## 1 Introduction  
The lifting installation method (LIM) is widely used to install subsea equipment in deep-sea oil and gas development [1, 2]. Instead of the traditional installation with drilling strings, the LIM method installs subsea equipment through crane lifting and lowering (see Figure 1). The installation with drilling strings may have some limitations of equipment size by the moonpool dimension, and is usually expensive, due to the high day rate of drilling platforms. By overcoming these disadvantages, the LIM method is much more cost-effective and versatile [3, 4]. Even so, it is still one of the more costly operations in offshore oilfield construction, mainly due to the high daily cost of dedicated vessels. To improve efficiency and reduce the cost, improvement of the methodology to give quicker and more accurate predictions of the system response and weather window is critical.  
The first studies of subsea equipment lowering were mainly through models in the frequency domain, by linearizing the hydrodynamic force on both the cable and the equipment [5]. Although the frequency domain methods were cost-effective during the simulation time, the simplification of the problem is not practical. Later, the problem was solved in the time domain with single-degree-of-freedom (SDOF) models [6, 7] and multi-degree-of-freedom (MDOF) models [8, 9, 10]. [PERSON] et al. [11] proposed a MDOF model with the discrete lumped-mass method, incorporating structural rope damping and nonlinear effects,such as snap loads and drag forces. These studies all analysed the problem with fixed rope length, which could save computational costs. Therefore, the fixed-length method is recommended in the widely-recognized recommended practice (DNV-RP-H103 [12]) for the modeling and analysis of offshore operations, and has been widely adopted for subsea structure lowering operations [13; 14; 15; 16]. However, this method does not take into account the effect that the lowering velocity may have on the dynamics of the system.  
To study the effect of lowering velocity on the system dynamics, [PERSON] [17] proposed a one-degree-of-freedom system model that incorporated the lowering velocity in the governing equation. However, it linearized the displacement through the entire rope, which could not reflect the exact tension in the rope. Although research on LIM analysis considering the lowering velocity is limited, some related works have been presented dealing with this effect. [PERSON] et al. [18] proposed a finite element model to analyse an underwater cable with time-dependent length. [PERSON] et al. [19] dealt with the dynamics of overhead crane cables during the load hoisting or lowering. [PERSON] et al. [20] proposed a novel analysis model with varying riser length to study the installation of a subsea production tree. These studies all solved the problem with the variable-domain method [21], in which the element length varied with time. Conversely, [PERSON] and [PERSON] [22] proposed a lumped-mass model for continuous lowering analysis, and solved it with variable element number. In the solution process, the number of discrete elements increases during the lowering operation. In general, there is still a lack of a model that fully incorporates the effects of the rope's structural damping and lowering velocity, as well as nonlinear effects like drag forces and snap loads [23].  
In the present study, a numerical model to analyse the subsea equipment lowering is proposed based on the lumped-mass method, which is reported to be more effective and widely recognized [24]. In the model, the effects of the rope's structural damping, lowering velocity, snap loads, and nonlinear drag forces are considered. By integrating all these factors into the model, it differs from previous models and shows some advantages. In addition, this model is solved with the variable-domain method. The present model is believed to provide a valuable tool for the subsea equipment installation analysis.