## 1 Introduction  
Flutter has gradually become one of the most critical factors affecting turbomachinery reliability as technology has advanced recently (higher pressure ratio, thinner blade, etc.) [1]. Due to the high cost and time requirements of flutter experiments, accurate numerical flutter prediction methods are indispensable in the development of high-performance turbomachinery with wide flutter margins [2]. In turbomachinery, the traveling wave model, which was developed by [PERSON], is generally used for flutter predictions [3]. This model supposes that the blades in a rotor row vibrate with a constant inter-blade phase angle (IBPA) and frequency. It was first employed in flutter predictions by [PERSON] [4], and subsequently named as an energy method by [PERSON] [5]. Due to its clear physical implications, it is widely used in numerical blade flutter analysis and design optimization [6; 7; 8]. In this way, the aeroelastic stability of a blade is revealed by aerodynamic damping, which is calculated using the unsteady aerodynamic work integrated into one vibration period with natural frequencies. General flutter analysis methods, including the energy method, are concerned with the effects of flow on the surrounding structure, and vice versa. The flow and structure behave as a single system, an idea which is based on the small fluid-structure coupling assumption, which is often adequate when only considering the onset of flutter. Furthermore, nonlinear flutter behavior, such as limit-cycle oscillation (LCO), is unpredictable unless the full aeroelastic system is solved, includingthe flow and structural equations. For the coupled approach, the simultaneous integration of the Navier-Stokes equation and the structure's motion in time allows for the accurate assessment of energy exchange between the fluid and the blade structure. [PERSON] et al. [9] applied a coupled method to study stall flutter in a linear cascade. A vortex and boundary layer method was used for incompressible flow, coupled with a spring model for the blade's motion. A similar structure model was used by [PERSON] and [PERSON] [10] to study the stall flutter of a transonic cascade. [PERSON] and [PERSON] [11] solved the Euler and Navier-Stokes equations using a transonic cascade coupled with a linear-spring structure model. The coupled results showed strong nonlinear effects for a range of mass flow rates, which was significantly different from results obtained by conventional energy methods. In addition, the effects of frequency mistuning on cascade flutter were studied using this coupled method in their following paper [12]. [PERSON] and [PERSON] [13] proposed a time-domain coupled fluid-structure method and applied it to the flutter analysis of Rotor 67. [PERSON] et al. [14] and [PERSON] et al. [15] adopted a two-way fluid-structure coupled method for aeroelastic analysis of a transonic axial compressor. The results showed that the blade vibration frequency deviated from the natural frequency due to the coupled influence. [PERSON] and [PERSON] et al. [16] applied fully coupled aero-hydro-servo-elastic time domain simulations to investigate the aerodynamic damping of a floating vertical axis wind turbine, which found the aerodynamic damping was almost independent of the rotor azimuth angle. In turbomachinery, a coupled fluid-structure analysis for flutter represents the actual physical processes more closely rather than separated fluid and structural models. In the decoupled method, flutter stability is determined by unsteady aerodynamic work on a given rotor blade during an oscillating cycle. However, in the coupled fluid-structure method, the unsteady aerodynamic work usually converges to zero if vibration is stable. Thus, the time-domain fluid-structure coupled method is often only employed to predict the onset of flutter. Aerodynamic damping calculation methods for coupled fluid-structure method are essential to quantify the flutter margin of turbomachinery.