Automatic Balancing of Twin Co-Planar Rotors

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
H. A. DeSmidt ◽  
D. Jung
Keyword(s):  
Equations Of Motion ◽  
Rotor System ◽  
Parameter Study ◽  
Valuable Insight ◽  
Dynamic Interactions ◽  
Torsional Mode ◽  
Rotor Imbalance ◽  
Nonlinear Equations Of Motion ◽  
Automatic Balancing ◽  
Twin Rotor

This paper explores the dynamics and stability of a twin rotor system fitted with passive automatic balancing devices (ABD). Essentially, autobalancers consist of several freely moving eccentric balancing masses. At certain speeds, the stable equilibrium position of the balancer masses is such that they naturally adjust to cancel the rotor imbalance. This “automatic balancing” phenomena occurs as a result of nonlinear dynamic interactions between the balancer masses and the rotor vibrations. Previous studies have explored automatic balancing of single rotors. In particular, ABDs are widely utilized for imbalance correction in computer optical disk and hard-drive applications. For such systems, automatic balancing occurs at supercritical operating speeds. While automatic balancing of single rotors is generally well understood, there has been only limited work on the topic of multirotor system automatic balancing. Therefore, this investigation considers a twin co-planar rotor system consisting of two symmetrically situated rotors mounted on a common flexible foundation structure. Both rotors are fitted with ABDs and the simultaneous autobalancing behavior of both rotors is investigated. Here, the nonlinear equations-of-motion of the twin-rotor/ABD system are derived and the asymptotic stability about the balanced condition is determined via a perturbation and floquet analysis. It is found that for the case of co-rotating rotors, automatic balancing is only achievable at supercritical speeds relative to the system torsional and lateral modes. However, for counter-rotating rotors, automatic balancing occurs at both subcritical and supercritical speeds relative to the foundation torsional mode. In this investigation, a dimensionless parameter study conducted to explore the effects of rotation speed, torsion and lateral mode placement, twin-rotor imbalance phasing, autobalancer mass, and damping for both the co- and counter-rotating cases. By considering the dynamic interactions between two rotor/ABD sub-systems, it is hoped that this study will provide valuable insight into the use of ABDs in multirotor system applications.

Hybrid Adaptive Rotor Imbalance Vibration Control via Passive Autobalancer and Active Bearing Actuation

2011 ◽  
Author(s):  
DaeYi Jung ◽  
Hans DeSmidt
Keyword(s):  
Steady State ◽  
Vibration Control ◽  
Vibration Isolation ◽  
Dynamic Interactions ◽  
Control Approach ◽  
Rotor Systems ◽  
Rotor Imbalance ◽  
Globally Attractive ◽  
Storage Batteries ◽  
Automatic Balancing

Many researchers have explored the use of active bearings, such as non-contact Active Magnetic Bearings for example, to control imbalance vibration in rotor systems. This paper develops a new hybrid adaptive imbalance vibration control approach based on an active bearing augmented with a passive automatic balancing device (ABD) to enhance the balancing and vibration isolation capabilities. Essentially, an ABD or “autobalance” 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. This “automatic balancing” phenomena occurs as a result of nonlinear dynamic interactions between the balancer and rotor wherein the balancer masses naturally synchronize with the rotor with appropriate phase to cancel the imbalance. Since the ABD acts directly on the rotor in the rotating frame, rotor whirl amplitudes are passively reduced without any forces transmitted between rotor and bearing. Therefore, this hybrid ABD/active bearing approach enables increased rotor balancing capability and reduced steady-state control power consumption. However, due to the inherent nonlinearity of the autobalancer, the potential for other, non-synchronous limit-cycle behavior exists. In such situations, the balancer masses do not reach their desired synchronous balanced steady-state equilibrium positions resulting in increased rotor vibration. To address this, a new adaptive active control algorithm for the rotor/bearing/ABD system is derived based on the Lyapunov approach which guarantees global asymptotic stability of the synchronous balanced condition. This approach enables the controller to cope with both the system nonlinearity introduced by the passive ABD and with the rotor imbalance uncertainty. Here, the controllability of system is established through an accessible distribution Lie bracket operational analysis. The simulation results demonstrate the advantages of the hybrid ABD/active bearing system. In particular, it is shown that the balanced equilibrium can be made globally attractive under the action of the adaptive bearing control law, and that the steady-state power levels are significantly reduced via the addition of the ABD. These findings are relevant to limited power applications such as in satellite reaction wheels or flywheel energy storage batteries.

