技术交流 › 刀具论坛 › 螺杆压缩机转子滚刀刀刃分离廓形的精确解析

HEATS 发表于 2010-9-12 10:59:14

螺杆压缩机转子滚刀刀刃分离廓形的精确解析

<STRONG><FONT face="Times New Roman">1</FONT>．引　言</STRONG>
<P align=left>　　螺杆转子是螺杆压缩机的关键零件，压缩机工作的可靠性和效率取决于螺杆转子的加工精度。我国规定的标准螺杆转子端面型线复杂，且廓形存在尖点，滚刀的设计制造困难，因此，螺杆转子滚削加工方法在我国尚未推广应用。<BR>　　国际上一些发达国家，如日本、英国、法国等均对螺杆转子的滚削加工进行了大量研究工作，他们在保证压缩机效率的前提下，通过改进螺杆转子端面型线，使之光滑流畅无尖点，从而简化滚刀的设计原理，实现了螺杆转子的滚削加工。<BR>　　本文以国家标准螺杆转子的滚削加工为对象，对压缩机螺杆转子滚刀的设计理论与滚刀制作进行了试验研究。</P>
<P align=left>　<STRONG>　<FONT face="Times New Roman">2</FONT>．刀刃分离廓形方程的求解</STRONG></P>
<P align=left>　　（<FONT face="Times New Roman">1</FONT>）阴转子端面型线和滚刀轴向刃形<BR>　　阴转子端面齿形采用单边非对称摆线圆弧组成（见图<FONT face="Times New Roman">1</FONT>），即阴转子齿形端面型线由直线<EM><FONT face="Times New Roman">ab</FONT></EM>段、圆弧<EM><FONT face="Times New Roman">bc</FONT></EM>段、延长外摆<EM><FONT face="Times New Roman">cd</FONT></EM>段和径向直线<EM><FONT face="Times New Roman">de</FONT></EM>段组成。已知阴转子螺杆参数：左旋齿数<FONT face="Times New Roman"><EM>Z</EM><SUB>2</SUB></FONT>＝<FONT face="Times New Roman">6</FONT>，杆长<EM><FONT face="Times New Roman">L</FONT></EM>＝<FONT face="Times New Roman">95mm</FONT>，导程<FONT face="Times New Roman"><EM>h</EM><SUB>2</SUB></FONT>＝<FONT face="Times New Roman">170</FONT>．<FONT face="Times New Roman">1mm</FONT>，节圆半径<FONT face="Times New Roman"><EM>r</EM><SUB>2</SUB></FONT>＝<FONT face="Times New Roman">30</FONT>．<FONT face="Times New Roman">24mm</FONT>，齿高半径<EM><FONT face="Times New Roman">R</FONT></EM>＝<FONT face="Times New Roman">12</FONT>．<FONT face="Times New Roman">915mm</FONT>。阴阳转子中心距<FONT face="Times New Roman"><EM>A</EM><SUB>D</SUB></FONT>＝<FONT face="Times New Roman">50</FONT>．<FONT face="Times New Roman">4mm</FONT>，阳转子节圆半径<EM><FONT face="Times New Roman">RR</FONT></EM>＝<FONT face="Times New Roman">20</FONT>．<FONT face="Times New Roman">16mm</FONT>。根据上述条件求出<EM><FONT face="Times New Roman">ab</FONT></EM>、<EM><FONT face="Times New Roman">bc</FONT></EM>、<EM><FONT face="Times New Roman">cd</FONT></EM>、<EM><FONT face="Times New Roman">de</FONT></EM>段的滚刀轴向刃形系列坐标点，由这些点可画出所设计滚刀的轴向刃形如图<FONT face="Times New Roman">2</FONT>所示。</P>
<P align=center><IMG src="http://www.chmcw.com/upload/news/RCL/13220_omdg3m20083515543.gif"></P>
<P align=center><STRONG>图<FONT face="Times New Roman">1</FONT>　阴转子端面截形</STRONG></P>
<P align=center><IMG src="http://www.chmcw.com/upload/news/RCL/13220_95mdok200835155423.gif"></P>
<P align=center><STRONG>图<FONT face="Times New Roman">2</FONT>　滚刀轴向刃形</STRONG></P>
