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The Schr dinger equation for a Kirchhoff elastic rod with noncircular cross section

Authors:
Xue Yun
Xue Yun
Xue Yun
  • This person is not on ResearchGate, or hasn't claimed this research yet.
Liu Yan-Zhu
Liu Yan-Zhu
Liu Yan-Zhu
  • This person is not on ResearchGate, or hasn't claimed this research yet.
Li-Qun Chen at Shanghai University
Li-Qun Chen
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Abstract

The extended Schrödinger equation for the Kirchhoff elastic rod with noncircular cross section is derived using the concept of complex rigidity. As a mathematical model of supercoiled DNA, the Schrödinger equation for the rod with circular cross section is a special case of the equation derived in this paper. In the twistless case of the rod when the principal axes of the cross section are coincident with the Frenet coordinates of the centreline, the Schrödinger equation is transformed to the Duffing equation. The equilibrium and stability of the twistless rod are discussed and a bifurcation phenomenon is presented.
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Ｔｈｅ　Ｓｃｈｒ￣Ｓｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎ　ｆｏｒ　ａ　Ｋ ｉｒｃｈｈｏｆｆ　
ｅｌａｓｔｉｃ　ｒｏｄ　ｗｉｔｈ　ｎｏｎｃｉｒｃｕｌａｒ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ　
Ｘｕｅ　Ｙｕｎ（￣ 纭）ａ）　，Ｌｉｕ　Ｙａｎ—ｚｈｕ（刘延柱）　，ａｎｄ　Ｃｈｅｎ　Ｌｉ－Ｑｕｎ（陈立群）　）　
）Ｉｎｓｔｉｔｕｔｅ　ｏｆ　Ａｐｐｌｉｅｄ　Ｍ ａｔｈｅｍａｔｉｃｓ　ａｎｄ　Ｍ ｅｃｈａｎｉｃｓ，Ｓｈａｎｇｈａｉ　Ｕｎｉｖｅｒｓｉｔｙ，Ｓｈａｎｇｈａｉ　２０００７２，Ｃｈｉｎａ　
ｂ）Ｄｅｐａｒｔｍ ｅｎｔ　ｏｆ　Ｍ ｅｃｈａｎｉｃａｌ　Ｅｎｇｉｎｅｅｒｉｎｇ，Ｓｈａｎｇｈａｉ　Ｉｎｓｔｉｔｕｔｅ　ｏｆ　Ｔｅｃｈｎｏｌｏｇｙ，Ｓｈａｎｇｈａｉ　２００２３３，Ｃｈｉｎａ　
）Ｄ ｅｐａｒｔｍｅｎｔ　ｏｆ　Ｅｎｇｉｎｅｅｒｉｎｇ　Ｍ ｅｃｈａｎｉｃｓ，Ｓｈａｎｇｈａｉ　Ｊｉａｏｔｏｎｇ　Ｕｎｉｖｅｒｓｉｔｙ，Ｓｈａｎｇｈａｉ　２０００３０，Ｃｈｉｎａ　
ｆＲｅｃｅｉｖｅｄ　２７　Ｊｕｎｅ　２００３；ｒｅｖｉｓｅｄ　ｍａｎｕｓｃｒｉｐｔ　ｒｅｃｅｉｖｅｄ　４　Ｎｏｖｅｍｂｅｒ　２００３）　
Ｔｈｅ　ｅｘｔｅｎｄｅｄ　ＳｃｈｒＳｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎ　ｆｏｒ　ｔｈｅ　Ｋｉｒｃｈｈｏｆｆ　ｅｌａｓｔｉｃ　ｒｏｄ　ｗｉｔｈ　ｎｏｎｃｉｒｃｕｌａｒ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ　ｉｓ　ｄｅｒｉｖｅｄ　ｕｓｉｎｇ　
ｔｈｅ　ｃｏｎｃｅｐｔ　ｏｆ　ｃｏｍｐｌｅｘ　ｒｉｇｉｄｉｔｙ．Ａ ｓ　ａ　ｍ ａｔｈｅｍ ａｔｉｃａｌ　ｍ ｏｄｅｌ　ｏｆ　ｓｕｐｅｒｃｏｉｌｅｄ　Ｄ ＮＡ．ｔｈｅ　ＳｃｈｒＳｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎ　ｆｏｒ　ｔｈｅ　ｒｏｄ　
ｗｉｔｈ　ｃｉｒｃｕｌａｒ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ　ｉｓ　ａ　ｓｐｅｃｉａｌ　ｃａｓ ｅ　ｏｆ　ｔｈｅ　ｅｑｕａｔｉｏｎ　ｄｅｒｉｖｅｄ　ｉｎ　ｔｈｉｓ　ｐａｐｅｒ．Ｉｎ　ｔｈｅ　ｔｗｉｓｔｌｅｓｓ　ｃａｓｅ　ｏｆ　ｔｈｅ　ｒｏｄ　ｗｈｅｎ　
ｔｈｅ　Ｄｒｉｎｃｉｐａｌ　ａｘｅｓ　ｏｆ　ｔｈｅ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ　ａｒｅ　ｃｏｉｎｃｉｄｅｎｔ　ｗ ｉｔｈ　ｔｈｅ　Ｆｒｅｎｅｔ　ｃｏｏｒｄｉｎａｔｅｓ　ｏｆ　ｔｈｅ　ｃｅｎｔｒｅｌｉｎｅ，ｔｈｅ　Ｓｃｈｒ６ｄｉｎｇｅｒ　