Limit-Cycle Analysis of Planar Rotor/Autobalancer System Influenced by Alford's Force

2016 ◽  
Vol 138 (2) ◽  
Author(s):  
DaeYi Jung ◽  
H. A. DeSmidt
Keyword(s):  
Steady State ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Journal Bearing ◽  
Dynamic Interactions ◽  
Rotor Imbalance ◽  
Synchronous Condition ◽  
Freely Moving ◽  
Automatic Balancing ◽  
Special Case

In recent years, there has been much interest in the use of so-called automatic balancing devices (ABDs) in rotating machinery. Essentially, ABDs or “autobalancers” consist of several freely moving eccentric balancing masses mounted on the rotor, which, at certain operating speeds, act to cancel rotor imbalance at steady-state. This “automatic balancing” phenomenon occurs as a result of nonlinear dynamic interactions between the balancer and rotor, wherein the balancer masses naturally synchronize with the rotor with appropriate phase and cancel the imbalance. However, due to inherent nonlinearity of the autobalancer, the potential for other, undesirable, nonsynchronous limit-cycle behavior exists. In such situations, the balancer masses do not reach their desired synchronous balanced steady-state positions resulting in increased rotor vibration. In this paper, an approximate analytical harmonic solution for the limit cycles is obtained for the special case of symmetric support stiffness together with the so-called Alford's force cross-coupling term. The limit-cycle stability is assessed via Floquet analysis with a perturbation. It is found that the stable balanced synchronous conditions coexist with undesirable nonsynchronous limit cycles. For certain combinations of bearing parameters and operating speeds, the nonsynchronous limit-cycle can be made unstable guaranteeing global asymptotic stability of the synchronous balanced condition. Additionally, the analytical bifurcation of the coexistence zone and the pure balanced synchronous condition is derived. Finally, the analysis is validated through numerical time- and frequency-domain simulation. The findings in this paper yield important insights for researchers wishing to utilize ABDs on rotors having journal bearing support.

On the Parametric Excitation of Pendulum-Type Vibration Absorber

1961 ◽  
Vol 28 (3) ◽  
pp. 330-334 ◽  
Author(s):  
Eugene Sevin
Keyword(s):  
Energy Transfer ◽  
Numerical Solution ◽  
Parametric Excitation ◽  
Equations Of Motion ◽  
Vibration Absorber ◽  
Single Cycle ◽  
Linearized System ◽  
Nonlinear Equations Of Motion ◽  
Beam Oscillation ◽  
Energy Interchange

The free motion of an undamped pendulum-type vibration absorber is studied on the basis of approximate nonlinear equations of motion. It is shown that this type of mechanical system exhibits the phenomenon of auto parametric excitation; a type of “instability” which cannot be accounted for on the basis of the linearized system. Complete energy transfer between modes is shown to occur when the beam frequency is twice the simple pendulum frequency. On the basis of a numerical solution, approximately 150 cycles of the beam oscillation take place during a single cycle of energy interchange.

