Abstract
We study the primary resonance of a parametrically damped Mathieu equation with direct excitation. Potential applications include wind-turbine blade vibration with cyclic stiffening and aeroelastic effects, which may induce parametric damping, and devices with designed cyclic damping for resonance manipulation. The parametric stiffness, parametric damping, and the direct forcing all have the same excitation frequency, with phase parameters between these excitation sources. The parametric amplification at primary resonance is examined by applying the second-order method of multiple scales. With parametric stiffness and direct excitation, it is known that there is a primary parametric resonance that is an amplifier under most excitation phases, but can be a slight suppressor in a small range of phases. The parametric damping is shown to interact with the parametric stiffness to further amplify, or suppress, the resonance amplitude relative to the resonance under parametric stiffness. The effect of parametric damping without parametric stiffness is to shift the resonant frequency slightly, while inducing less significant resonance amplification. The phase of the parametric damping excitation, relative to the parametric stiffness, has a strong influence on the amplification or suppression characteristics. There are optimal phases of both the direct excitation and the parametric damping for amplifying or suppressing the resonance. The effect of the strength of parametric damping is also studied. Numerical simulations validate the perturbation analysis.
Graphical Abstract Figure
References
1.
McCroskey
, W. J.
, 1981
, “The Phenomenon of Dynamic Stall
,” NASA Technical Memorandum 81264, NASA -81264, Ames Research Center, Moffett Field, CA.2.
Afzali
, F.
, Kapucu
, O.
, and Feeny
, B. F.
, 2016
, “Vibrational Analysis of Vertical-Axis Wind-Turbine Blades
,” ASME International Design Engineering Technical Conferences, 28th Conference on Vibration and Noise, Aug. 21–24, Charlotte, NC
, Paper No. IDETC2016-60374
.3.
Ishida
, Y.
, Inoue
, T.
, and Nakamura
, K.
, 2009
, “Vibration of a Wind Turbine Blade (Theoretical Analysis and Experiment Using a Single Rigid Blade Model)
,” J. Environment Eng.
, 4
(2
), pp. 443
–454
. 4.
Allen
, M. S.
, Sracic
, M. W.
, Chauhan
, S.
, and Hansen
, M. H.
, 2011
, “Output-Only Modal Analysis of Linear Time-Periodic Systems With Application to Wind Turbine Simulation Data
,” Mech. Syst. Signal. Process.
, 25
(4
), pp. 1174
–1191
. 5.
Inoue
, T.
, Ishida
, Y.
, and Kiyohara
, T.
, 2012
, “Nonlinear Vibration Analysis of the Wind Turbine Blade (Occurrence of the Superharmonic Resonance in the Out-of-Plane Vibration of the Elastic Blade)
,” ASME J. Vib. Acoust.
, 134
(3
), p. 031009
. 6.
Ramakrishnan
, V.
, and Feeny
, B. F.
, 2012
, “Resonances of the Forced Mathieu Equation With Reference to Wind Turbine Blades
,” ASME J. Vib. Acoust.
, 134
(6
), p. 064501
. 7.
Ikeda
, T.
, Harata
, Y.
, and Ishida
, Y.
, 2018
, “Parametric Instability and Localization of Vibrations in Three-Blade Wind Turbines
,” ASME J. Comput. Nonlinear. Dyn.
, 13
(7
), p. 071001
. 8.
Acar
, G. D.
, Acar
, M. A.
, and Feeny
, B. F.
, 2020
, “Parametric Resonances of a Three-Blade-Rotor System With Reference to Wind Turbines
,” ASME J. Vib. Acoust.
, 142
(2
), p. 021013
. 9.
Feynman
, R. P.
, Leighton
, R. B.
, and Sands
, M.
, 1977
, The Feynman Lectures on Physics
, Addison-Wesley Publishing Company
, Reading, MA
.10.
Bae
, J. S.
, Kwak
, M. K.
, and Inman
, D. J.
, 2004
, “Vibration Suppression of a Cantilever Beam Using Eddy Current Damper
,” J. Sound. Vib.
, 284
(3–5
), pp. 805
–824
. 11.
Sodano
, H. A.
