## 1 Introduction
Similarly, [PERSON] and [PERSON] [20] obtained the semi-analytical results for dynamic response in a container with a baffle by solving the Stokes-Joukowski potential function. However, the available analytical and numerical methods are based on the motion equation for continuous fluid. Tedious derivation and a great amount of calculation will be required when it comes to the application to the complicated tank-liquid systems. [PERSON] et al. [21; 22] proposed a mass-spring model for replacing continuous liquid analysis in a cylindrical container with an anti-sloshing baffle, which can be simply assembled with other complex systems with acceptable precision and small calculation cost. Ignoring effects of flexible soil on the superstructure can induce uncontrollable analysis errors in the practical engineering [23]. [PERSON] and [PERSON] [24] adopted the lumped-mass model for the container and finite element technique for soil to study seismic responses for containers considering soil-structure interaction (SSI). [PERSON] et al. [25] simulated the dynamic performance of cylindrical tanks on a surface foundation and pile group foundation under seismic excitation by centrifugal model tests. [PERSON] et al. [26] determined influences of SSI and sloshing mitigation on the development of axial stress by shake table experiments. [PERSON] et al. [27] analyzed dynamic responses for a cylindrical tank on the surface foundation through shaking table tests, showing that flexible soil amplifies the wall acceleration, however, diminishes the axial compressive stress of wall. For the soil-tank system, the numerical methods can establish the overall analysis model; however, the computational cost is great and at the same time the calculation accuracy is dependent on the discrete model. The restriction of shake table experimental investigations may exist due to the high cost for tests and the limit of the ultimate bearing capacity of the shaking table. In order to reasonably reflect the SSI effect and improve the calculation efficiency, the substructure method is applied to dynamics for the soil-tank system. As the key to the substructure method, dynamic impedance with frequency dependence is employed to portray the force-displacement relationship between foundation and soil. Exact data for flexibility coefficients of the circular surface foundation were presented by [PERSON] and [PERSON] [28] without assuming the distribution of the contact pressure. A lumped parameter model (LPM) is utilized to eliminate frequency dependence for impedances and can be employed in a time domain [29]. Considering the rocking motion and lateral translation, an analytical model for cylindrical tanks was utilized with a lumped parameter of soil [30]. [PERSON] and [PERSON] [31] established a nested LPM for unbounded soil; however, the precision of LPMs is remarkably influenced by the impedances employed in the fitting process. The numerical stability of rational approximation for the simple polynomial with a high degree cannot be guaranteed. The employment of the Chebyshev polynomials for a nested LPM has the advantage of reducing the numerical oscillation in the modelling while showing excellent convergence [32]. [PERSON] et al. [33] established a three lumped-mass model for sloshing in a container and a lumped parameter model for soil considering horizontal and rocking motions of foundation. According to the fluid partition approach, [PERSON] et al. [34] developed an analytical model for sloshing in a two-dimensional aqueduct with a baffle. Then, a dynamic analysis model for the soil-aqueduct system is constructed by using the LPM to describe the impedance for surface foundation. By employing the substructure method, [PERSON] et al. [35] analyzed seismic responses for a circular cylindrical tank resting on a circular surface foundation. Due to the multi-filed interaction for soil, structure and liquid, it is rare to apply the substructure method to theoretical dynamics for a storage container with multiple baffles and resting on soil foundation.