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id="tpnrxl7j" class="title"> <h3><a>數控加工時相貫線的解析式法<small></small></a></h3> </div> <!-- 頁面標題結束 --> <hr class="featurette-divider"> <!-- 中部開始 --> <div id="tpnrxl7j" class="container"> <!-- 正文 --> <div id="tpnrxl7j" class="lead" id="print"> <p> <span style="text-indent: 21pt; font-family: 宋體;">在我們的生活和生產中</span><span lang="EN-US" style="text-indent: 21pt;">, </span><span style="text-indent: 21pt; font-family: 宋體;">常常會遇到兩個或若干個立體相交的問題</span><span lang="EN-US" style="text-indent: 21pt;">(</span><span style="text-indent: 21pt; font-family: 宋體;">在畫法幾何中</span><span lang="EN-US" style="text-indent: 21pt;">, </span><span style="text-indent: 21pt; font-family: 宋體;">我們稱之為相貫</span><span lang="EN-US" style="text-indent: 21pt;">), </span><span style="text-indent: 21pt; font-family: 宋體;">比如在機械生產</span><span style="text-indent: 21pt;"> </span><span style="text-indent: 21pt; font-family: 宋體;">化工設備</span><span style="text-indent: 21pt;"> </span><span style="text-indent: 21pt; font-family: 宋體;">運輸管道</span><span style="text-indent: 21pt;"> </span><span style="text-indent: 21pt; font-family: 宋體;">鋼結構構件的連接中</span><span lang="EN-US" style="text-indent: 21pt;">, </span><span style="text-indent: 21pt; font-family: 宋體;">通常會遇到一些平面與曲面立體之間的截交以及曲面立體曲面立體之間的相貫問題</span><span lang="EN-US" style="text-indent: 21pt;">. </span><span style="text-indent: 21pt; font-family: 宋體;">而通常我們只用作圖法來求</span><span lang="EN-US" style="text-indent: 21pt;">, </span><span style="text-indent: 21pt; font-family: 宋體;">作圖法雖然形象直觀</span><span style="text-indent: 21pt;"> </span><span style="text-indent: 21pt; font-family: 宋體;">迅速</span><span lang="EN-US" style="text-indent: 21pt;">, </span><span style="text-indent: 21pt; font-family: 宋體;">但卻由于受到作圖操作和儀器工具的限制</span><span lang="EN-US" style="text-indent: 21pt;">, </span><span style="text-indent: 21pt; font-family: 宋體;">再加之作圖過程繁頊</span><span style="text-indent: 21pt;"> </span><span style="text-indent: 21pt; font-family: 宋體;">圖形繪制不準確</span><span lang="EN-US" style="text-indent: 21pt;">, </span><span style="text-indent: 21pt; font-family: 宋體;">往往給生產制作造成許多麻煩</span><span lang="EN-US" style="text-indent: 21pt;">, </span><span style="text-indent: 21pt; font-family: 宋體;">且精度更不能滿足現代技術與計算機輔助設計和數控制造技術的要求</span><span lang="EN-US" style="text-indent: 21pt;">, </span><span style="text-indent: 21pt; font-family: 宋體;">因此這我們采用圖解與計算相結合的方法</span><span lang="EN-US" style="text-indent: 21pt;">, </span><span style="text-indent: 21pt; font-family: 宋體;">通過投影圖上的幾何關系</span><span lang="EN-US" style="text-indent: 21pt;">, </span><span style="text-indent: 21pt; font-family: 宋體;">再利用解析方法</span><span lang="EN-US" style="text-indent: 21pt;">, </span><span style="text-indent: 21pt; font-family: 宋體;">建立相貫線上點的坐標方程或極坐標方程</span><span lang="EN-US" style="text-indent: 21pt;">. </span><span style="text-indent: 21pt; font-family: 宋體;">并根據所求的坐標方程或極坐標方程</span><span lang="EN-US" style="text-indent: 21pt;">, </span><span style="text-indent: 21pt; font-family: 宋體;">在</span><span lang="EN-US" style="text-indent: 21pt;"><a target="_blank"><span lang="EN-US" style="font-family:宋體;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">數控機床</span></a></span><span style="text-indent: 21pt; font-family: 宋體;">上利用切割工具</span><span lang="EN-US" style="text-indent: 21pt;">, </span><span style="text-indent: 21pt; font-family: 宋體;">切割出我們所需的各種相貫曲線</span><span lang="EN-US" style="text-indent: 21pt;">. </span><span style="text-indent: 21pt; font-family: 宋體;">為此建立相貫線上的點的坐標方程或極坐標方程是非常重要的</span><span lang="EN-US" style="text-indent: 21pt;">.</span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US">1 </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">兩圓柱體相貫</span><span lang="EN-US"><o:p></o:p></span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">如圖</span><span lang="EN-US">1 </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">所示為軸線相交的兩圓柱體相貫（正立投影）</span><span lang="EN-US"> , </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">現用作圖與解析相結合的方法來求相貫線</span><span lang="EN-US">.