The Analysis of Higher Order Nonlinear Vibrations of an Elastic Beam with the Extended Galerkin Method

Lian, Chencheng; Meng, Baochen; Jing, Huijuan; Wu, Rongxing; Lin, Ji; Wang, Ji

doi:10.1007/s42417-023-01011-6

Cited by 7 publications

(5 citation statements)

References 32 publications

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“…The exact solution to Equation ( 2) can be given in elliptic functions known as the analytical solution of an elastica [26,27], as has been shown in earlier solutions [43,45,46]. The vibrations of such a beam has been studied with the EGM also in a similar procedure [4].…”

Section: The Large Deformation Of An Elastic Beammentioning

confidence: 98%

“…As it was solved with the homotopy analysis method (HAM) in earlier studies [40][41][42][43], the EGM is now used for the approximate solutions as a new and effective alternative with great potential for the same problem [4,5,44]. It is found that using the sine function as the basis function in this study, relatively accurate solutions with explicit expressions of a few lower-order terms can be obtained in a simple and efficient procedure based on the Galerkin method with integration over the physical domain for the coefficients of series solutions with accuracy and efficiency [43,44].…”

Section: The Large Deformation Of An Elastic Beammentioning

confidence: 99%

“…Mathematical Problems in Engineering is the iterative process by the successive determination of coefficients one at each time for the successive approximation. Both approaches have been demonstrated as part of the solution procedures of the EGM [4,5,[44][45][46]. In this study, the approximation starts from the linear solution…”

Section: The Large Deformation Of An Elastic Beammentioning

confidence: 99%

“…The large deformation of a beam is frequently encountered in mechanical and structural engineering with critical importance in guaranteeing the safety and proper functioning of such essential components. The analysis based on the beam equations has been done with many methods and techniques for the deformation, stress, and vibration frequencies of an elastica [1][2][3][4]. In addition, many techniques have been developed and improved in solving problems with nonlinear equations, and the applications can be found in many fields and subjects with rich histories [5][6][7].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Approximate Solutions of Large Deflection of a Cantilever Beam under a Point Load

Meng,

Lian,

Zhang

et al. 2023

Mathematical Problems in Engineering

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By integrating the physical and time domains with the Galerkin method and adopting approximate solutions satisfying the boundary conditions, a set of algebraic equations of nonlinear nature is obtained from the nonlinear differential equation of an elastic beam for the undetermined coefficients of solutions of the deflection. These coefficients are to be used with the approximate basis functions for the asymptotic and explicit solutions of the nonlinear differential equation of flexure of a beam widely known as an elastica. Taking advantage of powerful tools for the symbolic manipulation of algebraic equations, such a novel method and procedure offer a new and efficient approach and option with known linear solutions in dealing with increasingly complex nonlinear problems in practical applications of both static and dynamic nature.

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Section: The Large Deformation Of An Elastic Beammentioning

confidence: 98%

Section: The Large Deformation Of An Elastic Beammentioning

confidence: 99%

Section: The Large Deformation Of An Elastic Beammentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Approximate Solutions of Large Deflection of a Cantilever Beam under a Point Load

Meng,

Lian,

Zhang

et al. 2023

Mathematical Problems in Engineering

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“…The micro-cantilever can be excited using different methods such as photoacoustic [7], magnetic [8], electrostatic [9], and piezoelectric [10] methods. When compared to monomodal operations, implementing multi-frequency excitation schemes for driving an Atomic Force Microscopy (AFM) micro-cantilever can yield higher observables sensitivity to tip-sample interaction force at the fundamental (first) or the higher eigenmodes [11,12]. Similarly, the micro-cantilevers on dynamic modes exhibit much more sensitivity to hydrodynamic loads owing to the surrounding fluids.…”

Section: Introductionmentioning

confidence: 99%

A multi-modal nonlinear dynamic model to investigate time-domain responses of a micro-cantilever in fluids

Yilmaz

2024

Eng. Res. Express

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In this current work, a new nonlinear dynamic model based on the forced Van der Pol oscillator is introduced to demonstrate the time-domain sensitivities of the micro-cantilever to the varying properties of the surrounding fluids. Effects of diverse multi-frequency excitations on the hydrodynamically forced displacements are investigated for the Glycerol-water solutions with different concentrations. Driving forces at the eigenmode frequencies are applied simultaneously to actuate the micro-cantilever in multi-modal operations. The hydrodynamic force induces notable variations in the observables of high-frequency steady-state vibrations. To illustrate, the frequency of the displacements decreases with increasing dynamic viscosity and density of the fluids (among 55% and 85% Glycerol-water solutions) in bimodal- and trimodal-frequency excitations. Essentially, the observable responses are often used to distinguish the surrounding fluids in which the micro-cantilever operates. In addition, steady-state observables are achieved at only particular eigenmodes in single- and multi-frequency operations. It is highlighted that the periodic oscillations are obtained for the first and second eigenmodes with the highest value of forced Van der Pol parameter (μ =1030). Clearly, higher eigenmodes require different values of the nonlinearity parameter to acquire periodic vibrations in multi-modal operations. In general, achieving steady-state observables is substantially critical in quantifying sensitivity to varying fluid properties. For instance, the vibration frequency of around 7.33 MHz and amplitude of around 0.03 pm are obtained at the first eigenmode for 75% Glycerol-water solution in tetra-modal operations. Note that femtometer amplitudes of deflections can be measured using quantum-enhanced AFM techniques. The frequency responses obtained in this work are compared with the measured ones in the literature and the results show satisfactory agreements. Therefore, a novel multi-modal nonlinear dynamic model enables to quantify observable sensitivity to micro-rheological properties at higher eigenmodes of the micro-cantilever.

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The Analysis of Bending of an Elastic Beam Resting on a Nonlinear Winkler Foundation with the Galerkin Method

Wei,

Jing,

Zhang

et al. 2024

Acta Mech. Solida Sin.

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The Analysis of Higher Order Nonlinear Vibrations of an Elastic Beam with the Extended Galerkin Method

Cited by 7 publications

References 32 publications

The Approximate Solutions of Large Deflection of a Cantilever Beam under a Point Load

The Approximate Solutions of Large Deflection of a Cantilever Beam under a Point Load

A multi-modal nonlinear dynamic model to investigate time-domain responses of a micro-cantilever in fluids

The Analysis of Bending of an Elastic Beam Resting on a Nonlinear Winkler Foundation with the Galerkin Method

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