ｅａｕａｔｉｏｎ　ｉｓ　ｔｒａｎｓｆｏｒｍ ｅｄ　ｔｏ　ｔｈｅ　Ｄ ｕｆｆｉ ｎｇ　ｅｑｕａｔｉｏｎ．Ｔｈｅ　ｅｑｕｉｌｉｂｒｉｕｍ 　ａｎｄ　ｓｔａｂｉｌｉｔｙ　ｏｆ　ｔｈｅ　ｔｗ ｉｓｔｌｅｓｓ　ｒｏｄ　ａｒｅ　ｄｉｓｃｕｓｓｅｄ，ａｎｄ　
ａ　ｂｉｆｕｒｃａｔｉｏｎ　ｐｈｅｎｏｍｅｎｏｎ　ｉｓ　ｐｒｅｓｅｎｔｅｄ．　
Ｋｅｙｗｏｒｄｓ：Ｋｉｒｃｈｈｏｆｆ　ｅｌａｓｔｉｃ　ｒｏｄ，ＳｃｈｒＳｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎ，ＤＮＡ　ｍ ｏｄｅｌ　
ＰＡ ＣＣ ：０３２０，０３４０Ｄ　
１．Ｉｎｔｒｏｄｕｃｔｉｏｎ　
Ｔｈｅ　ａｎａｌｙｓｉｓ　ｏｆ　ｔｈｅ　ｅｑｕｉｌｉｂｒｉｕｍ　ａｎｄ　ｓｔａｂｉｌｉｔｙ　ｏｆ　
ａ　ｔｈｉｎ　ｅｌａｓｔｉｃ　ｒｏｄ　ｈａｓ　ａ　１ｏｎｇ　ｔｒａｄｉｔｉｏｎ　ｉｎ　ｃｌａｓ ｓｉｃ　ｍｅ—　
ｃｈａｎｉｃｓ　ｓｉｎｃｅ　１７３０　ｖｉａ　Ｂｅｒｎｏｕｌｌｉ　ａｎｄ　Ｅｕｌｅｒ．Ｔｈｉｓ　ｔｈｅ－　
ｏｒｅｔｉｃａｌ　ｐｒｏｂｌｅｍ　ｈａｓ　ａｎ　ｉｍｐｏｒｔａｎｔ　ａｐｐｌｉｃａｔｉｏｎ　ｉｎ　ｔｈｅ　
ｓｔｕｄｙ　ｏｆ　ｃｈｅｍｉｃａｌ　ａｎｄ　ｂｉｏｌｏｇｉｃａｌ　ｆｉｂｒｅｓ　ｓｕｃｈ　ａｓ　ｔｈｅ　ｓｕ－　
ｐｅｒｃｏｉｌｅｄ　ｓｔｒｕｃｔｕｒｅ　ｏｆ　ＤＮＡ．Ｉｎ　１８５９　Ｋｉｒｃｈｈｏｆｆ　ｆｏｕｎｄ　
ｔｈｅ　ｄｙｎａｍｉｃ　ａｎａｌｏｇｕｅ　ｏｆ　ｅｑｕｉｌｉｂｒｉｕｍ　ｏｆ　ａ　ｔｈｉｎ　ｅｌａｓｔｉｃ　
ｒｏｄ　ａｎｄ　ｔｈｅ　ｍｏｔｉｏｎ　ｏｆ　ａ　ｒｉｇｉｄ　ｂｏｄｙ　ａｂｏｕｔ　ａ　ｆｉｘｅｄ　ｐｏｉｎｔ，　
ａｎｄ　ｃｏｎｓｔｉｔｕｔｅｄ　ｆｕｎｄａｍｅｎｔａｌ　ｅｑｕａｔｉｏｎｓ　ｆｏｒ　ｔｈｅ　ｅｑｕｉｌｉｂ—　
ｒｉｕｍ　ｏｆ　ｔｈｅ　ｔｈｉｎ　ｅｌａｓｔｉｃ　ｒｏｄ．【　，　 Ｉｎ　１９９４．ｏｎ　ｔｈｅ　ｂａｓｉｓ　
ｏｆ　ｔｈｅ　Ｋｉｒｃｈｈｏｆｆ　ｅｑｕａｔｉｏｎ．Ｓｈｉ　ａｎｄ　Ｈｅａｒｓｔ　ｄｅｒｉｖｅｄ　ｔｈｅ　
ＳｃｈｒＳｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎ　ｗｉｔｈ　ｔｈｅ　ｃｕｒｖａｔｕｒｅ　ａｎｄ　ｔｗｉｓｔ—　
ｉｎｇ　ａｓ　ｕｎｋｎｏｗｎ　ｖａｒｉａｂｌｅｓ．【０—６１　Ａｓ　ａ　ｎｅｗ　ｍ ａｔｈｅｍａｔｉｃａｌ　
ｍｏｄｅｌ　ｏｆ　ｓｕｐｅｒｃｏｉｌｅｄ　ＤＮＡ　ｔｈｅ　ＳｃｈｒＳｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎ　ｉｓ　
ｓｕｉｔａｂｌｅ　ｆｏｒ　ｔｈｅ　ｒｏｄ　ｗｉｔｈ　ｃｉｒｃｕｌａｒ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ　ｏｎｌｙ．Ｉｎ　
ｔｈｅ　ｐｒｅｓｅｎｔ　ｐａｐｅｒ，ｕｓｉｎｇ　ａ　ｃｏｎｃｅｐｔ　ｏｆ　ｃｏｍｐｌｅｘ　ｒｉｇｉｄｉｔｙ　
ｏｆ　ｔｈｅ　ｒｏｄ，ｗｅ　ｅｘｔｅｎｄ　ｔｈｅ　ＳｃｈｒＳｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎ　ｔｏ　ｔｈｅ　
ｍｏｒｅ　ｇｅｎｅｒａｌｉｚｅｄ　ｃａｓｅ　ｗｈｅｎ　ｔｈｅ　ｒｏｄ　ｈａｓ　ａ　ｎｏｎｃｉｒｃｕ—　
ｌａｒ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ． Ｔｈｅ　Ｓｃｈｒ６ｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎ　ｄｅｒｉｖｅｄ　
ｂｙ　Ｓｈｉ　ａｎｄ　Ｈｅａｒｓｔ　ｗａｓ　ａ　ｓｐｅｃｉａｌ　ｃａｓｅ　ｏｆ　ｔｈｅ　ｅｘｔｅｎｄｅｄ　
ＳｃｈｒＳｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎ．Ａｓ　ａｎｏｔｈｅｒ　ｓｐｅｃｉａｌ　ｃａｓｅ　ｏｆ　ｔｈｅ　
ｅｘｔｅｎｄｅｄ　Ｓｃｈｒ６ｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎ，ｔｈｅ　ｅｑｕｉｌｉｂｒｉｕｍ　ｏｆ　ａ　
ｔｗｉｓｔｌｅｓｓ　ｒｏｄ　ｉｓ　ｄｉｓｃｕｓｓｅｄ　ｗｈｅｎ　ｔｈｅ　ｐｒｉｎｃｉｐａｌ　ａｘｅｓ　ｏｆ　
ｔｈｅ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ　ａｒｅ　ｃｏｉｎｃｉｄｅｎｔ　ｗｉｔｈ　ｔｈｅ　Ｆｒｅｎｅｔ　ｃｏｏｒｄｉ－　
ｈｔｔｐ：／／ＷＷＷ．１ｏｐ．ｏｒｇ／ｊｏｕｒｎａｌｓ／ｃｐ　
ｎａｔｅｓ　ｏｆ　ｔｈｅ　ｃｅｎｔｒｅｌｉｎｅ，ａｎｄ　ｔｈｅ　ｅｘｔｅｎｄｅｄ　ＳｃｈｒＳｄｉｎｇｅｒ　
ｅｑｕａｔｉｏｎ　ｉｓ　ｔｒａｎｓｆｏｒｍｅｄ　ｔｏ　ｔｈｅ　Ｄｕｆｆｉ ｎｇ　ｅｑｕａｔｉｏｎ． Ａ　
ｑｕａｌｉｔａｔｉｖｅ　ａｎａｌｙｓｉｓ　ｃａｎ　ｂｅ　ｍａｄｅ　ｉｎ　ｔｈｉｓ　ｃａｓｅ　ｔｏ　ｄｉｓ—　
ＣＵＳＳ　ｔｈｅ　ｅｑｕｉｌｉｂｒｉｕｍ　ａｎｄ　ｓｔａｂｉｌｉｔｙ　ｏｆ　ｔｈｅ　ｒｏｄ，ａｎｄ　ａ　
ｐｉｔｃｈｆｏｒｋ　ｂｉｆｕｒｃａｔｉｏｎ　ｉｓ　ｐｒｅｓｅｎｔｅｄ．　
２．Ｄ ｅｒｉｖａｔｉｏｎ　ｏｆ　ｎｏｎｌｉｎｅａｒ　Ｓｃｈｒ６ｄ．　
ｉｎｇｅｒ　ｅｑｕａｔｉｏｎ　
％ ｔｒｅａｔ　ａ　ｔｈｉｎ　ｅｌａｓ ｔｉｃ　ｒｏｄ　ｗｉｔｈ　ｎｏｎｃｉｒｃｕｌａｒ　ｃｒｏｓｓ　
ｓｅｃｔｉｏｎ．Ｌｅｔ　Ｐ　ｂｅ　ａｎ　ａｒｂｉｔｒａｒｙ　ｐｏｉｎｔ　ｏｎ　ｔｈｅ　ｃｅｎｔｒｅｌｉｎｅ　
ｗｉｔｈ　ａｒｃ—ｃｏｏｒｄｉｎａｔｅ　ｓ． Ｅｓｔａｂｌｉｓｈ　ｔｈｅ　Ｆｒｅｎｅｔ　ｃｏｏｒｄｉ—　
ｎａｔｅ　ｆｌａｍｅ（Ｐ—ｎｂｔ），ｗｈｅｒｅ　ｎ一，ｂ－，ｔ－ａｘｅｓ　ａｒｅ　ｎｏｒｍａｌ，　
