Vibration Analysis of Geometrically Nonlinear and Fractional Viscoelastic Cantilever Beams

Abstract

We investigate the nonlinear vibration of a fractional viscoelastic cantilever beam, subject to base excitation, where the viscoelasticity takes the general form of a distributed-order fractional model, and the beam curvature introduces geometric nonlinearity into the governing equation. We utilize the extended Hamilton's principle to derive the governing equation of motion for specific material distribution functions that lead to fractional Kelvin-Voigt viscoelastic model. By spectral decomposition in space, the resulting governing fractional PDE reduces to nonlinear time-fractional ODEs. We use direct numerical integration in the decoupled system, in which we observe the anomalous power-law decay rate of amplitude in the linearized model. We further develop a semi-analytical scheme to solve the nonlinear equations, using method of multiple scales as a perturbation technique. We replace the expensive numerical time integration with a cubic algebraic equation to solve for frequency response of the system. We observe the super sensitivity of response amplitude to the fractional model parameters at free vibration, and bifurcation in steady-state amplitude at primary resonance.

Data-Driven Failure Prediction in Brittle Materials: A Phase-Field Based Machine Learning Framework

“…Future works will focus on addressing the noted issues. Besides, the focus of future studies will be on the evaluation of the developed framework with the inclusion of plasticity/visco-elasto-plasticity [45,46,50,48] in the damage and fatigue phase-field model, combined with efficient and stable long time integration schemes [47,56].…”

A Thermodynamically Consistent Fractional Visco-Elasto-Plastic Model with Memory-Dependent Damage for Anomalous Materials

“…Potential applications of the developed work could be, e.g., failure of polymers, bio-tissues, and ductile metals, where the fractional-orders β E , β K can be related to the evolving fractal-like microstructure [36]. The presented model could also be employed in the context of nonlinear dynamics of mechanical systems [52,56].…”

“…Database: ε ve and 2N − 1 coefficients b Compute ∆ε ve n+1 using(53), and form ∆ε ve * n+1 using(56). Compute the FFT F(∆ε ve * n+1 ).4: Compute c (β E ) n+1 using (55), using the known b (β E ) coefficients.…”

Anomalous Features in Internal Cylinder Flow Instabilities subject to Uncertain Rotational Effects

“…Uncertainty in modeling procedure and also inaccuracy of the measured data are two main factors in arising epistemic uncertainty. The uncertainty in modeling could be the result of a variety of possibilities including the effects of geometry [30][31][32][33][34] , constitutive laws [35][36][37][38][39][40][41][42][43][44][45][46][47][48] , rheological models [49][50][51] , low-fidelity and reduced-order modeling [52][53][54][55][56][57][58][59][60][61][62][63][64] , and random forcing sources in addition to the random field boundary/initial conditions [65][66][67][68][69][70][71] . In the current work, we seek to fill a gap in the rich literature of investigating flow instabilities inside rotating flow systems by emphasizing on the stochastic modeling of the fluid dynamics and later focusing on the anomalies in the anomalous transport features of such system through statistical and scaling analysis of the response.…”