Bend effect on vibration pattern<088/title> </head> <body BACKGROUND="color_yellow.gif"> <div align="center"><center> <table CELLSPACING="0" BORDER="0" CELLPADDING="7" WIDTH="679"> <tr> <td WIDTH="40%" VALIGN="TOP" HEIGHT="2"><strong><font face="Times New Roman" size="4"><b>SELF</b></font></strong></td> <td WIDTH="60%" VALIGN="TOP" HEIGHT="2"><p ALIGN="RIGHT"><font face="Times New Roman">88</font></td> </tr> <tr> <td VALIGN="TOP" COLSPAN="2" HEIGHT="2"><i><font SIZE="4" COLOR="#008000"><p ALIGN="CENTER"></font><font face="Times New Roman" size="4" color="#008000"><em>S.B. Karavashkin, O.N. Karavashkina</em></font></i></td> </tr> <tr> <td VALIGN="TOP" COLSPAN="2" HEIGHT="2"><p ALIGN="CENTER"><font face="Times New Roman"><img SRC="line.gif" WIDTH="553" HEIGHT="3"></font></td> </tr> <tr> <td VALIGN="TOP" COLSPAN="2" HEIGHT="2"><p ALIGN="JUSTIFY" style="text-indent: 12pt; text-align: justify; font-family: Times New Roman; font-size: 12pt"><font face="Times New Roman" size="3">Having now exact analytical solutions for elastic lumped and distributed systems [20]–[24] obtained without any matrix methods and fully determined in reference to the studied parameters of elastic systems, we gain the possibility to find exact analytical solutions also for the lines having one or more bends. Theproof of theorem much simplifies the finding of solutions and the analysis of vibration pattern for a wide class of elastic systems. In this paper we will consider some of most typical examples that can be investigated in frames of the theorem validity. </font></p> <b><font SIZE="4" COLOR="#008000"><p ALIGN="JUSTIFY"></font><font face="Times New Roman" size="4" color="#008000"><strong>2. Theorem of bend influence on the pattern of processes </strong></font></b></p> <p ALIGN="JUSTIFY" style="text-align: justify; font-family: Times New Roman; font-size: 12pt"><font face="Times New Roman" size="3">To investigate elastic bended lines, let us first prove the general theorem determining the degree of bend effect on the solution pattern.</font></p> <p ALIGN="JUSTIFY" style="text-align: justify; font-family: Times New Roman; font-size: 12pt"><font face="Times New Roman" size="3"><font color="#C5A378">THEOREM 1.</font> <i>A bend in an elastic line does not effect on the vibration pattern only in case, when the transverse and longitudinal stiffness coefficients of its constraints are equal. </i></font></p> <p ALIGN="JUSTIFY" style="text-indent: 12pt; text-align: justify; font-family: Times New Roman; font-size: 12pt"><font face="Times New Roman" size="3">To prove this theorem, consider some elastic line (see Fig. 1) having a bend at its <em>k</em>th element, and let in general case the masses of all mass elements be inequal. Suppose also that in the considered elastic line some wave process described by <em>x</em>- and <em>y</em>-components takes place. Conveniently introduce two reference systems (<em>x</em>,<em> y</em>) and (<img src="etacut.gif" width="10" height="11" alt="etacut.gif (842 bytes)">, <img src="ksicut.gif" width="9" height="13" alt="ksicut.gif (843 bytes)">) describing the vibration processes before and after the bend relatively. Their direction is shown in Fig. 1. </font></td> </tr> <tr> <td VALIGN="TOP" COLSPAN="2" HEIGHT="2"><p align="center" style="text-align: center"><img src="fig1.gif" width="364" height="165" alt="fig1.gif (3251 bytes)"></p> <p ALIGN="JUSTIFY" style="text-align: justify; font-family: Times New Roman; font-size: 12pt; padding-left: 40px; padding-right: 40px"><font face="Times New Roman" size="3">Fig. 1. The calculation scheme of an elastic line having a bend at the <em>k</em>th element with the angle <img src="alphacut.gif" width="11" height="8" alt="alphacut.gif (839 bytes)"></font></td> </tr> <tr> <td VALIGN="TOP" COLSPAN="2" HEIGHT="2"><p ALIGN="JUSTIFY" style="text-indent: 12pt; text-align: justify; font-size: 12pt"><font face="Times New Roman" size="3">Then the modelling