ｂｉｎｏｍｉａｌ　ａｎｄ　ｔａｎｇｅｎｔ　ｏｆ　ｔｈｅ　ｃｅｎｔｒｅｌｉｎｅ　ａｔ　ｐｏｉｎｔ　Ｐ　ｒｅ—　
ｓｐｅｃｔｉｖｅｌｙ．Ａ　ｐｒｉｎｃｉｐａｌ　ｃｏｏｒｄｉｎａｔｅ　ｆｌａｍｅ（Ｐ—ｘｙｚ）ｉｓ　
ｆｉｘｅｄ　ｏｎ　ｔｈｅ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ．ｗｈｅｒｅ　ｔｈｅ　ｚ—ａＸ　 ｉｓ　ａｌｏｎｇ　
ｔｈｅ　ｔａｎｇｅｎｔ　ｔ－ａｘｉｓ　ａｎｄ　Ｘ　ｉｓ　ｔｈｅ　ａｎｇｌｅ　ｂｅｔｗｅｅｎ　ｃｏｏｒ－　
ｄｉｎａｔｅ　ｐｌａｎｅｓ（ｎ，ｂ）ａｎｄ（ｘ，　）．Ｌｅｔ　ｅｌ，ｅ２，ｅ３，ａｎｄ　
ｅｎ，ｅｂ，ｅｔ　ｂｅ　ｂａｓｉｓ　ｖｅｃｔｏｒｓ　ｏｆ（Ｐ—ｘｙｚ）ａｎｄ（Ｐ—ｎｂｔ）；ｉｔ　
ｆｏｌｌｏｗｓ　ｔｈａｔ　
ｅｎ ‘ｅ１　 ｅｂ‘ｅ２　 ＣＯＳ）（，　
ｅｔ　 ｅ３　 （１）　
Ｄｅｎｏｔｅ　ｔｈｅ　ｃｕｒｖａｔｕｒｅ　ｖｅｃｔｏｒ　ｏｆ　ｔｈｅ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ　ａｔ　
ｐｏｉｎｔ　Ｐ　ｂｙ　 ，ｏｆ　ｗｈｉｃｈ　ｔｈｅ　ｃｏｍｐｏｎｅｎｔｓ　ｗｉ（ｉ＝１，２，　
３）ｉｎ（Ｐ—ｘｙｚ）ｃａｎ　ｂｅ　ｄｅｔｅｒｍｉｎｅｄ　ｂｙ　ｔｈｅ　ｃｕｒｖａｔｕｒｅ　，　
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Ｎｏ．６　 Ｔｈｅ　Ｓｃｈｒ６ｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎ　ｆｏｒ　ａ　Ｋｉｒｃｈｈｏｆｆ　ｅｌａｓｔｉｃ　ｒｏｄ　ｗｉｔｈ　 ７９５　
ｔｏｒｓｉｏｎ　７＿ａｎｄ　ｔｗｉｓｔｉｎｇ　ａｎｇｌｅ）（ａｓ　ｆｏｌｌｏｗｓ　
０３１　 ｓｉｎ ）（　
ａ）２　 ＣＯＳ）（　
０Ｊ３　 ７－＋ ）（，　
ｗｈｅｒｅ　ｔｈｅ　ｄｏｔ　ｍｅａｎｓ　ｔｈｅ　ｄｅｒｉｖａｔｉｖｅ　ｗｉｔｈ　ｒｅｓｐｅｃｔ　ｔｏ　
ｔｈｅ　ａｒｃ—ｃｏｏｒｄｉｎａｔｅ　８．Ｉｎｔｒｏｄｕｃｅ　ａ　ｃｏｍｐｌｅｘ　ｖａｒｉａｂｌｅ∈，　
ｄｅｆｉｎｅｄ　ａｓ　
∈＝　ｌ＋ｉ０Ｊ２＝　ｅｘｐ　（三一）（）　
Ｔｈｅ　ｒｏｄ　ｉｓ　ｓｕｂｊｅｃｔｅｄ　ｔｏ　ａ　ｓｅｌｆ－ｃｏｎｔａｃｔ　ｆｏｒｃｅ　Ｆ。　
ｅｎ＋　 ｅｂ，ｏｆ　ｗｈｉｃｈ　ｔｈｅ　ｃｏｍｐｏｎｅｎｔｓ　 （ｉ＝１，　
３）ｉｎ（Ｐ—ｘｙｚ）ｃａｎ　ｂｅ　ｗｒｉｔｔｅｎ　ａｓ　
１　＋　 ２）　
２一　 ０Ｊ１），　
ｗｈｅｒｅ　 ＝　 ．Ｄｅｎｏｔｅ　ｔｈｅ　ｒｅｓｕｌｔａｎｔ　ｆｏｒｃｅ　ａｎｄ　ｔｏｒｑｕｅ　
ｏｎ　ｔｈｅ　ｃｒＯＳＳ　ｓｅｃｔｉｏｎ　ａｔ　ｐｏｉｎｔ　Ｐ ，ｒｅｓｐｅｃｔｉｖｅｌｙ，ｂｙ　Ｆ 　ａｎｄ　
Ｍ　ｗｉｔｈ　ｃｏｍｐｏｎｅｎｔｓ　 ａｎｄ尬 （ｉ＝１，２，３）ｉｎ（Ｐ—ｘｙｚ），　
ａｎｄ　ａｓｓｕｍ ｅ　ｔｈａｔ　ｔｈｅ　ｒｏｄ　ｈａｓ　ａｎ　ｉｎｔｒｉｎｓｉｃ　ｔｗｉｓｔｉｎｇ　
ｉｎ　ｔｈｅ　ｒｅｌａｘｅｄ　ｓｔａｔｅ，ｔｈｅｎ　ｔｈｅ　ｆｏｌｌｏｗｉｎｇ　ｒｅｌａｔｉｏｎｓｈｉｐ　
ｓｈｏ１】１ｄ　ｂｅ　ｓａｔｊｓｆｉｅｄ：　
Ｍ １＝ Ａ　１，　
Ｍ ２＝ Ｂ　２，　
Ｍ３＝ｃ（　３一　 ），　 （５）　
ｗｈｅｒｅ　Ａ ，Ｂ ，Ｃ　ａｒｅ　ｔｈｅ　ｂｅｎｄｉｎｇ　ａｎｄ　ｔｗｉｓｔｉｎｇ　ｒｉｇｉｄｉｔｉｅｓ　
ｏｆ　ｔｈｅ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ　ａｂｏｕｔ　ｘ一，Ｙ一，ｚ—ａｘｅｓ，ｒｅｓｐｅｃｔｉｖｅｌｙ．　
Ｉｎｔｒｏｄｕｃｅ　ａ　ｃｏｍｐｌｅｘ　ｔｏｒｑｕｅ　Ｍ ＝ Ｍ ｌ＋ ｉＭ ２，ａｎｄ　ａ　
ｃｏｍ ｐｌｅｘ　ｒｉｇｉｄｉｔｙ　Ｄ　ａｓ　
Ｄ ＝ Ｄ１＋ ｉＤ２，　
Ｄ１＝ Ａ　ｓｉｎ　）（＋ Ｂ　ＣＯＳ　）（　
Ｄ２＝（Ｂ—Ａ）ｓｉｎ）（ＣＯＳ）（　
ｔｈｅｎ　ｔｈｅ　ｆｉｒｓｔ　ｔｗｏ　ｔｅｒｍｓ　ｏｆ　Ｅｑ．（５）ｃａｎ　ｂｅ　
ｃｏｍｐｌｅｘ　ｆｏｒｍ ，　
Ｍ ＝ Ｊ［）∈．　
ｗｒｉｔｔｅｎ　ｉｎ　
ｃｏｍｐｌｅｘ　ｆｏｒｃｅｓ　Ｆ ＝Ｒ ＋ｉＦ２　ａｎｄ　Ｆ。：　 ＋ｉ　 ，ｗｅ　
ｃａｎ　ｗｒｉｔｅ　ｔｈｅ　ｃｏｍｐｏｎｅｎｔｓ　ｏｆ　Ｋｉｒｃｈｈｏｆｆ　ｅｑｕａｔｉｏｎｓ（８ａ）　
ａｎｄ（８ｂ）ｏｎ　ｘ一，ｙ－ａｘｅｓ　ｉｎ　ｃｏｍｐｌｅｘ　ｆｏｒｍ，　
户　 ｗ３Ｆ－　 ）一ｉ　 Ｆ。＝。　
Ｍ ＋ｉ（Ｄｗ３一ＭａＸ＋ｉＦ＝０　
Ｔｈｅ　ｃｏｍｐｏｎｅｎｔｓ　ｏｆ　Ｋｉｒｃｈｈｏｆｆ　ｅｑｕａｔｉｏｎｓ　
ｏｎ　ｔｈｅ　ｚ—ａｘｉｓ　ｃａｎ　ｂｅ　ｗｒｉｔｔｅｎ　ａｓ　
＋Ｉｍ（Ｆ￣）：０　
ｃ　３＋ Ｄ２　 ＝ ０　
（１０）　
（８ａ）ａｎｄ（８ｂ）　
（１１）　
（１２）　
Ｉｎ　ｔｈｅ　ｃａｓｅ　ｗｉｔｈｏｕｔ　ｃｏｎｔａｃｔ　ｆｏｒｃｅ　ｗｈｅｎ　Ｆ。＝ ０．ｔｈｅ　
Ｋｉｒｃｈｈｏｆｆ　ｅｑｕａｔｉｏｎ　ｐｅｒｍ ｉｔｓ　ａ　Ｊａｃｏｂｉ　ｉｎｔｅｇｒａｌ　ａｓ　
去（Ｄ１ｎ２＋ｃ　；）＋　＝Ｈ　 （１３）　
ｗｈｅｒｅ　ｔｈｅ　ｃｏｎｓｔａｎｔ　Ｈ　ｉｓ　ｄｅｔｅｒｍ ｉｎｅｄ　ｂｙ　ｖａｌｕｅｓ　ｏｆ￣ｄｉ　
ａｎｄ　Ｆ３　ａｔ　ｉｎｉｔｉａｌｌｙ　ｇｉｖｅｎ　ａｒｃ—ｃｏｏｒｄｉｎａｔｅ　８．　
Ｗｅ　ｓｏｌｖｅ　Ｆ　ｆｒｏｍ　Ｅｑ．（１０）ａｎｄ　ｏｂｔａｉｎ　
Ｆ＝ｉＭ 一（Ｄｗ３一　 ）　 （１４）　
Ｓｕｂｓｔｉｔｕｔｉｎｇ　Ｅｑ．（１４）ａｎｄ　ｉｔｓ　ｄｅｒｉｖａｔｉｖｅ　ｗｉｔｈ　ｒｅｓｐｅｃｔ　ｔｏ　
８　ｉｎｔｏ　Ｅｑ．（９），ｉｎ　ｗｈｉｃｈ　Ｍ　ｉｓ　ｅｘｐｒｅｓｓｅｄ　ｂｙ　Ｅｑ．（７），ｗｅ　
