Analytical and Experimental Analysis of a Self-Compensating Dynamic Balancer in a Rotating Mechanism

1996 ◽  
Vol 118 (3) ◽  
pp. 468-475 ◽  
Author(s):  
Jongkil Lee ◽  
W. K. Van Moorhem
Keyword(s):  
Critical Speed ◽  
Equations Of Motion ◽  
Circular Disk ◽  
Experimental Results ◽  
Normal Operation ◽  
Rotating System ◽  
Balance System ◽  
Rotor Imbalance ◽  
Low Viscosity ◽  
The Stability

A theoretical and experimental approach was used to investigate the motion and effectiveness of a Self-Compensating Dynamic Balancer (SCDB). This is a device intended to minimize the effects of rotor imbalance and vibratory forces on a rotating system during normal operation. The basic concept of an automatic dynamic balancer has been described in many U.S. patents. The SCDB is composed of a circular disk with a groove containing massive balls and a low viscosity damping fluid. The objective of this research is to determine the motion of the balls and how this ball motion is related to the vibration of the rotating system using both theoretical and experimental methods. The equations of motion the balls were derived by the Lagrangian method. Static and dynamic solutions were derived from the analytic model. To consider dynamic stability of the motion, perturbation equations were investigated by two different methods: Floquet theory and direct computer simulation. On the basis of the results of the stability investigation, ball positions which result in a balance system are stable above the critical speed and unstable at critical speed and below critical speed. To determine the actual critical speed of the rotating system used in the experimental work, a modal analysis was conducted. Experimental results confirm the predicted ball positions. Based on the theoretical and experimental results, when the system operates below and near the first critical speed, the balls do not balance the system. However, when the system operates above the first critical speed the balls can balance the system.

scholarly journals An Analytical Study of Self-Compensating Dynamic Balancer with Damping Fluid and Ball

1995 ◽  
Vol 2 (1) ◽  
pp. 59-67 ◽  
Author(s):  
Jongkil Lee
Keyword(s):  
Dynamic Stability ◽  
Critical Speed ◽  
Analytical Study ◽  
Equations Of Motion ◽  
Circular Disk ◽  
Lagrangian Method ◽  
The Self ◽  
Rotating System ◽  
Stability Investigation ◽  
Low Viscosity

The self-compensating dynamic balancer (SCDB) is composed of a circular disk with a groove containing ball and a low viscosity damping fluid. The equations of motion of the rotating system with SCDB were derived by the Lagrangian method. To consider dynamic stability of the motion, perturbation equations were investigated. Based on the results of stability investigation, ball positions that result in a balanced system are stable above the critical speed with small damping (β′>3.8 case). At critical speed the perturbed motion is said to be stable for large damping (β′>2.3 case). However, below critical speed the balls cannot stabilize the system in any case.

Determination of Stability Derivatives by Impulse-Response Techniques

1977 ◽  
Vol 14 (02) ◽  
pp. 265-275
Author(s):  
Carl A. Scragg
Keyword(s):  
Memory Effect ◽  
Impulse Response ◽  
Traditional Method ◽  
Equations Of Motion ◽  
Experimental Results ◽  
Regular Motion ◽  
Stability Derivatives ◽  
The Stability ◽  
Derivatives Of

This paper presents a new method of experimentally determining the stability derivatives of a ship. Using a linearized set of the equations of motion which allows for the presence of a memory effect, the response of the ship to impulsive motions is examined. This new technique is compared with the traditional method of regular-motion tests and experimental results are presented for both methods.

scholarly journals Analytical study of auto-balancing within the framework of the flat model of a rotor and an auto-balancer with a single cargo

2021 ◽  
Vol 2 (7 (110)) ◽  
pp. 66-73
Author(s):  
Gennadiy Filimonikhin ◽  
Lubov Olijnichenko ◽  
Guntis Strautmanis ◽  
Antonina Haleeva ◽  
Vasyl Hruban ◽  
...  
Keyword(s):  
Differential Equations ◽  
Analytical Study ◽  
Rotation Axis ◽  
Equations Of Motion ◽  
The Third ◽  
Rotor Imbalance ◽  
Small Resistance ◽  
The Stability ◽  
Resistance Forces ◽  
Rotor Model

