The present work investigates the existence/nonexistence of instability regions for a parametrically excited linear gyroscopic system. To achieve this goal, a new approach is proposed to determine the boundaries of parametric instability regions. As long as the gyroscopic system's undamped equations of motion are derived, one can easily use this approach to determine its instability regions. For convenience, a rotor-bearing system with periodic axial loaded is used as a parametrically excited representative gyroscopic system. The approach rewrites the second order differential equations in the state space form. Then the generalized eigenvalue problem is solved to derive the left and right eigenvectors. They are used to decouple the governing equations and reduce the order. In the subsequent theoretical derivation, the multiple scale method is applied to obtain the analytical solutions of the boundaries of instability regions. The numerical simulation is also carried out to validate the analytical boundaries. Wherein the numerical instability regions are obtained by applying the discrete state transition matrix method. From the theoretical and numerical analysis, we find out: (1) the analytical boundaries match well with the numerical results; (2) only the sum type instability regions can be observed; (3) the primary difference type instability regions do not exist; (4) the secondary or higher order difference type instability regions also may not exist.

目前的工作研究了参数激励线性陀螺系统的不稳定区域的存在/不存在。为了实现这一目标，提出了一种确定参数不稳定区域边界的新方法。只要推导出陀螺系统的无阻尼运动方程，就可以很容易地使用这种方法来确定其不稳定区域。为方便起见，将具有周期性轴向载荷的转子轴承系统用作参数激励的代表性陀螺系统。该方法以状态空间形式重写了二阶微分方程。然后是广义特征值问题求解得到左右特征向量。它们用于解耦控制方程并降低阶数。在随后的理论推导中，应用多尺度方法得到不稳定区域边界的解析解。还进行了数值模拟以验证分析边界。其中数值不稳定区域是通过应用离散状态转移矩阵方法获得的。通过理论和数值分析，我们发现：（1）解析边界与数值结果吻合较好；(2) 只能观察到和型不稳定区域；(3) 不存在初级差型不稳定区；(4) 二级或更高阶差分型的不稳定区域也可能不存在。