ｏｂｔａｉｎ　ａ　ｃｏｍｐｌｅｘ　ｄｉｆｆｅｒｅｎｔｉａｌ　ｅｑｕａｔｉｏｎ　ａｓ　
Ｊ［）　 ∈＋　 一　 Ｆ。　
ｗｈｅｒｅ　ｋｌ　ａｎｄ　ｋ２　ａｒ ｅ　ｄｅｆｉｎｅｄ　ａｓ　
ｋ１＝２Ｄｗ３一　（　３一　 ）一２ｉ　，　
ｚ＝一Ｈ－Ｃｗ￣　ｓ＋（２３＿Ｃ＿Ｄ）　；　
＋Ｄ＋ｉ［２Ｄｗ３一（Ｃ—Ｊ［））　３】．　
（１５）　
（１６ａ）　
（１６ｂ）　
Ｅｑｕａｔｉｏｎ（１５）ｉｓ　ｔｈｅ　ＳｃｈｒＳｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎ　ｆｏｒ　ａｎ　ｅｌａｓ—　
ｔｉｃ　ｒｏｄ　ｗｉｔｈ　ｎｏｎ—ｃｉｒｃｕｌａｒ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ． Ｔｈｅ　ｓｅｔ　ｏｆ　
ｅｑｕａｔｉｏｎｓ（１５）ａｎｄ（１２）ｉｓ　ｃｌｏｓｅｄ　ａｂｏｕｔ　ｃｏｍｐｌｅｘ　ｖａｒｉ—　
ａｂｌｅ∈ａｎｄ　ｒｅａｌ　ｖａｒｉａｂｌｅ　０Ｊ３．　
Ｓｕｂｓｔｉｔｕｔｉｎｇ　ｔｈｅ　ｅｘｐｏｎｅｎｔｉａｌ　ｅｘｐｒｅｓｓｉｏｎ　ｏｆ∈ｉｎｔｏ　
ｃｏｍｐｌｅｘ　ｅｑｕａｔｉｏｎ（１５）ａｎｄ　ｓｅｐａｒａｔｉｎｇ　ｔｈｅｍ　ｉｎｔｏ　ｒｅａｌ　
ａｎｄ　ｉｍ ａｇｉｎａｒ ｙ　ｐａｒｔｓ，ｗｅ　ｏｂｔａｉｎ　
Ｄ１诧＋２（Ｄ　一Ｄ２７－）庀＋去Ｊ［）　３一　 ＝　，　
Ｊ［）２托＋ ＃２ｋ ＋　 ３　 ＝　 ，　
Ｔｈｅ　Ｋｉｒｃｈｈｏｆｆ　ｅｑｕａｔｉｏｎ　ｏｆ　ｔｈｅ　ｒｏｄ　ｉｎ　ｖｅｃｔｏｒ　ｆｏｒｍ　ｉｓ　 ．　
Ｗ ｎｅｒｅ　
Ｆ ＋　 ｘ　Ｆ ＋ Ｆ。 ＝ ０　
Ｍ ＋　 ｘ　Ｍ ＋ ｅｔ　ｘ　Ｆ ＝ ０，　
（８ａ）　
（８ｂ）　
ｉｎ　ｗｈｉｃｈ　ｔｈｅ　ｄｅｒｉｖａｔｉｖｅｓ　ｗｉｔｈ　ｒｅｓｐｅｃｔ　ｔｏ　ｔｈｅ　ａｒｃ—　
ｃｏｏｒｄｉｎａｔｅ　８　ａｒｅ　ｐｅｒｆｏｒｍｅｄ　ｉｎ　ｔｈｅ（Ｐ—ｘｙｚ）．Ｄｅｆｉｎｉｎｇ　
（１７ａ）　
（１７ｂ）　
ｌ＝２［日＋Ｄ２亍一　 ｌ＋（ｃ　 ＋２Ｄ２—２ｃ２）７－１　
一（３ｃ一２Ｄｌ　７－　一Ｃ２　，　
２＝ｃ（　 一７＿一戈）＋２（Ｄｌ　７＿＋　２），　
３＝（２Ｄ１一Ｄ２７－）７－＋Ｄｌ亍＋Ｄ２一ｃ（亍＋戈）．（１８）　
＝ 乏　
，●　
一　 一　
矸　
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７９６　 Ｘｕｅ ’　 ｎ　ｅｔ　ａｌ　 Ｖ０１．１３　
Ｅｑｕａｔｉｏｎ（１２）ｃａｎ　ｂｅ　ｗｒｉｔｔｅｎ　ａｓ　
（于＋戈）＋Ｄ２　 ＝０　
Ｔｈｅ　ｓｅｔ　ｏｆ　ｒｅａｌ　ｄｉｆｆｅｒｅｎｔｉａｌ　ｅｑｕａｔｉｏｎｓ（１７）ａｎｄ（１９）ｉｓ　
ｃｌｏｓｅｄ　ａｂｏｕｔ　ｔｈｅ　ｃｕｒｖａｔｕｒｅ　 ．ｔｏｒｓｉｏｎ　７－ａｎｄ　ｔｗｉｓｔｉｎｇ　
ａｎｇｌｅ）（．　
３．Ｓｐｅｃｉａｌ　ｃａｓｅｓ　
３．１．Ａ　ｒｏｄ　ｗ ｉｔｈ　ｃｉｒｃｕｌａｒ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ　
Ｆｏｒ　ａ　ｒｏｄ　ｗｉｔｈ　ｃｉｒｃｕｌａｒ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ，ｌｅｔ　Ａ ＝ Ｂ，　
ｔｈｅｎ　ＤＩ＝Ａ，Ｄ２＝０，ａｎｄ　Ｅｑ．（１２）ｏｒ　Ｅｑ．（１９）ｈａｓ　ａ　
ｆｉｒｓｔ　ｉｎｔｅｇｒａｌ　ａｓ　
０３３　 ＣＯ３０　ｏｒ　）（　 ０３３０一Ｔ　 （２０）　
ｗｈｉｃｈ　ｍｅａｎｓ　ｔｈａｔ　ｔｈｅ　ｔｗｉｓｔｉｎｇ　ｄｅｎｓｉｔｙ　０３３　ｏｆ　ｔｈｅ　ｒｏｄ　
ｉｓ　ｃｏｎｓｔａｎｔ．Ｔｈｅ　ｃｏｅｆｆｉ ｃｉｅｎｔｓ　ｏｆ　ＳｃｈｒＳｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎｓ　
（１５）ａｎｄ（１７）ｂｅｃｏｍｅ　
ｌ＝２Ｄｗ３一Ｃ（０３３０　
，＝ 一日　
１　
２　
３　
ｕ　ｕ３ｏ＋　
ｕ　）　
（兰　
２（Ｈ＋Ａｒ　）一　ｕ；ｏ　
（ｕ　一ｕ３ｏ）＋２ＡＴ，　
Ａ彳－．　
Ｄ１　ｕ；。，　｝　
Ｓｕｂｓｔｉｔｕｔｉｎｇ　ｔｈｅｓｅ　ｉｎｔｏ　Ｅｑｓ．（１５）ａｎｄ（１７），ｗｅ　
ｔｈｅ　Ｓｃｈｒ６ｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎ　ｄｅｒｉｖｅｄ　ｂｙ　Ｓｈｉ　ａｎｄ　
ｆｏｒ　ａ　ｒｏｄ　ｗｉｔｈ　ｃｉｒｃｕｌａｒ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ．［。，　，。］　
ｏ ｒ　
（２１）　
ｏｂｔａｉｎ　
Ｈｅａｒｓｔ　
≤＋　１　 一　 （２２）　
恙一　 Ｔ －－
２
＿ｃｎ＋　 ３　
２忘 Ｔ －－ 詈）　 ，　（２３ａ）　
ｗｈｅｒｅ　ｉ＝ｋｉ／Ａ （ｉ＝１，２），Ｆ　＝Ｆ　／Ａ，ａｎｄ　
Ｃ　
Ｑ　
Ｈ　 Ｑ　
Ｂ　 ４　 （罢　
Ａ（ｗ３ｏ一０３ｏ）　
ｗｈｅｒｅ　 ＝Ｃ／Ａ　ｉｓ　ｔｈｅ　ｒａｔｉｏ　ｏｆ　ｒｉｇｉｄｉｔｉｅｓ　
３．２．Ｔｗｉｓｔｌｅｓｓ　ｒｏｄ　
（２３ｂ）　
（２４）　
Ｉｎ　ｔｈｅ　ｃａｓｅ　ｏｆ　ａ　ｒｏｄ　ｗｉｔｈ　ｎｏｎｃｉｒｃｕｌａｒ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ　
ｗｅ　ｒｅｗｒｉｔｅ　Ｅｑ．（１２）ａｓ　
３　 Ｂ）　 ＣＯＳ）（ｓｉｎ）（　 （２５）　
Ａｓ　Ａ ≠ Ｂ．ｔｈｅ　ｔｗｉｓｔｉｎｇ　ｄｅｎｓｉｔｙ　０３３　ｉｓ　ｃｏｎｓｔａｎｔ　ｏｎｌｙ　