This paper reports the analytically established conditions for the onset of auto-balancing for the case of a flat rotor model on isotropic elastic-viscous supports and an auto-balancer with a single load. The rotor is statically unbalanced, the rotation axis is vertical. The auto-balancer has a single cargo – a pendulum, a ball, or a roller. The balancing capacity of the cargo is equal to the rotor imbalance. The physical-mathematical model of the system is described. The differential equations of motion are recorded in dimensionless form relative to the coordinate system that rotates synchronously with the rotor. The so-called main movement has been found; in it, the cargo synchronously rotates with the rotor and balances it. The differential equations of motion are linearized in the neighborhood of the main movement. A characteristic equation has been constructed. It helped investigate the stability of the main movement (an auto-balancing mode) for the cases of the absence and presence of resistance forces in the system. It was established that in the absence of resistance forces in the system: – the rotor has three characteristic rotational speeds, and the first always coincides with the resonance frequency; – auto-balancing occurs when the rotor rotates at speeds between the first and second ones, and above the third characteristic speed; – the value of the second and third characteristic speeds is significantly influenced by the ratio of weight to the mass of the system; – the second and third characteristic speeds monotonously increase with an increase in the ratio of cargo weight to the mass of the system. Resistance forces significantly affect both the values of the second and third characteristic speeds and the conditions of their existence. Small resistance forces do not change the quality behavior of the system. With high resistance forces, the number of characteristic speeds decreases to one. The paper reports the results applicable to an auto-balancer with many cargoes when it balances the imbalance that equals the balancing capacity of the auto-balancer

On the Response of a Two-Degree-of-Freedom Rigid Disk With a Moving Massive Load

1973 ◽  
Vol 40 (1) ◽  
pp. 114-120 ◽  
Author(s):  
K. J. Stahl ◽  
W. D. Iwan
Keyword(s):  
Critical Speed ◽  
Equations Of Motion ◽  
Circular Disk ◽  
Instability Region ◽  
Degree Of Freedom ◽  
Viscous Damping ◽  
Elastically Supported ◽  
Two Degree Of Freedom ◽  
Limiting Value ◽  
Mathieu Equations

An analysis of the dynamic response of an elastically supported two-degree-of-freedom rigid circular disk excited by a moving massive load is presented. The equations of motion of the system are solved by transforming a set of coupled Hill-Mathieu equations into an ordinary eigenvalue problem. Two types of system instability are observed. A stiffness instability region exists above the critical speed of the disk and a terminal instability region exists for all load speeds exceeding a certain limiting value. The introduction of viscous damping may destabilize the system.

scholarly journals The Stability of an Asymmetric Rotor in Damped Supports

1978 ◽  
Author(s):  
A. J. Smalley ◽  
J. M. Tessarzik ◽  
R. H. Badgley
Keyword(s):  
Critical Speed ◽  
Equations Of Motion ◽  
General Purpose ◽  
Coordinate Frame ◽  
Flexible Rotor ◽  
Complex Eigenvalues ◽  
Tilting Pad ◽  
Resonant Behavior ◽  
Asymmetric Rotor ◽  
The Stability

A general-purpose method of evaluating the stability of an asymmetric flexible rotor, mounted in symmetric damped bearings, is defined. This method evaluates the complex eigenvalues of the rotor system by solving the equations of motion in a rotating coordinate frame. The application of this method to a rotor mounted in tilting-pad bearings is demonstrated. The observed behavior of a number of different rotor configurations is compared with corresponding predictions of stability. For the configurations predicted to be unstable, a distinct and unnegotiable threshold of instability is encountered. The sharpness of this threshold is emphasized by careful balancing at speeds fractionally below the threshold. In a final configuration predicted to be marginally stable, lightly damped resonant behavior, negotiable by balancing, is encountered in the region of the first critical speed.