, Bae
, J.
, Inman
, D. J.
, and Belvin
, W. K.
, 2005
, “Concept and Model of Eddy Current Damper for Vibration Suppression of a Beam
,” J. Sound. Vib.
, 288
(4–5
), pp. 1177
–1196
. 12.
Xing
, X.
, and Feeny
, B. F.
, 2017
, “Experimental Study on Complex Modes of an End-Damped Continuous Beam
,” ASME J. Vib. Acoust.
, 139
(6
), p. 061014
. 13.
Rugar
, D.
, and Grutter
, P.
, 1991
, “Mechanical Parametric Amplification and Thermomechanical Noise Squeezing
,” Phys. Rev. Lett.
, 67
(6
), pp. 699
–702
. 14.
Rand
, R.
, 2005
, “Lecture Notes on Nonlinear Vibration,” Ithaca, NY, http://audiophile.tam.cornell.edu/randdocs/nlvibe52.pdf.15.
Nayfeh
, A. H.
, and Mook
, D. T.
, 1979
, Nonlinear Oscillations
, Wiley Interscience Publications, John Wiley and Sons
, New York
.16.
McLachlan
, N.
, 1964
, Theory and Application of Mathieu Functions
, Dover Publications
, New York
.17.
Acar
, G.
, and Feeny
, B. F.
, 2016
, “Floquet-Based Analysis of General Responses of the Mathieu Equation
,” ASME J. Vib. Acoust.
, 138
(4
), p. 041017
. 18.
Hartono
, and van der Burgh
, A. H. P.
, 2004
, “An Equation With a Time-Periodic Damping Coefficient: Stability Diagram and an Application
,” J. Eng. Math.
, 49
(2
), pp. 99
–112
.19.
Afzali
, F.
, Acar
, G. D.
, and Feeny
, B. F.
, 2021
, “A Floquet-Based Analysis of Parametric Excitation Through the Damping Coefficient
,” ASME J. Vib. Acoust.
, 143
(4
), p. 041003
. 20.
Afzali
, F.
, and Feeny
, B. F.
, 2020
, “Response Characteristics of Systems With Parametric Excitation Through Damping and Stiffness
,” ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 32nd Conference on Vibration and Noise, (Originally St. Louis), Aug. 16–19, Paper No. DETC2020-22457
.21.
Rhoads
, J. F.
, Shaw
, S. W.
, Turner
, K. L.
, Moehlis
, J.
, DeMartini
, B. E.
, and Zhang
, W.
, 2006
, “Generalized Parametric Resonance in Electrostatically Actuated Microelectromechanical Oscillators
,” J. Sound. Vib.
, 296
(4–5
), pp. 797
–829
. 22.
Rhoads
, J. F.
, and Shaw
, S. W.
, 2010
, “The Impact of Nonlinearity on Degenerate Parametric Amplifiers
,” Appl. Phys. Lett.
, 96
(23
), p. 234101
. 23.
Zalalutdinov
, M.
, Olkhovets
, A.
, Zehnder
, A.
, Ilic
, B.
, Czaplewski
, D.
, Craighead
, H. G.
, and Parpia
, J. M.
, 2001
, “Optically Pumped Parametric Amplification for Micromechanical Oscillators
,” Appl. Phys. Lett.
, 78
(20
), pp. 3142
–3144
. 24.
Rhoads
, J. F.
, Miller
, N. J.
, Shaw
, S. W.
, and Feeny
, B. F.
, 2008
, “Mechanical Domain Parametric Amplification
,” ASME J. Vib. Acoust.
, 130
(6
), p. 061006
. 25.
Li
, D. H.
, and Shaw
, S. W.
, 2020
, “The Effects of Nonlinear Damping on Degenerate Parametric Amplification
,” Nonlinear Dyn.
, 102
(4
), pp. 2433
–2452
. 26.
Mohamad
, M. A.
, and Sapsis
, T.
, 2016
, “Probabilistic Response and Rare Events in Mathieu’s Equation Under Correlated Parametric Excitation
,” Ocean. Eng.
, 120
, pp. 289
–297
. 27.
Ecker
, H.