ｗｈｅｎ）（三 ｊｒｒ／２（ｊ＝Ｏ，１，．．．），ｉ．ｅ．ｗｈｅｎ　ｔｈｅ　ｐｒｉｎｃｉ—　
ｐａｌ　ａｘｅｓ　ｏｆ　ｔｈｅ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ　ａｒｅ　ｃｏｉｎｃｉｄｅｎｔ　ｗｉｔｈ　ｔｈｅ　
Ｆｒｅｎｅｔ　ｃｏｏｒｄｉｎａｔｅｓ　ｏｆ　ｔｈｅ　ｃｅｎｔｒｅｌｉｎｅ．Ｗ ｉｔｈｏｕｔ　ｌｏｓｓ　ｏｆ　
ｇｅｎｅｒａｌｉｔｙ，ｌｅｔ）（三 丌／２，ｔｈｅｎ　０３１＝　 ，０３２＝ ０　ａｎｄ　
Ｄ ｌ＝ Ａ，Ｄ２＝ ０．Ｓｉｎｃｅ　ｔｈｅｒｅ　ｉｓ　ｎｏ　ｔｗｉｓｔｉｎｇ　ｒｅｌａｔｉｖｅ　ｔｏ　
ｔｈｅ　Ｆｒｅｎｅｔ　ｃｏｏｒｄｉｎａｔｅｓ　ｆｒａｍｅ，戈＝ ０，ａｎｄ　ｔｈｅ　ｔｗｉｓｔｉｎｇ　
ｄｅｎｓｉｔｙ　０３３　ｉｓ　ｅｑｕｉｖａｌｅｎｔ　ｔｏ　ｔｈｅ　ｔｏｒｓｉｏｎ　７－．Ｌｅｔ　
０３ ３　 Ｔ ＝ ０３３０　
ａｎｄ　ｔｈｅ　ｃｏｅｆｆｉ ｃｉｅｎｔ　 ｌ　ｉｓ　ｓｉｍｐｌｉｆｉｅｄ　ｔｏ　
１　 ２（Ｈ＋Ｃ０３￣０３３ｏ）　 （３Ｃ一２Ａ）０３￣ｏ　
（２６）　
（２７）　
Ｓｕｂｓｔｉｔｕｔｉｎｇ　Ｅｑ．（２ｒ）ｉｎｔｏ　Ｅｑ．（１７ａ）ａｎｄ　ｌｅｔｔｉｎｇ　 ＝　
＝０｝ｗｅ　ｏｂｔａｉｎ　ａ　Ｄｕｆｆｉｎｇ　ｅｑｕａｔｉｏｎ　ｏｆ　ｃｕｒｖａｔｕｒｅ　Ｋ　
一 　 ．　 １　 ３　
—ｃ　 ＋ 一　’　 ２　
ｗｈｅｒｅ　ｔｈｅ　ｃｏｅｍｃｉｅｎｔ　ｃ　ｉｓ　ｄｅｆｉｎｅｄ　ａｓ　
Ｃ　 Ｈ＋　 ｕ。。＋ １　 ；　）　
（２８）　
（２９）　
４．Ｓｔａｂｉｌｉｔｙ　ａｎａｌｙｓｉｓ　ａｎｄ　ｂｉｆｕｒｃａ－　
ｔｉｏｎ　
Ｔｈｅ　ｃｏｅｆｆｉｃｉｅｎｔ　ｃ　ｏｆ　Ｄｕｆｆｉｎｇ　ｅｑｕａｔｉｏｎ（２８）ｃａｎ　ｂｅ　
ｗｒｉｔｔｅｎ　ａｓ　
Ｃ　 （　 １）ｕ；ｏ　 （３０）　
ｗｈｅｒｅ　 ｉｓ　ａ　ｄｉｍ ｅｎｓｉｏｎｌｅｓｓ　ｐａｒａｍ ｅｔｅｒ，ｄｅｐｅｎｄｉｎｇ　ｏｎ　
ｔｈｅ　ｉｎｉｔｉａｌ　ｃｏｎｄｉｔｉｏｎ　ｏｆ　ｔｈｅ　ｒｏｄ　
Ｌｅｔｔｉｎｇ 忍＝ ０　ｌｅａｄｓ　ｔｏ　ａｎ　ｅｑｕａｔｉｏｎ　ｏｆ　ｃｏｎｓｔａｎｔ　ｃｕｒｖａ－　
ｔｕｒｅ　 ｓ　ａｓ　
ｌ（３Ａ一２）（１　 ）ｕ；ｏ＋　２］二０　 （３２）　
Ｌｅｔ　 ＞ ２／３　ａｎｄ　 １，ｔｈｅｎ　Ｅｑ．（３２）ｈａｓ　ｏｎｌｙ　ａ　
ｔｒｉｖｉａｌ　ｓｏｌｕｔｉｏｎ　 ｓ ＝ ０，ｃｏｒｒｅｓｐｏｎｄｉｎｇ　ｔｏ　ａ　ｓｔｒａｉｇｈｔ　
ｅｑｕｉｌｉｂｒｉｕｍ　ｏｆ　ｔｈｅ　ｒｏｄ． Ｗ ｈｅｎ　 ＞ １，ａｓ　ｗｅｌｌ　ａｓ　ｔｈｅ　
ｔｒｉｖｉａｌ　ｓｏｌｕｔｉｏｎ．ｔｈｅｒｅ　ｅｘｉｓｔ　ｔｗｏ　ｎｏｎｔｒｉｖｉａｌ　ｓｏｌｕｔｉｏｎｓ　
＝ －［－０３３０、／／（３　一２）（　一１），ｃｏｒｒｅｓｐｏｎｄｉｎｇ　ｔｏ　ａ　ｈｅ—　
ｌｉｃａｌ　ｅｑｕｉｌｉｂｒｉｕｍ ． Ｔｈｅ　ｃｕｒｖａｔｕｒｅ　 ｏｆ　ｔｈｅ　ｒｏｄ　ｈａｓ　ａ　
ｐｉｔｃｈｆｏｒｋ　ｂｉｆｕｒｃａｔｉｏｎ　ａｔ　 ＝ １，ａｎｄ　ａ　ｓｔａｂｉｌｉｔｙ　ｃｒｉｔｅｒｉｏｎ　
ｏｆ　ｔｈｅ　ｓｔｒａｉｇｈｔ　ｅｑｕｉｌｉｂｒｉｕｍ　ｏｆ　ｔｈｅ　ｒｏｄ　ｃａｎ　ｂｅ　ｄｅｒｉｖｅｄ　
ａｃｃｏｒｄｉｎｇ　ｔｏ　ｔｈｅ　ｔｈｅｏｒｙ　ｏｆ　Ｐｏｉｎｃａｒ６　ａｓ　
１： ｓｔａｂｌｅ，　
＞１：ｕｎｓｔａｂｌｅ　 （３３）　
重；　
＋　
一　
＝　
２ 　
＼、●／ 　
Ｏ　
＋　
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ｒ 『　
Ｎ ｏ．６　 Ｔｈｅ　ＳｃｈｒＳｄｉｎｇｅｒ　ｅｑｕａｔｉｏｎ　ｆｏｒ　ａ　Ｋ ｉｒｃｈｈｏｆｆ　ｅｌａｓｔｉｃ　ｒｏｄ　ｗｉｔｈ　 ７９７　
Ｉｎ　ｔｈｅ　ｃａｓｅ　ｏｆ　ｓｔｒａｉｇｈｔ　ｅｑｕｉｌｉｂｒｉｕｍ ，ｌｅｔ　 ｓ ＝ ０，　
＝Ｆ０　ａｎｄ　０３３０＝ Ｍｏ／Ｃ，ｔｈｅ　ｃｏｎｓｔａｎｔ　Ｈ　ｃａｎ　ｂｅ　
ｃａｌｃｕｌａｔｅｄ　ｆｒｏｍ　Ｅｑ．（１３）；ａｎｄ　ｔｈｅｎ　ｓｕｂｓｔｉｔｕｔｉｎｇ　ｉｎｔｏ　
Ｅｑ．（３１），ｔｈｅ　ｓｔａｂｌｅ　ｃｏｎｄｉｔｉｏｎ　 １　ｃａｎ　ｂｅ　ｏｂｔａｉｎｅｄ：　
等（　一　Ａ一　ｃｄ３０）　
Ｉｎ　ａ　ｓｐｅｃｉａｌ　ｃａｓｅ　ｗｈｅｎ　Ｃ ＝ ２Ａ，ｕ　 ＝ ０　
（３４）ｉｓ　ｓｉｍｐｌｉｆｉｅｄ　ｔｏ　ｔｈｅ　ｓａｍｅ　ｆｏｒｍ　ａｓ　
Ｒｅｆｅｒｅｎｃｅｓ　
（３４）　
ｔｈｅ　ｃｏｎｄｉｔｉｏｎ　
ｔｈｅ　Ｇｒｅｅｎｈｉｌ１　
ｆｏｒｍｕｌａ．［。］　
４Ａ．　 （３５）　
Ｉｔ　ｉｓ　ｓｈｏｗ ｎ　ｔｈａｔ　ｔｈｅ　ｓｔａｂｉｌｉｔｙ　ｃｒｉｔｅｒｉｏｎ　ｆｏ ｒ　ａ　ｓｔｒａｉｇｈｔ　
ｒｏｄ　ｗｉｔｈ　ｃｉｒｃｕｌａｒ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ　ｕｎｄｅｒ　ｅｘｔｅｎｓｉｏｎ　ａｎｄ　
ｔｏｒｓｉｏｎ　ｉｓ　ｅｆｆｅｃｔｉｖｅ　ｆｏｒ　ａ　ｔｗｉｓｔｌｅｓｓ　ｓｔｒａｉｇｈｔ　ｒｏｄ　ｗｉｔｈ　
ｎｏｎｃｉｒｃｕｌａｒ　ｃｒｏｓｓ　ｓｅｃｔｉｏｎ　ｗｈｅｎ　ｔｈｅ　ｒｉｇｉｄｉｔｉｅｓ　ｓａｔｉｓｆｙ　ｔｈｅ　
Ｋｏｖａｌｅｖｓｋａｙａ　ｃｏｎｄｉｔｉｏｎ　Ｃ ＝ ２Ａ．　
『１１　Ｋｉｒｃｈｈｏｆｆ　Ｇ　１８５９　Ｊ．Ｒｅｉｎｅ　Ａｎｇｅｗ．Ｍ ａｔｈ．５６　２８５　
［２】Ｌｏｖｅ　Ａ　Ｅ　Ｈ　１９４４　Ａ　Ｔｒｅａｔｉｓｅ　Ｏｎ　Ｍａｔｈｅｍａｔｉｃａｌ　Ｔｈｅｏｒｙ　ｏｆ　
Ｅｌａｓｔｉｃｉｔｙ　４ｔｈ　ｅｄｎ（Ｎｅｗ　Ｙｏｒｋ：Ｄｏｖｅｒ）ｐ３８１　
［３】Ｓｈｉ　Ｙ　ａｎｄ　Ｈｅａｒｓｔ　Ｊ　Ｅ　１９９４　Ｊ．Ｃｈｅｍ．Ｐｈｙｓ．１０１　５１８６　
