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The purpose of this study is to provide some references about applying the semi-active particle damper to enhance the stability of the pipe structure.

This paper establishes the dynamical models of semi-active particle damper based on traditional dynamical theory and fractional-order theory, respectively. The semi-active particle damping vibration isolation system applied in a pipe structure is proposed, and its analytical solution compared with G-L numerical solution is solved by the averaging method. The quantitative relationships of fractional-order parameters (a and kp) are confirmed and their influences on the amplitude-frequency response of the vibration isolation system are analyzed. A fixed point can be obtained from the amplitude-frequency response curve, and the optimal parameter used for improving the vibration reduction effect of semi-active particle damper can be calculated based on this point. The nonlinear phenomenon caused by nonlinear oscillators is also investigated.

The results show that the nonlinear stiffness parameter p will cause the jump phenomenon while p is close to 87; with the variation of nonlinear damping parameter μ, the pitchfork bifurcation phenomenon will occur with an unstable branch after the transient response; with the change of fractional-order coefficient k_p, a segmented bifurcation phenomenon will happen, where an interval that k_p between 18.5 and 21.5 has no bifurcation phenomenon.

This study establishes a mathematical model of the typical semi-active particle damping vibration isolation system according to fractional-order theory and researches its nonlinear characteristics.