<o:p></o:p></span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">設兩圓柱的底面半徑分別為</span><span lang="EN-US">r1 </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">與</span><span lang="EN-US">r2 (</span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">均已知</span><span lang="EN-US">),</span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">兩圓柱軸線的交點為</span><span lang="EN-US">o</span><span style="mso-bidi-font-size:10.5pt;font-family:宋體; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">它們之間的夾角為</span><span lang="EN-US"><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter" /> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0" /> <v:f eqn="sum @0 1 0" /> <v:f eqn="sum 0 0 @1" /> <v:f eqn="prod @2 1 2" /> <v:f eqn="prod @3 21600 pixelWidth" /> <v:f eqn="prod @3 21600 pixelHeight" /> <v:f eqn="sum @0 0 1" /> <v:f eqn="prod @6 1 2" /> <v:f eqn="prod @7 21600 pixelWidth" /> <v:f eqn="sum @8 21600 0" /> <v:f eqn="prod @7 21600 pixelHeight" /> <v:f eqn="sum @10 21600 0" /> </v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect" /> <o:lock v:ext="edit" aspectratio="t" /> </v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:10.5pt; height:15pt'> <v:imagedata src="file:///E:\QQPCmgr\Temp\msohtml1\01\clip_image001.png" o:title="" /> </v:shape><![endif]--><!--[if !vml]--><img border="0" width="14" height="20" src="file:///E:/QQPCmgr/Temp/msohtml1/01/clip_image002.jpg" v:shapes="_x0000_i1025" alt="" /><!--[endif]--> (</span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">已知</span><span lang="EN-US">),</span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">以</span><span lang="EN-US">o</span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">為原點</span><span lang="EN-US">, </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">分別建立各自的三維坐標體系</span><span lang="EN-US">, y1 </span><span style="mso-bidi-font-size:10.5pt;font-family:宋體; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">軸垂直于</span><span lang="EN-US">x1 oz1 </span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">平面</span><span lang="EN-US">y2 </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">軸垂直于</span><span lang="EN-US">x2 oz2 </span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">平面</span><span lang="EN-US">,</span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">而</span><span lang="EN-US">x1 oz1 </span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">和</span><span lang="EN-US">x2 oz2 </span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">正好在同一平面內</span><span lang="EN-US">, </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">所以</span><span lang="EN-US">y1 </span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">與</span><span lang="EN-US">y2 </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">為同一條軸</span><span lang="EN-US">,</span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">其坐標我們均計為</span><span lang="EN-US">y, </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">由圖</span><span lang="EN-US"> 1 </span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">的幾何關系可得下面解析式</span><span lang="EN-US">:<o:p></o:p></span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><img src="/upload/images/201406/14033650815373.jpg" alt="" /></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">在制作該相貫線時</span><span lang="EN-US">, </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">可將圓柱面展開成板</span><span lang="EN-US">, </span><span style="mso-bidi-font-size:10.5pt;font-family:宋體; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">橫坐標為</span><span lang="EN-US"> , </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">縱坐標為</span><span lang="EN-US"> , </span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">在數控機床</span><span lang="EN-US">[5]</span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">上用數控線切割機按坐標便可加工出所需曲線</span><span lang="EN-US">.