Stability Analysis of a Rotating System Due to the Effect of Ball Bearing Waviness

2002 ◽  
Vol 125 (1) ◽  
pp. 91-101 ◽  
Author(s):  
G. H. Jang ◽  
S. W. Jeong
Keyword(s):  
Parametric Excitation ◽  
Ball Bearing ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Fourier Series Expansion ◽  
Time Varying ◽  
Algebraic Equations ◽  
Rotating System ◽  
Mathieu’S Equation ◽  
The Stability

This research presents an analytical model to investigate the stability due to the ball bearing waviness in a rotating system supported by two ball bearings. The stiffness of a ball bearing changes periodically due to the waviness in the rolling elements as the rotor rotates, and it can be calculated by differentiating the nonlinear contact forces. The linearized equations of motion can be represented as a parametrically excited system in the form of Mathieu’s equation, because the stiffness coefficients have time-varying components due to the waviness. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as the simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving Hill’s infinite determinant for these algebraic equations. The validity of this research is proven by comparing the stability chart with the time responses of the vibration model suggested by prior research. This research shows that the waviness in the ball bearing generates the time-varying component of the stiffness coefficient, whose frequency is called the frequency of the parametric excitation. It also shows that the instability takes place from the positions in which the ratio of the natural frequency to the frequency of the parametric excitation corresponds to i/2 i=1,2,3,….

Dynamic Behavior of Unbalanced Rigid Shaft Supported on Turbulent Journal Bearings—Theory and Experiment

1990 ◽  
Vol 112 (2) ◽  
pp. 404-408 ◽  
Author(s):  
H. Hashimoto ◽  
S. Wada
Keyword(s):  
Dynamic Behavior ◽  
Equations Of Motion ◽  
Journal Bearings ◽  
Operating Conditions ◽  
Experimental Results ◽  
Fluid Film ◽  
Force Components ◽  
The Stability ◽  
Nonlinear Equations Of Motion ◽  
Analytical Expressions

In this paper, the combined effects of turbulence and fluid film inertia on the dynamic behavior of an unbalanced rigid shaft supported horizontally on two identical aligned short journal bearings are investigated theoretically and experimentally. Utilizing analytical expressions for the dynamic fluid film force components considering the effects of turbulence and fluid film inertia, the nonlinear equations of motion for the rotor-bearing systems are solved by the improved Euler’s forward integration method. The journal center trajectories with unbalance eccentricity ratio of εμ = 0, 0.1 and 0.2 are examined theoretically for Reynolds number of Re = 2750, 4580, and 5500, and the theoretical results are compared with experimental results. From the theoretical and experimental results, it was found that the fluid film inertia improves the stability of unbalanced rigid shaft under certain operating conditions.

Stability of the Unison Response for a Rotating System With Multiple Tautochronic Pendulum Vibration Absorbers

1997 ◽  
Vol 64 (1) ◽  
pp. 149-156 ◽  
Author(s):  
Chang-Po Chao ◽  
S. W. Shaw ◽  
Cheng-Tang Lee
Keyword(s):  
Equations Of Motion ◽  
Vibration Absorbers ◽  
Rotating System ◽  
Total Moment ◽  
Equivalent Viscous Damping ◽  
Future Paper ◽  
The Stability ◽  
The Individual ◽  
Special Case ◽  
Selection Of

Due to spatial and balancing considerations, the implementation of centrifugal pendulum absorbers (CPVA’s) invariably requires that the total absorber inertia be divided into several absorber masses and stationed about the center of rotation. To achieve the designed-for performance, the CPVA’s are expected to move in exact unison, since the selection of the total absorber mass is made by assuming an equivalent single absorber mass. In this paper, we determine the conditions under which the unison motion of a system of several identical CPVA’s is dynamically stable. This is done for the special case of tautochronic absorbers subjected to a purely harmonic torque. The stability criterion is obtained by an asymptotic method that exploits certain symmetries in the equations of motion and is based on the assumption that total moment of inertia of the absorbers is much smaller than that of the entire rotating system—an assumption that is almost always satisfied in practice. It is expressed in terms of a critical torque level that is proportional to the square root of the equivalent viscous damping of the individual absorbers. The result is verified by numerical simulations of the system near the critical parameter conditions. A future paper will consider the post-critical response of the system.

