Asymptotic analysis of linearly elastic shells: ‘Generalized membrane shells’

We consider a family of linearly elastic shells indexed by their half-thickness , all having the same middle surface % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadofacqGH9aqpcqaHvpGAcaGGOaGafqyYdCNbaebacaGGPaaa% aa!4317!\[S = \varphi (\bar \omega )\], with % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabew9aQjaacQdacuaHjpWDgaqeaiabgkOimlaadkfadaahaaWc% beqaaiaaikdaaaGccqGHsgIRcaWGsbWaaWbaaSqabeaacaaIZaaaaa% aa!4812!\[\varphi :\bar \omega \subset R^2 \to R^3 \], and clamped along a portion of their lateral face whose trace on S is % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabew9aQjaacIcacqaHZoWzdaWgaaWcbaGaaGimaaqabaGccaGG% Paaaaa!41EB!\[\varphi (\gamma _0 )\], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacIcacqaHZoWzdaWgaaWcbaGaaGimaaqabaGccaGGPaaaaa!401F!\[(\gamma _0 )\] is a fixed portion of with length % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacIcacqaHZoWzdaWgaaWcbaGaaGimaaqabaGccaGGPaGaeyOp% a4JaaGimaaaa!41E1!\[(\gamma _0 ) > 0\]. Let % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacIcacqaHZoWzdaWgaaWcbaGaeqySdeMaeqOSdigabeaakiaa% cIcacqaH3oaAcaGGPaGaaiykaaaa!45AA!\[(\gamma _{\alpha \beta } (\eta ))\] be the linearized strain tensor of S. We make an essential geometric and kinematic assumption, according to which the semi-norm % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacYhacqGHflY1caGG8bWaa0baaSqaaiabeM8a3bqaaiaad2ea% aaaaaa!4345!\[| \cdot |_\omega ^M \] defined by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacYhacqaH3oaAcaGG8bWaa0baaSqaaiabeM8a3bqaaiaad2ea% aaGccqGH9aqpdaGadeqaamaaqababaGaaiiFaiaacYhaaSqaaiabeg% 7aHfrbbjxAHXgaiuaacaWFSaGaeqOSdigabeqdcqGHris5aOGaeq4S% dCMaeqySdeMaeqOSdiMaaiikaiabeE7aOjaacMcacaGG8bGaaiiFam% aaDaaaleaacaWGmbWaaWbaaWqabeaacaaIYaaaaSGaaiikaiabeM8a% 3jaacMcaaeaacaaIYaaaaaGccaGL7bGaayzFaaWaaWbaaSqabeaaca% aIXaGaai4laiaaikdaaaaaaa!61F1!\[|\eta |_\omega ^M = \left\{ {\sum\nolimits_{\alpha ,\beta } {||} \gamma \alpha \beta (\eta )||_{L^2 (\omega )}^2 } \right\}^{1/2} \] is a norm over the space % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadAfacaGGOaGaeqyYdCNaaiykaiabg2da9maacmqabaGaeq4T% dGMaeyicI4SaamisamaaCaaaleqabaGaaGymaaaakiaacIcacqaHjp% WDcaGGPaGaai4oaiabeE7aOjabg2da9iaab+gacaqGUbGaeq4SdC2a% aSbaaSqaaiaabcdaaeqaaaGccaGL7bGaayzFaaaaaa!5361!\[V(\omega ) = \left\{ {\eta \in H^1 (\omega );\eta = {\text{on}}\gamma _{\text{0}} } \right\}\], excluding however the already analyzed membrane shells, where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabeo7aNnaaBaaaleaacaqGWaaabeaakiabg2da9iabgkGi2kab% eM8a3baa!42F8!\[\gamma _{\text{0}} = \partial \omega \] and S is elliptic. This new assumption is satisfied for instance if % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabeo7aNnaaBaaaleaacaqGWaaabeaakiabgcMi5kabgkGi2kab% eM8a3baa!43B9!\[\gamma _{\text{0}} \ne \partial \omega \] and S is elliptic, or if S is a portion of a hyperboloid of revolution.We then show that, as 0, the averages % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaalaaabaGaaGymaaqaaiaaikdadaahaaWcbeqaaiabew7aLbaa% aaGcdaWdXaqaaiaadwhadaqhaaWcbaGaamyAaaqaaiabew7aLbaaki% aabsgacaWG4bWaa0baaSqaaiaaiodaaeaacqaH1oqzaaaabaGaeyOe% I0IaeqyTdugabaGaeqyTduganiabgUIiYdaaaa!4E28!\[\frac{1}{{2^\varepsilon }}\int_{ - \varepsilon }^\varepsilon {u_i^\varepsilon {\text{d}}x_3^\varepsilon } \] across the thickness of the shell of the covariant components u _i of the displacement of the points of the shell strongly converge in the completion V _M ^#() of V() with respect to the norm % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacYhacqGHflY1caGG8bWaa0baaSqaaiabeM8a3bqaaiaad2ea% aaaaaa!4345!