應用Chebyshev偽譜法調合共振梁及高加速度衝擊試驗

Use of Chebyshev Pseudospectral Method to Tune Resonant Beam for High-g Shock Test

許君平(Chun-Ping Hsu)

指導教授 ： 洪振發

國立臺灣大學/工學院/工程科學及海洋工程學研究所/博士(2019年)

https://doi.org/10.6342/NTU201902613

摘要

本文以彈擊式高加速度衝擊試驗裝備的研製為主軸，提出以改良式的Chebyshev偽譜法求解尤拉-伯努利梁模型自由振動模態問題，其次，應用操作模態分析與共振梁等實驗設計，配合部分夾持邊界之尤拉梁解析法，提出一套減少試誤次數的共振梁邊界調整試驗與分析流程。
一般梁結構自由振動問題可使用解析與數值方法求解其自然頻率與模態振形，但遇到梁的邊界條件屬非典型邊界、端點質量、梁外形屬於非棱柱形、階段結構或求解甚高階的模態(大於10階以上)時，都必須採用個案的近似解或數值方法處理，否則容易造成系統矩陣求解精確度不足與高階模態參數發散等現象。本文提出Chebyshev偽譜法與零空間轉換法，針對非棱柱形且帶有邊界值問題的梁元素進行模態參數求解。Chebyshev偽譜法以微分化矩陣為基礎，搭配零空間轉換可將複雜的邊界、梁外形與高低階混合模態等條件所構成的系統矩陣，侷限在有界的數值內，進而搭配MATLAB與Chebfun等工具箱的使用，大大增加求解的面向與速度。此外，對部分夾持型態的棱柱形尤拉伯努利梁之高階模態參數也提出近似解，以上兩種求解過程不僅對目前梁問題都可得到高精度的模態參數，並可應用在本文後續探討彈擊式高加速度衝擊試驗裝備之共振梁模態參數估算。此外，本文針對操作模態分析提出理論說明與實際應用於結構補強或振動控制問題改善的實例，證實大型結構透過操作模態分析所獲得的操作模態參數或響應放大倍率等資訊，對結構設計、壽命評估或試驗再現性極有助益。
最後本文將提出多種不同型態的尤拉-柏努利梁模型，以本研究方法進行自由振動模態數值分析驗證其模擬之精確度，並以共振梁結構搭配彈擊式氣壓系統進行實際衝擊量測，以衝擊響應譜方式呈現規劃試驗流程的正確性，後續執行此型態高加速度衝擊試驗時，更能加速相關試驗參數調校與組件的選用，以期達成試驗高試驗重複性與規格要求。

關鍵字

Chebyshev偽譜法 ； 零空間轉換 ； 共振梁 ； 操作模態分析 ； 高加速度衝擊試驗 ； 衝擊響應譜

並列摘要

In this paper, the main purpose is development of impact equipment with a pneumatic projectile launcher and a length adjustable resonant beam using for generating a high-g shock environment. The modified Chebyshev pseudospectral method is proposed to solve the free vibration of Euler-Bernoulli beam for modelling the resonant beam, and the experimental design of the operational modal analysis will be applied. Analytical method of beam element for clamping boundary, a set of resonance beam boundary adjustment test and analysis flow to reduce the number of trial and error times is proposed.
Free vibration modal analysis of the general Euler-Bernoulli beam can be solved by analytical and numerical methods for natural frequencies and mode shapes. For non-classical boundary conditions, non-prismatic or stepped shape, tip-massed at both end, and higher order modal solutions of Euler-Bernoulli beam, will cause an ill-conditioned system matrix of eigen-value problems, it must be treated by the approximate solution or numerical method case by case. A Chebyshev pseudospectral method with a null space approach is proposed for investigating the boundary-value problem of a non-prismatic Euler-Bernoulli beam with generalized boundary or interfacial conditions. It is shown that, with few vital improvements, the Chebfun toolbox introduced by Trefethen et al. can be systematically applied to modeling non-prismatic Euler-Bernoulli beams with eigenvalue embedded tip-massed boundary conditions as well as the jump conditions that appear at the stepped interfaces. This study also presents a numerical stable asymptotic modal solution for the higher-order modes of a partially clamped beam and show that the proposed approach validates the robust higher-order modal solutions. Through a sequence of four increasingly complicated examples, using the proposed approach with higher-order modes, generalized boundary conditions and interface jump conditions of non-prismatic beams, the results are in excellent agreement with those reported in the literature using various other approaches. Based on the presented analytical beam model, we apply our approach to a mechanical high-g shock machine by tuning the resonant frequencies and clamping stiffness of the beam. In addition, this paper presents theoretical examples and practical examples for the improvement of structural reinforcement or vibration control problems for operational modal analysis, and confirms the operational modal parameters or response magnification obtained by large-scale structures through operational modal analysis. Life assessment or test reproducibility is extremely helpful.
Finally, this paper presents numerical solutions to verify the accuracy for free vibration analysis of different types of Euler-Bernoulli beam. Furthermore, with operational modal analysis of a shock machine, this study also builds the empirical curves for clamping position and stiffness, and shows an optimized procedure that may determine the feasible parameters to alleviate the need for trial and error. Finally, two experiments are conducted to verify the selected parameters, with a pneumatic projectile launcher and a length adjustable beam use for generating a high-g shock environment. With the proposed resontant beam tunning procedure, the measured shock responses are within the tolerance of required specification of shock serponse spectrum.

並列關鍵字

Chebyshev pseudospectral method ； null space approach ； resonant beam ； operational modal analysis ； high-g shock test ； shock response spectrum

參考文獻

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