## 1 Introduction  
Deepwater pipe is the key equipment for the exploitation of offshore oil and gas resources and is the only channel for connecting the subsea wellhead/equipment to the surface floating body. Under the action of ocean currents, vortices are generated and alternately shed from the sides of the pipe, and periodic vortex shedding will produce periodic hydrodynamic force around the structure. When the frequency of hydrodynamic force is close to a certain order of the natural frequency of the riser, a vibration will be induced in the cross-flow (CF) and in-line (IL) directions in the riser, which is termed vortex-induced vibration (VIV). VIV can lead to severe fatigue damage to the riser. Therefore, an accurate prediction of VIV and the induced fatigue damage is of great significance for the design of the riser. In order to study the force characteristics of VIV in the early stage, researchers extracted a rigid cylinder segment from a slender flexible riser based on strip theory and measured the VIV force acting on the cylinder segment synchronously by forcing the rigid cylinder segment to perform harmonic forced oscillation in a steady flow field. [PERSON] [1] of MIT and [PERSON] [2] of NTNU carried out unidirectional forced oscillation experimentsin the CF and IL directions, respectively, in which the VIV forces in both the CF and IL directions were divided into the excitation force in the same phase with velocity and the added mass force in the same phase with acceleration, and the corresponding force coefficient was excitation coefficient and added mass coefficient. In the experiment, the rigid cylinder was forced to move according to different oscillation amplitudes and frequencies, the excitation force coefficients and added mass coefficients under the corresponding dimensionless amplitude and frequency were obtained, and the vortex-induced force coefficient libraries in the CF direction and IL direction were established. At present, the vortex-induced force coefficient models in the semi-empirical frequency-domain prediction software shear7 and VIVANA are widely used in the world, which are simplified from these two vortex-induced force coefficient libraries. However, for real marine flexible risers, VIV in the CF direction and IL direction exists simultaneously, and it has been shown that they are coupled with each other. This coupling exists in both vibration response and vortex-induced force [3; 4]. In order to systematically study the vortex-induced force characteristics of the riser in the CF direction and IL direction under the coupling of VIV, [PERSON] [5] carried out rigid cylindrical bidirectional forced oscillation experiments under different Reynolds numbers (Re), amplitudes in the CF direction, amplitude ratios in the CF and IL directions, dimensionless frequencies in the CF direction, frequency ratios in the CF and IL directions and displacement phase differences in the MIT towing tank. It was found that the force coefficients under the bidirectional coupling of VIV are closely related to the displacement phase difference in both directions. The flow field of the rigid cylindrical segmental forced oscillation experiment/numerical simulations has significant two-dimensional characteristics. Meanwhile, in practice, when VIV occurs, the flow field of the marine slender flexible riser, especially steel catenary risers, flexible riser covered with buoyancy modules and helical strakes, has strong three-dimensional characteristics. Therefore, researchers proposed a tank model experiment based on the VIV of a slender flexible riser with three-dimensional flow field characteristics to invert the vortex-induced force on the flexible riser. The main idea of this method is that the response of the structure can be obtained by using the finite element method when the forces on the structure are known.