一种高精度的圆分度测量原理
<STRONG><FONT face="Times New Roman">1</FONT></STRONG>．<STRONG>引言</STRONG><P> 在圆分度测量系统中，目前多采用平均读数原理来提高测量精度，其基本方法是通过在度盘圆周上均布多个读数头，利用平均读数有规律地消除读数中的部分谐波误差。但这种方法存在一定局限性，一是读数中仍然残留读数头个数整数倍的读数谐波误差，使测量精度难以进一步提高（尤其当读数头个数较少时）；二是应用平均读数原理提高圆分度测量系统精度必须建立在多个读数头的特性完全一致的前提下，这就必然增加圆分度测量系统的制造难度和制造成本（尤其当读数头个数较多时）。<BR> 本文提出一种可有效提高圆分度测量系统精度的新的测量原理。实际测量证明，应用本原理的圆分度测量系统精度高于采用平均读数原理的圆分度测量系统，同时还具有测量系统结构简单、测量易于实现等优点。<BR> <STRONG><FONT face="Times New Roman">2</FONT>．基本原理</STRONG></P>
<P> 应用本测量原理的圆分度测量系统如图<FONT face="Times New Roman">1</FONT>所示。被测度盘<FONT face="Times New Roman">2</FONT>与标准度盘<FONT face="Times New Roman">1</FONT>同轴安装，它们可绕共同的回转中心<EM><FONT face="Times New Roman">O</FONT></EM>作同步转动或相对转动，读数头<FONT face="Times New Roman">3</FONT>、<FONT face="Times New Roman">4</FONT>分别对标准度盘进行细分读数，瞄准显微镜<FONT face="Times New Roman">5</FONT>实现对被测度盘的瞄准。测量时，首先相对移动两度盘，使瞄准显微镜瞄准被测度盘的“<FONT face="Times New Roman">0</FONT>”刻线时读数<FONT face="Times New Roman">4</FONT>的读数为<FONT face="Times New Roman">0</FONT>，并设定此时主轴回转角度为<FONT face="Times New Roman">0</FONT>。当主轴回转角度为<EM>θ</EM>时，瞄准显微镜瞄准被测度盘的<EM>θ</EM><FONT face="Times New Roman"><SUB>i</SUB></FONT>刻线，读数头<FONT face="Times New Roman">4</FONT>、<FONT face="Times New Roman">3</FONT>的读数分别为<FONT face="Times New Roman"><EM>a</EM><SUB>i</SUB></FONT>，<FONT face="Times New Roman"><EM>b</EM><SUB>i</SUB></FONT>，在圆周上测得<EM><FONT face="Times New Roman">N</FONT></EM>组数据（<EM>θ</EM><FONT face="Times New Roman"><SUB>i</SUB></FONT>，<FONT face="Times New Roman"><EM>a</EM><SUB>i</SUB></FONT>，<FONT face="Times New Roman"><EM>b</EM><SUB>i</SUB></FONT>）（<EM><FONT face="Times New Roman">i</FONT></EM>＝<FONT face="Times New Roman">0</FONT>，<FONT face="Times New Roman">1</FONT>，<FONT face="Times New Roman">2</FONT>，…，<FONT face="Times New Roman"><EM>N</EM>-1</FONT>）。</P>
<P align=center><IMG src="http://www.chmcw.com/upload/news/RCL/13220_e92hdq200841916852.gif"></P>
<P align=center>1．标准度盘 2．被测度盘 <BR>3，4．读数头 5．瞄准显微镜<BR><STRONG>图</STRONG></P>
<P align=left> 圆分度测量系统存在三种主要误差：标准度盘的刻划误差、主轴晃动误差和度盘的安装偏心误差，它们均会引起读数头的读数误差，所以读数头<FONT face="Times New Roman">3</FONT>、<FONT face="Times New Roman">4</FONT>的读数中实际包含了被测度盘的刻划误差<EM>Δ</EM><FONT face="Times New Roman"><SUB>i</SUB></FONT>和上述三种主要误差引起的读数误差，其中读数误差是一以<FONT face="Times New Roman">2</FONT><EM>π</EM>为周期的函数误差。若以<EM><FONT face="Times New Roman">f</FONT></EM>（<EM>θ</EM>）表示此周期函数误差，则有</P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_hw9zhk200841916932.gif"> （<FONT face="Times New Roman">1</FONT>）
<P align=left>则读数头<FONT face="Times New Roman">4</FONT>的读数为</P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_zqjxot200841916947.gif"> （<FONT face="Times New Roman">2</FONT>）
<P align=left>若测量过程中按顺时针方向转动，由于读数头<FONT face="Times New Roman">3</FONT>的位置与读数头<FONT face="Times New Roman">4</FONT>相差<EM>β</EM>角，故读数头<FONT face="Times New Roman">3</FONT>的读数同样包含读数误差，只是在相位上超前了<EM>β</EM>角，则有</P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_zfyqdo200841916100.gif"> （<FONT face="Times New Roman">3</FONT>）
<P align=left> 在读数信号（<FONT face="Times New Roman"><EM>a</EM><SUB>i</SUB></FONT>，<FONT face="Times New Roman"><EM>b</EM><SUB>i</SUB></FONT>）中，不能直接分离出被测度盘的刻划误差<EM>Δ</EM><FONT face="Times New Roman"><SUB>i</SUB></FONT>，如果将读数头<FONT face="Times New Roman">3</FONT>的读数减去读数头<FONT face="Times New Roman">4</FONT>的读数，得到综合信号<FONT face="Times New Roman"><EM>d</EM><SUB>i</SUB></FONT>为</P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_md1npk2008419161014.gif"> （<FONT face="Times New Roman">4</FONT>）
<P align=left>经过适当处理后有</P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_9gzpom2008419161030.gif"> （<FONT face="Times New Roman">5</FONT>）
