study Into Rotational-oscillatory Vibrations of a Vibration Machine Platform Excited by the Ball Auto-balancer

We have experimentally investigated the rotational-oscillatory vibrations of vibratory machine platform excited by the ball auto-balancer.The law of change in the vibration accelerations at a platform was studied using the accelerometer sensors, a board of the analog-to-digital converter with an USB interface and a PC. The amplitude of rapid and slow vibratory displacements of the platform was investigated employing a laser beam.It was established that the resonance frequency (frequency of natural oscillations) of the platform is: 62.006 rad/s for the platform with a mass of 2,000 gm; 58.644 rad/s ‒ of 2,180 gm; 55.755 rad/s ‒ of 2,360 gm. An error in determining the frequencies does not exceed 0.2 %.The ball auto-balancer excites almost perfect dual-frequency vibrations of a vibratory machine platform. Slow frequency corresponds to the rotational speed of the center of balls around the longitudinal axis of the shaft, while the Fast one ‒ to the shaft rotation speed, with the unbalanced mass attached to it. A dual-frequency mode occurs in a wide range of change in the parameters and it is possible to alter its basic characteristics by changing the mass of balls and the unbalanced mass, the angular velocity of shaft rotation.It has been established experimentally that the balls get stuck at a frequency that is approximately 1 % lower than the resonance frequency of platform oscillations.Assuming the platform executes the dual frequency oscillations, we employed the software package for statistical analysis Statistica to select coefficients for the respective law. It was found that:– the process for determining the magnitudes of coefficients is steady (robust); coefficients almost do no change when altering the time interval for measuring the law of a platform motion;– the amplitude of accelerations due to the low oscillations is directly proportional to the total mass of the balls and the square of the frequency at which balls get stuck;– the amplitude of rapid oscillations is directly proportional to the unbalanced mass at the auto-balancer's casing and to the square of angular velocity of shaft rotation.The discrepancy between the law of motion, obtained experimentally, and the law, obtained using the methods of statistical analysis, is less than 3 %. The results obtained add relevance to both the analytical studies into dynamics of the examined vibratory machine and to the creation of the prototype a vibratory machine

An Increase of the Balancing Capacity of Ball or Roller-type Auto-balancers with Reduction of TIME of Achieving Auto-balancing

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

Search for the Dual­frequency Motion Modes of a Dual­mass Vibratory Machine with a Vibration Exciter in the Form of Passive Auto­balancer

We analytically investigated dynamics of the vibratory machine with rectilinear translational motion of platforms and a vibration exciter in the form of a ball, a roller, or a pendulum auto-balancer.The existence of steady-state motion modes of the vibratory machine was established, which are close to the dual-frequency regimes. Under 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 serve as the first vibration exciter, inducing vibrations with the frequency at which loads get stuck. The second vibration exciter is formed by the unbalanced mass on the casing of the auto-balancer. The mass rotates at rotor speed and excites Faster vibrations of this frequency. 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 %.A dual-frequency vibratory machine has two oscillation eigenfrequencies. Loads can get stuck only at speeds close to the eigenfrequencies of vibratory machine's oscillations, or to the rotor rotation frequency.The vibratory machine has always one, and only one, frequency at which loads get stuck, which is slightly lower than the rotor speed.At low rotor speeds, there is only one frequency at which loads get stuck.In the case of small viscous resistance forces in the supports, at an increase in the rotor speed, the quantity of frequencies at which loads get stuck in a vibratory machine increases, first, to 3, then to 5. In this case, new frequencies at which loads get stuck:– occur in pairs in the vicinity of each eigenfrequency of the vibratory machine's oscillations;– one of the frequencies is slightly lower, while the other is slightly higher, than the eigenfrequency of vibratory machine's oscillations.Arbitrary viscous resistance forces in the supports may interfere with the emergence of new frequencies at which loads get stuck. That is why, in the most general case, the quantity of such frequencies can be 1, 3, or 5, depending on the rotor speed and the magnitudes of viscous resistance forces in supports