A principle of similarity for nonlinear vibration absorbers

Habib, Giuseppe; Kerschen, Gaëtan

doi:10.1016/j.physd.2016.06.001

Cited by 62 publications

(54 citation statements)

References 31 publications

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“…It is worth noting that the vectorŨ 0 is unknown and is normalized by the third equation so thatŨ 0 -0. At this point, we can summarize the number of unknowns and equations of (22). We have:…”

Section: Limit Point Continuationmentioning

confidence: 99%

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Numerical antiresonance continuation of structural systems

Renault

Thomas

Mahé

2019

Mechanical Systems and Signal Processing

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a b s t r a c tTuned dynamic absorbers are usually used to counteract vibrations at a given frequency. Presence of non-linearities causes energy-dependent relationship of their resonance and antiresonance frequencies at large amplitude of motion, which consequently leads to a detuning of the absorber from the targeted frequency. This paper presents a procedure to track an extremum point (minimum or maximum) of nonlinear frequency responses, based on a numerical continuation technique coupled to the harmonic balance method to follow periodic solutions in forced steady-state. It thus enable to track a particular antiresonance. The procedure is tested and applied on some application cases to highlight the resonance and antiresonance behavior in presence of geometrically non-linear and/or inertial interactions.

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Section: Limit Point Continuationmentioning

confidence: 99%

“…For instance, Habib and Kerschen use in Ref. [22] a perturbation method coupled to the HBM in order to analytically predict resonance peaks and establish some design rules about of the NLTVA. Some numerical methods are proposed to accurately predict amplitude resonance peak of a nonlinear system.…”

Section: Introductionmentioning

confidence: 99%

Numerical antiresonance continuation of structural systems

Renault

Thomas

Mahé

2019

Mechanical Systems and Signal Processing

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“…Febbo and Machado [12] explored the potential of using a nonlinear absorber with a saturation nonlinearity for vibration mitigation of a nonlinear primary oscillator possessing a cubic stiffness. Habib et al [13], Detroux et al [14] and Habib and Kerschen [15] stated that nonlinear https://doi.org/10.1016/j.ymssp.2019.07.005 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved. stiffness k 2 is placed in parallel to the passive mount.…”

Section: Introductionmentioning

confidence: 99%

“…Although it exhibits a simple form, a variety of physical examples can be dynamically characterised by the Duffing equation such as pendulum dynamics [29], beam buckling [30], cable dynamics [31] and nonlinear isolators [32,33]. Using the NPPF controller for vibration mitigation of a Duffing oscillator can be considered as an active anti-vibration approach developed based on the principle of similarity as proposed in [13][14][15], since the NPPF controller possesses the same nonlinearity as the primary structure. Therefore, the optimisation process is sequentially performed in two steps.…”

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confidence: 99%

Nonlinear positive position feedback control for mitigation of nonlinear vibrations

Zhao

Paknejad

Raze

et al. 2019

Mechanical Systems and Signal Processing

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a b s t r a c tThis paper investigates the potential of using a nonlinear positive position feedback controller for vibration mitigation of a Duffing oscillator. The proposed controller is designed based on the principle of similarity which states that anti-vibration devices should be governed by the same equations as those of the host structure. Closed-form expressions for the H 1 optimal control parameters that minimise the maximal response of the structure are firstly derived for the linear positive position feedback controller and then extended to the nonlinear counterpart. The harmonic balance method is employed to approximate the analytical solutions. Both numerical simulations and experimental validations are performed to demonstrate the proposed control strategy.vibration absorbers can be designed as a mirror of the primary structures i.e. nonlinear vibration absorbers should possess the same nonlinearities as those in the primary systems. This design principle is also referred to as the principle of similarity. It was reported that nonlinear primary systems attached with nonlinear vibration absorbers designed based on the principle of similarity behave in a similar fashion as their linear counterpart. Although this concept is promising, it may become cumbersome and expensive to realise them in practice using passive means for complex nonlinear primary systems.On the other hand, in an active approach it is attempted to introduce the desired nonlinear control forces using sensors and actuators. This may yield a nonlinear anti-vibration system that is less complex. Various types of active controllers have been investigated for vibration attenuation of nonlinear systems [16][17][18][19][20][21][22][23][24][25]. Among them, linear positive position feedback (LPPF) and nonlinear positive position feedback (NPPF) controllers are found to be particularly effective if they are aimed to damp one particular structural vibration mode. This type of controller is implemented by feeding the structural position directly to a linear or nonlinear compensator, whose output is then fed through a fixed gain positively back to drive the actuator. In this context, they would be well suited for the applications where piezoelectric sensors and actuators are employed for vibration damping. This is because the voltage from the sensor is proportional to the strain of the attached structure, which can be directly measured to drive the strain-based piezoelectric actuators. Warminski et al. [21] compared the control performance of a LPPF controller with three other controllers, namely proportional position feedback, cubic position feedback and nonlinear saturation feedback, for suppression of nonlinear composite beam vibrations. It was found that the LPPF controller is only effective for weakly nonlinear systems and the nonlinear saturation controller was concluded to be superior for the nonlinear primary structure under consideration. The performance of the LPPF controller was also investigated in [22], but on a four-degree-of-freedo...