<P align=left>综合信号<FONT face="Times New Roman"><EM>d</EM><SUB>i</SUB></FONT>消除了被测度盘的刻划误差<EM>Δ</EM><FONT face="Times New Roman"><SUB>i</SUB></FONT>，只包含测量系统的读数误差。<FONT face="Times New Roman"><EM>d</EM><SUB>i</SUB></FONT>也是一周期为<FONT face="Times New Roman">2</FONT><EM>π</EM>的谐波函数信号，故可应用信号分析技术对<FONT face="Times New Roman"><EM>d</EM><SUB>i</SUB></FONT>进行谱分析。若圆周上的读数个数为<EM><FONT face="Times New Roman">N</FONT></EM>时，利用<FONT face="Times New Roman">FT</FONT>分析可分离出<FONT face="Times New Roman"><EM>d</EM><SUB>i</SUB></FONT>中的前<EM><FONT face="Times New Roman">N</FONT></EM>／<FONT face="Times New Roman">2</FONT>阶谐波误差，即</P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_hd3ohl2008419161044.gif"> （<FONT face="Times New Roman">6</FONT>）
<P align=left>从而可得到前<EM><FONT face="Times New Roman">N</FONT></EM>／<FONT face="Times New Roman">2</FONT>阶谐波的幅值和相位（<FONT face="Times New Roman"><EM>D</EM><SUB>k</SUB></FONT>，<EM>ψ</EM><FONT face="Times New Roman"><SUB>k</SUB></FONT>）。比较式（<FONT face="Times New Roman">5</FONT>）和式（<FONT face="Times New Roman">6</FONT>），可得读数头的读数谐波误差信号的幅值和相位（<FONT face="Times New Roman"><EM>A</EM><SUB>k</SUB></FONT>，<EM>φ</EM><FONT face="Times New Roman"><SUB>k</SUB></FONT>）与（<FONT face="Times New Roman"><EM>D</EM><SUB>k</SUB></FONT>，<EM>ψ</EM><FONT face="Times New Roman"><SUB>k</SUB></FONT>）的关系为</P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_8ykgh62008419161057.gif"> （<FONT face="Times New Roman">7</FONT>）
<P align=left>则被测度盘的刻划误差为</P><IMG src="http://www.chmcw.com/upload/news/RCL/13220_kldnru2008419161121.gif"> （<FONT face="Times New Roman">8</FONT>）
<P align=left> 式（<FONT face="Times New Roman">8</FONT>）即为被测度盘的测量结果。为了有效地分离出读数误差中的前<EM><FONT face="Times New Roman">N</FONT></EM>／<FONT face="Times New Roman">2</FONT>阶谐波误差，必须合理确定读数头<FONT face="Times New Roman">3</FONT>、<FONT face="Times New Roman">4</FONT>之间的夹角<EM>β</EM>。<EM>β</EM>的确定原则是尽可能避免<EM><FONT face="Times New Roman">k</FONT>β</EM>／<FONT face="Times New Roman">2</FONT>＝<EM><FONT face="Times New Roman">j</FONT>π</EM>（<EM><FONT face="Times New Roman">k</FONT></EM>＝<FONT face="Times New Roman">1</FONT>，<FONT face="Times New Roman">2</FONT>，…，<EM><FONT face="Times New Roman">N</FONT></EM>／<FONT face="Times New Roman">2</FONT>，<EM><FONT face="Times New Roman">j</FONT></EM>为整数）。见式（<FONT face="Times New Roman">5</FONT>），当<EM><FONT face="Times New Roman">k</FONT>β</EM>／<FONT face="Times New Roman">2<EM>j</FONT>π</EM>时，<FONT face="Times New Roman">sin</FONT>（<EM><FONT face="Times New Roman">k</FONT>β</EM>／<FONT face="Times New Roman">2</FONT>）＝<FONT face="Times New Roman">0</FONT>，信号<FONT face="Times New Roman"><EM>d</EM><SUB>i</SUB></FONT>中将不包含第<EM><FONT face="Times New Roman">k</FONT></EM>阶谐波误差，对<FONT face="Times New Roman"><EM>d</EM><SUB>i</SUB></FONT>进行<FONT face="Times New Roman">FT</FONT>分析时就不能分离出第<EM><FONT face="Times New Roman">k</FONT></EM>阶谐波误差信号的幅值和相位（<FONT face="Times New Roman"><EM>A</EM><SUB>k</SUB></FONT>，<EM>φ</EM><FONT face="Times New Roman"><SUB>k</SUB></FONT>），则在式（<FONT face="Times New Roman">8</FONT>）中将包含读数误差的第<EM><FONT face="Times New Roman">k</FONT></EM>阶谐波误差的影响，使测量结果的精度下降。例如，当<EM><FONT face="Times New Roman">N</FONT></EM>＝<FONT face="Times New Roman">24</FONT>时，可取<EM>β</EM>＝<FONT face="Times New Roman">56°</FONT>或<EM>β</EM>＝<FONT face="Times New Roman">62°</FONT>，但切不可取<EM>β</EM>＝<FONT face="Times New Roman">60°</FONT>，因为当<EM><FONT face="Times New Roman">k</FONT></EM>＝<FONT face="Times New Roman">6</FONT>时，<EM><FONT face="Times New Roman">k</FONT>β</EM>／<FONT face="Times New Roman">2</FONT>＝<FONT face="Times New Roman">180°</FONT>，<FONT face="Times New Roman"><EM>d</EM><SUB>i</SUB></FONT>中将不含第<FONT face="Times New Roman">6</FONT>阶谐波信号，在式（<FONT face="Times New Roman">8</FONT>）中只能消除<FONT face="Times New Roman">1</FONT>～<FONT face="Times New Roman">5</FONT>阶和<FONT face="Times New Roman">7</FONT>～<FONT face="Times New Roman">12</FONT>阶谐波误差，残留了第<FONT face="Times New Roman">6</FONT>阶谐波误差，从而影响测量结果的精度。</P>