<o:p></o:p></span><img src="/upload/images/201406/1403365091268.jpg" alt="" /></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"> </p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US">2 </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">圓柱體與圓錐體相貫</span><span lang="EN-US"><o:p></o:p></span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman"">如圖</span><span lang="EN-US"> 2 </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">所示為軸線相交的圓柱體與圓錐體的相貫</span><span lang="EN-US">[1 2], </span><span style="mso-bidi-font-size:10.5pt;font-family:宋體; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">圓錐頂角為</span><img src="/upload/images/201406/14033651882693.jpg" alt="" /></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US">, </span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">圓柱體的中心軸線與</span><span lang="EN-US">x </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">軸的夾角為</span><img src="/upload/images/201406/14033651992741.jpg" alt="" /></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><span lang="EN-US">, </span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">且其中心軸線與圓錐中心軸線相交</span> <span style="mso-bidi-font-size:10.5pt; font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">坐標如圖</span><span lang="EN-US"> 2 </span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman"">所示</span><span lang="EN-US">, </span><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">則圓柱曲面方程</span><span lang="EN-US">[4]</span><span style="mso-bidi-font-size:10.5pt;font-family:宋體; mso-ascii-font-family:"Times New Roman";mso-hansi-font-family:"Times New Roman"">為</span><span lang="EN-US">:<o:p></o:p></span></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><img src="/upload/images/201406/1403365102055.jpg" alt="" /></p> <p class="MsoNormal" style="text-indent:21.0pt;mso-char-indent-count:2.0"><img src="/upload/images/201406/14033651159185.jpg" alt="" /></p> <p class="MsoNormal" align="left" style="text-indent: 21pt;"><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-family:宋體;mso-font-kerning:0pt">上式即為相貫線上任一點的極坐標方程</span> <span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-bidi-font-family:宋體;mso-font-kerning: 0pt">在制作時將圓錐面展開成板</span> <span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-bidi-font-family: 宋體;mso-font-kerning:0pt">用數控線切割機按該極坐標方程即可加工成形對于任何平面與立體截交以及立體與立體相貫</span> <span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-bidi-font-family:宋體;mso-font-kerning: 0pt">都可求出其解析方程</span> <span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-bidi-font-family: 宋體;mso-font-kerning:0pt">使我們在</span><span lang="EN-US"><a target="_blank"><span lang="EN-US" style="font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman"">數控機床</span></a></span><span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-bidi-font-family:宋體;mso-font-kerning: 0pt">的生產中</span> <span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-bidi-font-family:宋體;mso-font-kerning: 0pt">可以根據相貫線的曲線方程來控制切割工具的運動走向</span> <span style="mso-bidi-font-size:10.5pt; font-family:宋體;mso-ascii-font-family:"Times New Roman";mso-hansi-font-family: "Times New Roman";mso-bidi-font-family:宋體;mso-font-kerning:0pt">致使在機床上加工出我們所需的各種曲線</span> <span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family:"Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-bidi-font-family:宋體;mso-font-kerning: 0pt">且克服了由于作圖而產生的誤差</span> <span style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-ascii-font-family: "Times New Roman";mso-hansi-font-family:"Times New Roman";mso-bidi-font-family: 宋體;mso-font-kerning:0pt">并能更好地滿足日益發展的計算機輔助設計和制造技術的需要</span><span lang="EN-US" style="mso-bidi-font-size:10.5pt;font-family:宋體;mso-bidi-font-family:宋體; mso-font-kerning:0pt"><o:p></o:p></span></p> <p class="MsoNormal" align="center" style="margin-top:4.5pt;margin-right:0cm; margin-bottom:7.5pt;margin-left:0cm;text-align:center;line-height:13.5pt; mso-pagination:widow-orphan;mso-outline-level:1"><b><span style="mso-bidi-font-size: 10.5pt;font-family:宋體;mso-bidi-font-family:宋體;color:#666666;mso-font-kerning: 0pt">本文由<span lang="EN-US"> <a ><span lang="EN-US" style="color:#333333"><span lang="EN-US">伯特利數控</span></span></a></span>文章<span lang="EN-US"> </span>整理發表，文章來自網絡僅參考學習，本站不承擔任何法律責任。</span></b><span lang="EN-US" style="font-family: 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