<P align=left>　　根据已知条件，计算出与阴转子<FONT face="Times New Roman">cd</FONT>段相啮合的轴向刃形<FONT face="Times New Roman"><EM>c</EM><SUB>1</SUB><EM>d</EM><SUB>1</SUB></FONT>，对应于工件上<EM><FONT face="Times New Roman">d</FONT></EM>点的刀刃上<FONT face="Times New Roman"><EM>d</EM><SUB>1</SUB></FONT>点坐标为<EM><FONT face="Times New Roman">d</FONT></EM>（<FONT face="Times New Roman">44</FONT>．<FONT face="Times New Roman">0837</FONT>，<FONT face="Times New Roman">6</FONT>．<FONT face="Times New Roman">203069</FONT>）；计算出与阳转子<EM><FONT face="Times New Roman">de</FONT></EM>段相啮合的滚刀上<EM><FONT face="Times New Roman">d</FONT></EM>′<FONT face="Times New Roman"><SUB>1</SUB></FONT>点坐标为<EM><FONT face="Times New Roman">d</FONT></EM>′<FONT face="Times New Roman"><SUB>1</SUB><EM>e</EM><SUB>1</SUB></FONT>，对应于工件上<EM><FONT face="Times New Roman">d</FONT></EM>点，刀刃上<EM><FONT face="Times New Roman">d</FONT></EM>′<FONT face="Times New Roman"><SUB>1</SUB></FONT>点坐标为<EM><FONT face="Times New Roman">d</FONT></EM>′<FONT face="Times New Roman"><SUB>1</SUB></FONT>（<FONT face="Times New Roman">37</FONT>．<FONT face="Times New Roman">6313</FONT>，<FONT face="Times New Roman">5</FONT>．<FONT face="Times New Roman">02781</FONT>）。不难看出，在<FONT face="Times New Roman"><EM>c</EM><SUB>1</SUB><EM>d</EM><SUB>1</SUB></FONT>段与<EM><FONT face="Times New Roman">d</FONT></EM>′<FONT face="Times New Roman"><SUB>1</SUB><EM>e</EM><SUB>1</SUB> </FONT>段之间出现了一段分离的曲线。出现上述分离现象是由于阴转子端面齿形上的<EM><FONT face="Times New Roman">cd</FONT></EM>段与<EM><FONT face="Times New Roman">de</FONT></EM>段的交点<EM><FONT face="Times New Roman">d</FONT></EM>非光滑，存在尖点的缘故。因此，为了设计制造出正确的刀刃廓形，从而加工出正确的工件廓形（即不致使阴转子廓形上的尖点<EM><FONT face="Times New Roman">d</FONT></EM>被切掉），有必要精确计算出滚刀刃形上的这段分离曲线。<BR>　　（<FONT face="Times New Roman">2</FONT>）刀刃分离廓形的精确设计<BR>　　此种设计是利用公共齿条的概念，把空间啮合转化为平面啮合来求解。已知滚刀基本蜗杆与工件的啮合，通过工件的端面齿形求出与工件端面上尖点<FONT face="Times New Roman">d</FONT>相啮合的齿条上的分离段曲线，再通过这段分离曲线求出与之相啮合的滚刀上相应的分离段曲线。<BR>　　设滚刀基本蜗杆与左旋阴转子啮合的相互位置如图<FONT face="Times New Roman">3</FONT>所示。当滚刀蜗杆<FONT face="Times New Roman">1</FONT>转过<EM>φ</EM><FONT face="Times New Roman"><SUB>1</SUB></FONT>角时，工件<FONT face="Times New Roman">2</FONT>相应地转过<EM>φ</EM><FONT face="Times New Roman"><SUB>2</SUB></FONT>角。</P>
<P align=center><IMG src="http://www.chmcw.com/upload/news/RCL/13220_lrngwm200835155445.gif"></P>
<P align=center>图<FONT face="Times New Roman">3</FONT>　蜗杆与工件的相对位置</P>
<P align=left>　　由齿形法线定理可得到与工件相啮合的齿条方程为</P>
<P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_i5erko20083515552.gif"></P>
<P align=left>　　将阴转子端面齿形方程的<EM><FONT face="Times New Roman">cd</FONT></EM>段、<EM><FONT face="Times New Roman">de</FONT></EM>段代入上式，求得工件端面齿条齿形如图<FONT face="Times New Roman">4</FONT>所示。