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Research Papers

Tuning for Performance and Stability in Systems of Nearly Tautochronic Torsional Vibration Absorbers

[+] Author and Article Information
Steven W. Shaw1

Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824shawsw@msu.edu

Bruce Geist

Powertrain Virtual Engineering, Chrysler Group LLC, 800 Chrysler Drive, Auburn Hills, MI 48326bruce.geist@chrysler.com

Multiple branches arise due to the fact that multiple eigenvalues become simultaneously unstable. Furthermore, if the absorbers are not perfectly identical, the details of the behavior near these conditions can be extremely complicated.

1

Corresponding author.

J. Vib. Acoust 132(4), 041005 (May 25, 2010) (11 pages) doi:10.1115/1.4000840 History: Received December 10, 2008; Revised December 03, 2009; Published May 25, 2010; Online May 25, 2010

This paper considers the dynamic response and performance characteristics of a special class of centrifugal pendulum torsional vibration absorbers. The absorbers of interest are designed by selection of the path that their center of mass follows, such that their dynamics are linear or nearly so, out to large amplitudes of motion, thereby avoiding the nonlinear-induced detuning that typically accompanies such responses. These order-tuned, tautochronic or isochronic, absorbers have been the subject of previous investigations, including analyses of the synchronous and certain nonsynchronous responses of systems comprised of a set of identical absorbers. The analysis and experiments have demonstrated that the synchronous response of such absorber systems can experience an instability that results in nonsynchronous responses in which a subset of absorbers have significantly larger amplitude than the corresponding synchronous response. In this study, we present results that generalize these stability results to include absorbers whose dynamics differ slightly from tautochronic by varying the absorber path such that both linear and nonlinear perturbations of perfect tuning are included. It is shown by analysis and verified by simulations that the perfect tuning case is quite special, specifically that the instability described above occurs for tunings very close to ideal and that the synchronous response can be made stable over the entire feasible operating range by employing small levels of linear and/or nonlinear detuning. Such detuning is known to have the additional benefit of resulting in smaller absorber responses and an attendant larger operating range albeit at the expense of absorber performance in terms of attenuating rotor torsional vibrations. The main conclusion of these results is that one can select a very small amount of detuning to avoid this undesirable instability and that such detuning does not have a significant effect on absorber effectiveness. The analytical results derived also provide a quantitative means of predicting synchronous absorber response amplitudes and the associated rotor torsional vibration levels, as well as the stability properties of these responses, results are very useful for the design of absorber systems.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 7

The diagram on the left shows three tautochronic paths. The radius of curvature of each path at its vertex is ρ0=R0/(1+ñ2), where ñ=1.5, 1.55, and 1.6. Enforcing tautochronicity implies that for each of these curves, λ=λe(ñ)≡ñ2/(1+ñ2). The diagram on the right shows three curves that produce the same linear tuning, ñ=1.53, only one of which is tautochronic. For the circular-path, λ=0; the middle curve is tautochronic, with λ=λe≈.8371; the third curve is cycloidal, with λ=1. Assuming R0=1, the γ that produces O(s6) approximations for these three curves is the fourth-order series coefficient in the series expansion below for r2; that is, γ=112(ñ2+1)3(λe2−λ2). All noncircular curves are plotted between path cusps, over an arc-length range between ±ρ0/λ.

Grahic Jump Location
Figure 1

N identical absorbers on a rotor. All parameters are indicated in the figure as dimensional quantities. The path radius r(s) in Eq. 1 corresponds to R/R0. The equations of motion 4,5 are written in terms of scaled arc-length si=Si/R0, i=1,…,N. T0/IdΩ2 corresponds to ϵΓ0 and Tθ/IdΩ2 corresponds to ϵΓ cos(nθ), where Ω is the nominal steady-state rotor speed and Id is the moment of inertia of the rotor about its axis of rotation.

Grahic Jump Location
Figure 2

System response versus torque amplitude: absorber amplitude a, absorber phase ϕ, and order n rotor angular acceleration |w′|. The linear tuning is set to ñ=1.48 and nonlinear tuning takes on values γ=0,1,2,3, varying from neutral toward increased softening. In the plot of a the largest amplitude is for the most softening path, γ=3, and decreasing γ reduces the amplitude. The other curves can be correlated by noting stability and termination points, which occur when the absorber reaches the cusp. The dotted line in the |w′| plot depicts the rotor torsional vibration reference line. The parameter values are ϵ=0.1, n=1.5, μ=0.3, and N=2. Note that the γ=0 cusp amplitude is given by acusp,0=0.3798.

Grahic Jump Location
Figure 3

System response versus torque amplitude: absorber amplitude a, absorber phase ϕ, and order n rotor angular acceleration |w′|. The nonlinear tuning is set to γ=0 and the linear tuning takes on values ñ=1.50,1.51,1.52,1.53, varying from perfect tuning toward increased overtuning. The topmost a curve is for perfect tuning, ñ=1.5 and overtuning extends the response out to larger torques. The other curves can be correlated by considering the termination points. The dotted line in the |w′| plot depicts the rotor torsional vibration reference line. The parameter values are ϵ=0.1, n=1.5, μ=0.3, and N=2. Note that the γ=0 cusp amplitude is given by acusp,0=0.3798.

Grahic Jump Location
Figure 4

Simulation results versus the rotor angle θ. The parameter values are ϵ=0.1, n=1.5, μ=0.3, N=2, Γ=0.679, and γ=0. The first panel shows 12(s1+s2) along with two reference lines: the cusp amplitude and the steady-state amplitude predicted by averaging. The second panel shows 12(s1−s2), indicating the stability of the synchronous response. The third and fourth panels show w′, first over the entire run and then a segment of the steady-state from simulation and reconstructed in the two ways described in the text.

Grahic Jump Location
Figure 5

Simulation results versus the rotor angle θ. The parameter values are ϵ=0.1, n=1.5, μ=0.3, N=2, Γ=0.813, and γ=0. The first panel shows 12(s1+s2) along with two reference lines: the cusp amplitude and the steady-state amplitude predicted by averaging. The second panel shows 12(s1−s2), indicating the instability of the synchronous response. The third and fourth panels show w′, first over the entire run and then a segment from the end of the run, from simulation and reconstructed in the two ways described in the text. The last panel shows the two individual absorbers near the end of the run, as they drift apart and one approaches the cusp amplitude.

Grahic Jump Location
Figure 6

Stability zones in path parameter space, γ versus ñ, for a<amax=αacusp with α=0.9 for excitation orders n=3/2 (left panel) and n=2 (right panel). White regions are stable, gray regions experience NS instability to nonsynchronous response and black regions experience J instabilities corresponding to jumps. The parameter values are ϵ=0.1, μ=0.3, and N=2.

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