2:1内共振条件下变转速预变形叶片的非线性动力学响应

NONLINEAR DYNAMIC RESPONSE OF PRE-DEFORMED BLADE WITH VARIABLE ROTATIONAL SPEED UNDER 2:1 INTERNAL RESONANCE

在工程实际中,涡轮机叶片的转速在很多应用场景下不是一个定常值,比如发动机在启动、变速、停机等工况下,转子输入与输出功率失衡,伴随产生扭振,产生速度脉冲.另外,由于服役环境、安装误差等因素会引起叶片在所难免的预变形.本文主要研究预变形叶片,在变转速条件下的非线性动力学行为.考虑叶片转速由一定常转速和一简谐变化的微小扰动叠加而成.应用拉格朗日原理得到变转速叶片的动力学控制方程,并采用假设模态法将偏微分方程转为常微分方程,通过引入无量纲,使方程更具有一般性.运用多尺度方法求解了该参激振动系统,得到了在2:1内共振情形下的平均方程,进而获得系统的稳态响应.详细研究温度梯度、阻尼以及转速扰动幅值等系统参数对叶片动力学响应的影响规律,同时考察了立方项在2:1内共振下对方程的影响.对原动力方程进行正向、反向扫频积分来观察其跳跃现象,并对解析解进行验证.结果发现参数的变化对叶片均有不同程度影响,在2:1内共振下立方项对系统响应的影响很小,解析解与数值解吻合很好.

In the engineering practice, the rotating speed of turbine blade is not a constant value during many application scenarios, for example, during the start-up, the speed varying and the outage of engines, the input and output power of the rotor are out of balance, usually along with the generation of torsional vibration and resulting in velocity pulse. At the same time, the pre-deformation of the blades, caused by some factors including service environment and the installation imperfection, is often inevitable. Nonlinear dynamic behavior of pre-deformed blade with the varying rotating speed is studied in this paper. Considering the rotating speed is consisted of a constant speed and small perturbation, the dynamic governing equation is obtained by Lagrange principle. The partial differential equation is transformed into ordinary differential equation by using assumed mode method. For the sake of generality, a set of dimensionless parameters are introduced. The method of multiple scales is exploited to solve the excitation system. The average equation is derived in the case of 2:1 internal resonance. After that the steady-state response of the system is obtained. The influences of rotating speed, temperature gradient and damping on the dynamic behavior of the blade are studied in detail. Meanwhile,we clarify the effects of cubic nonlinear terms on the steady state response of the blade in the case of the 2:1 internal resonance. The original dynamic equation is integrated numerically in forward and backward frequency sweep direction to observe the jump phenomenon, and to verify the analytical solution. The results show that the changes of parameters have different influences on the dynamic behavior of blade. In the case of the 2:1 internal resonance, the cubic nonlinear terms have little influence on the dynamic response of the system. The analytical solutions are in good agreement with the numerical solutions.