## 5 Conclusions  
This paper focuses on the dynamic modeling and transient response characteristics of a complicated gas generator rotor in a turboshaft engine with blade off. It is proposed that the finite element method with cylindrical and conical elements combined with the reduction of system degrees of freedom is used to establish the dynamic model of the complex rotors, in which the complex structural characteristics, sudden unbalance excitation, and inertia asymmetry can be considered. Then, the validity of the modeling method proposed for complicated rotors is verified through experiments with a general complex rotor. On the basis of the established dynamical model used for blade off transient response analysis, the vibration mechanism of blade off for the actual rotating machinery is studied by time-frequency analysis. Furthermore, parametric studies including the rotational speed and blade off location are carried out in detail. The main conclusions are as follows:  
1. The validity of the rotor modeling method is verified by a test. The proposed dynamic model can reflect the dynamic characteristics of the complicated rotor well. It is suitable for vibration analysis because the complex rotor structure is characterized with fewer degrees of freedom. Based on the dynamic model established, the natural characteristics and transient response evolution, including the time-domain, frequency-domain, and orbit, of a high-speed gas generator rotor with blade off are revealed. 2. The impact effect due to blade off has time-dependent and location-dependent characteristics. When the blade off occurs, the amplitude of transient response increases suddenly and reaches to a peak point soon and then oscillates downward to a stable value with several revolutions. The instantaneous impact force due to blade off is the fundamental reason for the sharp increase in vibration response, and the generated large unbalance after blade off and system damping are the reason for the finally stable amplitude reached. The blade off excites some lower-order natural modes, such that the transient response contains some lower natural frequency components. 3. The responses of the rotor with blade off occurring at different rotational speeds have similar characteristics, while the rotational speed has a significant influence on the impact factor, i.e., the higher the speed, the greater the impact factor. 4. The parameter of the blade off location will not only have a significant effect on the impact factor, but also on the frequency spectrum characteristics at each node of the rotor. The transient response excited by the blade off of 1 AC is greater than that caused by the loss of the 1 GT blade. The sudden unbalance excitation at 1 AC can  
Figure 23: Unbalance response comparison obtained from simulation and test data. excite the first three-order natural modes simultaneously (\(f_{n1},f_{n2-},f_{n2+},f_{n3-},f_{n3+}\)), while the excitation at 1 GT can only excite the first two modes (\(f_{n1},f_{n2-},f_{n2+}\)). 5. When blade off occurs at 1 AC, 1# bearing and its supporting structure are most likely to be damaged and some safety design needs to be adopted because it has the largest transient and steady-state amplitude and a high impact factor. 6. The above-mentioned transient response time series, frequency spectrum difference, and impact factor distribution characteristics between two blade off cases can provide a good theoretical basis for the fault recognition of these complex rotors suffering from blade off. At the same time, the modeling method can be applied to other types of rotors. L.W., Writing--original draft, Investigation, Methodology, Data visualization, Validation. Y.Y. : Validation, Writing--review and editing. A.W. : Editing, Funding acquisition. X.H. : Data curation. M.J.: Data visualization, Funding acquisition. All authors have read and agreed to the published version of the manuscript. This work was supported by the Major State Basic Research Development Program of China (No.