\[| \cdot |_\omega ^M \], toward the solution of a generalized membrane shell problem. This convergence result also justifies the recent formal asymptotic approach of D. Caillerie and E. Sanchez-Palencia.The limit problem found in this fashion is sensitive, according to the terminology recently introduced by J.L. Lions and E. Sanchez-Palencia, in the sense that it possesses two unusual features: it is posed in a space that is not necessarily contained in a space of distributions, and its solution is highly sensitive to arbitrarily small smooth perturbations of the data.Under the same assumption, we also show that the average % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaalaaabaGaaGymaaqaaiaaikdadaahaaWcbeqaaiabew7aLbaa% aaGcdaWdXaqaaiaadwhadaqhaaWcbaGaamyAaaqaaiabew7aLbaaki% aabsgacaWG4bWaa0baaSqaaiaaiodaaeaacqaH1oqzaaaabaGaeyOe% I0IaeqyTdugabaGaeqyTduganiabgUIiYdaaaa!4E28!\[\frac{1}{{2^\varepsilon }}\int_{ - \varepsilon }^\varepsilon {u_i^\varepsilon {\text{d}}x_3^\varepsilon } \] where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadwhadaahaaWcbeqaaiabew7aLbaakiabg2da9iaacIcacaWG% 1bWaa0baaSqaaiaadMgaaeaacqaH1oqzaaGccaGGPaaaaa!452C!\[u^\varepsilon = (u_i^\varepsilon )\], and the solution % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabe67a4naaCaaaleqabaGaeqyTdugaaOGaeyicI4SaamOvamaa% BaaaleaacaWGlbaabeaakiaacIcacqaHjpWDcaGGPaaaaa!465B!\[\xi ^\varepsilon \in V_K (\omega )\] of Koiter's equations have the same principal part as 0 in the same space V _M () as above. For such generalized membrane shells, the two-dimensional shell model of W.T. Koiter is thus likewise justified.We also treat the case where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacYhacqGHflY1caGG8bWaa0baaSqaaiabeM8a3bqaaiaad2ea% aaaaaa!4345!\[| \cdot |_\omega ^M \] is no longer a norm over V(), but is a norm over the space % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadAfadaWgaaWcbaGaam4saaqabaGccaGGOaGaeqyYdCNaaiyk% aiabg2da9maacmqabaGaeq4TdGMaeyypa0JaaiikaiabeE7aOnaaBa% aaleaacaWGPbaabeaakiaacMcacqGHiiIZcaWGibWaaWbaaSqabeaa% caaIXaaaaOGaaiikaiabeM8a3jaacMcacqGHxdaTcaWGibWaaWbaaS% qabeaacaaIYaaaaOGaaiikaiabeM8a3jaacMcacaGG7aGaeq4TdG2a% aSbaaSqaaiaadMgaaeqaaOGaeyypa0JaeyOaIy7aaSbaaSqaaiaadA% haaeqaaOGaeq4TdG2aaSbaaSqaaiaaiodaaeqaaOGaeyypa0JaaGim% aiGac+gacaGGUbGaeq4SdC2aaSbaaSqaaiaaicdaaeqaaaGccaGL7b% GaayzFaaaaaa!68B8!\[V_K (\omega ) = \left\{ {\eta = (\eta _i ) \in H^1 (\omega ) \times H^2 (\omega );\eta _i = \partial _v \eta _3 = 0\operatorname{on} \gamma _0 } \right\}\], thus also excluding the already analyzed flexural shells. Then a convergence theorem can still be established, but only in the completion of the quotient space V()/V ₀() with repect to % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaacYhacqGHflY1caGG8bWaa0baaSqaaiabeM8a3bqaaiaad2ea% aaaaaa!4345!\[| \cdot |_\omega ^M \], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadAfadaWgaaWcbaGaaGimaaqabaGccaGGOaGaeqyYdCNaaiyk% aiabg2da9maacmqabaGaeq4TdGMaeyicI4SaamOvaiaacIcacqaHjp% WDcaGGPaGaai4oaiabeo7aNjabeg7aHjabek7aIjaacIcacqaH3oaA% caGGPaGaeyypa0JaaeimaiaabMgacaqGUbGaeqyYdChacaGL7bGaay% zFaaaaaa!5997!\[V_0 (\omega ) = \left\{ {\eta \in V(\omega );\gamma \alpha \beta (\eta ) = {\text{0in}}\omega } \right\}\].These convergence results, together with those that we already obtained for membrane and flexural shells, jointly with B. Miara in the second case, thus constitute an asymptotic analysis of linearly elastic shells in all possible cases.     

