NDLI: Secondary resonances of a quadratic nonlinear oscillator following two-to-one resonant Hopf bifurcations

Content Provider	Springer Nature Link
Author	Ji, J. C.
Copyright Year	2014
Abstract	Stable bifurcating solutions may appear in an autonomous time-delayed nonlinear oscillator having quadratic nonlinearity after the trivial equilibrium loses its stability via two-to-one resonant Hopf bifurcations. For the corresponding non-autonomous time-delayed nonlinear oscillator, the dynamic interactions between the periodic excitation and the stable bifurcating solutions can induce resonant behaviour in the forced response when the forcing frequency and the frequencies of Hopf bifurcations satisfy certain relationships. Under hard excitations, the forced response of the time-delayed nonlinear oscillator can exhibit three types of secondary resonances, which are super-harmonic resonance at half the lower Hopf bifurcation frequency, sub-harmonic resonance at two times the higher Hopf bifurcation frequency and additive resonance at the sum of two Hopf bifurcation frequencies. With the help of centre manifold theorem and the method of multiple scales, the secondary resonance response of the time-delayed nonlinear oscillator following two-to-one resonant Hopf bifurcations is studied based on a set of four averaged equations for the amplitudes and phases of the free-oscillation terms, which are obtained from the reduced four-dimensional ordinary differential equations for the flow on the centre manifold. The first-order approximate solutions and the nonlinear algebraic equations for the amplitudes and phases of the free-oscillation terms in the steady state solutions are derived for three secondary resonances. Frequency-response curves, time trajectories, phase portraits and Poincare sections are numerically obtained to show the secondary resonance response. Analytical results are found to be in good agreement with those of direct numerical integrations.
Starting Page	2161
Ending Page	2184
Page Count	24
File Format	PDF
ISSN	0924090X
Journal	Nonlinear Dynamics
Volume Number	78
Issue Number	3
e-ISSN	1573269X
Language	English
Publisher	Springer Netherlands
Publisher Date	2014-07-24
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Quadratic nonlinear oscillator Resonant Hopf bifurcations resonances Time delay Two-to-one internal resonance Vibration, Dynamical Systems, Control Mechanics Mechanical Engineering Automotive Engineering
Content Type	Text
Resource Type	Article
Subject	Ocean Engineering Applied Mathematics Control and Systems Engineering Mechanical Engineering Electrical and Electronic Engineering Aerospace Engineering

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Parametric resonances in the two-to-one resonant beams on elastic foundation

Internal, combinational and sub-harmonic resonances of a nonlinear asymmetrical rotating shaft

Nonlinear vibration and of the Duffing oscillator to parametric excitation with time delay feedback

Erratum to: Hopf bifurcations through delay in pilot reaction in a longitudinal flight

Double Hopf Bifurcations and Chaos of a Nonlinear Vibration System

Amplitude reduction of primary resonance of nonlinear oscillator by a dynamic vibration absorber using nonlinear coupling

Primary and secondary resonance analyses of clamped–clamped micro-beams

Effect of External Excitations on a Nonlinear System with Time Delay

Bifurcations of backbone curves for systems of coupled nonlinear two mass oscillator

Secondary resonances of a quadratic nonlinear oscillator following two-to-one resonant Hopf bifurcations

Similar Documents

Parametric resonances in the two-to-one resonant beams on elastic foundation

Internal, combinational and sub-harmonic resonances of a nonlinear asymmetrical rotating shaft

Nonlinear vibration and of the Duffing oscillator to parametric excitation with time delay feedback

Erratum to: Hopf bifurcations through delay in pilot reaction in a longitudinal flight

Double Hopf Bifurcations and Chaos of a Nonlinear Vibration System

Amplitude reduction of primary resonance of nonlinear oscillator by a dynamic vibration absorber using nonlinear coupling

Primary and secondary resonance analyses of clamped–clamped micro-beams

Effect of External Excitations on a Nonlinear System with Time Delay

Bifurcations of backbone curves for systems of coupled nonlinear two mass oscillator

Secondary resonances of a quadratic nonlinear oscillator following two-to-one resonant Hopf bifurcations