<P align=left> <STRONG><FONT face="Times New Roman">3</FONT>．数据处理流程和测量结果比对</STRONG></P>
<P align=left> 依据上述原理，数据处理流程如下<BR> （<FONT face="Times New Roman">1</FONT>）在圆周上获得<FONT face="Times New Roman">N</FONT>组测量数据（<EM>θ</EM><FONT face="Times New Roman"><SUB>i</SUB></FONT>，<FONT face="Times New Roman"><EM>a</EM><SUB>i</SUB></FONT>，<FONT face="Times New Roman"><EM>b</EM><SUB>i</SUB></FONT>）（<EM><FONT face="Times New Roman">i</FONT></EM>＝<FONT face="Times New Roman">0</FONT>，<FONT face="Times New Roman">1</FONT>，<FONT face="Times New Roman">2</FONT>，…，<FONT face="Times New Roman"><EM>N</EM>-1</FONT>）；<BR> （<FONT face="Times New Roman">2</FONT>）求出综合信号<FONT face="Times New Roman"><EM>d</EM><SUB>i</SUB></FONT>＝<FONT face="Times New Roman"><EM>b</EM><SUB>i</SUB></FONT>－<FONT face="Times New Roman"><EM>a</EM><SUB>i</SUB></FONT>（<EM><FONT face="Times New Roman">i</FONT></EM>＝<FONT face="Times New Roman">0</FONT>，<FONT face="Times New Roman">1</FONT>，<FONT face="Times New Roman">2</FONT>，…，<FONT face="Times New Roman"><EM>N</EM>-1</FONT>）；<BR> （<FONT face="Times New Roman">3</FONT>）调用<FONT face="Times New Roman">FT</FONT>谱分析软件，获得综合信号<FONT face="Times New Roman"><EM>d</EM><SUB>i</SUB></FONT>中前<EM><FONT face="Times New Roman">N</FONT></EM>／<FONT face="Times New Roman">2</FONT>阶谐波的幅值和相位（<FONT face="Times New Roman"><EM>D</EM><SUB>k</SUB></FONT>，<EM>ψ</EM><FONT face="Times New Roman"><SUB>k</SUB></FONT>）（<EM><FONT face="Times New Roman">k</FONT></EM>＝<FONT face="Times New Roman">1</FONT>，<FONT face="Times New Roman">2</FONT>，…，<EM><FONT face="Times New Roman">N</FONT></EM>／<FONT face="Times New Roman">2</FONT>）；<BR> （<FONT face="Times New Roman">4</FONT>）按式（<FONT face="Times New Roman">7</FONT>）分离出读数谐波误差的幅值和相位（<FONT face="Times New Roman"><EM>A</EM><SUB>k</SUB></FONT>，<EM>φ</EM><FONT face="Times New Roman"><SUB>k</SUB></FONT>）（<EM><FONT face="Times New Roman">k</FONT></EM>＝<FONT face="Times New Roman">1</FONT>，<FONT face="Times New Roman">2</FONT>，…，<EM><FONT face="Times New Roman">N</FONT></EM>／<FONT face="Times New Roman">2</FONT>）；<BR> （<FONT face="Times New Roman">5</FONT>）按式（<FONT face="Times New Roman">8</FONT>）修正读数头<FONT face="Times New Roman">4</FONT>的读数中的前<EM><FONT face="Times New Roman">N</FONT></EM>／<FONT face="Times New Roman">2</FONT>阶谐波误差，得到被测度盘的刻划误差值<EM>Δ</EM><FONT face="Times New Roman"><SUB>i</SUB></FONT>（<EM><FONT face="Times New Roman">i</FONT></EM>＝<FONT face="Times New Roman">0</FONT>，<FONT face="Times New Roman">1</FONT>，<FONT face="Times New Roman">2</FONT>，…，<FONT face="Times New Roman"><EM>N</EM>-1</FONT>）；<BR> （<FONT face="Times New Roman">6</FONT>）被测度盘的零起刻线误差为<IMG src="http://www.chmcw.com/upload/news/RCL/13220_juhpjc2008419161152.gif"><FONT face="Times New Roman"><SUB>i</SUB></FONT>＝<EM>Δ</EM><FONT face="Times New Roman"><SUB>i</SUB></FONT>－<EM>Δ</EM><FONT face="Times New Roman"><SUB>0</SUB></FONT>（<EM><FONT face="Times New Roman">i</FONT></EM>＝<FONT face="Times New Roman">0</FONT>，<FONT face="Times New Roman">1</FONT>，<FONT face="Times New Roman">2</FONT>，…，<FONT face="Times New Roman"><EM>N</EM>-1</FONT>），最大间距误差为<EM><FONT face="Times New Roman">f</FONT></EM>＝<FONT face="Times New Roman">max</FONT>（<EM>Δ</EM><FONT face="Times New Roman"><SUB>i</SUB></FONT>）－<FONT face="Times New Roman">min</FONT>（<EM>Δ</EM><FONT face="Times New Roman"><SUB>i</SUB></FONT>）（<EM><FONT face="Times New Roman">i</FONT></EM>＝<FONT face="Times New Roman">0</FONT>，<FONT face="Times New Roman">1</FONT>，<FONT face="Times New Roman">2</FONT>，…，<FONT face="Times New Roman"><EM>N</EM>-1</FONT>）。<BR> 为验证本文提出的圆分度测量原理分离和修正测量系统读数谐波误差的有效性，笔者分别进行了采用本测量原理和采用平均读数原理的模拟测量计算，并对二者进行比较。