中文摘要：

将线性随机振动系统中通常的简谐势阱推广为更一般的幂函数型势阱,得到幂函数型单势阱非线性随机振动系统。利用随机情形下的二阶Runge-Kutta算法研究了噪声强度、势阱参数和周期激励参数对系统稳态响应的一阶矩振幅和系统响应的稳态方差的影响。对决定势阱形状的势阱参数之一b历经b〈2, b〉2以及相当于简谐势阱的b=2等全部情况的研究表明：随噪声强度D的变化,系统稳态响应的一阶矩振幅可以在b〈2时出现非单调变化,即发生广义随机共振现象,而对通常的b=2简谐势阱以及b〉2的情况,则无该现象发生；随势阱参数的变化,系统稳态响应的一阶矩振幅以及系统响应的稳态方差也可以发生非单调变化。

英文摘要：

To generalize the harmonic potential of the linear random vibration system, a more general power type potential is presented, and the corresponding power function type nonlinear single-well random vibration system is obtained. The first moment of the system steady-state response and the stationary variance of the system response, which are influenced by noise strength, parameters of the potential and the periodic excitation, are studied by using the second order stochastic Runge-Kutta algorithm. The parameter b, which determines the shape of the potential, goes through b〈2, b〉2 and b=2 （harmonic potential）, and it is shown that varying the noise strength, if b〈2, the first moment of the system steady-state response can be non-monotonic （generalized stochastic resonance）, if b=2 （harmonic potential） or b 〉 2, this phenomenon does not occur; varying the parameters of the potential, the first moment of the system steady-state response and the stationary variance of the system response can also be non-monotonic.