The dynamic system described by a finite number of first-order differential equations is considered. The right side of each equation is a sum of a slow, deterministic and, in the general case, nonlinear function of dynamic variables and a stochastic excitation. The stochastic action is a superposition of a finite number of independent random processes with coefficients depending on dynamic variables and slow time. The problem statement is oriented to applications in the field of driven systems. The analysis is based on the concept of vibration mechanics proposed by I. I. Blekhman. The modified method of direct separation of slow and fast motions uses the explicit introduction of a small parameter and some ideas of the two-scale technique. The general formulas for vibrational forces (or fluxes) are obtained. These additional terms appear in the resulting system for averaged motion instead of the stochastic terms to make the averaged system equivalent to the initial stochastic system with respect to slow motions and, in particular, to low-frequency resonances. As an example, the model of a vibration machine for bulk material processing is considered. The stochastic effect is caused by random oscillations of the bulk material mass. It is transformed into a modification of the machine's frequency characteristics leading to a specific stochastic resonance. This article is part of the theme issue ‘Vibrational and stochastic resonance in driven nonlinear systems (part 1)’.

中文翻译：

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高频随机作用对动态系统低频行为的影响

考虑由有限数量的一阶微分方程描述的动态系统。每个方程的右侧是一个缓慢的、确定性的，在一般情况下，动态变量的非线性函数和随机激励的总和。随机作用是有限数量的独立随机过程的叠加，其系数取决于动态变量和慢时间。问题陈述面向驱动系统领域的应用。该分析基于 II Blekhman 提出的振动力学概念。改进的慢速运动直接分离方法使用了一个小参数的显式引入和两尺度技术的一些思想。获得了振动力（或通量）的一般公式。这些附加项出现在平均运动的结果系统中，而不是随机项，以使平均系统等效于慢速运动的初始随机系统，特别是低频共振。例如，考虑用于散装材料加工的振动机模型。随机效应是由散装材料质量的随机振荡引起的。它被转化为机器频率特性的修改，导致特定的随机共振。本文是主题问题“驱动非线性系统中的振动和随机共振（第 1 部分）”的一部分。尤其是低频共振。例如，考虑用于散装材料加工的振动机模型。随机效应是由散装材料质量的随机振荡引起的。它被转化为机器频率特性的修改，导致特定的随机共振。本文是主题问题“驱动非线性系统中的振动和随机共振（第 1 部分）”的一部分。尤其是低频共振。例如，考虑用于散装材料加工的振动机模型。随机效应是由散装材料质量的随机振荡引起的。它被转化为机器频率特性的修改，导致特定的随机共振。本文是主题问题“驱动非线性系统中的振动和随机共振（第 1 部分）”的一部分。