Investigation on Solving Vibrations of a Plate with Internal Crack via the Ritz Method
|關鍵字:||Ritz 法;內部裂縫;矩形(薄或厚)板;振動分析;應力奇異;Ritz method;internal crack;rectangular (thin/thick) plate;vibration analysis;stress singularity|
定於所使用允許函數(admissible functions)之恰當性。結構桿件(structural components)常
於第一年(97 年8 月~98 年7 月)，以薄板理論分析具內部裂縫矩形薄板振動。以
於第二年(98 年8 月~99 年7 月)，以Mindlin 板理論分析具內部裂縫矩形中厚板振
於第三年(99 年8 月~100 年7 月)，以Reddy 板理論分析具內部裂縫矩形厚板振動。
The Ritz method has been frequently applied to study the vibration behaviors of a structure. The efficiency of the numerical solution mainly depends on the appropriately chosen admissible functions. Plate as an important structural component may have an internal crack due to overloaded condition, fatigue, or material defect. Consequently, from a practical view point, it is needed to investigate the vibration behaviors of a plate with an internal crack. It is very scarce to find the works of applying the Ritz method to study the vibration behaviors of a plate with an internal crack in the literature. The main reason is that it is very difficult to establish a set of admissible functions suitable for that type of problems, and there is no a systematic procedure to develop such admissible functions. The main purpose of this three-year project is to develop a systematic procedure of establishing admissible functions for a plate with an internal crack, according to different plate theories. Although only rectangular plates are under consideration here, the proposed admissible functions can be applied to other plates with different shapes. In the first year, the classical plate theory will be used for studying the vibration behaviors of a thin plate. Two sets of admissible functions will be used in the Ritz method. One is a mathematical complete set of polynomials. The other set is established by modifying the asymptotic solutions proposed by Williams (1952a). Several approaches on using the asymptotic solutions will be proposed. Comprehensive convergence studies will be carried out to find out which approach is the best. Then, that set of admissible functions will be used to determine the natural frequencies of internally cracked plates. The effects of the orientation and the length of a crack on vibration behaviors of a rectangular plate with an internal crack will be thoroughly studied. In the second year, Mindlin plate theory will be applied to study the vibration behaviors of a moderately thick plate. A suitable set of admissible functions will be proposed based on an asymptotic solution for Mindlin plate theory developed by the writer (Huang, 2003). The asymptotic solution will be appropriately modified according to the experience of the study in the first year. Then, it is expected to obtain an accurate solution, based on the Ritz method, for vibrations of a Mindlin plate with an internal crack. Again, the effects of the orientation and the length of a crack on vibration behaviors of a rectangular plate with an internal crack will be thoroughly studied. In the third year, the third-order shear deformation plate theory proposed by Reddy will be used for studying the vibration behaviors of a thick plate. The corresponding asymptotic solution proposed by the writer (Huang, 2002) will be modified to establish a set of admissible functions suitable for a thick plate with an internal crack. Then, accurate results will be obtained for the vibration frequencies and nodal patterns of vibration modes for a thick plate with an internal having different lengths and orientations. The studies performed in this project have not been seen in the literature. Academically, the success of the work will make a big contribution to considerably expand the applicability of the Ritz method on solving a problem whose solution has a jump behavior. Practically, this work will also provide a lot of vibration results for a plate with an internal crack having different lengths and orientations, so that an engineer will get a feeling how an internal crack changes the vibration behaviors of a plate.