## 1 Introduction  
In past years, many mathematical models have been developed for analyzing ground seismic liquefaction. For the simulation target for the cyclic triaxial test, previous examples considered pore water pressure. Research [1, 2] mentioned that simplified mathematical models can predict soil liquefaction phenomena under dynamic load. However, the correctness of the pore water pressure in the model will affect the results. Three different models accompanied by parameters were used to simulate pore water pressure [3, 4, 5]. The most significant difficulty in this related research is that the formulas developed are difficult to derive, and the parameters of the mathematical model are not easy to obtain [6, 7, 8, 9, 10, 11, 12].  
The mass-spring-damping system has been used [6, 7] to design the foundation of the hammer and press base and the structural analysis between the piers and supports, both of which were based on the mass-spring-damping system. Design and analysis were commonly seen in structural dynamics-related research but less conducted in geotechnical engineering.  
The soil shear modulus (G) and damping ratio (D) can be determined with cyclic triaxial tests [13, 14, 15]. However, [PERSON] et al. [16] found that the damping ratio D and shear modulus G of the first five cycles will decrease as the cyclic number increases. These two parameters also relate to the damping coefficient (c) and spring coefficient (k) in the mass-spring-damping system. Its trend can be used for more in-depth research and discussion. The parameters are easy to understand and obtain, showing the extensibility of this model, so the mass-spring-damper system could be chosen to simulate the strain of cyclic triaxial tests. It was also found that the damping ratio may increase with shear strain but reverse under large shear strains [17]. The damping ratio will decrease when the specimen increases with the dynamic strain [18].  
As for the waste soil materials, the damping ratio first decreased and then increased as the shear strain increased using a cyclic triaxial instrument [19]. Some researchers [20; 21; 22; 23; 24] also used a viscoelastic-plastic model for soil stiffness degradation and damping ratio and applied it to related engineering. As for regional studies, [PERSON] and [PERSON] [25] discussed Kasai River sand in India using a model. [PERSON] et al. [26] analyzed and compared the model and test results of undrained repeated triaxial tests on densely saturated sand with different particle sizes and fine content. They also used the equivalent granular state parameter to discuss the model parameters.  
The repeated strain of soil is generally obtained through laboratory liquefaction cyclic triaxial tests, and relevant research is conducted under the conditions of many influencing factors such as material type, particle size, gradation, etc. This paper attempts to develop a mathematical model using the mass-spring-damper system, discusses whether both the spring coefficient and damping coefficient are variables, and studies the values of various parameters of the system, including mass m, spring coefficient k, damping coefficient c, and external force Q\({}_{0}\), calculated using Mathematica software. Finally, the original thin-tube specimens from four locations in the Yunlin area of Taiwan were selected to conduct liquefaction cyclic triaxial tests. Then, the test results were compared with the model to investigate pattern correctness.