本文考慮工件為彈性體，受刀具壓迫後，產生形變。將工件視為尤拉柏努利樑，使用二階偏微分方程式模擬工件側向強制振動，並用二階常微分方程式模擬刀具振動，再由工件與刀具之相互作用力將兩式結合。在切削力計算上，忽略刀腹之力，採用薄剪面切削模式及正交切削，令切削力與切削深度及寬度成正比。在顫振機制上，考慮其為自激性振動中之再生性顫振。對兩動態方程式取拉普拉斯轉換後，求系統特徵方程式，並用頻域響應分析及奈氏圖作穩定性分析，可得到各種材質、尺寸工件在不同轉速下之臨界切屑寬度方程式。

　　在加工方式上，考慮切斷加工，分別討論使用頂心與不使用頂心兩種加工方法；在切削力分析上，分別討論不考慮前一圈撓度與考慮前一圈撓度兩種方法。

　　在模擬過程中，無論是否使用頂心或是否考慮前一圈撓度，均有以下結果：
1.當撓度越大時，則臨界切屑寬度越大。
2.當自然共振頻率越大時，則臨界切屑寬度越小。
3.青銅之臨界切屑寬度大於鋼材之臨界切屑寬度。

　　在交叉比對中，使用頂心工件之臨界切屑寬度均小於不使用頂心工件之臨界切屑寬度；考慮前一圈撓度例子之臨界切屑寬度均大於不考慮前一圈撓度例子之臨界切屑寬度。

　In this study, the workpiece was assumed to be elastic and deformable under external force generated from the cutting tool. By considering the workpiece under investigation as an Euler-Bernoulli beam, then a second order partial differential equation was established to simulate the lateral forced vibration. Additionally, a model for the cutting tool vibration is also developed in a second order ordinary differential equation formulation. These developed models permitted the full analysis and discussion of the interaction between the workpiece and the tool. For the calculation of cutting force, the modes of “shear surface was a plane” and of orthogonal cutting were utilized. For the chatter mechanism, considering the regenerative chatter of the self-excited vibration, system characteristic equation could be yielded from the Laplace transform of the dynamic equations. The stability analyses by frequency domain responses and Nyquist charts could imply the governing equations of critical chip width for different materials, sizes and spindle speeds.

　In the machining process, considering the slot cutting, two cases of with or without the tailstock were both discussed. In the cutting force analysis, two cases of with or without the previous cycle deflection were also be discussed.

　In the simulation, the differences of the critical chip widths were studied under different conditions. It was found that, no matter using the tailstock or not; and no matter considering the previous cycle deflection or not, the following conclusions were always obtained.
1.The greater the deflection and the bigger the critical chip width.
2.The smaller the natural frequency and the larger the critical chip width.
3.The critical chip width of bronze is greater than the steel.

　Besides the above results, it was also found that the critical chip width when using the tailstock was always smaller than the case without using the tailstock. Also, the critical chip width with the previous cycle deflection was always greater than the without case.

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