Nonlinear dynamic response of open and breathing cracked functionally graded beam under single and multi-frequency excitation

• Published in: Engineering structures 2021-09, Vol.242, p.112437, Article 112437
• Main Authors: Panigrahi, B., Pohit, G.
• Format: Article
• Language: English
• Subjects: Beams (structural); Breathing crack; Cracked beam; Cracks; Differential equations; Dynamic response; Dynamical systems; Equations of motion; Euler-Lagrange equation; Excitation; FGM; Functionally gradient materials; Iterative methods; Material properties; Mathematical models; Multi frequency excitation; Nonlinear differential equations; Nonlinear dynamics; Nonlinear response; Nonlinear systems; Nonlinear vibration; Nonlinearity; Periodic variations; Stiffness; Super harmonic response; Time dependence; Time integration; Timoshenko beams
• ISSN: 0141-0296; 1873-7323
• DOI: 10.1016/j.engstruct.2021.112437

•A time domain nonlinear modeling of cracked FG Timoshenko beam is performed.•An open crack model is developed and methodology is extended for breathing cracks.•A novel MDOF discrete model is developed for the breathing crack in FGM beam.•A relatively easier methodology with significantly good accuracy is proposed.•An exhaustive study on various crack, material and loading parameters are reported. The present work investigates a time domain nonlinear modeling and response analysis of cracked functionally graded Timoshenko beam. Response of the system influenced by various sources of nonlinearity is studied in detail. Properties of functionally graded material (FGM) are assumed to vary nonlinearly in thickness direction by an exponential gradation relation. Initially, an open crack model is considered and subsequently, methodology is extended for breathing cracks assuming periodic variation in the stiffness of beam on opening and closing of cracks. It is to be highlighted that hardly any work has been carried out to model and analyze breathing crack in continuous or discrete multi degree of freedom (MDOF) models for FGM beams. In addition, present methodology seems to be computationally easier and can effectively solve dynamic problems with coupled nonlinearity, one arises from time dependent stiffness (nonlinearity due to crack) and another due to geometric aspect (nonlinear strain-displacement), for isotropic as well as nonlinear material property variation of beam models, leading to wide ranges of applications. Nonlinear differential equations of motion are derived using Lagrange’s equation. These equations are solved using an implicit direct time integration method, known as Newmark-β method, in conjunction with an iterative technique. Accuracy of the solution is confirmed by comparing some of the results with that of available results. Subsequently, an exhaustive parametric study on effects of cracks, material properties, multi frequency excitation, influence of super harmonics on periodic as well as quasi periodic nonlinear dynamic response of thick and thin beams are performed.