Generalization of Den Hartog’s Equal-Peak Method for nonlinear primary systems

Habib, Giuseppe; Detroux, Thibaut; Kerschen, Gaëtan

doi:10.1051/matecconf/20141601005

Cited by 5 publications

(4 citation statements)

References 17 publications

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“…Similarly, the appearance of an “island” in the response of a single forced oscillator has been predicted for subharmonic resonances, when the driving force frequency is close to one third the natural oscillation frequency [ 15 ]. Very recently, analogous effects have been found in related systems such as vibrating electroelastic crystals [ 32 ] and velocity-coupled nonlinear oscillators [ 33 ].…”

Section: Discussionmentioning

confidence: 71%

Internal Resonance in a Vibrating Beam: A Zoo of Nonlinear Resonance Peaks

Mangussi¹,

Zanette²

2016

PLoS ONE

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In oscillating mechanical systems, nonlinearity is responsible for the departure from proportionality between the forces that sustain their motion and the resulting vibration amplitude. Such effect may have both beneficial and harmful effects in a broad class of technological applications, ranging from microelectromechanical devices to edifice structures. The dependence of the oscillation frequency on the amplitude, in particular, jeopardizes the use of nonlinear oscillators in the design of time-keeping electronic components. Nonlinearity, however, can itself counteract this adverse response by triggering a resonant interaction between different oscillation modes, which transfers the excess of energy in the main oscillation to higher harmonics, and thus stabilizes its frequency. In this paper, we examine a model for internal resonance in a vibrating elastic beam clamped at its two ends. In this case, nonlinearity occurs in the form of a restoring force proportional to the cube of the oscillation amplitude, which induces resonance between modes whose frequencies are in a ratio close to 1:3. The model is based on a representation of the resonant modes as two Duffing oscillators, coupled through cubic interactions. Our focus is put on illustrating the diversity of behavior that internal resonance brings about in the dynamical response of the system, depending on the detailed form of the coupling forces. The mathematical treatment of the model is developed at several approximation levels. A qualitative comparison of our results with previous experiments and numerical calculations on elastic beams is outlined.

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Section: Discussionmentioning

confidence: 71%

Internal Resonance in a Vibrating Beam: A Zoo of Nonlinear Resonance Peaks

Mangussi¹,

Zanette²

2016

PLoS ONE

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“…The equal-peak method proposed by Den Hartog (1985) is used to solve the optimal stiffness and damping. Habib et al (2015) developed a nonlinear generalization of Den Hartog’s equal-peak method which is widely used for designing linear vibration absorbers. And the parameters of the system can be confirmed by the following analytical formulas…”

Section: Approximate Analytical Solution and Its Verificationmentioning

confidence: 99%

Dynamic analysis and vibration control of two-degree-of-freedom boring bar with fractional-order model of magnetorheological fluid

Hou

Niu

Shen

et al. 2021

Journal of Vibration and Control

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The dynamic analysis and vibration control of a kind of damping boring bar system with magnetorheological fluid absorber are studied. A two-degree-of-freedom dynamic model of boring bar with magnetorheological fluid dynamic vibration absorber is established, in which the magnetorheological fluid damper adopts the fractional-order Bingham model. The forced vibration of the system is studied, and the approximate analytical solution is obtained by the averaging method. By comparing the analytical solution with the numerical solution, the accuracy of the analytical solution is proved by their high degree of fit. And the parameters of boring bar system are optimized by using the analytical solution. To reduce the amplitude of the system, a semi-active control strategy of modified relative velocity method is used to suppress the resonance response of the boring bar. Under the semi-active relative velocity control, the approximate analytical solution of the system is also obtained by means of averaging method. The stability of the system under semi-active control is analyzed by using the Lyapunov method. Compared with passive control, the semi-active relative velocity control has a very great effect on the vibration amplitude suppression. The results can provide a reference for the study of dynamics and vibration control of the systems with magnetorheological fluid control.

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“…Equation (11) is equivalent to Equation (5). The idea of resonating a mechanical system is equivalent to designing a vibration isolator [26,27] that is attempting to minimize the vibration response of the main structure such as an airplane engine. An exception is the WEC resonator design which intentionally excites the mechanical system to increase power/energy capture.…”

Section: Mechanical Systemsmentioning

confidence: 99%

Extending Complex Conjugate Control to Nonlinear Wave Energy Converters

Wilson

Robinett

Bacelli

et al. 2020

JMSE

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This paper extends the concept of Complex Conjugate Control (CCC) of linear wave energy converters (WECs) to nonlinear WECs by designing optimal limit cycles with Hamiltonian Surface Shaping and Power Flow Control (HSSPFC). It will be shown that CCC for a regular wave is equivalent to a power factor of one in electrical power networks, equivalent to mechanical resonance in a mass-spring-damper (MSD) system, and equivalent to a linear limit cycle constrained to a Hamiltonian surface defined in HSSPFC. Specifically, the optimal linear limit cycle is defined as a second-order center in the phase plane projection of the constant energy orbit across the Hamiltonian surface. This concept of CCC described by a linear limit cycle constrained to a Hamiltonian surface will be extended to nonlinear limit cycles constrained to a Hamiltonian surface for maximum energy harvesting by the nonlinear WEC. The case studies presented confirm increased energy harvesting which utilizes nonlinear geometry realization for reactive power generation.

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Generalization of Den Hartog’s Equal-Peak Method for nonlinear primary systems

Cited by 5 publications

References 17 publications

Internal Resonance in a Vibrating Beam: A Zoo of Nonlinear Resonance Peaks

Internal Resonance in a Vibrating Beam: A Zoo of Nonlinear Resonance Peaks

Dynamic analysis and vibration control of two-degree-of-freedom boring bar with fractional-order model of magnetorheological fluid

Extending Complex Conjugate Control to Nonlinear Wave Energy Converters

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