The formation of dynamic variable order fractional differential equation

In this paper, the formation of variable order (VO) model is established for continuous order fractional model. We review the definitions and properties of VO operators given by many researchers. We use the VO operator to define the new transfer function and analyze the model of a dynamic viscoelastic oscillator.

Response of viscoelastic damping system modeled by fractional viscoelastic oscillator

The fractional model considering geometric factor of viscoelastic damping systems is proposed by adopting fractional viscoelastic oscillator. To obtain dynamic responses of the fractional model, a numerical method is derived based on matrix function theory and Grumwald–Letnikov discrete form of fractional derivative. As a special engineering application example, the vibration response of the viscoelastic suspension installed in heavy crawler-type vehicles is studied through the proposed model. Furthermore, the parameter influence on the vibration control capability of the viscoelastic suspension is researched. The results indicate that the fractional viscoelastic oscillator is a favorable choice to characterize the dynamic behavior of viscoelastic damping structures. Additionally, the parameters in fractional viscoelastic oscillator namely geometric factor and fractional order exert considerable impact on the dynamic response of viscoelastic damping structures.

On the Selection and Meaning of Variable Order Operators for Dynamic Modeling

We review the application of differential operators of noninteger order to the modeling of dynamic systems. We compare all the definitions of Variable Order (VO) operators recently proposed in literature and select the VO operator that has the desirable property of continuous transition between integer and non-integer order derivatives. We use the selected VO operator to connect the meaning of functional order to the dynamic properties of a viscoelastic oscillator. We conclude that the order of differentiation of a single VO operator that represents the dynamics of a viscoelastic oscillator in stationary motion is a normalized phase shift. The normalization constant is found by taking the difference between the order of the inertial term (2) and the order of the spring term (0) and dividing this difference by the angular phase shift between acceleration and position in radians (), so that the normalization constant is simply .