, 2011
, “Beneficial Effects of Parametric Excitation in Rotor Systems,” IUTAM Symposium on Emerging Trends in Rotor Dynamics, Vol. 1011 of IUTAM Bookseries
, K.
Gupta
, ed., Springer
, New York
, pp. 361
–371
.28.
Tchokogoué
, D.
, Mu
, M.
, Feeny
, B. F.
, Geist
, B. K.
, and Shaw
, S. W.
, 2021
, “The Effects of Gravity on the Response of Centrifugal Pendulum Vibration Absorbers
,” ASME J. Vib. Acoust.
, 143
(6
), p. 061011
. 29.
Arvin
, H.
, Arena
, A.
, and Lacarbanara
, W.
, 2020
, “Nonlinear Vibration Analysis of Rotating Beams Undergoing Parametric Instability: Lagging-Axial Motion
,” Mech. Syst. Signal. Process.
, 144
, p. 106892
. 30.
Arrowsmith
, D.
, and Mondragón
, R.
, 1999
, “Stability Region Control for a Parametrically Forced Mathieu Equation
,” Meccanica
, 34
(6
), pp. 401
–410
. 31.
Latalski
, J.
, and Warminski
, J.
, 2022
, “Primary and Combined Multi-frequency Parametric Resonances of a Rotating Thin-Walled Composite Beam Under Harmonic Base Excitation
,” J. Sound. Vib.
, 523
, p. 116680
. 32.
Song
, Y.
, Sato
, H.
, Iwata
, Y.
, and Komatsuzaki
, T.
, 2003
, “The Response of a Dynamic Vibration Absorber System With a Parametrically Excited Pendulum
,” J. Sound. Vib.
, 259
(4
), pp. 747
–759
. 33.
Warminski
, J.
, and Kecik
, K.
, 2009
, “Instabilities in the Main Parametric Resonance Area of a Mechanical System With a Pendulum
,” J. Sound. Vib.
, 322
(3
), pp. 612
–628
. 34.
Gupta
, A.
, and Tai
, W.-C.
, 2022
, “The Response of an Inerter-Based Dynamic Vibration Absorber System With a Parametrically Excited Centrifugal Pendulum
,” ASME J. Vib. Acoust.
, 144
(4
), p. 041011
. 35.
Gupta
, A.
, and Tai
, W.-C.
, 2023
, “Ocean Wave Energy Conversion With a Spar-Floater System Using a Nonlinear Inerter Pendulum Vibration Absorber
,” ASME International Design Engineering Technical Conferences, Paper No. IDETC2023–117069
.36.
Belhaq
, M.
, and Houssni
, M.
, 1999
, “Quasi-Periodic Oscillations, Chaos and Suppression of Chaos in a Nonlinear Oscillator Driven by Parametric and External Excitations
,” Nonlinear Dyn.
, 18
(1
), pp. 1
–24
. 37.
Pandey
, M.
, Rand
, R. H.
, and Zehnder
, A. T.
, 2008
, “Frequency Locking in a Forced Mathieu-van Der Pol-Duffing System
,” Nonlinear Dyn.
, 54
(1–2
), pp. 3
–12
. 38.
Ng
, L.
, and Rand
, R. H.
, 2002
, “Bifurcations in a Mathieu Equation With Cubic Nonlinearities
,” Chaos Solitons Fractals
, 14
(2
), pp. 173
–181
. 39.
Marghitu
, D. B.
, Sinha
, S. C.
, and Boghiu
, D.
, 1998
, “Stability and Control of a Parametrically Excited Rotating System. Part 1: Stability Analysis
,” Dyn. Control
, 8
(1
), pp. 7
–20
. 40.
Tondl
, A.
, and Ecker
, H.
, 2003
, “On the Problem of Self-Excited Vibration Quenching by Means of Parametric Excitation
,” Appl. Mech.
, 72
(11–12
), pp. 923
–932
. 41.
Month
, L.
, and Rand
, R.
, 1982
, “Bifurcation of 4-1 Subharmonics in the Non-Linear Mathieu Equation
,” Mech. Res. Commun.
, 9
(4
), pp. 233
–240
.42.
Zounes
, R. S.