设对一已知零起刻线误差真值的标准度盘进行测量，给定测量系统读数误差的各阶谐波幅值和相位（<FONT face="Times New Roman"><EM>A</EM><SUB>k</SUB></FONT>，<EM>φ</EM><FONT face="Times New Roman"><SUB>k</SUB></FONT>）（<EM><FONT face="Times New Roman">k</FONT></EM>＝<FONT face="Times New Roman">0</FONT>，<FONT face="Times New Roman">1</FONT>，<FONT face="Times New Roman">2</FONT>，…，<FONT face="Times New Roman">17</FONT>），对度盘圆周上均布的<EM><FONT face="Times New Roman">N</FONT></EM>＝<FONT face="Times New Roman">24</FONT>个测点进行测量。采用本测量原理的两读数头<FONT face="Times New Roman">3</FONT>、<FONT face="Times New Roman">4</FONT>之间的夹角<EM>β</EM>＝<FONT face="Times New Roman">56°</FONT>。应用平均读数原理的模拟测量采用圆周上均布<FONT face="Times New Roman">5</FONT>个读数头的读数方式。读数头<FONT face="Times New Roman">4</FONT>、<FONT face="Times New Roman">3</FONT>的读数值（<FONT face="Times New Roman"><EM>a</EM><SUB>i</SUB></FONT>，<FONT face="Times New Roman"><EM>b</EM><SUB>i</SUB></FONT>）、被测标准度盘零起刻线误差真值<IMG src="http://www.chmcw.com/upload/news/RCL/13220_juhpjc2008419161152.gif"><FONT face="Times New Roman"><SUB>io</SUB></FONT>以及采用两种原理的误差计算结果列于下表（因篇幅所限，未列出平均读数原理<FONT face="Times New Roman">5</FONT>个读数头的读数值）。<BR> 由表可知，与采用平均读数原理的测量结果<IMG src="http://www.chmcw.com/upload/news/RCL/13220_juhpjc2008419161152.gif"><FONT face="Times New Roman">pi</FONT>相比，采用本测量原理得出的测量结果<IMG src="http://www.chmcw.com/upload/news/RCL/13220_juhpjc2008419161152.gif"><FONT face="Times New Roman"><SUB>i</SUB></FONT>更接近于被测度盘零起刻线误差真值<IMG src="http://www.chmcw.com/upload/news/RCL/13220_juhpjc2008419161152.gif"><FONT face="Times New Roman"><SUB>oi</SUB></FONT>，其最大间距误差结果也更接近于被测度盘真值。由此可见，本测量原理能有效分离和修正圆分度测量系统的读数谐波误差，其测量精度高于均布<FONT face="Times New Roman">5</FONT>个读数头的平均读数原理测量方法。理论上，本测量原理能消除全部前<EM><FONT face="Times New Roman">N</FONT></EM>／<FONT face="Times New Roman">2</FONT>＝<FONT face="Times New Roman">12</FONT>阶谐波读数误差，而平均读数原理只能消除非读数头个数整数倍的各阶谐波读数误差（<FONT face="Times New Roman">1</FONT>、<FONT face="Times New Roman">2</FONT>、<FONT face="Times New Roman">3</FONT>、<FONT face="Times New Roman">4</FONT>、<FONT face="Times New Roman">6</FONT>、<FONT face="Times New Roman">7</FONT>、<FONT face="Times New Roman">8</FONT>、<FONT face="Times New Roman">9</FONT>、<FONT face="Times New Roman">11</FONT>、<FONT face="Times New Roman">12</FONT>阶），而残留了读数头个数整数倍的各阶谐波读数误差（<FONT face="Times New Roman">5</FONT>、<FONT face="Times New Roman">10</FONT>阶）。由于随着谐波阶数的增高（<EM><FONT face="Times New Roman">k</FONT></EM>＞<FONT face="Times New Roman">15</FONT>），读数谐波误差的幅值将减小到<FONT face="Times New Roman">10</FONT><SUP>－<FONT face="Times New Roman">2</FONT></SUP>秒数量级，因此可认为本测量原理基本上可以完全消除读数谐波误差。</P>
<P align=right><STRONG>表 原始数据和计算结果</STRONG> <STRONG> </STRONG> <STRONG> </STRONG>（单位：秒）</P>
<CENTER>
<TABLE border=1>
<TBODY>
<TR>
<TD align=middle rowSpan=2><FONT size=2>序号</FONT></TD>
<TD align=middle rowSpan=2><FONT size=2>度盘零起刻线<BR>误差真值<IMG src="http://www.chmcw.com/upload/news/RCL/13220_juhpjc2008419161152.gif"><FONT face="Times New Roman"><SUB>oi</SUB></FONT></FONT></TD>
<TD align=middle colSpan=2><FONT size=2>原始数据</FONT></TD>
<TD align=middle colSpan=2><FONT size=2>度盘零起刻线误差计算结果</FONT></TD></TR>
<TR>
<TD align=middle><FONT size=2>读数头<FONT face="Times New Roman">4</FONT>读数值<FONT face="Times New Roman"><EM>a</EM><SUB>i</SUB></FONT></FONT></TD>
<TD align=middle><FONT size=2>读数头<FONT face="Times New Roman">3</FONT>读数值<FONT face="Times New Roman"><EM>b</EM><SUB>i</SUB></FONT></FONT></TD>
<TD align=middle><FONT size=2>本测量原理<IMG src="http://www.chmcw.com/upload/news/RCL/13220_juhpjc2008419161152.gif"><FONT face="Times New Roman"><SUB>i</SUB></FONT></FONT></TD>
