Free Response of a Rotational Nonlinear Energy Sink Coupled to a Linear Oscillator: Fractality, Riddling, and Initial-Condition Sensitivity at Intermediate Initial Displacements

2021 ◽  
pp. 1-59
Author(s):  
Ke Ding ◽  
Arne Pearlstein
Keyword(s):  
Initial Conditions ◽  
Nonlinear Energy Sink ◽  
Initial Energy ◽  
Linear Oscillator ◽  
Rectilinear Motion ◽  
Free Response ◽  
Time Harmonic ◽  
Wide Range ◽  
Energy Sink ◽  
Nonlinear Energy

Abstract Free response of a rotational nonlinear energy sink (NES) inertially coupled to a linear oscillator is investigated for dimensionless initial rectilinear displacements ranging from just above the smallest amplitude at which nonrotating, harmonically rectilinear motion is unstable absent direct rectilinear damping, up to the next-largest amplitude at which such motion is orbitally stable. With motionless initial conditions (MICs), i.e., initial velocity of the primary mass and initial angular velocity of the NES mass both zero, predicted behavior for two previously investigated combinations of the dimensionless parameters (characterizing rotational damping, and coupling of rectilinear and rotational motions) differs strongly from that found at smaller initial displacements (2021, J. Appl. Mech. 88, 011005). For both combinations, a wide range of MICs leads to solutions displaying transient chaos and depending sensitively on initial conditions, giving rise to fractality and riddling in the relationship between initial conditions and asymptotic solutions. Absent direct rectilinear damping of the linear oscillator, for one combination of parameters there exists a wide range of MICs with trajectories leading to time-harmonic, orbitally stable “quot;special”quot; solutions with a single amplitude, but no MICs are found for which all initial energy is dissipated. For the other combination, no such special solutions are found, but there exist MICs for which all initial energy is dissipated. With direct rectilinear damping, sensitivity extends to a measure of settling time, which can be extremely sensitive to initial conditions. A statistical approach to this sensitivity is discussed, along with implications for design and implementation.

scholarly journals Free Response of a Rotational Nonlinear Energy Sink: Complete Dissipation of Initial Energy for Small Initial Rectilinear Displacements

2020 ◽  
Vol 88 (1) ◽  
Author(s):  
Ke Ding ◽  
Arne J. Pearlstein
Keyword(s):  
Coupling Parameter ◽  
Equations Of Motion ◽  
Nonlinear Energy Sink ◽  
Initial Energy ◽  
Rectilinear Motion ◽  
Initial Condition ◽  
Free Response ◽  
Primary Mass ◽  
Energy Sink ◽  
Nonlinear Energy

Abstract For two combinations of a dimensionless rotational damping parameter and a dimensionless inertial coupling parameter, we consider free response of a rectilinearly vibrating linearly sprung primary mass inertially coupled to damped rotation of a second mass, for which Gendelman et al. (2012, “Dynamics of an Eccentric Rotational Nonlinear Energy Sink,” ASME J. Appl. Mech. 79(1), 011012) developed equations of motion in the context of a rotational nonlinear energy sink (NES) with no direct damping of the rectilinear motion. For dimensionless initial rectilinear displacements comparable with those considered by Gendelman et al., we identify a region in the motionless projection of the initial condition space (i.e., for zero values of the initial rectilinear and rotational velocities) in which every initial condition leads to a previously unrecognized zero-energy solution, with all initial energy dissipated by rotation. We also show that the long-time nonrotating, rectilinear solutions of the type found by Gendelman et al. are (orbitally) stable only in limited ranges of amplitude. Finally, we show how direct viscous damping of rectilinear motion of the primary mass affects dissipation, and that results with no direct rectilinear dissipation provide excellent guidance for performance when direct rectilinear dissipation occurs. Some applications are discussed.

Tuned pendulum as nonlinear energy sink for broad energy range

2016 ◽  
Vol 23 (3) ◽  
pp. 373-388 ◽  
Author(s):  
Maor Farid ◽  
Oleg V Gendelman
Keyword(s):  
Initial Conditions ◽  
Nonlinear Energy Sink ◽  
Primary System ◽  
Targeted Energy Transfer ◽  
Broad Energy Range ◽  
Wide Range ◽  
Mass Damper ◽  
Energy Sink ◽  
Nonlinear Energy ◽  
Broad Energy

Nonlinear energy sinks (NES) are widely studied as a possible engineering solution for mitigation of steady-state, impulsive and transient broadband excitations. Current work is devoted to the applicability of common pendulum as the NES for mitigation of impulsive excitations. It turns out that the pendulum NES can overcome one of the main shortcomings of more traditional NES designs, since it is able to mitigate excitation of a primary system in a relatively wide range of initial energies. This is because the pendulum can be captured into a resonance with primary oscillator both for rotational and oscillatory responses. If parameters are chosen properly, for small energies the pendulum responds almost as a common tuned mass damper. However, at higher energies, the pendulum acts as rotational NES. Thus, relatively broad diapason of initial energies can be covered. This paper presents numeric evidence for the efficiency of this design of the NES and discusses its optimal tuning. Another important finding is that the NES’s efficiency exhibits rather broad deviations for different realizations of the initial conditions with the same energy. We present a theoretical analysis of the damped targeted energy transfer into the pendulum NES from the primary mass with an account of corrections caused by the effect of gravity.