, and Rand
, R. H.
, 2002
, “Subharmonic Resonance in the Non-linear Mathieu Equation
,” Int. J. Non-Linear Mech.
, 37
(1
), pp. 43
–73
. 43.
Szabelski
, K.
, and Warminski
, J.
, 1995
, “Self-Excited System Vibrations With Parametric and External Excitations
,” J. Sound. Vib.
, 187
(4
), pp. 595
–607
. 44.
Sharma
, A.
, 2020
, “A Re-Examination of Various Resonances in Parametrically Excited Systems
,” ASME J. Vib. Acoust.
, 142
(3
), p. 031010
. 45.
Aghamohammadi
, A.
, Sorokin
, V.
, and Mace
, B.
, 2022
, “Dynamic Analysis of the Response of Duffing-Type Oscillators Subject to Interacting Parametric and External Excitations
,” Nonlinear Dyn.
, 107
(1
), pp. 99
–120
. 46.
Chakraborty
, S.
, and Sarkar
, A.
, 2013
, “Parametrically Excited Non-linearity in Van Der Pol Oscillator: Resonance, Anti-Resonance and Switch
,” Physica D: Nonlinear Phenomena
, 254
, pp. 24
–28
. 47.
Afzali
, F.
, Kharazmi
, E.
, and Feeny
, B. F.
, 2023
, “Resonances of a Forced Van Der Pol Equation With Parametric Damping
,” Nonlinear Dyn.
, 111
(6
), pp. 5269
–5285
. 48.
Náprstek
, J.
, and Fischer
, C.
, 2019
, “Super and Sub-Harmonic Synchronization in Generalized Van Der Pol Oscillator
,” Comput. Struct.
, 224
, p. 106103
. 49.
Shariati
, A.
, Hosseini
, S. H. S.
, Ebrahimi
, F.
, and Toghroli
, A.
, 2021
, “Nonlinear Dynamics and Vibration of Reinforced Piezoelectric Scale-Dependent Plates as a Class of Nonlinear Mathieu-Hill Systems: Parametric Excitation Analysis
,” Eng. Comput.
, 37
(3
), pp. 2285
–2301
. 50.
Ramakrishnan
, V.
, and Feeny
, B. F.
, 2022
, “Primary Parametric Amplification in a Weakly Forced Mathieu Equation
,” J. Vib. Acoust. ASME
, 144
(5
), p. 051006
. 51.
Nayfeh
, A. H.
, 1986
, “Perturbation Methods in Nonlinear Dynamics,” Lecture Notes in Physics
, Vol. 247
, M.
Jowett
, M.
Month
, S.
Turner
, eds., Springer-Verlag
, Berlin
, pp. 238
–314
.52.
Sayed
, M.
, and Hamed
, Y. S.
, 2011
, “Stability and Response of a Nonlinear Coupled Pitch-Roll Ship Model Under Parametric and Harmonic Excitations
,” Nonlinear Dyn.
, 64
(3
), pp. 207
–220
. 53.
Sapmaz
, A.
, and Feeny
, B. F.
, 2018
, “Second-Order Perturbation Analysis of In-Plane Blade-Hub Dynamics of Horizontal-Axis Wind Turbines
,” ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 30th Conference on Vibration and Noise, Quebec City
. Aug. 26–29, Paper No. DETC2018-88203
.54.
Ramakrishnan
, V.
, and Feeny
, B. F.
, 2023
, “Responses of a Strongly Forced Mathieu Equation. Part 1: Cyclic Loading
,” ASME J. Vib. Acoust.
, 145
(3
), p. 031010
. 55.
Acar
, G.
, and Feeny
, B. F.
, 2018
, “Bend-Bend-Twist Vibrations of a Wind Turbine Blade
,” Wind Energy
, 21
(1
), pp. 15
–28
. 56.
Sapmaz
, A.
, and Feeny
, B. F.
, 2020
, “Parametric Stiffness in Large-Scale Wind-Turbine Blades and the Effects on Resonance and Speed Locking
,” ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 32nd Conference on Vibration and Noise. On-Line (originally St. Louis), Aug. 16–19, Paper No. DETC2020-27717
.
