The forced vibration of a single-degree-of-freedom piecewise linear system containing fractional time-delay feedback was investigated. The approximate analytical solution of the system was obtained by employing an averaging method. A frequency response equation containing time delay was obtained by studying a steady-state solution. The stability conditions of the steady-state solution, the amplitude–frequency results, and the numerical solutions of the system under different time-delay parameters were compared. Comparison results indicated a favorable goodness of fit between the two parameters and revealed the correctness of the analytical solution. The effects of the time-delay and fractional parameters, piecewise stiffness, and piecewise gap on the principal resonance and bifurcation of the system were emphasized. Results showed that fractional time delay occurring in the form of equivalent linear dampness and stiffness under periodic variations in the system and influenced the vibration characteristic of the system. Moreover, piecewise stiffness and gap induced the nonlinear characteristic of the system under certain parameters.

研究了包含分数阶时滞反馈的单自由度分段线性系统的强迫振动。通过使用平均方法获得系统的近似解析解。通过研究稳态解，获得了包含时间延迟的频率响应方程。比较了系统在不同时滞参数下的稳态解的稳定性条件，幅频结果以及数值解。比较结果表明两个参数之间的拟合度良好，并表明了分析溶液的正确性。强调了时延和分数参数，分段刚度和分段间隙对系统主共振和分叉的影响。结果表明，分数时间延迟在系统的周期性变化下以等效线性阻尼和刚度的形式出现，并影响了系统的振动特性。此外，在某些参数下，分段刚度和间隙会引起系统的非线性特性。