Dynamics Of Nonlinear Oscillator With Fractional Operator

Fractional derivative can be considered as the Volterra's integral of Abel kernelfunction. Its value is not only associated with current value, but also associated withthe whole numerical history. So, the time effect of the materials can be perfectlydescribed by applying fractional derivative analysis to some viscoelastic material, forinstance, many high molecular synthetic materials. Compared with the classicalmodels, the fractional derivative viscoelastic constitutive model can describe theconstitutive relations of the viscoelastic material as well as its related mechanicsproperties accurately in a wider frequency range with less model parameters.The main purpose of this dissertation is to explore the influence of fractionalderivative damping to the nonlinear oscillator system:(1) A new algorithm for nonlinear fractional derivative is deduced and errorestimation is provided. All these preparations are the foundation to approach thenonlinear dynamics investigation of the nonlinear oscillator systems, whichintroducing the fractional differential damping. The numerical simulation example tononlinear differential equation with nonlinear fractional derivatives indicates that: bycombining Newmark algorithm with our algorithm, the numerical solution of thefractional derivative nonlinear differential equations can be well achieved. Moreover,the algorithm is of better convergence, preferable stability and higher computationprecision.(2) Several nonlinear fractional differential equations for oscillator, which containfractional derivative damping, are established. The background of the study is: It hasbeen proved by both experimental research and the theoretical study that for highmolecular synthetic materials, the material damping which is described by fractionalderivative, could display the dissipation of the system more accurately. Therefore, it isdefinitely necessary to study the mechanism of the mechanics properties of thesesystems. The three oscillator systems, which are fractional differential oscillatorcontaining quadratic nonlinear terms, the Duffing fractional differential oscillator andthe Van der pol fractional differential oscillator, are studied. Furthermore, the freevibration and forced vibration are investigated separately.(3) The study to the nonlinear characteristic of the nonlinear fractional differentialoscillator indicates that when the nonlinear fractional differential oscillator plays thedominant role, the linear damping increases as the fractional order increases, or as the linear term coefficient increases, or as the nonlinear coefficient increases. And, theoscillator attenuation period reduces as the elasticity coefficient increases. Thefractional order, the nonlinear term coefficient, linear term coefficient, the excitementforce in the forced vibration is of positive correlation with the nonlinear degree of theoscillator.(4) The study to fractional Duffing oscillator indicates that: With the decreasingof the damping fractional order, the time period of oscillator changes and further goesto chaos. Under the external excitation forces of different frequencies, the strangeattractor can be found when at a lower frequency. The fractional Duffing oscillatorcomes into chaotic state earlier than the integral classical Duffing oscillator. Thesmaller the fractional order is, the smaller is the critical excitation needed to becomechaos.(5) The study of the Van der pol oscillator fractional differential equationindicates that the fractional Van der pol oscillator, as an atypical self-excited system,can delay the occurrence of stable self-excited vibration more than that the typicalintegral order damping could do. The bigger the fractional order is, the more likely thefractional differential Van der pol oscillator's nonlinear characteristic is to the typicalVan der pol oscillator. The study to the forced vibration of Van der pol fractionaldifferential oscillator indicates that with some parameters, the smaller fractionalderivative drawbacks the stability of the oscillator, and may even cause chaoticmovement.The study indicates that the fractional nonlinear oscillators, which contain fractionalderivatives, have lots of mechanics properties that are different from classical integraloscillators. The kinds of fractional differential nonlinear oscillators analyzed in thisdissertation can not be found yet at least in the domestic literature. The research hasstarted an explorative research of and provided a solid foundation for further study tothe fractional nonlinear system.