其中两分离点<EM><FONT face="Times New Roman">d</FONT></EM>′<FONT face="Times New Roman"><SUB>2</SUB></FONT>和<FONT face="Times New Roman"><EM>d</EM><SUB>2</SUB></FONT>在<FONT face="Times New Roman">o<SUB>t1</SUB></FONT>系中的坐标分别为<EM><FONT face="Times New Roman">d</FONT></EM>′<FONT face="Times New Roman"><SUB>2</SUB></FONT>（<FONT face="Times New Roman">6.628252</FONT>，－<FONT face="Times New Roman">7.008753</FONT>）和<FONT face="Times New Roman"><EM>d</EM><SUB>2</SUB></FONT>（－<FONT face="Times New Roman">8.32824</FONT>，－<FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">626718</FONT>）。<BR>　　为了精确求解滚刀刃形上的分离段曲线，必须先求出与工件相啮合的工件端面齿条上的分离段曲线。显然，图<FONT face="Times New Roman">4</FONT>上的<FONT face="Times New Roman"><EM>d</EM><SUB>2</SUB><EM>d</EM></FONT>′<FONT face="Times New Roman"><SUB>2</SUB></FONT>曲线是由工件相对齿条作啮合运动，工件齿形上尖点<EM><FONT face="Times New Roman">d</FONT></EM>的运动在齿条齿形上形成的轨迹。</P>
<P align=center><IMG src="http://www.chmcw.com/upload/news/RCL/13220_xfwpxi200835155531.gif"></P>
<P align=center><STRONG>图<FONT face="Times New Roman">4</FONT>　工件端面齿条齿形</STRONG></P>
<P align=left>　　由图<FONT face="Times New Roman">5</FONT>可知，工件与齿条的啮合相当于工件节圆在齿条节线上作纯滚动，当工件由<FONT face="Times New Roman">o</FONT>点滚到<FONT face="Times New Roman">o</FONT>′点时，<EM><FONT face="Times New Roman">d</FONT></EM>点在<FONT face="Times New Roman">o<SUB>t1</SUB></FONT>系中的运动轨迹方程为</P>
<P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_6hgrub200835155550.gif"></P>
<P>式中　<EM>ρ</EM>＝<FONT face="Times New Roman">0</FONT>，<FONT face="Times New Roman"><EM>d</EM><SUB>2</SUB></FONT>＝<FONT face="Times New Roman">29</FONT>．<FONT face="Times New Roman">925</FONT>，<EM>φ</EM><FONT face="Times New Roman"><SUB>4</SUB></FONT>＝<FONT face="Times New Roman">15°53</FONT>′<FONT face="Times New Roman">64</FONT>″<BR>　　由式（<FONT face="Times New Roman">1</FONT>）求得<FONT face="Times New Roman"><EM>d</EM><SUB>2</SUB></FONT>和<EM><FONT face="Times New Roman">d</FONT></EM>′<FONT face="Times New Roman"><SUB>2</SUB></FONT>点坐标，并将其代入式（<FONT face="Times New Roman">2</FONT>），即可求得<EM>θ</EM><FONT face="Times New Roman"><SUB>1</SUB></FONT>对应于<EM><FONT face="Times New Roman">d</FONT></EM>′<FONT face="Times New Roman"><SUB>2</SUB></FONT>和<FONT face="Times New Roman"><EM>d</EM><SUB>2</SUB></FONT>点的值，它们分别为<FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">40459</FONT>和－<FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">1329434</FONT>（<FONT face="Times New Roman">rad</FONT>）。故对应于<FONT face="Times New Roman"><EM>d</EM><SUB>2</SUB><EM>d</EM></FONT>′<FONT face="Times New Roman"><SUB>2</SUB></FONT>曲线，方程中<EM>θ</EM><FONT face="Times New Roman"><SUB>1</SUB></FONT>的取值范围为－<FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">1329434</FONT>≤<EM>θ</EM><FONT face="Times New Roman"><SUB>1</SUB></FONT>≤<FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">40459</FONT>。<BR>　　为了求出与齿条上<FONT face="Times New Roman"><EM>d</EM><SUB>2</SUB><EM>d</EM></FONT>′<FONT face="Times New Roman"><SUB>2</SUB></FONT>曲线相啮合的滚刀刃形上的分离段廓形<FONT face="Times New Roman"><EM>d</EM><SUB>2</SUB><EM>d</EM></FONT>′<FONT face="Times New Roman"><SUB>2</SUB></FONT>，必须先求出齿条在滚刀蜗杆端面的方程。<BR>　　由图<FONT face="Times New Roman">6</FONT>中的几何关系可知，齿条的法向坐标与齿条在工件端面坐标之间的关系为<BR>　　<IMG src="http://www.chmcw.com/upload/news/RCL/13220_qrbxhh200835155558.gif"></P>
