On methods for continuous systems with quadratic and cubic nonlinearities

Nayfeh, Ali H.; Nayfeh, Jamal; Mook, Dean T.

doi:10.1007/bf00118990

Cited by 231 publications

(262 citation statements)

References 2 publications

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“…(22) and (23), a solvability condition must be satisfied for this nonhomogenous equation (see details in Refs. Nayfeh 37,38 ). Applying the solvability condition for Eqs.…”

Section: Et Al: Nonlinear Transverse Vibrations Of a Slightly Curvedmentioning

confidence: 99%

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Nonlinear Transverse Vibrations of a Slightly Curved Beam resting on Multiple Springs

Özkaya¹,

Sarıgül²,

Boyacı³

2016

IJAV

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In this study, nonlinear vibrations of a slightly curved beam of arbitrary rise functions is handled in case it rests on multiple springs. The beam is simply supported on both ends and is restricted in longitudinal directions using the supports. Thus, the equations of motion have nonlinearities due to elongations during vibrations. The method of multiple scales (MMS), a perturbation technique, is used to solve the integro-differential equation analytically. Primary and 3 to 1 internal resonance cases are taken into account during steady-state vibrations. Assuming the rise functions are sinusoidal in numerical analysis, the natural frequencies are calculated exactly for different spring numbers, spring coefficients, and spring locations. Frequency-amplitude graphs and frequency-response graphs are plotted by using amplitude-phase modulation equations.

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“…(22) and (23), a solvability condition must be satisfied for this nonhomogenous equation (see details in Refs. Nayfeh 37,38 ). Applying the solvability condition for Eqs.…”

Section: Et Al: Nonlinear Transverse Vibrations Of a Slightly Curvedmentioning

confidence: 99%

“…If the solvability condition (see Nayfeh for further details 37,38 ) is applied in order to solve Eqs. (37), (38) and (39) the following equations were obtained:…”

Section: Et Al: Nonlinear Transverse Vibrations Of a Slightly Curvedmentioning

confidence: 99%

Nonlinear Transverse Vibrations of a Slightly Curved Beam resting on Multiple Springs

Özkaya¹,

Sarıgül²,

Boyacı³

2016

IJAV

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“…Hence, the primary thrust of this component has been to examine if oscillations of a buckled system can be used to explain experimentally observed nonlinear motions. Representative results obtained through the analysis [9,10,11] are shown in Figure 3. These results are in good agreement with the corresponding experimental observations.…”

Section: Dynamic Buckling Of Composite Micro-scale Structures: Nonlinmentioning

confidence: 99%

Nonlinear Oscillations of Microscale Piezoelectric Resonators and Resonator Arrays

Balachandran¹

2006

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Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden to Washington Headquarters Service, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302, and PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)8 AFRL-SR-AR-TR-07_007 SUPPLEMENTARY NOTES ABSTRACTOscillations of clamped-clamped and free-free micro-electromechanical resonators and resonator arrays have been studied in this effort. Piezoelectric actuation is used to excite these resonator structures on the input side and piezoelectric sensing is carried out on the output side. Composite structural models have been developed for these filters, and analyses has been carried out to explain experimental observations of nonlinear phenomena as well as to guide the design of these filters are presented. Semi-analytical design tools for microelectromechanical resonators and micro-electromechanical resonator arrays have been developed. The phenomenon of intrinsic localized modes in resonator arrays has also been studied, and it is shown that these modes can be explained as forced nonlinear vibration modes. Oscillations of clamped-clamped and free-free micro-electromechanical resonators and resonator arrays have been studied in this effort. Piezoelectric actuation is used to excite these resonator structures on the input side and piezoelectric sensing is carried out on the output side. Composite structural models have been developed for these filters, and analyses has been carried out to explain experimental observations of nonlinear phenomena as well as to guide the design of these filters are presented. Semi-analytical design tools for micro-electromechanical resonators and micro-electromechanical resonator arrays have been developed. The phenomenon of intrinsic localized modes in resonator arrays has also been studied, and it is shown that these modes can be explained as forced nonlinear vibration modes. The research findings can open the doors to new resonator array designs.Keywords: microscale resonators, buckling, nonlinear phenomena, axial-load effects, piezoelectric actuation, Doiffing oscillator, resonator arrays, intrinsic localized modes. IntroductionTwo types of piezoelectrically actuated micro-scale resonators, which are attractive for communication and signal-processing applications [1], are considered in this research effort. One type of resonators will be referred to as the AIGaAs resonator, and the other type of resonators will be referred to as the PZT resonator. Both types of resonators are composite structures, and the PZT resonators have asymmetric cross-sections, as shown in Figure 1. During the fabrication of the reson...

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“…With its effect on the controlling system, dynamic system model directly affects the images resolution. Most of the mathematical models which have been used until the date are lumped-mass spring models (Sarid et al, 1997;Pishkenari et al, 2008), while it has been proved (Nayfeh et al, 1992) that nonlinear lumped mass models which approximate continuous dynamic systems with the nonlinear boundary conditions may encounter substantial errors. NMC vibrating analysis has been studied at the specific attachment of piezoelectric (throughout layer) with simple rectangular cantilever and considering the continuous beam model in non-contact mode (Wolf and Gottlieb, 2002;Fung and Huang, 2001).…”

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confidence: 99%

Nonlinear mathematical modeling of vibrating motion of nanomechanical cantilever active probe

Ghaderi

2014

Lat. Am. j. solids struct.

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Nonlinear vibration response of nanomechanical cantilever (NMC) active probes in atomic force microscope (AFM) application has been studied in the amplitude mode. Piezoelectric layer is placed piecewise and as an actuator on NMC. Continuous beam model has been chosen for analysis with regard to the geometric discontinuities of piezoelectric layer attachment and NMC's cross section. The force between the tip and the sample surface is modeled using Leonard-Jones potential. Assuming that cantilever is inclined to the sample surface, the effect of nonlinear force on NMC is considered as a shearing force and the concentrated bending moment is regarded at the end. Nonlinear frequency response of NMC is obtained close to the sample surface using the dynamic modeling. It is then become clear that the distance and angle of NMC, the probe length, and the geometric dimensions of piezoelectric layer can affect frequency response bending of the curve.

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On methods for continuous systems with quadratic and cubic nonlinearities

Cited by 231 publications

References 2 publications

Nonlinear Transverse Vibrations of a Slightly Curved Beam resting on Multiple Springs

Nonlinear Transverse Vibrations of a Slightly Curved Beam resting on Multiple Springs

Nonlinear Oscillations of Microscale Piezoelectric Resonators and Resonator Arrays

Nonlinear mathematical modeling of vibrating motion of nanomechanical cantilever active probe

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