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Research Papers

Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers—Part I: Theory

[+] Author and Article Information
Ryan J. Monroe

Air and Missile Defense Department,
The Johns Hopkins University
Applied Physics Laboratory,
Laurel, MD 20723
e-mail address: Ryan.Monroe@jhuapl.edu

Steven W. Shaw

Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824

The results obtained are applicable to the synchronous response of systems that use multiple identical absorbers.

Additionally, the mass of the rollers typically used in the suspension of a CPVA can be included in the analysis by adding the additional kinematic terms derived in [15].

The tuning order for a circular path absorber with a single point suspension is n˜/ β (see Eq. (3)).

This can be verified by evaluating the near-tautochronic averaged equations in the limit γ0, which results in the general path averaged equations, except for the differences in the definitions of χt and χc.

We note that the results shown here break down when the mean torque is Γ0>2Γ, which is expected since both torques are scaled in the same manner.

Recall that for general path absorbers, the absorber amplitude is scaled by ε1/2, and the resulting perturbation equations break down for large amplitudes.

The viscous damping coefficient μ is related to the damping ratio ζ by μ=2ζn˜.

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received February 8, 2012; final manuscript received July 15, 2012; published online February 4, 2013. Assoc. Editor: Philip Bayly.

J. Vib. Acoust 135(1), 011017 (Feb 04, 2013) (10 pages) Paper No: VIB-12-1029; doi: 10.1115/1.4007561 History: Received February 08, 2012; Revised July 15, 2012

Transient dynamics of centrifugal pendulum vibration absorbers, which are used for reducing torsional vibrations in rotating machines, are investigated using analysis and simulations of a dynamical model. These absorbers are being implemented in automotive engines to smooth vibrations and aid with fuel saving technologies, such as cylinder deactivation and torque converter lockup. In order for the absorbers to operate effectively with minimal mass, they must be designed to accommodate large amplitude, nonlinear responses, and in automotive engines they will experience a variety of transient environments. Here we consider the most severe transient environment, that of sudden activation near resonance, which leads to beating behavior of a nonlinear oscillator coupled to a driven rotor. An approximate method for predicting the percent overshoot of the beating transient response is derived, based on perturbation analysis of the system equations of motion. The main result is expressed in terms of the system and excitation parameters, and is found to accurately predict results from direct simulations of the model equations of motion. It is shown that absorbers with near-tautochronic paths behave much like linear absorbers, and when lightly damped and start from small initial conditions, they have an overshoot close to 100%. For absorbers with softening paths, such as the commonly used circular path absorbers, the overshoot can reach up to 173%, depending on system and input parameters, far exceeding predictions from linear analysis. These results provide a useful tool for design of absorbers to meet transient response specifications. In the following companion paper an experimental investigation is used to verify the analytical predictions.

