미분구적법(DQM)을 이용한 단면적이 선형적으로 변하는 비대칭 곡선보의 내평면 신장 진동해석

In-Plane Extensional Vibration Analysis of Asymmetric Curved Beams with Linearly Varying Cross-Section Using DQM

Abstract

빌딩, 자동차, 선박, 항공기 등에서의 곡선보 사용 증가로 인해 이러한 구조물의 동적거동해석에 있어 괄목할만한 성과가 있어 왔다. 탄성곡선보의 안정성 거동 해석분야는 많은 연구자들의 관심분야였다. 전통적으로 미분방정식의 해법은 유한차분법으로 해결해왔다. 이러한 방법들은 복잡한 기하학적 구조 및 하중에 따른 격자점의 증가로 많은 계산시간을 요구한다. 편미분방정식의 해를 구하기 위한 효율적인 방법 중의 하나는 미분구적법이다. 복잡한 기하학적 구조 및 하중으로 인한 과도한 컴퓨터 용량의 사용과 복합알고리즘 프로그램의 어려움을 극복하기 위하여 미분구적법(DQM)이 많은 분야에 적용되어왔다. 본 연구에서는 선형적으로 단면적이 변하는 비대칭 곡선보에 대하여 DQM을 적용하여 아크축 신장을 고려한 내 평면 진동해석을 수행하였다. 다양한 매개변수 비, 경계조건, 그리고 열림 각에 따른 기본진동수를 계산하였다. DQM 결과는 활용 가능한 다른 엄밀해와 비교하였다. 다양한 매개변수 비, 경계조건, 그리고 열림 각에 따른 기본진동수를 계산하였으며 DQM 결과를 활용 가능한 다른 엄밀해와 비교하였다. 해석결과에 따르면 DQM은, 적은 격자점을 사용하고도, 엄밀해 결과와 일치함을 보여주었다.

The increasing use of curved beams in buildings, vehicles, ships, and aircraft has results in considerable effort being directed toward developing an accurate method for analyzing the dynamic behavior of such structures. The stability behavior of elastic curved beams has been the subject of a large number of investigations. Solutions of the relevant differential equations have traditionally been obtained by the standard finite difference. These techniques require a great deal of computer time as the number of discrete nodes becomes relatively large under conditions of complex geometry and loading. One of the efficient procedures for the solution of partial differential equations is the method of differential quadrature. The differential quadrature method(DQM) has been applied to a large number of cases to overcome the difficulties of the complex algorithms of programming for the computer, as well as excessive use of storage due to conditions of complex geometry and loading. In this study, the in-plane extensional vibration for asymmetric curved beams with linearly varying cross-section is analyzed using the DQM. Fundamental frequency parameters are calculated for the member with various parameter ratios, boundary conditions, and opening angles. The results are compared with the result by other methods for cases in which they are available. According to the analysis of the solutions, the DQM, used only a limited number of grid points, gives results which agree very well with the exact ones.

Fig. 1. Coordinates for a curved beam

Fig. 2. Forces on a curved beam

Table 1. Fundamental frequency parameter, λ=(mr⁴ω²/EI₀)^1/2, for in-plane extensional vibration of asymmetric curved beams with simply-simply supported ends and η=0.1

Table 2. Fundamental frequency parameter, λ=(mr⁴ω²/EI₀)^1/2, for in-plane extensional vibration of asymmetric curved beams with simply-simply supported ends and η=0.4

Table 3. Fundamental frequency parameter, λ=(mr⁴ω²/EI₀)^1/2, for in-plane extensional vibration of asymmetric curved beams with fixed-fixed ends and η=0.1

Table 4. Fundamental frequency parameter, λ=(mr⁴ω²/EI₀)^1/2, for in-plane extensional vibration of asymmetric curved beams with fixed-fixed ends and η=0.4

Table 5. Fundamental frequency parameter, λ=(mr⁴ω²/EI₀)^1/2, for in-plane extensional vibration of asymmetric curved beams with fixed-simply supported ends and η=0.1

Table 6. Fundamental frequency parameter, λ=(mr⁴ω²/EI₀)^1/2, for in-plane extensional vibration of asymmetric curved beams with fixed-simply supported ends and η=0.4

Table 7. Fundamental frequency parameter, λ=(mr⁴ω²/EI₀)^1/2, for in-plane extensional vibration of asymmetric curved beams with simply supported-fixed ends and η=0.1

Table 8. Fundamental frequency parameter, λ=(mr⁴ω²/EI₀)^1/2, for in-plane extensional vibration of asymmetric curved beams with simply supported-fixed ends and η=0.4

Table 9. Fundamental frequency parameter, λ=(mr⁴ω²/EI₀)^1/2, for in-plane extensional vibration of uniform and non-uniform curved beams with fixed-fixed ends

Table 10. Fundamental frequency parameter, λ=(mr⁴ω²/EI₀)^1/2, for in-plane vibrations of curved beams with fixed-fixed ends