Less=Moor: A Time-Efficient Computational Tool to Assess the Behavior of Moored Ships in Waves

2017 ◽  
Author(s):  
Yijun Wang ◽  
Alex van Deyzen ◽  
Benno Beimers
Keyword(s):  
Dynamic Response ◽  
Research Effort ◽  
Equations Of Motion ◽  
Line Tension ◽  
Mooring Line ◽  
Induced Response ◽  
Wave Forces ◽  
Computational Tool ◽  
The Time Domain ◽  
Nonlinear Equations Of Motion

In the field of port design there is a need for a reliable but time-efficient method to assess the behavior of moored ships in order to determine if further detailed analysis of the behavior is required. The response of moored ships induced by gusting wind and/or waves is dynamic. Excessive motion response may cause interruption of the (un)loading operation. High line tension may cause lines to snap, introducing dangerous situations. A (detailed) Dynamic Mooring Analysis (DMA), however, is often a time-consuming and expensive exercise, especially when responses in many different environmental conditions need to be assessed. Royal HaskoningDHV has developed a time-efficient computational tool in-house to assess the wave (sea or swell) induced dynamic response of ships moored to exposed berths. The mooring line characteristics are linearized and the equations of motion are solved in the frequency domain with both the 1st and 2nd wave forces taken into account. This tool has been termed Less=Moor. The accuracy and reliability of the computational tool has been illustrated by comparing motions and mooring line forces to results obtained with software that solves the nonlinear equations of motion in the time domain (aNySIM). The calculated response of a Floating Storage and Regasification Unit (FSRU) moored to dolphins located offshore has been presented. The results show a good comparison. The computational tool can therefore be used to indicate whether the wave induced response of ships moored at exposed berths proves to be critical. The next step is to make this tool suitable to assess the dynamic response of moored ships with large wind areas, e.g. container ships, cruise vessels, RoRo or car carriers, to gusting wind. In addition, assessment of ship responses in a complicated wave field (e.g. with reflected infra-gravity waves) also requires more research effort.

Simulation of Engine Vibration on Nonlinear Hydraulic Engine Mounts

2005 ◽  
Author(s):  
A. R. Ohadi ◽  
G. Maghsoodi
Keyword(s):  
Equations Of Motion ◽  
Low Frequency ◽  
Governing Equations ◽  
Engine Mounts ◽  
Engine Vibration ◽  
Engine Mount ◽  
Rotational Motions ◽  
The Time Domain ◽  
Nonlinear Equations Of Motion ◽  
Four Cylinders

In this paper, vibration behavior of engine on nonlinear hydraulic engine mount including inertia track and decoupler is studied. In this regard, after introducing the nonlinear factors of this mount (i.e. inertia and decoupler resistances in turbulent region), the vibration governing equations of engine on one hydraulic engine mount are solved and the effect of nonlinearity is investigated. In order to have a comparison between rubber and hydraulic engine mounts, a 6 degree of freedom four cylinders V-shaped engine under inertia and balancing masses forces and torques is considered. By solving the time domain nonlinear equations of motion of engine on three inclined mounts, translational and rotational motions of engines body are obtained for different engine speeds. Transmitted base forces are also determined for both types of engine mount. Comparison of rubber and hydraulic mounts indicates the efficiency of hydraulic one in low frequency region.

Consistent Tangent Stiffness of a Three-Dimensional Wheel–Rail Interaction Element

2021 ◽  
Author(s):  
Quan Gu ◽  
Jinghao Pan ◽  
Yongdou Liu
Keyword(s):  
Finite Element ◽  
Quadratic Convergence ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Light Rail ◽  
Software Framework ◽  
Tangent Stiffness ◽  
Rail Vehicle ◽  
Interaction Element ◽  
Nonlinear Equations Of Motion

Consistent tangent stiffness plays a crucial role in delivering a quadratic rate of convergence when using Newton’s method in solving nonlinear equations of motion. In this paper, consistent tangent stiffness is derived for a three-dimensional (3D) wheel–rail interaction element (WRI element for short) originally developed by the authors and co-workers. The algorithm has been implemented in finite element (FE) software framework (OpenSees in this paper) and proven to be effective. Application examples of wheelset and light rail vehicle are provided to validate the consistent tangent stiffness. The quadratic convergence rate is verified. The speeds of calculation are compared between the use of consistent tangent stiffness and the tangent by perturbation method. The results demonstrate the improved computational efficiency of WRI element when consistent tangent stiffness is used.