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References

Wachs, M. A., 1973, “The Main Rotor Bifilar Absorber and Its Effect on Helicopter Reliability/Maintainability,” SAE Technical Paper No. 730894. [CrossRef]
Nester, T. M., Haddow, A. G., Shaw, S. W., Brevick, J. E., and Borowski, V. J., 2003, “Vibration Reduction in Variable Displacement Engines Using Pendulum Absorbers,” Proceedings of the SAE Noise and Vibration Conference and Exhibition, Traverse City, MI, May 5, SAE Technical Paper 2003-01-1484. [CrossRef]
Chao, C. P., Lee, C. T., and Shaw, S. W., 1997, “Stability of the Unison Response for a Rotating System With Multiple Centrifugal Pendulum Vibration Absorbers,” ASME J. Appl. Mech., 64(1), pp. 149–156. [CrossRef]
Shaw, S. W., and Geist, B. K., 2010, “Tuning for Performance and Stability in Systems of Nearly Tautochronic Torsional Vibration Absorbers,” ASME J. Vib. Acoust., 132(4), p. 041005. [CrossRef]
Alsuwaiyan, A. S., and Shaw, S. W., 2003, “Steady-State Responses in Systems of Nearly Identical Torsional Vibration Absorbers,” ASME J. Vib. Acoust., 125(1), pp. 80–87. [CrossRef]
Alsuwaiyan, A. S., and Shaw, S. W., 2002, “Performance and Dynamic Stability of General-Path Centrifugal Pendulum Vibration Absorbers,” J. Sound Vib., 252(5), pp. 791–815. [CrossRef]
Chao, C. P., Lee, C. T., and Shaw, S. W., 1997, “Non-Unison Dynamics of Multiple Centrifugal Pendulum Vibration Absorbers,” J. Sound Vib., 204(5), pp. 769–794. [CrossRef]
Nester, T. M., Haddow, A. G., Schmitz, P. M., and Shaw, S. W., 2004, “Experimental Observations of Centrifugal Pendulum Vibration Absorbers,” 10th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery (ISROMAC-10), Honolulu, HI, March 7–11, Paper No. ISROMAC10-2004-043.
Haddow, A. G., and Shaw, S. W., 2003, “Centrifugal Pendulum Vibration Absorbers: An Experimental and Theoretical Investigation,” Nonlinear Dyn., 34(3–4), pp. 293–307. [CrossRef]
Nester, T., Haddow, A. G., and Shaw, S. W., 2003, “Experimental Investigation of a System With Multiple Nearly Identical Centrifugal Pendulum Vibration Absorbers,” Proceedings of the ASME 19th Biennial Conference on Mechanical Vibration and Noise, Chicago, IL, September 2–6, ASME Paper No. DETC2003/VIB-48410, pp. 913–921. [CrossRef]
Shaw, S. W., Schmitz, P. M., and Haddow, A. G., 2006, “Tautochronic Vibration Absorbers for Rotating Systems,” ASME J. Comput. Nonlinear Dyn., 1(1), pp. 283–293. [CrossRef]
United States Environmental Protection Agency, and United States Department of Transportation, 2010, “Light-Duty Vehicle Greenhouse Gas Emission Standards and Corporate Average Fuel Economy Standards; Final Rule,” Federal Register, 75(8), pp. 25324–25728 (May 7, 2010).
Falkowski, A. G., McElwee, M. R., and Bonne, M. A., 2004, “Design and Development of the DaimlerChrysler 5.7L HEMI® Engine Multi-Displacement Cylinder Deactivation System,” SAE Technical Paper 2004-01-2106. [CrossRef]
Monroe, R. J., and Shaw, S. W., 2013, “Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers—Part II: Experiment,” ASME J. Vib. Acoust.135(1), p. 011018. [CrossRef]
Monroe, R. J., Shaw, S. W., Haddow, A. G., and Geist, B. K., 2011, “Accounting for Roller Dynamics in the Design of Bifilar Torsional Vibration Absorbers,” ASME J. Vib. Acoust., 133(6), p. 061002. [CrossRef]
Monroe, R. J., and Shaw, S. W., 2012, “On the Transient Response of Forced Nonlinear Oscillators,” Nonlinear Dyn., 67(4), pp. 2609–2619. [CrossRef]
Manevitch, L. I., Kovaleva, A. S., Manevitch, E. L., and Shepelev, D. S., 2011, “Limiting Phase Trajectories and Non-Stationary Resonance Oscillations of the Duffing Oscillator. Part 1. A Non-Dissipative Oscillator,” Commun. Nonlinear Sci. Numer. Simul., 16(2), pp. 1089–1097. [CrossRef]
Manevitch, L. I., Kovaleva, A. S., Manevitch, E. L., and Shepelev, D. S., 2011, “Limiting Phase Trajectories and Nonstationary Resonance Oscillations of the Duffing Oscillator. Part 2. A Dissipative Oscillator,” Commun. Nonlinear Sci. Numer. Simul., 16(2), pp. 1098–1105. [CrossRef]
Denman, H. H., 1992, “Tautochronic Bifilar Pendulum Torsion Absorbers for Reciprocating Engines,” J. Sound Vib., 159(2), pp. 251–277. [CrossRef]
Monroe, R. J., Shaw, S. W., Haddow, A. G., and Geist, B. K., 2009, “Accounting for Roller Dynamics in the Design of Bifilar Torsional Vibration Absorbers,” Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (IDETC/CIE2009), Vol. 1: 22nd Biennial Conference on Mechanical Vibration and Noise, ASME Paper No. DETC2009-87431, pp. 1225–1236. [CrossRef]
Chao, C. P., Shaw, S. W., and Lee, C. T., 1997, “Non-Unison Dynamics of Multiple Centrifugal Pendulum Vibration Absorbers,” J. Sound Vib., 204(5), pp. 769–794. [CrossRef]
Shaw, S. W., Orlowski, M. B., Haddow, A. G., and Geist, B., 2008, “Transient Dynamics of Centrifugal Pendulum Vibration Absorbers,” Proceedings of the 12th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery (ISROMAC-12), Honolulu, HI, February 17–22, pp. 816–825.
Hartog, J. P. D., 1985, Mechanical Vibrations, Dover, New York.

Figures

Grahic Jump Location
Fig. 1

Depiction of the CPVA system model showing three possible paths for the absorber COM. The outermost (circular) and innermost (cycloidal) paths bound the two-parameter family under consideration; the middle path corresponds to the tautochrone. The paths are shown enlarged, and not to scale relative to the rotor, to exaggerate the differences in the paths.

Grahic Jump Location
Fig. 2

A damped zero initial condition transient trajectory, IC 1, obtained from simulations of the general path averaged equations, indicating the peak and steady-state amplitudes: (a) shown in the phase space, and (b) as the absorber response versus θ. System parameter values: Dc = 0.30 and χc = 0.10. Also shown in (a) is the trajectory from the unstable manifold of the saddle that tends toward the desired steady state.

