載具沿滑軌運動之傾墜分析

Abstract

本研究考量載具與滑軌間的結構互制行為，發展半解析方法分析載具沿著滑軌運 動的傾墜反應。對於飛彈發射系統設計而言，精確的計算其傾墜反應是極為重要的課題，飛彈可視為本研究之載具。本文提出R.E. 與E.E. 兩種模型進行動態反應分析。其中R.E. 模型係將載具視為剛性梁，滑軌視為彈性梁，而在E.E. 模型中將載具與滑軌皆視為彈性梁。在上述分析模型的解析時，皆考慮了慣性力、科氏力與離心力的影響，同時載具與滑軌之間皆透過兩個剛性滑腳來接觸。

本研究基於Euler-Bernoulli 梁理論，採用振態疊加法與拉格蘭日乘子法來推導載具與滑軌運動的控制方程式。此系統的運動方程式是由一組非線性的微分方程式所組成，其數值解採用Petzold-Gear 的後向微分法（Backward Differentiation Formula, BDF）來求算此動態系統的數值模擬結果，所需花費時間較少於一般傳統逐步數值積分的方法，對於載具與滑軌的分析模擬經與文獻實際特殊案例的結果比較，可呈現本研究所提出之方法的優點。

利用此分析方法進一步探討滑軌的長度、載具滑腳的距離、載具與滑軌的質量比 與剛性比等各參數對載具傾墜反應的影響，經由本研究之成果，於載具發射系統設計時，可以提供非常有價值的資訊。

A semi-analytical solution for analyzing the tip-off responses of a vehicle moving along a guideway is developed by considering the dynamic interaction between the vehicle and the guideway. Accurately determined tip-off responses are important for designing a launch system for a missile, in which the missile can be treated as the vehicle in the present study. Two models are proposed to determine those dynamic responses, namely, R.E. model and E.E. model. In the R.E. model, the vehicle is assumed rigid and its guideway is modeled as a flexible beam, while both of the vehicle and guideway are modeled as flexible beams in the E.E model. The inertia, Coriolis, and centrifugal forces are considered in these models. The vehicle contacts with its guideway through two rigid shoes.

Equations for governing the motions of the vehicle and the guideway are developed using the Lagrangian approach and the modal superposition method, on the basis of the Euler-Bernoulli beam theory. The governing equations, which are a set of nonlinear differential equations, are solved by the Petzold-Gear backward differentiation formula numerical method. It takes time lower than the traditional step-by-step numerical integration methods. Comparisons of the presented solutions with those based on different published models for the vehicle and guideway reveal the advantages of the present approach.

The solutions are further employed to investigate the effects of the length of the guideway, distance between the shoes of the vehicle, and mass and rigidity ratios of the vehicle to the guideway on the tip-off responses of the vehicle. The results presented herein provide valuable information for designing vehicle launch systems.