## 1 Introduction
[6], which identified the nonlinear damping by solving the nonlinear Volterra integral equation of the first kind. All abovementioned studies and methods were based on the differential equation of roll with nonlinear damping models. In fact, in the ship engineering field, sometimes the nonlinear damping is approximated to an equivalent linear one for simplicity. Normally, the equivalent linear roll damping coefficient depends upon not only the roll frequency and forward speed, but also on the roll amplitude. Therefore, it is vital to correctly determine the equivalent linear roll damping coefficient for the large-amplitude roll analysis. Currently, there are two main ways to obtain the equivalent roll damping coefficient. One is from carrying out ship model tank tests, which can provide reliable resulting data. The other is computational fluid dynamics (CFD) methods, which are preferred due to their lower cost and shorter time consumption. [PERSON] et al. [13] experimentally and numerically investigated hydrodynamic coefficients of various forced rolling sections and observed that viscous effects were significantly present for sharply edged sections. [PERSON] and [PERSON] [14] compared the roll moment from the potential theory with that from using Reynolds-averaged Navier-Stokes equations (RANS) and found that the shear roll moment was negligible, while the vortex effect was identified to have significant influence on the phase and magnitude. [PERSON] et al. [15] demonstrated that the unsteady RANS (URANS) calculations are in a good agreement with experimental results under large amplitudes and shallow draft conditions. [PERSON] et al. [16] clearly captured the vortex shedding around a forced rolling hull section with sharp keel by using the RANS solver in OpenFOAM, which suggests that the viscous effects mainly originate from the vortex shedding. Similar results were achieved by using the mesh-free numerical model [17]. Actually, the vortex shedding around sharp edges is the major source of viscous effects in the roll damping. However, CFD methods still require significant computational resources, which might not meet the engineering demand. An alternative approach is to exploit semi-empirical methods. One of the most common semi-empirical methods is [PERSON]'s estimation method [18], which divides the total equivalent linear roll damping coefficient into skin friction damping, eddy damping, wave damping, lift damping, and bilge keel damping. Amongst these, only wave damping and lift damping can be calculated using potential theories, while the rest are associated with fluid viscous effects and can be obtained using empirical approaches. Beyond [PERSON]'s estimation method, [PERSON] and [PERSON] [19] incorporated the Direct Vortex Method (DVM) into a potential theory to model viscous effects within an inviscid model, in which the roll center of bilge keels needs to be estimated. Based on the work of [PERSON] and [PERSON] [19], [PERSON] et al. [20] developed a method using DVM to assess the roll damping of ships. More recently, [PERSON] et al. [21] first proposed a linear time-domain Green's function method (TGF_V) for assessing the interaction between floating bodies and water waves with viscous dissipation effects, which can accurately evaluate the hydrodynamic coefficients of sharply edged floating bodies with small roll amplitude. The TGF_V [21] can consider the full viscous dissipation effects of water, while other similar Green's functions [22; 23; 24] can only take the partial viscous dissipation effects into account. However, the linear TGF_V or other [PERSON]'s functions cannot be employed to evaluate the equivalent damping coefficient of large-amplitude roll motion, which is known to be highly nonlinear. In this paper, a body-nonlinear time-domain [PERSON]'s function method (TGF_N_V) is first developed for evaluating the equivalent damping of large-amplitude roll motion for two-dimensional floating bodies with consideration of viscous dissipation effects of water.