應用Ritz法求解具內部裂縫板振動之探討

Abstract

Ritz 法已被廣泛應用於結構之振動分析。該方法分析結構振動之成功與否，主要決 定於所使用允許函數(admissible functions)之恰當性。結構桿件(structural components)常 因過載或疲勞而產生裂縫，故分析具內部裂縫(internal crack)板振動有其實際應用性。但 在應用Ritz 法求解各種結構桿件振動問題之相關文獻中，非常少之文章求解具裂縫之問 題。此現象是可以理解的；因為要找一組適當描述裂縫之允許函數是不容易的，且目前 並沒有一套系統化方法建構此允許函數。本研究擬以三年發展一套系統可架構適用於求 解具內部裂縫板振動問題之允許函數，分析案例則祇考量矩形板： 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 Williams 所發展之漸進解為基礎，發展適用於求解具內部裂縫板振動問題之各種允許函 數。透過詳細之收斂性分析，確認可提供滿意結果之允許函數。並進一步探討裂縫位置 與幾何(大小、orientation)對該板振動頻率與模態之影響。 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 動。以吾人以前所發展之Mindlin 板漸進解為基礎，並利用第一年研究經驗，發展適用 於求解具內部裂縫矩形中厚板振動問題之允許函數。並進一步探討板厚、裂縫位置與幾 何(大小、orientation)對該板振動頻率與模態之影響。 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。 以吾人以前所發展之Reddy 板理論漸進解為基礎，並利用前二年研究經驗，發展適用於 求解具內部裂縫矩形厚板振動問題之允許函數。並進一步探討板厚、裂縫位置與幾何(大 小、orientation)對該板振動頻率與模態之影響。 以上所探討者，均未見於現有文獻；故本研究之成功將使得Ritz 法之應用有突破 性之發展，於學術及應用上必有相當貢獻。

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.