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<h1 id="第一離隙角避讓">第一離隙角避讓</h1>
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<p>本文所述離隙角皆為第一離隙角，即離切削區域最近的離隙角(Relief Angle)。或稱清角(Clearance Angle)。</p>
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<p>銑刀刀刃在切削過程中為擺線運動，為下圖紅色擺線。藍色圓為刀具包絡圓。</p>
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<p>紅色擺線之內(即朝刀具包絡圓圓心方向)為已切削區域；紅色擺線之外為未切削區域，如果離隙面在該處則會與未切削材料產生碰撞。
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所以<strong>標示RA(Relief Angle的縮寫)處為最小所需離隙角。</strong></p>
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<p><img src="relief-angle.png" alt="relief-angle.png"></p>
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<p>如果實際刀具離隙角小於最小離隙角，則清角面會壓到未切削工件，造成刀具及工件受力增加，使得刀具振動加大且工件表面回彈，表粗驟升，會降低刀具壽命。</p>
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<h2 id="最小所需離隙角計算">最小所需離隙角計算</h2>
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<p>於定軸加工，最小所需離隙角可由進給、轉速、刀具半徑算出；於多軸同動加工，需由程式對每個接觸點批量計算。</p>
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<p>以下簡述定軸加工的最小所需離隙角計算方式。</p>
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<p><span class="math">\(\vec r_p = \left(t\cdot v-R\cdot\sin\left(t\cdot w\right),R-R\cdot\cos\left(t\cdot w\right)\right)\)</span></p>
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<p><span class="math">\(\vec r_b = \left(a\cdot v-R\cdot\cos\left(s\right),R-R\cdot\sin\left(s\right)\right)\)</span></p>
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<p>其中
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<span class="math">\(\vec r_p\)</span> 為紅色擺線位置向量；
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<span class="math">\(\vec r_b\)</span> 為藍色圓位置向量；
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<span class="math">\(R\)</span> 為刀具半徑(mm)；
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<span class="math">\(w\)</span> 為轉速(rad/s)；
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<span class="math">\(v\)</span> 為進給(mm/s)；
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<span class="math">\(t\)</span> 為時間；
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<span class="math">\(s = t\cdot w\)</span> ；
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<span class="math">\(a\)</span> 為指定時間，作為常數使用。</p>
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<p>令速度向量為</p>
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<div class="math">
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\[
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\vec v_p = \frac{d\vec r_p}{dt} ,
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\vec v_b = \frac{d\vec r_b}{ds}
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\]</div>
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<p><span class="math">\(\vec v_p\)</span> 、 <span class="math">\(\vec v_b\)</span> 兩向量夾角為最小所需離隙角。</p>
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