Nonlinear Dynamics of a Cantilever Beam Actuated by Piezoelectric Layers

Wolf, Kai; Gottlieb, O.

doi:10.1115/detc2001/vib-21489

Cited by 9 publications

(4 citation statements)

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“…Later, Ray [16] also presented the closed form solution for the optimal control of flexural vibration of a simply supported symmetric laminated plate. Wolf and Gottlieb [17] derived the non-linear equations of motion for a cantilever beam covered by piezoelectric ceramic material (PZT) layers in symmetric and asymmetric configurations. By defining a non-linear enthalpy function with non-linear strain and applying the Hamilton's principle, they derived the equations of motion and solved by perturbation analysis and multiple scales method.…”

Section: Introductionmentioning

confidence: 99%

A mathematical model in three-dimensional piezoelectric continuum to predict non-linear responses of piezoceramic materials

Samal

Seshu

Wagner

et al. 2008

Proceedings of the Institution of Mechanical Engineers, Part C:

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It has been experimentally observed that the piezoceramic materials exhibit different types of non-linearities under different combinations of electrical and mechanical fields. When excited near resonance in the presence of weak electric fields, they exhibit typical non-linearities similar to a Duffing oscillator such as jump phenomena and the presence of superharmonics in the response spectra. In this work, these non-linearities have been modelled for a generalized three-dimensional piezoelectric continuum using higher-order quadratic and cubic terms in the electric enthalpy density function and the virtual work. The identification of the parameters of the model requires a closed form solution for non-linear response of a simplified geometry. A simple proportional damping formulation has been used in the model. Experiments have been conducted on rectangular and cylindrical geometries of piezoceramic PIC 181 at different magnitudes of applied electric fields and results have been compared with those of simulation.

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Section: Introductionmentioning

confidence: 99%

A mathematical model in three-dimensional piezoelectric continuum to predict non-linear responses of piezoceramic materials

Samal

Seshu

Wagner

et al. 2008

Proceedings of the Institution of Mechanical Engineers, Part C:

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“…PZT materials are usually glued on elastic structures and both can present nonlinear behaviors. The PZT material nonlinearities were studied to understand their dynamics (Abdelkefi et al, 2012; Guyomar et al, 1997, 2011; Parashar and Wagner, 2004; Von Wagner and Hagedorn, 2002; Wolf and Gottlieb, 2001), but if deriving a proper nonlinear PZT law, thermodynamically consistent, has been already addressed, obtaining the values of the coefficients for a practical application seems to be still an open field of investigation. Moreover, the aforementioned publications use a classical electric enthalpy function which is a high-order (smooth) polynomial in the strain, whereas in Leadenham and Erturks (2015), a low-order function that includes the absolute value of the strain (it is, thus, nonsmooth), is proposed and seems to be closer to what is measured in practice.…”

Section: Introductionmentioning

confidence: 99%

Theoretical and experimental investigation of a 1:3 internal resonance in a beam with piezoelectric patches

Guillot

Givois

Thomas

et al. 2020

Journal of Vibration and Control

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Experimental and theoretical results on the nonlinear dynamics of a homogeneous thin beam equipped with piezoelectric patches, presenting internal resonances, are provided. Two configurations are considered: a unimorph configuration composed of a beam with a single piezoelectric patch and a bimorph configuration with two collocated piezoelectric patches symmetrically glued on the two faces of the beam. The natural frequencies and mode shapes are measured and compared with those obtained by theoretical developments. Ratios of frequencies highlight the realization of 1:2 and 1:3 internal resonances, for both configurations, depending on the position of the piezoelectric patches on the length of the beam. Focusing on the 1:3 internal resonance, the governing equations are solved via a numerical harmonic balance method to find the periodic solutions of the system under harmonic forcing. A homodyne detection method is used experimentally to extract the harmonics of the measured vibration signals, on both configurations, and exchanges of energy between the modes in the 1:3 internal resonance are observed. A qualitative agreement is obtained with the model.

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“…It has been seen that the nonlinearity added to the vibration absorber should be of the same form as the host structure (Habib and Kerschen, 2016). Beams present geometrical and inertial nonlinearities which can correspond to a cubic nonlinearity (Wolf and Gottlieb, 2001). In recent works (Lossouarn et al, 2018; Silva et al, 2018; Soltani and Kerschen, 2015; Zhou et al, 2014), designed electrical circuits which are used to present cubic nonlinearity and to improve the vibration mitigation.…”

Section: Introductionmentioning

confidence: 99%

Study of an electromechanical nonlinear vibration absorber: Design via analytical approach

Guillot

Savadkoohi

Lamarque

2020

Journal of Intelligent Material Systems and Structures

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This article deals with the behavior and design of a homogeneous beam linked to an electrical absorber, via a piezoelectric material patched on the beam. The beam presents a cubic nonlinearity, which is consistent with geometrical nonlinearities for clamped-free beams. A cubic nonlinearity is added to the electrical circuit for similarity purposes. The coupled electromechanical equations are reduced to a two degrees of freedom system. The equation representing the mechanical response of the system is seen as the main system, whereas the equation coming from the electrical circuit would represent a nonlinear absorber. An analytical treatment at different time scales is endowed to identify electrical coefficients allowing the design of the electromechanical vibration absorber behavior. The equilibrium and singular points are identified, allowing the definition of ranges of forcing amplitudes leading to periodic and modulated regimes. The analytical results are compared with those obtained from direct numerical integration of the two degree of freedom system.

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Nonlinear Dynamics of a Cantilever Beam Actuated by Piezoelectric Layers

Cited by 9 publications

References 0 publications

A mathematical model in three-dimensional piezoelectric continuum to predict non-linear responses of piezoceramic materials

A mathematical model in three-dimensional piezoelectric continuum to predict non-linear responses of piezoceramic materials

Theoretical and experimental investigation of a 1:3 internal resonance in a beam with piezoelectric patches

Study of an electromechanical nonlinear vibration absorber: Design via analytical approach

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