［４】Ｔｉａｎ　Ｑ　ａｎｄ　Ｍａ　Ｂ　Ｋ　１９９９　Ａｃｔａ　Ｐｈｙｓ．Ｓｉｎ．４８　２１２５（ｉｎ　
Ｃｈｉｎｅｓｅ）　
［５】Ｌｉｕ　Ｊ　Ｂ　ａｎｄ　Ｃａｉ　Ｘ　Ｐ　２００１　Ａｃｔａ　Ｐｈｙｓ．Ｓｉｎ．５０　８２０（ｉｎ　
Ｃｈｉｎｅｓｅ）　
【６】Ｒｕａｎ　Ｈ　Ｙ　ａｎｄ　Ｃｈｅｎ　Ｙ　Ｘ　２００１　Ａｃｔａ　Ｐｈｙｓ．Ｓｉｎ．５０　５８６（ｉｎ　
Ｃｈｉｎｅｓｅ）　
［７】Ｓｈｉ　Ｙ　Ｂｏｒｏｖｉ　Ａ　Ｅ　ａｎｄ　Ｈｅａｒｓｔ　Ｊ　Ｅ　１９９５　Ｊ．Ｃｈｅｍ．Ｐｈｙｓ．　
１０３　３１６６　
［８】Ｓｈｉ　Ｙ，Ｈｅａｒｓｔ　Ｊ　Ｅ　ｅｔ　ａｌ　１９９８　Ｊ．Ｃｈｅｍ．Ｐｈｙｓ．１０９　２９５９　
［９】Ｇｒｅｅｎｈｉｌｌ　Ａ　Ｇ　１８８３　Ｐｒｏｃ．１ｎｓｔ．Ｍｅｃｈ．Ｅｎｇ．（Ｌｏｎｄｏｎ）　
维普资讯 http://www.cqvip.com
... Shi et al [7][8] derived a class of one-dimensional fixed-state nonlinear Schrödinger equation by introducing complex vectors and complex bending moments, and gave a closed solution of the DNA molecular centerline. Xue et al. [9] further extended the results to the general case of non-circular cross sections. Wang et al. [10][11][12] used symmetry to derive some conserved quantities of elastic thin rods. ...
... Since there is no friction, the contact torque is m = 0. The elastic rod equation (3) can be expressed as complex curvature equation [9]. ...
... class of nonlinear Schrödinger equation for elastic thin rod equation is given by Ref.[8,9] but there was no further discussion, and the Euler-Lagrangian equation expressed by curvature is given. Because it contains the derivative of curvature cubed, we can not use Schrödinger equation analogy method. ...
A new solution of thin elastic rod by dynamic analogy
Article
Full-text available
  • Aug 2020
Inspired by Kirchhoff dynamic analogy, we write the Kirchhoff equation of thin elastic rod in form of curvature expression. Compared it with nonlinear Schrödinger equation, we extend a Jacobi elliptic function analogy solution to elastic rod equation and give a new alternative way to solve the Kirchhoff equation.
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... Shi and Hearst [15] derived a time-independent, one-dimensional nonlinear Schrödinger equation for the stationary state configurations of supercoiled DNA. Xue et al. [16] extended the Schrödinger equation to fit the noncircular Kirchhoff elastic rod by using the complex rigidity. Wang et al. [17,18] rebuilt the initial Kirchhoff equations in a complex style to suit the character of obvious asymmetry and the periodically varying bending coefficients, which is embodied on the cross-section by considering the mathematical background of DNA double helix, and introduced a complex form variable solution of the torque to obtain a simplified second ordinary differential equation 2 Mathematical Problems in Engineering with single variable. ...
