Analysis Of Linear Fractional Damping Vibration System

Some of the viscoelastic materials,such as polymer materials,have many forms of constitutive equations. All these equations which called classical descriptions are often expressed with complex parameters, but the use of fractional derivatives forms can make it both concise and accurate. Therefore, it could become meaningful when the fractional viscoelastic constitutive model is applied to the vibration in engineering.In the first chapter, the background of fractional derivatives and its development are described.In the second chapter, some preliminary knowledge are introduced, including the main definitions of fractional calculus and the common three kinds of fractional viscoelastic models, which will be the basis of the study of the later chapters.For the third chapter, the single-degree-of-freedom damped forced vibration is considered. We use the Laplace transform and the inverse Laplace transform methods to the vibration equation in the case of the initial values obtained the expression for the response function under the general excitation, and the correctness is verified by numerical solution. After this, a detailed analysis of the free vibration and the characteristics of fractional vibration were demonstrated.In the fourth chapter, we bring the fractional derivatives into the two-degree-of-freedom vibration equation. The model is based on the viscoelastic suspension system of the vehicle. We mainly studied the viscoelastic suspension under the harmonic excitation, got some of the basic characteristics of the steady-state response, including the amplitude and phase angle.In the fifth chapter, we considered the vibration characteristics of continuous systems, including the longitudinal vibration of fractional viscoelastic rods and the lateral vibration of fractional viscoelastic beams. The solution about rods was solved by introducing Mittag-Leffler function. For the viscoelastic beam, the steady-state response solution and numerical simulation were given.In the last chapter, we summarized the whole paper and prospected the fractional derivatives in the fractional vibration.