The present paper deals with the investigation of dynamic responses of a rotating pre-deformed blade in four cases of resonance, including two subharmonic resonances and two combination resonances. The dimensionless gas excitation amplitude is assumed to share the same order with the dimensionless vibration displacement. Four cases of resonance are confirmed by examining the secular terms. The theoretical analysis framework is established for each resonance case based on the method of multiple scales. The original dynamic system is integrated numerically by the Runge–Kutta method. The frequency components and phases obtained from fast Fourier transform of the numerical response are used to verify the theoretical results. For the purpose of contrast, modulation equations are also integrated numerically. In all four resonance cases, the theoretical results agree well with the numerical simulation. Parameter studies are conducted to clarify the effects of system parameters on the perturbation curves. Various results are obtained for the rotating blade. A quasi-saturation phenomenon occurs in both combination resonances of summed type and difference type, and the corresponding limit value of the second-mode response can be reduced by decreasing the external detuning parameter. The quasi-saturation phenomenon of rotating blade only appears with high gas pressure. The subharmonic resonance of second mode and the combination resonance of summed type are hard to excite in practice compared with the other two cases.

中文翻译：

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高压下旋转预变形叶片的次谐波和组合共振

本文研究了旋转预变形叶片在四种共振情况下的动力响应，包括两种次谐波共振和两种组合共振。假设无量纲的气体激发振幅与无量纲的振动位移共享相同的阶数。通过检查世俗术语可以确定共振的四种情况。基于多尺度方法，为每个共振情况建立了理论分析框架。原始的动态系统通过Runge-Kutta方法进行数值集成。从数值响应的快速傅立叶变换获得的频率分量和相位用于验证理论结果。为了对比，还对调制方程进行了数值积分。在所有四种共振情况下，理论结果与数值模拟吻合良好。进行参数研究以阐明系统参​​数对摄动曲线的影响。对于旋转刀片获得各种结果。在求和型和差分型的组合谐振中都出现准饱和现象，并且可以通过减小外部失谐参数来减小第二模式响应的相应极限值。旋转叶片的准饱和现象仅在高气压下出现。与其他两种情况相比，在实践中很难激发第二模式的次谐波谐振和求和型组合谐振。对于旋转刀片获得各种结果。在求和型和差分型的组合谐振中都出现准饱和现象，并且可以通过减小外部失谐参数来减小第二模式响应的相应极限值。旋转叶片的准饱和现象仅在高气压下出现。与其他两种情况相比，在实践中很难激发第二模式的次谐波谐振和求和型组合谐振。对于旋转刀片获得各种结果。在求和型和差分型的组合谐振中都出现准饱和现象，并且可以通过减小外部失谐参数来减小第二模式响应的相应极限值。旋转叶片的准饱和现象仅在高气压下出现。与其他两种情况相比，在实践中很难激发第二模式的次谐波谐振和求和型组合谐振。旋转叶片的准饱和现象仅在高气压下出现。与其他两种情况相比，在实践中很难激发第二模式的次谐波共振和求和型的组合共振。旋转叶片的准饱和现象仅在高气压下出现。与其他两种情况相比，在实践中很难激发第二模式的次谐波谐振和求和型组合谐振。