<TD align=middle><FONT size=2>平均读数原理<IMG src="http://www.chmcw.com/upload/news/RCL/13220_juhpjc2008419161152.gif"><FONT face="Times New Roman"><SUB>pi</SUB></FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>0</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">00</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">00</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">90</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">00</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">00</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>1</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">50</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">96</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">92</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">11</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">94</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>2</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">20</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">89</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">94</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">03</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">53</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>3</FONT></TD>
<TD align=middle><FONT size=2>（<FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">70</FONT>）</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">96</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">45</FONT></FONT></TD>
<TD align=middle><FONT size=2>（<FONT face="Times New Roman">-1</FONT>．<FONT face="Times New Roman">19</FONT>）</FONT></TD>
<TD align=middle><FONT size=2>（<FONT face="Times New Roman">-1</FONT>．<FONT face="Times New Roman">93</FONT>）</FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>4</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">00</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">48</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">09</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">84</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">95</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>5</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">80</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">20</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">3</FONT>．<FONT face="Times New Roman">65</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">20</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">85</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>6</FONT></TD>
<TD align=middle><FONT size=2>（<FONT face="Times New Roman">3</FONT>．<FONT face="Times New Roman">60</FONT>）</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">5</FONT>．<FONT face="Times New Roman">58</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">5</FONT>．<FONT face="Times New Roman">08</FONT></FONT></TD>
<TD align=middle><FONT size=2>（<FONT face="Times New Roman">3</FONT>．<FONT face="Times New Roman">62</FONT>）</FONT></TD>
<TD align=middle><FONT size=2>（<FONT face="Times New Roman">4</FONT>．<FONT face="Times New Roman">08</FONT>）</FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>7</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">60</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">11</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">93</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">79</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">48</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>8</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">3</FONT>．<FONT face="Times New Roman">20</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">3</FONT>．<FONT face="Times New Roman">90</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">95</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">3</FONT>．<FONT face="Times New Roman">35</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">32</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>9</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">80</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">69</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-1</FONT>．