用偏微分方程式研討圓型軸一端荷重時震動之狀況

Abstract

桿之一端荷重受力而振動時，其縱向及橫向之振動狀態，可用一偏微分方程式及其所附之初時及邊界值條件表示之。當變數未能分離時，用拉普拉斯轉換法(Laplace transform method)，不難求得其解，當變數分離時，R.E.Langer 與R.V.Churchill，曾用拉普拉斯轉換法，求得此方程式之非正交特徵函數(non-Orthogonal characteristic functions)之展開式，惟在其文獻中，未見詳解之過程，亦未提出在何條件下，可用其法解之。因此，本文之目的，一方面用常用之拉普拉斯轉換法解之，且參考R.E.Langer 與 R.V.Churchill 之解，解方程式，再用格陵式法(Green's method)，調和分析法(Harmonic analysis method)，及另二方法解之，並在文中提出詳細之解法以探討此類方程式可用之方法。

The behavious of the longitudinal and transverse vibrations of a rod loaded on one end were studied via two new techniques which involve the proper ways of choosing boundary conditions and change of variables. The same problem was also treated by using the Laplace transform and non-orthogonal expansions as suggested by Langer and later Churchill for Comparison.The merits of all those four methods of solutions were examined to a certain extent.