The signal transmission phenomenon in an underdamped fractional system composed of two harmonically coupled particles, but only the first particle, which is driven by a periodic force, is studied. We obtain the analytical expressions for the steady-state response of the two coupled particles by applying the stochastic averaging method and define the signal transmission factor to characterize signal transmission efficiency. We analyze the impact of noise on signal transmission efficiency and provide the discriminant criterion for the emergence of the signal transmission enhancement phenomenon. We also analyze the resonance behavior of the signal transmission factor and provide the discriminant criterion for the emergence of different types of resonance behavior based on system parameters. We draw the phase diagrams for varying resonance behavior versus different parameters and study the influences of the parameters on resonance behavior. It is found that the coupling strength and fractional-order exert important influences on the signal transmission efficiency and resonance behavior, and the fractional coupled system exhibits complicated dynamic behaviors than the integer coupled system. Lastly, numerical simulations for the signal transmission factor \(\eta \), and the output signal-to-noise ratio (SNR) transmission gain \(\text {SNR}_{\eta }\) are performed. We can control signal transmission efficiency and resonance behavior within a certain range by understanding the effects of system parameters on the underdamped fractional coupled system, and it has potential applications in modern science.

研究了由两个谐波耦合粒子组成的欠阻尼分数阶系统中的信号传输现象，但仅研究了由周期性力驱动的第一个粒子。通过应用随机平均方法，我们获得了两个耦合粒子的稳态响应的解析表达式，并定义了信号传输因子来表征信号传输效率。我们分析了噪声对信号传输效率的影响，并为信号传输增强现象的出现提供了判别标准。我们还分析了信号传输因子的共振行为，并为基于系统参数的不同类型的共振行为的出现提供了判别标准。我们绘制了针对不同参数变化的谐振行为的相位图，并研究了参数对谐振行为的影响。结果表明，耦合强度和分数阶对信号传输效率和共振行为有重要影响，分数耦合系统比整数耦合系统表现出复杂的动态行为。最后，对信号传输系数进行数值模拟\（\ eta \），并执行输出信噪比（SNR）传输增益\（\ text {SNR} _ {\ eta} \）。通过了解系统参数对阻尼不足的分数耦合系统的影响，可以将信号传输效率和谐振行为控制在一定范围内，在现代科学中具有潜在的应用前景。