... As a coarse-grained description, a DNA can be approximately regarded as a thin flexible and inextensible rod or string [12,15,16]. The classical theory of elasticity describes the geometry of an elastic rod in terms of its center line R = R( ) = ( ( ), ( ), ( )), three-dimensional curve parameterized by its arc-length . ...
... Phase space of(16). ...
Equilibrium Configurations of the Noncircular Cross-Section Elastic Rod Model with the Elliptic KB Method
Article
Full-text available
  • Mar 2015
  • MATH PROBL ENG
The mechanical deformation of DNA is very important in many biological processes. In this paper, we consider the reduced Kirchhoff equations of the noncircular cross-section elastic rod characterized by the inequality of the bending rigidities. One family of exact solutions is obtained in terms of rational expressions for classical Jacobi elliptic functions. The present solutions allow the investigation of the dynamical behavior of the system in response to changes in physical parameters that concern asymmetry. The effects of the factor on the DNA conformation are discussed. A qualitative analysis is also conducted to provide valuable insight into the topological configuration of DNA segments.
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... [14,15] Liu and Xue [11,16] studied thin elastic rod nonlinear mechanics of the DNA model. Xue et al. [17,18] developed the analytical mechanic theory of elastic rods. Furrer et al. [19] studied the relationship between the intrinsic shape and the existence of multiple stable equilibria within the context of DNA rings based on the Kirchhoff elastic rod model. ...
Growth induced buckling of morphoelastic rod in viscous medium
Article
Full-text available
  • May 2020
Biological growth is a common phenomenon in nature, and some organisms such as DNA molecules and bacterial filaments grow in viscous media. The growth induced instability of morphoelastic rod in a viscous medium is studied in this paper. Based on the Kirchhoff kinetic analogy method, the mechanical model for growing elastic thin rod in the viscous medium is established. A perturbation analysis is used to analyze the stability of the growing elastic rod in the viscous medium. We apply the results into planar growing ring and get its criterion of instability. Take the criterion into DNA ring to discuss the influence of viscous resistance on its instability.
