## 1 Introduction  
In the engine room and stern adjacent to the main excitation force of the ship, there are many fuel and fresh water tank structures required for ship operation which are always exposed to vibration. If a vibration problem occurs after construction or delivery, there is a large loss in terms of ship quality, such as reinforcement cost and delivery delay. Therefore, it is necessary to review the anti-vibration design to prevent such vibration problems in the design stage, and for this reason, each shipyard has developed and used an approximate vibration calculation program according to the circumstances of the design. In the case of this approximate analytical calculation method, the Bernoulli-Euler beam theory [1] is mainly used as a mode function. Since the computational process is complicated, many studies using polynomials with beam properties have been conducted to simplify it. [PERSON] [2,3] defined the assumed mode function by combining the Euler beam mode function with the same boundary condition and performed vibration analysis for the plate. [PERSON] [4] performed normal mode analysis in consideration of the effects of rotational inertia and shear deformation of plates and beams using a polynomial with [PERSON]'s beam properties. [PERSON] [5] considered the rotational elastic constraints at both ends of the structure in order to provide an appropriate boundary condition. In recent studies, mode functions are defined using Euler beams [6], Legendre polynomials [7] and orthogonality [8] and are applied not only to calculation of natural frequencies but also to calculations of hull deformation when a ship is impacted by waves while in operation. In addition, a study on the method of searching for the crack point of a beam using the Euler beam theory [9] was also carried out and vibration analysis of a wedge beam was carried out [10]. Many studies such as this are being conducted using the characteristics of Euler beams. For vibration analysis of the connection structure, it is important to define the function of each member and the constraint conditions of the connection part. Many researchers, such as [PERSON], defined and used the equation using Euler beam characteristics for the dynamic analysis of aircraft wings [11,12]. For vibration analysis, such as panel structure, a method of deriving a polynomial with [PERSON]'s beam property is presented. Furthermore, [PERSON] [13] uses a polynomial as mode functions. However, all of the above studies are used given elastic boundary conditions and no studies on deriving appropriate boundary conditions have been found. This is the biggest reason why there has been no study of normal mode analysis using the assumed mode function over the past 10 years. Many local structures, such as tank edges and equipment foundations, consist of connected structures, and there is a need to find the suitable elastic boundary condition to find the normal mode frequencies. Vibration analysis of many local structures in ships as Figure 1, such as tanks and supports for equipment, can be simplified by breaking them up into smaller subsystems which are related through geometrical conditions and natural conditions at junctions [14,15]. We have established a method to calculate the actual boundary condition at the junction of the connection structure at the tank wall. Figure 1: Arrangement of fresh water and fuel tanks in ship: (**a**) the arrangement shape of the tank; (**b**) 1700 TEU Container fresh water tank wall damage. To do this, firstly, we formalize the polynomials for simple support and fixed support that were proposed to represent each subsystem, and in order to verify the validity of the polynomials, we performed the normal mode calculation using component mode synthesis. The purpose of this study was to propose a method of satisfying the actual boundary condition by combining the mode function including the simple support and the fixed support boundary condition in the plate structure.