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“…An attempt to eliminate this undesired phenomenon demonstrated that IRCs can be avoided if the absorber has a properly-tuned piecewise-quadratic damping characteristic [40]. Similarly, the nonlinear tuned vibration absorber (NLTVA), possessing both a linear and a nonlinear elastic force characteristic, can present an IRC that limits its range of operation [41,42]. A numerical procedure exploiting bifurcation tracking allowed to define regions of appearance of the IRC [43].…”

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confidence: 99%

Isolated resonances and nonlinear damping

2018

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We analyze isolated resonance curves (IRCs) in a single-degree-of-freedom system with nonlinear damping. The adopted procedure exploits singularity theory in conjunction with the harmonic balance method. The analysis unveils a geometrical connection between the topology of the damping force and IRCs. Specifically, we demonstrate that extremas and zeros of the damping force correspond to the appearance and merging of IRCs.al. [16] and Bureau et al. [17] numerically and experimentally illustrated the generation of IRCs caused by impact and subharmonic resonances. Atomic force microscopies in tapping-mode operation can also undergo IRCs related to a discontinuity [18,19]. Nayfeh and Mook [20] illustrated the presence of IRCs related to the interaction of subharmonic and superharmonic resonances in a two-DoF system. showed the existence of IRCs in a softening single-DoF system in correspondence of a superharmonic resonance. Rega [22] denoted an elongated IRC in the vicinity of the 1/3 subharmonic region of a suspended cable. Lenci and Ruzziconi [23] showed the appearance of an IRC far from any resonance, while studying a single-DoF model of a suspended bridge, which includes quadratic and cubic hardening nonlinearity. DiBerardino and Dankowicz [24] identified IRCs related to symmetry breaking in a two-DoF system; adopting isola singularity identification, in combination with a multiple scale approach, they managed to predict the outbreak of an IRC. In [25], an IRC related to an internal resonance was encountered, while in [26] the phenomenon was explained through a frequency gap due to phase locking.Isolated branches of periodic solutions can also be related to self-excited oscillations [27][28][29][30]. Apart from mechanical systems, this is a typical and well-studied phenomenon in chemical reactors [31][32][33][34], but also in biological models [35,36]. However, in these cases they are not necessarily related to a resonance between external excitation and system response, therefore the phenomenon is qualitatively different from the one considered here.In the last 15 years numerous studies about nonlinear vibration absorbers appeared. Although very different types of vibration absorbers exist, most of them exploit internal resonances, making them prone to generation of IRCs. The nonlinear energy sink (NES), consisting of a purely nonlinear resonator, if attached to single-or multi-DoF primary systems, presents IRCs [37,38], whose existence was verified also experimentally [39]. An attempt to eliminate this undesired phenomenon demonstrated that IRCs can be avoided if the absorber has a properly-tuned piecewise-quadratic damping characteristic [40]. Similarly, the nonlinear tuned vibration absorber (NLTVA), possessing both a linear and a nonlinear elastic force characteristic, can present an IRC that limits its range of operation [41,42]. A numerical procedure exploiting bifurcation tracking allowed to define regions of appearance of the IRC [43]. Alexander and Schilder [44] demonstrated that the existence of...

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A principle of similarity for nonlinear vibration absorbers

Cited by 62 publications

References 31 publications

Numerical antiresonance continuation of structural systems

Numerical antiresonance continuation of structural systems

Nonlinear positive position feedback control for mitigation of nonlinear vibrations

Isolated resonances and nonlinear damping

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