A Fractional Calculus for Nonlinear Energy Sink Used in Vibration Absorption System

2011 ◽  
Vol 42 (10) ◽  
pp. 62-67
Author(s):  
Song Li ◽  
Bo Fang ◽  
Tianzhi Yang ◽  
Wenhu Huang
Keyword(s):  
Fractional Calculus ◽  
Nonlinear Energy Sink ◽  
Small Mass ◽  
Degree Of Freedom ◽  
Linear Oscillator ◽  
Order System ◽  
Good Correspondence ◽  
Vibration Absorption ◽  
Energy Sink ◽  
Nonlinear Energy

The phenomenon of energy pumping, in which vibratory energy is transferred irreversibly within a nonlinear, multi-degree-of-freedom system with the goal of reducing the transient response of the primary substructure, has recently been investigated analytically and through numerical simulations. The dynamics of single degree of freedom linear subsystem with attached nonlinear energy sink is investigated. The response of a linear oscillator attached to nonlinear energy sink with relatively small mass under external forcing in a vicinity of main resonance is studied analytically and numerically. It is possible that targeted energy could transfer from linear oscillators to the nonlinear energy sink in this system. Analytical model is verified numerically and a fairly good correspondence is observed. Fractional calculus offers a powerful tool to describe the dynamic behavior of real vibration absorption. A version of the fractional derivative models is presented and investigated in this paper for analyzing vibration absorption behavior of nonlinear energy sink. It is shown that the fractional-order system is in a stronger position than the traditional nonlinear energy sink coupled to the linear oscillator.

Numerical Study of a Single-Sided Vibro-Impact Track Nonlinear Energy Sink Considering Horizontal and Vertical Dynamics

2019 ◽  
Vol 141 (6) ◽  
Author(s):  
Wenke Li ◽  
Nicholas E. Wierschem ◽  
Xinhui Li ◽  
Tiejun Yang ◽  
Michael J. Brennan
Keyword(s):  
Primary Structure ◽  
Numerical Study ◽  
Nonlinear Energy Sink ◽  
Horizontal Direction ◽  
Impact Surface ◽  
Wide Range ◽  
Quartic Polynomial ◽  
Energy Sink ◽  
The Impact ◽  
Nonlinear Energy

Abstract In this paper, the single-sided vibro-impact track nonlinear energy sink (SSVI track NES) is studied. The SSVI track NES, which is attached to a primary structure, has nonlinear behavior caused by the NES mass moving on a fixed track and impacting on the primary structure at an impact surface. Unlike previous studies of the SSVI track NES, both the horizontal and vertical dynamics of the primary structure are considered. A numerical study is carried out to investigate the way in which energy is dissipated in this system. Assuming a track shape with a quartic polynomial, an optimization procedure that considers the total energy dissipated during a time period is carried out, to determine the optimum NES mass and track parameter. It is found that there is dynamic coupling between the horizontal and vertical directions caused by the SSVI track NES motion. The vibrational energy, originally in the structure in the horizontal direction, is transferred to the vertical motion of the structure where it is dissipated. Considering that many civil and mechanical systems are particularly vulnerable to extreme loads in the horizontal direction, this energy transformation can be beneficial to prevent or limit damage to the structure. The effect on energy dissipation of the position of the impact surface in the SSVI track NES and the ratio of the vertical to horizontal stiffness in the primary structure are discussed. Numerical results demonstrate a robust and stable performance of the SSVI track NES over a wide range of stiffness ratios.