Nonlinear Analysis of a Rigid Rotor on Magnetic Bearings

1995 ◽  
Author(s):  
C. Nataraj
Keyword(s):  
Nonlinear Analysis ◽  
Equations Of Motion ◽  
Forced Response ◽  
Magnetic Bearings ◽  
Rigid Rotor ◽  
Stability Criteria ◽  
Safe Operation ◽  
Qualitative And Quantitative ◽  
Quantitative Results ◽  
Nonlinear Equations Of Motion

A simple model of a rigid rotor supported on magnetic bearings is considered. A proportional control architecture is assumed, the nonlinear equations of motion are derived and some essential nondimensional parameters are identified. The free and forced response of the system is analyzed using techniques of nonlinear analysis. Both qualitative and quantitative results are obtained and stability criteria are derived for safe operation of the system.

Linear and Nonlinear Dynamics of Cantilevered Cylinders in Axial Flow

2014 ◽  
Author(s):  
Ahmad Jamal ◽  
Michael P. Païdoussis ◽  
Luc G. Mongeau
Keyword(s):  
Equations Of Motion ◽  
Axial Flow ◽  
Experimental System ◽  
Vital Role ◽  
Viscous Force ◽  
Flexible Cylinder ◽  
Force Coefficients ◽  
Precise Calculation ◽  
Linear And Nonlinear ◽  
Nonlinear Equations Of Motion

Understanding and prediction of the dynamics of slender flexible cylinders in axial flow is of interest for the design and safe operation of heat exchangers and nuclear reactors, specifically that of heat exchanger tubes, nuclear fuel elements, control rods, and monitoring tubes. In such fluid-structure interaction problems, the fluid forces acting on the flexible structure play a vital role in defining its dynamics. Therefore, a precise calculation of the coefficients associated to these forces, such as the longitudinal and normal viscous force coefficients, and base drag coefficient in the equation of motion is imperative. The present work is aimed at (i) calculating these force coefficients for a cantilevered slender flexible cylinder, fitted with an ogival end-piece, in axial flow and (ii) conducting experiments on the same system. In the calculation of these force coefficients, the parameters of the experimental system are used, so that the theoretically predicted dynamics would be representative of the actual physical system. These calculated force coefficients are then incorporated in the linear and nonlinear equations of motion and the predicted dynamics are compared with those of the experiments. The comparison shows good agreement between the theoretical and experimental results.

Stability analysis of rotor whirl under nonlinear internal friction by a general averaging approach

2016 ◽  
Vol 23 (5) ◽  
pp. 808-826 ◽  
Author(s):  
Francesco Sorge
Keyword(s):  
Internal Friction ◽  
Equations Of Motion ◽  
Translational Motion ◽  
Stable Limit ◽  
Rotor Shaft ◽  
Shaft System ◽  
Flexible Supports ◽  
Asymmetric Rotor ◽  
Nonlinear Equations Of Motion ◽  
Shrink Fit

The two main sources of internal friction in a rotor-shaft system are the shaft structural hysteresis and the possible shrink-fit release of the assembly. The internal friction tends to destabilize the over-critical rotor running, but a remedy against this effect may be provided by a proper combination of some external damping in the supports and an anisotropic arrangement of the support stiffness, or at most by the support damping alone, depending on the system geometry. The present analysis reported here considers a general asymmetric rotor-shaft system, where the rotor is perfectly rigid and is constrained by viscous–flexible supports having different stiffnesses on two orthogonal planes. The internal friction is modelled by nonlinear Coulombian forces, which counteract the translational motion of the rotor relative to a frame rotating with the shaft ends. The nonlinear equations of motion are dealt with using an averaging approach based on the Krylov-Bogoliubov method with some adaptation to address the multi-degree-of-freedom nature of the problem. Stable limit cycles may be attained by the overcritical whirling motions, whose amplitudes are inversely proportional to the external dissipation applied by the supports. A noteworthy result is that the stiffness anisotropy of the supports is recognized as beneficial in reducing the natural whirl amplitudes, albeit mainly in the symmetric configuration of the rotor at the mid span and, to a rather lesser extent, in the asymmetric configuration, which then requires a stronger damping action in the supports.

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