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Research Papers

Automatic Balancing of Twin Co-Planar Rotors

[+] Author and Article Information
H. A. DeSmidt

606 Dougherty Bldg., Mechanical Aerospace and Biomedical Engineering Dept.,  University of Tennessee, Knoxville, TN 37996-2210hdesmidt@utk.edu

D. Jung

606 Dougherty Bldg., Mechanical Aerospace and Biomedical Engineering Dept.,  University of Tennessee, Knoxville, TN 37996-2210

J. Vib. Acoust 134(1), 011007 (Dec 28, 2011) (16 pages) doi:10.1115/1.4004668 History: Received July 31, 2010; Revised March 06, 2011; Accepted April 11, 2011; Published December 28, 2011; Online December 28, 2011

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.

Copyright © 2012 by American Society of Mechanical Engineers
Topics: Rotors , Stability , Damping
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References

Figures

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Figure 1

Twin rotor/ABD system

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Figure 2

Twin rotor/ABD system degrees-of-freedom and coordinates

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Figure 3

Schematic of balanced condition for twin rotor/ABD system

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Figure 4

Effect of torsional natural frequency and rotor speed ratios on stable automatic balancing regions for several lateral frequency ratios,

unstable, co-rotating case, ABD mass ratio μb  = 0.025, ABD damping c¯b=1×10-4, rotor imbalance [ηA=0.7, ηB=0.7, θiA=0∘, θiB=90∘]; lateral frequency ratio (a) fw=1.0, (b) fw=4.0, (c) fw=7.0, (d) fw=10.0.

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Figure 5

Effect of torsional natural frequency and rotor speed ratios on stable automatic balancing regions for several lateral frequency ratios,

unstable, counter-rotating case, ABD mass ratio μb  = 0.025, ABD damping c¯b1×10-4, rotor imbalance [ηA=0.7, ηB=0.7, θiA=0∘, θiB=90∘]; lateral frequency ratio (a) fw=1.0, (b) fw=4.0, (c) fw=7.0, (d) fw=10.0.

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Figure 6

Effect of torsional natural frequency and rotor speed ratios on stable automatic balancing regions for several ABD mass ratios; —— μb  = 0.025,

μb  = 0.05,– – – μb  = 0.075, ABD damping c¯b1×10-4, lateral frequency ratio fw=1.0, rotor imbalance [ηA=0.7, ηB=0.7, θiA=0∘, θiB=90∘]; (a) co-rotating case (b) counter-rotating case.

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Figure 7

Effect of torsional natural frequency and rotor speed ratios on critical ABD damping c¯bcrit,

unstable, lateral frequency ratio fw=0.25, ABD mass ratio μb  = 0.025, rotor imbalance [ηA=0.7, ηB=0.7, θiA=0∘, θiB=90∘]; (a) counter-rotating case, (b) co-rotating case.

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Figure 8

Effect of torsional natural frequency and rotor speed ratios on critical ABD damping c¯bcrit,

unstable, counter-rotating case, ABD mass ratio μb  = 0.025, rotor imbalance [ηA=0.7, ηB=0.7, θiA=0∘, θiB=90∘]; lateral frequency ratio (a) fw=4.0, (b) fw=7.0.

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Figure 9

Effect of ABD mass and rotor speed ratios on critical ABD damping c¯bcrit,

unstable, torsional frequency ratio ft=4.0, lateral frequency ratio fw=0.25, rotor imbalance [ηA=0.7, ηB=0.7, θiA=0∘, θiB=90∘]; (a) counter-rotating case, (b) co-rotating case.

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Figure 10

Effect of rotor imbalance ratios on critical ABD damping c¯bcrit, co-rotating case, ABD mass ratio μb  = 0.025; (a) fw=0.25, ft=0.5, Ω¯=2.0, (b) fw=0.25, ft=0.2, Ω¯=0.5.

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Figure 11

Effect of rotor imbalance ratios on critical ABD damping c¯bcrit, counter-rotating case, ABD mass ratio μb  = 0.025; (a) fw=0.25, ft=0.5, Ω¯=2.0, (b) fw=0.25, ft=4.0, Ω¯=2.0.

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Figure 12

Twin rotor/ABD system imbalance scenario; (a) rotor A imbalance force ratio, (b) rotor A imbalance phase, (c) rotor B imbalance force ratio, (d) rotor B imbalance phase.

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Figure 13

Twin rotor system foundation response with and without autobalancers, counter-rotating case I

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Figure 14

Autobalancer ball angular positions, counter-rotating case I; (a) ABD rotor A, (b) ABD rotor B.

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Figure 15

Twin rotor system foundation response with and without autobalancers, co-rotating case I

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Figure 16

Autobalancer ball angular positions, co-rotating case I; (a) ABD rotor A, (b) ABD rotor B.

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Figure 17

Twin rotor system foundation response with and without autobalancers, counter-rotating case II

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Figure 18

Autobalancer ball angular speeds, counter-rotating case II; (a) ABD rotor A, (b) ABD rotor B.

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