Nonlinear Dynamics Of Arch

Fractional derivative can be considered as the Volterra's integral of Abel kernel function. Its value is not only associated with current value, but also associated with the whole numerical history. So, the time effect of the materials can be perfectly described by applying fractional derivative analysis to some viscoelastic material, for instance, many high molecular synthetic materials. Compared with the classical models, the fractional derivative viscoelastic constitutive model can describe the constitutive relations of the viscoelastic material as well as its related mechanics properties accurately in a wider frequency range with less model parameters.the fractional derivative constitutive relations are applied to study the dynamical problems of viscoelastic arch in theoretical analysis and numerical simulation.the main results as follow:(1) a numerical method for the fractional-derivative is proposed, and its discretization error is estimated. the algorithm for nonlinear fractional derivative was derivated .the method can be used to simulate the integro-differential equations numerically for a long time . Several numerical examples are presented . The numerical results are compared with the exact solutions, it is shown that the new numerical method has high accuracy and good stability.(2) the background and significance of the study of oscillator with fractional operator is elaborated. The influence of The load parameter and fractional derivative parameters on the displacement are investigated. It is indicated that the effect of fractional derivative parameter the load parameter and on the behavior of the beam are distinct. The influences of the load parameter p and the fractional derivative parameter q on the displacement of the oscillator are considered respectively. With the increasing of the load parameter, the motion states of the systems is changed from periodic motion with period 1 to complex motions, such as mint-periodicity, quasi-periodicity or chaos. With the increasing of the fractional derivative parameter, namely the damping of the viscoelastic material is adding, the motions of the systems are changed from muti-periodicity, quasi-periodicity or chaos to one-periodicity. So the increasing of the fractional derivative parameter benefits the stability of the systems(3) The motion equations governing the dynamical behavior of a viscoelastic arch is derived and simplified by Galerkin method. The equation is nonlinear integro-differential one with weakly singularity about the displacement. A new numerical method for the nonlinear integro-differential with fractional derivative is presented, the method combine with the numerical method for the fractional-derivative is to solve the equation .The numerical methods in nonlinear dynamics are synthetically applied to reveal plenty and complex dynamical behaviors of the arch.