A Non-Traditional Variant Nonlinear Energy Sink: Transient Responses

2019 ◽  
Author(s):  
Youzuo Jin ◽  
Kefu Liu ◽  
Deli Li ◽  
Liuyang Xiong ◽  
Lihua Tang
Keyword(s):  
Energy Harvester ◽  
Vibration Suppression ◽  
System Modeling ◽  
Nonlinear Energy Sink ◽  
Initial Energy ◽  
Transient Responses ◽  
Continuous Contact ◽  
Magnet Coil ◽  
Energy Sink ◽  
Nonlinear Energy

Abstract In this paper, a non-traditional variant nonlinear energy sink (NES) is developed for simultaneous vibration suppression and energy harvesting in a broad frequency band. The non-traditional variant NES consists of a cantilever beam attached by a pair of magnets at its free end, a pair of the so-called continuous-contact blocks, and a pair of coils. The beam is placed between the continuous-contact blocks. The constraint of the continuous-contact blocks forces the beam to deflect nonlinearly. Each of the magnet-coil pairs forms an electromagnetic energy harvester. Different from a traditional way that attaches the coils to the primary mass, the developed setup has the coils fixed to the base. First, the developed apparatus is described. Subsequently, the system modeling and parameter identification are addressed. The performance of the apparatus under transient responses is examined by using computer simulation. The results show that the proposed apparatus behaves similarly as the NES with the following features: 1:1 resonance, targeted energy transfer, initial energy dependence, etc.

scholarly journals Design Optimisation of a Nonlinear Energy Sink Embedded on a Harmonically Forced Linear Oscillator: Theoretical and Experimental Developments

2012 ◽  
Author(s):  
Etienne Gourc ◽  
Guilhem Michon ◽  
Sébastien Seguy ◽  
Alain Berlioz
Keyword(s):  
Invariant Manifolds ◽  
Relaxation Oscillation ◽  
Design Procedure ◽  
Nonlinear Energy Sink ◽  
Experimental Results ◽  
Linear Oscillator ◽  
Mapping Procedure ◽  
Modulation Equations ◽  
Energy Sink ◽  
Nonlinear Energy

In this paper, the dynamic response of a harmonically forced Linear Oscillator (LO) strongly coupled to a Nonlinear Energy Sink (NES) is investigated theoretically and experimentally. The system studied comprises a linear oscillator subject to an imposed displacement with an embedded, purely cubic, NES. The behavior of the system is analyzed in the vicinity of 1:1 resonance. The complexification averaging technique is used to obtain modulation equations and the associated fixed points. These modulation equations are analyzed using asymptotic expansion to study the regimes related as relaxation oscillation of the slow flow called Strongly Modulated Response (SMR). The zones where SMR occur are computed using a mapping procedure. The Slow Invariant Manifolds (SIM) is used to derive a proper optimization procedure. It is shown that there exist an optimal zone in the parameter plane forcing amplitude–nonlinear stiffness, where SMR occurs without having a high amplitude detached resonance tongue. An experimental setup exhibits a strong mass asymmetry (mass ratio ≈ 1%). The cubic stiffness is realized geometrically with two linear spring that extend axially and are free to rotate. Using the previous optimized stiffness of the NES, different frequency response curves and associated zones of SMR are obtained for various forcing amplitude. Good agreement between theoretical and experimental results is observed. The reported experimental results confirm the design procedure, and the possible application of NES for vibration mitigation under periodic forcing.

Quasi-Periodic Response Regimes of Linear Oscillator Coupled to Nonlinear Energy Sink Under Periodic Forcing

2006 ◽  
Vol 74 (2) ◽  
pp. 325-331 ◽  
Author(s):  
O. V. Gendelman ◽  
Yu. Starosvetsky
Keyword(s):  
Nonlinear Energy Sink ◽  
Small Mass ◽  
Linear Instability ◽  
Linear Oscillator ◽  
Analytic Model ◽  
Periodic Response ◽  
State Regime ◽  
Energy Sink ◽  
Nonlinear Normal ◽  
Nonlinear Energy

Quasi-periodic response of a linear oscillator attached to nonlinear energy sink with relatively small mass under external sinusoidal forcing in a vicinity of main (1:1) resonance is studied analytically and numerically. It is shown that the quasi-periodic response is exhibited in well-defined amplitude-frequency range of the external force. Two qualitatively different regimes of the quasi-periodic response are revealed. The first appears as a result of linear instability of the steady-state regime of the oscillations. The second one occurs due to interaction of the dynamical flow with invariant manifold of damped-forced nonlinear normal mode of the system, resulting in hysteretic motion of the flow in the vicinity of this mode. Parameters of external forcing giving rise to the quasi-periodic response are predicted by means of simplified analytic model. The model also allows predicting that the stable quasi-periodic regimes appear for certain range of damping coefficient. All findings of the simplified analytic model are verified numerically and considerable agreement is observed.

scholarly journals Chaotic characteristic of a linear oscillator coupled with vibro-impact nonlinear energy sink

2017 ◽  
Vol 91 (4) ◽  
pp. 2319-2330 ◽  
Author(s):  
Tao Li ◽  
Claude-Henri Lamarque ◽  
Sébastien Seguy ◽  
Alain Berlioz
Keyword(s):  
Nonlinear Energy Sink ◽  
Linear Oscillator ◽  
Chaotic Characteristic ◽  
Energy Sink ◽  
Nonlinear Energy
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