<P align=center><IMG src="http://www.chmcw.com/upload/news/RCL/13220_3jzejd200835155613.gif"></P>
<P align=center><STRONG>图<FONT face="Times New Roman">5</FONT>　工件端面齿条形上<FONT face="Times New Roman"><EM>d</EM><SUB>1</SUB><EM>d</EM><SUB>2</SUB></FONT>曲线的形成</STRONG></P>
<P align=left>将<FONT face="Times New Roman"><EM>x</EM><SUB>tn</SUB></FONT>、<FONT face="Times New Roman"><EM>y</EM><SUB>tn</SUB></FONT>换算到滚刀蜗杆的端剖面得</P>
<P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_y3izsc200835155630.gif"></P>
<P align=left>式中<EM>β</EM><FONT face="Times New Roman"><SUB>1</SUB></FONT>、<EM>β</EM><FONT face="Times New Roman"><SUB>2</SUB></FONT>分别为滚刀蜗杆<FONT face="Times New Roman">1</FONT>与工件<FONT face="Times New Roman">2</FONT>在其节圆柱上的螺旋角。</P>
<P align=center><IMG src="http://www.chmcw.com/upload/news/RCL/13220_oqpjhv200835155647.gif"></P>
<P align=center><STRONG>图<FONT face="Times New Roman">6</FONT>　公共齿条在工件端面、滚刀端面及法剖面截形</STRONG></P>
<P align=left>　　由图<FONT face="Times New Roman">3</FONT>可知，当齿条上<EM><FONT face="Times New Roman">M</FONT></EM>′（<EM><FONT face="Times New Roman">x</FONT></EM>，<EM><FONT face="Times New Roman">y</FONT></EM>）点进入啮合时，按齿形法线定理，则过<EM><FONT face="Times New Roman">M</FONT></EM>′点处的齿形法线应通过啮合节点<EM><FONT face="Times New Roman">P</FONT></EM>，故<EM><FONT face="Times New Roman">M</FONT></EM>′点处的法线方程在<FONT face="Times New Roman"><EM>O</EM><SUB>t</SUB></FONT>系中为<BR>　　（<FONT face="Times New Roman"><EM>X</EM><SUB>t</SUB></FONT>－<FONT face="Times New Roman"><EM>x</EM><SUB>t</SUB></FONT>）<FONT face="Times New Roman">cos</FONT><EM>μ</EM><FONT face="Times New Roman"><SUB>t</SUB></FONT>＋（<FONT face="Times New Roman"><EM>Y</EM><SUB>t</SUB></FONT>－<FONT face="Times New Roman"><EM>y</EM><SUB>t</SUB></FONT>）<FONT face="Times New Roman">sin</FONT><EM>μ</EM><FONT face="Times New Roman"><SUB>t</SUB></FONT>＝<FONT face="Times New Roman">0<BR></FONT>式中　<FONT face="Times New Roman"><EM>X</EM><SUB>t</SUB></FONT>，<FONT face="Times New Roman"><EM>Y</EM><SUB>t</SUB></FONT>——过<EM><FONT face="Times New Roman">M</FONT></EM>′点齿形法线上任意点的坐标<BR>　　将<EM><FONT face="Times New Roman">P</FONT></EM>点在<FONT face="Times New Roman">o<SUB>t</SUB></FONT>系中的坐标（<FONT face="Times New Roman"><EM>r</EM><SUB>1</SUB></FONT><EM>φ</EM><FONT face="Times New Roman"><SUB>1</SUB></FONT>，<FONT face="Times New Roman">o</FONT>）代入法线方程得</P>