system of differential equations will have the following form:</font></p> <p ALIGN="JUSTIFY"><font face="Times New Roman" size="3">for the <em>x</em>-component</font></td> </tr> <tr> <td VALIGN="top" HEIGHT="2" style="padding-left: 20px" width="90%"><p ALIGN="JUSTIFY"><font face="Times New Roman"><img SRC="Image520.gif" WIDTH="388" HEIGHT="133"> </font></td> <td VALIGN="middle" HEIGHT="2" style="text-indent: 12pt; text-align: right" width="10%"><font face="Times New Roman" size="3">(1)</font></td> </tr> <tr> <td VALIGN="TOP" COLSPAN="2" HEIGHT="2"><p ALIGN="JUSTIFY" style="text-align: justify; font-family: Times New Roman; font-size: 12pt"><font face="Times New Roman" size="3">where <em>m<sub>k</sub></em> is the mass of related element of an elastic line; <em>s</em> is the stiffness coefficient, and <img src="deltabig.gif" width="12" height="10" alt="deltabig.gif (843 bytes)"><sub><em>xk</em></sub> is the shift of <em>k</em>th element along the axis <em>x</em>;</font></p> <p ALIGN="JUSTIFY" style="text-align: justify; font-family: Times New Roman; font-size: 12pt"><font face="Times New Roman" size="3">for the <img src="etacut.gif" width="10" height="11" alt="etacut.gif (842 bytes)"> -component</font></td> </tr> <tr> <td VALIGN="top" HEIGHT="2" style="padding-left: 20px"><p ALIGN="JUSTIFY"><font face="Times New Roman"><img SRC="Image524.gif" WIDTH="404" HEIGHT="154"> </font></td> <td VALIGN="middle" HEIGHT="2" style="text-indent: 12pt; text-align: right" width="10%"><font face="Times New Roman" size="3">(2)</font></td> </tr> <tr> <td VALIGN="TOP" COLSPAN="2" HEIGHT="2"><p ALIGN="JUSTIFY"><font face="Times New Roman" size="3">for the <em>y</em>-component</font></td> </tr> <tr> <td VALIGN="top" HEIGHT="2" style="padding-left: 20px"><p ALIGN="JUSTIFY"><font face="Times New Roman"><img SRC="Image525.gif" WIDTH="386" HEIGHT="136"> </font></td> <td VALIGN="middle" HEIGHT="2" style="text-indent: 12pt; text-align: right" width="10%"><font face="Times New Roman" size="3">(3)</font></td> </tr> <tr> <td VALIGN="TOP" COLSPAN="2" HEIGHT="2"><p ALIGN="JUSTIFY"><font face="Times New Roman" size="3">for the <img src="ksicut.gif" width="9" height="13" alt="ksicut.gif (843 bytes)">-component</font></td> </tr> <tr> <td VALIGN="top" HEIGHT="2" style="padding-left: 20px"><font face="Times New Roman" SIZE="2"><img SRC="Image528.gif" WIDTH="413" HEIGHT="154"> </font></td> <td VALIGN="middle" HEIGHT="2" style="text-indent: 12pt; text-align: right; font-family: Times New Roman; font-size: 12pt" width="10%"><font face="Times New Roman" size="3">(4)</font></td> </tr> </table> </center></div> <p align="center"><strong><font size="4" face="Times New Roman"><b><a href="../../contents2_1.html#bend">Contents:</a> / <a href="../bend086/bend086.html">86</a> / <a href="../bend087/bend087.html">87</a> / <font color="#FF0000">88</font> / <a href="../bend089/bend089.html">89</a> / <a href="../bend090/bend090.html">90</a> / <a href="../bend091/bend091.html">91</a> / <a href="../bend092/bend092.html">92</a> / <a href="../bend093/bend093.html">93</a> / <a href="../bend094/bend094.html">94</a> / <a href="../bend095/bend095.html">95</a> / <a href="../bend096/bend096.html">96</a> / <a href="../bend097/bend097.html">97</a> / <a href="../bend098/bend098.html">98</a> / <a href="../bend099/bend099.html">99</a> / <a href="../bend100/bend100.html">100</a> /</b></font></strong></p>  <script src="//mc.yandex.ru/resource/watch.js" type="text/javascript"></script> <script type="text/javascript"> try { var yaCounter234892 = new Ya.Metrika(234892); } catch(e){} </script> <noscript><div style="position: absolute;"><img src="//mc.yandex.ru/watch/234892" alt="" /></div></noscript>  <div align="center"><a href="http://www.ucoz.ru/" title="Создать сайт бесплатно"><img style="margin:0;padding:0;border:0;" alt="Hosted by uCoz" src="http://s207.ucoz.net/img/cp/10.gif" width="80" height="15" title="Hosted by uCoz" /></a><br /></div> </body> </html>