Grahic Jump Location
Fig. 3

Simulations of the full EOM (Eqs. (1) and (2)) for ε = 0.03 and n = 1.5, and three sets of system parameter values that yield χc = 0.112, with a predicted overshoot of 122%. (a) Γc = 1.171, n˜ = 1.52 (σc = -4.263), λ = 0 (ξc = -4.22), simulated percent overshoot = 119%; (b) Γc = 0.799, n˜ = 1.51 (σc = -3.253), λ = 0.1 (ξc = -4.03), simulated percent overshoot = 121%; and (c) Γc = 0.477, n˜ = 1.5 (σc = -2.25), λ = 0.2 (ξc = -3.732), simulated percent overshoot = 124%. Additional parameters used to simulate the full EOM include: Γ0 = Γ/2, α = 0, and β = 1.

Grahic Jump Location
Fig. 4

The percent overshoot of an absorber system subject to zero initial conditions, computed using the pseudoenergy method for Di = 0, and using the averaged equations for Di≠0; for (a) general path and (b) near-tautochronic path absorbers

Grahic Jump Location
Fig. 5

The rotor and absorber response to a step input of sinusoidal torque. (a) Rotor and (b) absorber response to input +Γsin(nθ), resulting in absorber overshoot of 118.56%; (c) rotor and (d) absorber response to input -Γsin(nθ), resulting in absorber overshoot of 118.30%. Simulation data: ε = 0.07, Γc = 0.71, n˜ = 1.51, λ = 0, ζ = 0.002, Γ0 = Γ/2, α = 0, and β = 1. For these parameters, χc = 0.161, Dc = 0.10, and the pseudoenergy theory predicts absorber overshoot of 117.30%.

Grahic Jump Location
Fig. 6

Comparison of the general path pseudoenergy method with simulations of the full EOM for μ = 0, n = 1.5, ε = 0.03, Γ0 = Γ/2, α = 0, and β = 1. Solid line is the pseudoenergy prediction. Simulation data: sweep of χc by varying λ from 0.80 to 0 with εΓ = 0.005, and σc = -3.25 (n˜ = 1.51). Sweep of χc by varying εΓ from 0.001 to 0.006 with ξc = -4.09(λ = 0), and σc = -3.25 (n˜ = 1.51). Sweep of χc by varying n˜ from 1.58 to 1.501 with εΓ = 0.003 (note that in this case varying n˜ results in sweeping ξc from − 5 to −4 for a fixed λ = 0).

Grahic Jump Location
Fig. 7

Comparison of the general path pseudoenergy method with simulations of the full EOM for μ = 0, n = 1.5, Γ0 = Γ/2, α = 0, and β = 1. Solid line is the pseudoenergy prediction. (a) ε = 0.07 with simulation data: sweep of χc by varying λ from 0.80 to 0 with εΓ = 0.012, and σc = -2.68 (n˜ = 1.51). Sweep of χc by varying εΓ from 0.003 to 0.013 with ξc = -4.09 (λ = 0), and σc = -2.68 (n˜ = 1.51). Sweep of χc by varying n˜ from 1.58 to 1.501 with εΓ = 0.01 (note that in this case varying n˜ results in varying ξc from −5 to −4 for a fixed λ = 0) (b) ε = 0.10 with simulation data: sweep of χc by varying λ from 0.80 to 0 with εΓ = 0.019, and σc = -2.55 (n˜ = 1.51). Sweep of χc by varying εΓ from 0.005 to 0.021 with ξc = -4.09 (λ = 0), and σc = -2.55 (n˜ = 1.51). Sweep of χc by varying n˜ from 1.58 to 1.501 with εΓ = 0.016 (note that in this case varying n˜ results in sweeping ξc from −5 to −4 for a fixed λ = 0).

Grahic Jump Location
Fig. 8

Comparison of the general path pseudoenergy method with simulations of the full EOM for Dc = 0.05 (ζ = 0.001) and Dc = 0.10 (ζ = 0.002). Simulation data: sweep of χc by varying λ from 0 to 0.80 with εΓ = 0.013. Sweep of χc by varying εΓ from 0.003 to 0.013 with ξc = -4.09 (λ = 0). Other parameter values used in simulations are: n˜ = 1.51, n = 1.5, ε = 0.07, Γ0 = Γ/2, α = 0, and β = 1.

Grahic Jump Location
Fig. 9

Comparison of the near-tautochronic path pseudoenergy method with simulations of the full EOM for λ = λe, μ = 0, n = 1.5, ε = 0.07, Γ0 = Γ/2, α = 0, and β = 1. (a) Simulation data: sweep of χt by varying εΓ from 0.009 to 0.033 and σt = -2.68 (n˜ = 1.51). (b) Simulation data: sweep of χt by varying n˜ from 1.67 to 1.50 with εΓ = 0.028.

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