Test Response and Nonlinear Analysis of a Turbocharger Supported on Floating Ring Bearings

2005 ◽  
Vol 127 (2) ◽  
pp. 107-115 ◽  
Author(s):  
Chris Holt ◽  
Luis San Andre´s ◽  
Sunil Sahay ◽  
Peter Tang ◽  
Gerry La Rue ◽  
...  
Keyword(s):  
Nonlinear Analysis ◽  
Equations Of Motion ◽  
Forced Response ◽  
Experimental Results ◽  
Rotor Speed ◽  
Rotor Imbalance ◽  
Feed Pressure ◽  
Nonlinear Responses ◽  
Reaction Forces ◽  
Bearing Force

Measurements of casing acceleration on an automotive turbocharger running to a top speed of 115 krpm and driven by ambient temperature pressurized air are reported. Waterfall acceleration spectra versus rotor speed show the effects of increasing lubricant inlet pressure and temperature on turbocharger rotordynamic response. A comprehensive analysis of the test data shows regimes of speed operation with two subsynchronous whirl motions (rotordynamic instabilities). Increasing the lubricant feed pressure delays the onset speed of instability for the most severe subsynchronous motion. However, increasing the lubricant feed pressure also produces larger synchronous displacements. The effect of lubricant feed temperature is minimal on the onset and end speeds of rotordynamic instability. Nevertheless, operation with a cold lubricant exhibits lower amplitudes of motion, synchronous and subsynchronous. The experimental results show the subsynchronous frequencies of motion do not lock (whip) at system natural frequencies but continuously track the rotor speed. No instabilities (subsynchronous whirl) remain for operating speeds above 90 krpm. Linear and nonlinear analysis results for the operation of a small automotive turbocharger supported on floating ring bearings are presented. A comprehensive fluid film bearing model predicting the forced response of floating ring bearings is also described. The linear rotordynamic model predicts well the rotor free–free modes and onset speed of instability using linearized bearing force coefficients. The nonlinear model incorporating instantaneous bearing reaction forces in the numerical integration of the rotor equations of motion predicts the limit cycle amplitudes with two fundamental subsynchronous whirl frequencies. Comparisons of both models to experimental results follow. The predictions evidence two unstable whirl ratios at approximately 12 ring speed and 12 ring speed plus 12 journal speed. The transient nonlinear responses reveal the importance of rotor imbalance in suppressing the subsynchronous instabilities at large rotor speeds as also observed in the experiments.

scholarly journals Dynamic Analysis of a Pendulum Dynamic Automatic Balancer

2007 ◽  
Vol 14 (2) ◽  
pp. 151-167 ◽  
Author(s):  
Jin-Seung Sohn ◽  
Jin Woo Lee ◽  
Eun-Hyoung Cho ◽  
No-Cheol Park ◽  
Young-Pil Park
Keyword(s):  
Stability Analysis ◽  
Dynamic Stability ◽  
Critical Speed ◽  
Equations Of Motion ◽  
Polar Coordinates ◽  
Response Analysis ◽  
Rotating Speed ◽  
Time Response Analysis ◽  
The Stability ◽  
Nonlinear Equations Of Motion

The automatic dynamic balancer is a device to reduce the vibration from unbalanced mass of rotors. Instead of considering prevailing ball automatic dynamic balancer, pendulum automatic dynamic balancer is analyzed. For the analysis of dynamic stability and behavior, the nonlinear equations of motion for a system are derived with respect to polar coordinates by the Lagrange's equations. The perturbation method is applied to investigate the dynamic behavior of the system around the equilibrium position. Based on the linearized equations, the dynamic stability of the system around the equilibrium positions is investigated by the eigenvalue analysis.The stability analysis provides the design requirements for the pendulum automatic dynamic balancer to achieve a balancing of the system. The efficiency of ball automatic dynamic balancer, for reducing the total vibration of the system, is better than one of pendulum automatic balancer, if the rotating speed is above critical speed. However, pendulum automatic dynamic balancer can achieve balancing even if the rotating speed is below critical speed.The time response analysis demonstrates the stability analysis from computing the radial displacement of the rotating system and the positions of pendulums. Furthermore, in order to confirm the theoretical analysis, various experiments are made on pendulum automatic dynamic balancer.

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