<FONT face="Times New Roman">43</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">03</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">76</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>10</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">90</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">92</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">55</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">84</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">03</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>11</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">20</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">16</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">41</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">65</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">61</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>12</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">60</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-3</FONT>．<FONT face="Times New Roman">19</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">48</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">45</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">15</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>13</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">40</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">02</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">77</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">87</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">87</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>14</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">20</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">3</FONT>．<FONT face="Times New Roman">28</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">32</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">96</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">16</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>15</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">80</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">97</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">3</FONT>．<FONT face="Times New Roman">63</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">48</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">03</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>16</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">60</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">22</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">3</FONT>．<FONT face="Times New Roman">43</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">16</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">78</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>17</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">50</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">30</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">42</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">21</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">08</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>18</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">20</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">16</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">72</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">53</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">47</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>19</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">40</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">3</FONT>．<FONT face="Times New Roman">53</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">90</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">01</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">37</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>20</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">70</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">1</FONT>．<FONT face="Times New Roman">89</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-1</FONT>．<FONT face="Times New Roman">27</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-1</FONT>．<FONT face="Times New Roman">00</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">36</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>21</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">50</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">44</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">44</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">95</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-0</FONT>．<FONT face="Times New Roman">70</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>22</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">20</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">21</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">2</FONT>．<FONT face="Times New Roman">65</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">08</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">-1</FONT>．<FONT face="Times New Roman">29</FONT></FONT></TD></TR>
<TR>
<TD align=middle><FONT face="Times New Roman" size=2>23</FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">80</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">77</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">62</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">11</FONT></FONT></TD>
<TD align=middle><FONT size=2><FONT face="Times New Roman">0</FONT>．<FONT face="Times New Roman">68</FONT></FONT></TD></TR></TBODY></TABLE></CENTER>
<TABLE width="90%" border=0>
<TBODY>
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<TD><STRONG>
<P><BR><FONT size=2> <FONT face="Times New Roman">4</FONT>．有关问题的讨论</FONT></P>
<P></STRONG><FONT size=2> 本测量原理能有效提高圆分度测量系统精度的关键在于谱分析技术，其提高测量精度的有效性取决于频谱分析的精度。众所周知，在频谱分析中存在频率混叠问题。当圆周测量点个数<FONT face="Times New Roman">N</FONT>较小时，<FONT face="Times New Roman">FT</FONT>分析只能分离出较少阶次的谐波信号，读数信号中实际存在的其余高阶谐波将使分离出的前<EM><FONT face="Times New Roman">N</FONT></EM>／<FONT face="Times New Roman">2</FONT>阶谐波信号失真，这种失真将随着测量点个数的减小和实际信号中包含的谐波成分阶次的增高而增大，从而使谐波读数误差的修正精度下降。通过适当增加测量点个数虽然可解决这一问题，但测量点太多将使噪声增大，同样会引起谐波读数误差的修正精度下降。原则上，解决频率混叠问题的根本方法是在进行<FONT face="Times New Roman">FT</FONT>分析之前进行数字低通滤波，因此，截止频率的确定和数字滤波器的设计将成为决定本测量原理提高圆分度测量系统精度有效性的关键。<BR> 为了能有效分离出前<EM><FONT face="Times New Roman">N</FONT></EM>／<FONT face="Times New Roman">2</FONT>阶谐波误差，在测量时必须针对不同的读数点个数适当调整两读数头之间的夹角<EM>β</EM>，<EM>β</EM>角的调整误差是否会对测量精度产生影响呢？如果两读数头间的夹角<EM>β</EM>有一调整误差<EM>Δβ</EM>，由式（<FONT face="Times New Roman">3</FONT>）～（<FONT face="Times New Roman">6</FONT>）可知，读数信号<FONT face="Times New Roman"><EM>b</EM><SUB>i</SUB></FONT>中会加入一个常量<EM>Δβ</EM>，综合信号<FONT face="Times New Roman"><EM>d</EM><SUB>i</SUB></FONT>中也会加入一个常量<EM>Δβ</EM>，在频谱分析时，该<EM>Δβ</EM>只会使综合信号中的<FONT face="Times New Roman">0</FONT>阶（直流）分量改变，并不会对各阶谐波信号的分离产生影响，故读数头夹角的调整误差并不会影响圆分度测量精度，这将有利于应用本测量原理的测量装置的实现。<BR> 本测量原理能有效分离和修正圆分度测量系统的读数谐波误差对测量精度的影响，但与平均读数原理一样，这并不意味着对主轴回转精度和被测度盘安装精度没有要求，在测量中，主轴径向晃动和被测度盘安装偏心引起的瞄准误差会对测量精度产生直接影响。但在测量多面棱体时，如果采用自准直仪对被测多面棱体进行瞄准，则主轴晃动不会对测量精度产生影响，此时可以认为采用本测量原理能够以较低精度的圆分度测量系统实现对高精度多面棱体的测量。<BR> 由于本测量原理能实时、有效地分离出测量系统中前几十阶具有较大幅值的谐波误差，并消除其对测量精度的影响，因此其实际意义在于降低了对标准度盘精度和标准度盘安装精度的要求，使圆分度测量系统的制造难度和制造成本下降，并使测量易于实现。<BR> 在角度测量中，本测量原理虽不能实时地消除测量系统的读数谐波误差对测量精度的影响，但可以在进行角度测量前先进行上述测量过程，求得测量系统的读数谐波误差后，再对角度测量结果进行读数谐波误差的修正，从而达到提高角度测量精度的目的。</FONT></P></TD></TR></TBODY></TABLE>
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