<P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_rqtlrw20083515577.gif"></P>
<P>　　从<FONT face="Times New Roman"><EM>o</EM><SUB>t</SUB></FONT>系到<FONT face="Times New Roman"><EM>o</EM><SUB>3</SUB></FONT>系的变换式为</P>
<P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_sn5nac200835155720.gif"></P>
<P align=left>联解式（<FONT face="Times New Roman">4</FONT>）、（<FONT face="Times New Roman">5</FONT>），可求得滚刀的端面刃形。求得滚刀蜗杆的端面刃形方程后，令其绕滚刀蜗杆的轴线作螺旋运动，即可得到滚刀蜗杆的齿面方程式。<BR>　　如图<FONT face="Times New Roman">7</FONT>所示，设与滚刀蜗杆固联的辅助坐标系为<EM><FONT face="Times New Roman">o</FONT></EM>′<FONT face="Times New Roman"><SUB>3</SUB><EM>x</EM></FONT>′<FONT face="Times New Roman"><SUB>3</SUB><EM>y</EM></FONT>′<FONT face="Times New Roman"><SUB>3</SUB><EM>z</EM></FONT>′<FONT face="Times New Roman"><SUB>3</SUB></FONT>，在初始位置时，原点<FONT face="Times New Roman"><EM>o</EM><SUB>3</SUB></FONT>与<EM><FONT face="Times New Roman">o</FONT></EM>′<FONT face="Times New Roman"><SUB>3</SUB></FONT>重合，<FONT face="Times New Roman"><EM>x</EM><SUB>3</SUB></FONT>、<FONT face="Times New Roman"><EM>y</EM><SUB>3</SUB></FONT>轴分别与<EM><FONT face="Times New Roman">x</FONT></EM>′<FONT face="Times New Roman"><SUB>3</SUB></FONT>、<EM><FONT face="Times New Roman">y</FONT></EM>′<FONT face="Times New Roman"><SUB>3</SUB></FONT>轴重合。使滚刀蜗杆不动，然后令<FONT face="Times New Roman"><EM>o</EM><SUB>3</SUB></FONT>系及与其固联的滚刀端面刃形一起绕<EM><FONT face="Times New Roman">z</FONT></EM>′<FONT face="Times New Roman"><SUB>3</SUB></FONT>轴作螺旋参数<FONT face="Times New Roman"><EM>P</EM><SUB>1</SUB></FONT>的螺旋运动，从而形成螺杆的螺旋齿面，由<FONT face="Times New Roman"><EM>o</EM><SUB>3</SUB></FONT>系到<EM><FONT face="Times New Roman">o</FONT></EM>′<FONT face="Times New Roman"><SUB>3</SUB></FONT>系的坐标变换式为</P>
<P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_bs2cwz200835155733.gif"></P>
<P align=left>　　上式即为滚刀蜗杆的齿面方程式，令<FONT face="Times New Roman">x</FONT>′<FONT face="Times New Roman"><SUB>3</SUB></FONT>＝<FONT face="Times New Roman">0</FONT>，即可得到滚刀的轴向刃形为</P>
<P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_cs2e1l200835155749.gif"></P>
<P align=left>　　联解式（<FONT face="Times New Roman">5</FONT>）、（<FONT face="Times New Roman">6</FONT>），即可得到滚刀轴剖面上分离廓形的坐标点。</P>
<P align=center><IMG src="http://www.chmcw.com/upload/news/RCL/13220_ei1xmi200835155754.gif"></P>
<P align=center><STRONG>图<FONT face="Times New Roman">7</FONT>　滚刀蜗杆端面及轴向刃形</STRONG></P>
<P><STRONG>　　<FONT face="Times New Roman">3</FONT>．结论</STRONG></P>
<P>　　通过无锡压缩机股份公司计量处的检测验证，本文提出的滚刀分离部分廓形计算原理正确，保证了尖点<EM><FONT face="Times New Roman">d</FONT></EM>不致被切掉。通过一次性走刀即可包络加工出包括尖点在内的四段曲线。</P>
               

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