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New effective bending rigidity and structural instability analysis of noncircular cross-section elastic rod model
Article
Full-text available
  • Dec 2021
Many flexible settings characterized by long, thin, and twisted structures exist in the natural world. They consistently follow some inherent principles of configuration transition, which are implicitly embodied in their structures. To understand such basic principles, the elastic rod model plays a key role in the mathematical analysis of a structure’s generic instabilities. In this article, we present a novel effective bending rigidity of the noncircular cross-section elastic rod model and an analytical strategy to reveal those inherent conformation principles. First, a transformation parameter indicates the correlation between bending and twisting variables on the cross-section, which converts the original system into a lower order system. Second, the new effective rigidity reflects the conformation information and improves consistency with numerical results. Third, a reduced-form Kirchhoff equation is obtained, which is coherent with the original system but expressed in a more compact form. Finally, bifurcation and stability analyses reveal the trivial and buckling conformations of the rod. These results will benefit further study of conformation analysis for noncircular rod models using analytical methods and possible biological applications under generic instabilities.
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The conserved quantities in general theorems ofthe Cosserat elastic rod dynamics
Article
  • Jan 2011
  • ACTA PHYS SIN-CH ED
Conserved quantities of the Cosserat elastic rod dynamics are studied according to the general theorems of dynamics. The rod dynamical equation takes the cross section of the rod as its objective of study and is expressed by two independent variables, the arc coordinate of the rod and the time, so the conserved quantities are written in the integral forms and there exist the arc coordinate conservation and the time conservation. The existence conditions and the formulas of conservations of momentum and moment of momentum are derived from the theorem of momentum and the theorem of moment of momentum respectively, which contain two cases of conserved quanties, one is the time and the other is arc coordinate. Also existence conditions and formulas of conservations of energy about time and are coordinate, which contain mechanical energy conservation, are derived from energy equations about the time and arc coordinate of the rod respectively. All of conservative motions of the rod are explained by examples. The conserved quantities in the integral form are of practical significance in both theoretical and numerical analysis for the Cosserat elastic rod dynamics.
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The analytical reduction of the Kirchhoff thin elastic rod model with asymmetric cross section
Article
  • Jan 2012
  • ACTA PHYS SIN-CH ED
The Kirchhoff thin elastic rod models and related systems are always the important basis to research the topology and stability of the flexible structures in not only the macroscopic but also microscopic scale. Firstly the initial Kirchhoff equations are rebuilt in a complex style to suit the character of obvious asymmetry embodied on the cross section by considering the mathematical background of DNA double helix. Then we introduce a complex form variable solution of the torque, and extend the knowledge of effective bending coefficients as well as its facility in the high dimensional system by using the complicated system. As the result, a simplified second order ordinary differential equation with single variable is obtained. Furthermore the periodically varying bending coefficients of the DNA molecular are considered as the appended components to the effective bending coefficients. The whole reduction process makes the numerical simulation become not solely the exclusively eligible approach, and produces adaptable channel to quantitative analysis.
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Dynamic instability of super-long elastic rod in viscous fluid
Article
  • May 2017
  • ACTA PHYS SIN-CH ED
The external environment affects the structural form of biological system. Many biological systems are surrounded by cell solutions, such as DNA and bacteria. The solution will offer a viscous resistance as the biological system moves in the viscous fluid. How does the viscous resistance affect the stability of biological system and what mode will be selected after instability? In this paper, we establish a super-long elastic rod model which contains the viscous resistance to model this phenomenon. The stability and instability of the super-long elastic rod in the viscous fluid are studied. The dynamic equations of motion of the super-long elastic rod in viscous fluid are given based on the Kirchhoff dynamic analogy. Then a coordinate basis vector perturbation scheme is reviewed. According to the new perturbation method, we obtain the first order perturbation representation of super-long elastic rod dynamic equation in the viscous fluid, which is a group of the second order linear partial differential equations. The stability of the super-long elastic rod can be determined by analyzing the solutions of the second order linear partial differential equations. The results are applied to a twisted planar DNA ring. The stability criterion of the twisted planar DNA ring and its critical region are obtained. The results show that the viscous resistance has no effect on the stability of super-long elastic rod dynamics, but affects its instability. The mode selection and the influence of the viscous resistance on the instability of DNA ring are discussed. The amplitude of the elastic loop becomes smaller under the influence of the viscous resistance, and a bifurcation occurs. The mode number of instability of DNA loop becomes bigger with the increase of viscous resistance.
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Mei symmetries and the Noether conserved quantities of super-thin elastic rod
Article
  • Nov 2009
  • ACTA PHYS SIN-CH ED
Based on Kirchhoff's analogue, generalized Hamilton canonical equations for the dynamics of super-thin elastic rod are analyzed. The Mei symmetry transformations and the theorem are introduced. And the condition and theorem of Noether conserved quantity are deduced directly by Mei symmetry for the super-thin elastic rod system. An example is given to illustrate the application of the result.
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The analytical reduction of the Kirchhoff thin elastic rod model with asymmetric cross section
Article
  • Mar 2012
The Kirchhoff thin elastic rod models and related systems are always the important basis to research the topology and stability of the flexible structures in not only the macroscopic but also microscopic scale. Firstly the initial Kirchhoff equations are rebuilt in a complex style to suit the character of obvious asymmetry embodied on the cross section by considering the mathematical background of DNA double helix. Then we introduce a complex form variable solution of the torque, and extend the knowledge of effective bending coefficients as well as its facility in the high dimensional system by using the complicated system. As the result, a simplified second order ordinary differential equation with single variable is obtained. Furthermore the periodically varying bending coefficients of the DNA molecular are considered as the appended components to the effective bending coefficients. The whole reduction process makes the numerical simulation become not solely the exclusively eligible approach, and produces adaptable channel to quantitative analysis.
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Topological configuration of Kirchhoff thin elastic ring rod with asymmetric cross section
Article
  • Jan 2013
Considering the forces and torques acted on the Kirchhoff thin elastic ring rod with asymmetric cross section, equivalent initial torque was introduced as a new concept to reveal the quantitative relationship between the state properties and parameters of torque dynamic model, which determines the configuration of the ring rod. As there exists formally a similarity between the potential energy density function and the Hamiltonian function, the quantitative relationship was obtained, which connects the equivalent initial torque, Hamiltonian function and initial link number. With DNA ring molecule as the physical background, this paper made the nonlinear dynamic analysis of torque model, and used undecided fundamental frequency method to obtain the asymptotic expressions for steady state periodic solutions around the practical equilibrium point. When the equivalent initial torque took different values, the graph of arc length-torque and phase portrait were drawn. The curvature could be expressed by the torque function with the equivalent bending stiffness considered, which provides a new methodological way to understand and describe the stable topological configuration of DNA ring under the influences of biological enzymes.
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Elastic rod model incorporating shear and extension, generalized nonlinear Schrödinger equations, and novel closed-form solutions for supercoiled DNA
Article
Full-text available
  • Aug 1995
We have derived a generalized one-dimensional time-independent nonlinear Schrödinger equation for the stationary state configurations of supercoiled DNA, based on an elastic rod model which includes deformations of bending, twisting, shear, and extension. Closed-form solutions for the axis of DNA have been obtained in terms of elliptic functions and elliptic integrals. These solutions describe the stationary state configurations of supercoiled DNA. © 1995 American Institute of Physics.
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The Kirchhoff elastic rod, the nonlinear Schr??dinger equation, and DNA supercoiling
Article
Full-text available
  • Sep 1994
We have derived a time‐independent, one‐dimensional nonlinear Schrödinger equation for the stationary state configurations of supercoiled DNA. The effect of DNA self‐contact has been included analytically. For the cases of non‐self‐contact and periodic boundary conditions, closed‐form solutions have been obtained which describe the stationary state configurations of supercoiled DNA.
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