Date of download: 9/26/2017 Copyright © ASME. All rights reserved. From: Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers—Part I: Theory J. Vib. Acoust. 2013;135(1):011017-011017-10. doi:10.1115/1.4007561 Figure Legend: 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.

Date of download: 9/26/2017 Copyright © ASME. All rights reserved. From: Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers—Part I: Theory J. Vib. Acoust. 2013;135(1):011017-011017-10. doi:10.1115/1.4007561 Figure Legend: 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.

Date of download: 9/26/2017 Copyright © ASME. All rights reserved. From: Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers—Part I: Theory J. Vib. Acoust. 2013;135(1):011017-011017-10. doi:10.1115/1.4007561 Figure Legend: 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.

Date of download: 9/26/2017 Copyright © ASME. All rights reserved. From: Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers—Part I: Theory J. Vib. Acoust. 2013;135(1):011017-011017-10. doi:10.1115/1.4007561 Figure Legend: 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

Date of download: 9/26/2017 Copyright © ASME. All rights reserved. From: Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers—Part I: Theory J. Vib. Acoust. 2013;135(1):011017-011017-10. doi:10.1115/1.4007561 Figure Legend: 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%.

Date of download: 9/26/2017 Copyright © ASME. All rights reserved. From: Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers—Part I: Theory J. Vib. Acoust. 2013;135(1):011017-011017-10. doi:10.1115/1.4007561 Figure Legend: 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).

Date of download: 9/26/2017 Copyright © ASME. All rights reserved. From: Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers—Part I: Theory J. Vib. Acoust. 2013;135(1):011017-011017-10. doi:10.1115/1.4007561 Figure Legend: 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).

Date of download: 9/26/2017 Copyright © ASME. All rights reserved. From: Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers—Part I: Theory J. Vib. Acoust. 2013;135(1):011017-011017-10. doi:10.1115/1.4007561 Figure Legend: 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.

Date of download: 9/26/2017 Copyright © ASME. All rights reserved. From: Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers—Part I: Theory J. Vib. Acoust. 2013;135(1):011017-011017-10. doi:10.1115/1.4007561 Figure Legend: 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.