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TECHNICAL PAPERS

Parametric Instability of a Rotating Circular Ring With Moving, Time-Varying Springs

[+] Author and Article Information
Sripathi Vangipuram Canchi

Department of Mechanical Engineering, Ohio State University, 650 Ackerman Road, Columbus, OH 43202

Robert G. Parker1

Department of Mechanical Engineering, Ohio State University, 650 Ackerman Road, Columbus, OH 43202parker.242@osu.edu

1

Corresponding author.

J. Vib. Acoust 128(2), 231-243 (Aug 30, 2005) (13 pages) doi:10.1115/1.2159040 History: Received March 17, 2005; Revised August 30, 2005

Parametric instabilities of in-plane bending vibrations of a rotating ring coupled to multiple, discrete, rotating, time-varying stiffness spring-sets of general geometric description are investigated in this work. Instability boundaries are identified analytically using perturbation analysis and given as closed-form expressions in the system parameters. Ring rotation and time-varying stiffness significantly affect instability regions. Different configurations with a rotating and nonrotating ring, and rotating spring-sets are examined. Simple relations governing the occurrence and suppression of instabilities are discussed for special cases with symmetric circumferential spacing of spring-sets. These results are applied to identify possible conditions of ring gear instability in example planetary gears.

FIGURES IN THIS ARTICLE
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Copyright © 2006 by American Society of Mechanical Engineers
Topics: Springs , Stiffness , Gears
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References

Figures

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Figure 1

(a) Schematic of a rotating ring on multiple rotating spring-sets and (b) Definition of reference frames

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Figure 2

Variation of the natural frequencies with ring speed for the inextensible vibrations of a rotating ring: solid lines, natural frequency associated with einθ; dashed lines, natural frequency associated with e−inθ

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Figure 3

Parametric instability regions for one radial rotating spring with ε=1, k11=0, k21=k, β1=0deg: (a) νsp versus νr with  *** denoting numerical solution and the dotted-dashed line denoting stability boundaries without ring stiffening; (b) νrel versus νr for instabilities of the first kind; (c) νrel versus νr for instabilities of the second kind; and (d) instability width variation with νr (solid lines, principal and combination instabilities of the first kind; dashed lines, combination instabilities of the second kind)

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Figure 4

Parametric instability regions for principal and combination instabilities of the first kind for two identical, equally spaced, radial rotating springs with ε=1, k1j=0, k2j=k, βj=0deg: (a) νsp versus νr and (b) νrel versus νr ( *** denotes numerical solution)

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Figure 6

Parametric instability boundaries for changing spring stiffness variation frequency for a stationary ring with one rotating radial spring with ε=1, k11=0, k21=k(1+cosωmt+cos2ωmt): (a) principal instability boundaries and (b) combination instability boundaries (solid lines, instabilities of the first kind; dashed lines, instabilities of the second kind)

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Figure 7

Parametric instability boundaries for changing spring stiffness variation frequency for rotating ring νr=1 with one rotating radial spring with ε=1, k11=0, k21=k(1+cosωmt+cos2ωmt): (a) principal instability boundaries and (b) combination instability boundaries (solid lines, instabilities of the first kind; dashed lines, instabilities of the second kind)

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Figure 5

Parametric instability regions for principal and combination instabilities of the first kind for two identical, but unequally spaced, radial rotating springs with ε=1, k1j=0, k2j=k, βj=0deg; ϕ1=0deg, ϕ2=160deg: (a) νsp versus νr and (b) νrel versus νr ( *** denotes numerical solution)

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Figure 8

Parametric instability boundaries for changing spring stiffness variation frequency for a stationary ring with two equally spaced, identically time-varying rotating radial springs with ε=1, k1j=0, k2j=k(1+cosωmt+cos2ωmt), βj=0, ϕ1=0deg, ϕ2=180deg. No combination instabilities occur for this case.

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Figure 9

Parametric instability boundaries for changing spring stiffness variation frequency for a stationary ring with two unequally spaced, identically time varying rotating radial springs with ε=1, k1j=0, k2j=k(1+cosωmt+cos2ωmt), βj=0, ϕ1=0deg, ϕ2=120deg: (a) principal instability boundaries and (b) combination instability boundaries (solid lines, instabilities of the first kind; dashed lines, instabilities of the second kind)

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Figure 10

Parametric instability boundaries considering two nodal diameter modes for one rotating radial spring with ε=1, k11=0, k21=k(1+cosωmt+cos2ωmt)

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Figure 13

Analytically predicted parametric instability regions for aircraft gear: (a) P: primary instabilities, S: secondary instabilities and (b) zoomed portion showing secondary instabilities. Narrow regions between the labeled lines marked as “U” are unstable.

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