A non-linear seales method for strongly non-linear oscillators

A non-linear seales method is presented for the analysis of strongly non-linear oseillators of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbdiqb-Hha4zaadaGaey4kaSIa% am4zaiaacIcacqWF4baEcaGGPaGae8xpa0JaeqyTduMaamOzaiaacI% cacqWF4baEcqWFSaalcuWF4baEgaGaaiaabMcaaaa!4FEC!\[\ddot x + g(x) = \varepsilon f(x,\dot x{\text{)}}\], where g(x) is an arbitrary non-linear function of the displacement x. We assumed that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbdiab-Hha4jaacIcacqWF0baD% cqWFSaalcqaH1oqzcaGGPaGaeyypa0Jae8hEaG3aaSbaaSqaaiaaic% daaeqaaOGaaiikaiabe67a4jaacYcacqaH3oaAcaGGPaGaey4kaSYa% aabmaeaacqaH1oqzdaahaaWcbeqaaiaad6gaaaaabaGaamOBaiabg2% da9iaaigdaaeaacaWGTbGaeyOeI0IaaGymaaqdcqGHris5aOGae8hE% aG3aaSbaaSqaaiab-5gaUbqabaGccaGGOaGaeqOVdGNaaiykaiabgU% caRiaad+eacaGGOaGaeqyTdu2aaWbaaSqabeaacaWGTbaaaOGaaiyk% aaaa!67B9!\[x(t,\varepsilon ) = x_0 (\xi ,\eta ) + \sum\nolimits_{n = 1}^{m - 1} {\varepsilon ^n } x_n (\xi ) + O(\varepsilon ^m )\], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaabsgacqaH+oaEcaGGVaGaaeizaiaadshacqGH9aqpdaaeWaqa% aiabew7aLnaaCaaaleqabaGaamOBaaaaaeaacaWGUbGaeyypa0JaaG% ymaaqaaiaad2gaa0GaeyyeIuoakiaadkfadaWgaaWcbaGaamOBaaqa% baGccaGGOaGaeqOVdGNaaiykaaaa!4FFC!\[{\text{d}}\xi /{\text{d}}t = \sum\nolimits_{n = 1}^m {\varepsilon ^n } R_n (\xi )\], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaabsgacqaH3oaAcaGGVaGaaeizaiaadshacqGH9aqpdaaeWaqa% aiabew7aLnaaCaaaleqabaGaamOBaaaaaeaacaWGUbGaeyypa0JaaG% imaaqaaiaad2gaa0GaeyyeIuoakiaadofadaWgaaWcbaGaamOBaaqa% baGccaGGOaGaeqOVdGNaaiilaiabeE7aOjaacMcaaaa!5241!\[{\text{d}}\eta /{\text{d}}t = \sum\nolimits_{n = 0}^m {\varepsilon ^n } S_n (\xi ,\eta )\], and R _n,S _nare to be determined in the course of the analysis. This method is suitable for the systems with even non-linearities as well as with odd non-linearities. It can be viewed as a generalization of the two-variable expansion procedure. Using the present method we obtained a modified Krylov-Bogoliubov method. Four numerical examples are presented which served to demonstrate the effectiveness of the present method.     

Nonlinear combination resonances in cantilever composite plates

We experimentally investigated nonlinear combination resonances in two graphite-epoxy cantilever plates having the configurations (90/30/-30/-30/30/90)_s and (-75/75/75/-75/75/-75)_s. As a first step, we compared the natural frequencies and modes shapes obtained from the finite-element and experimental-modal analyses. The largest difference in the obtained frequencies for both plates was 6%. Then, we transversely excited the plates and obtained force-response and frequency-response curves, which were used to characterize the plate dynamics. We acquired time-domain data for specific input conditions using an A/D card and used them to generate time traces, power spectra, pseudo-state portraits, and Poincaré maps. The data were obtained with an accelerometer monitoring the excitation and a laser vibrometer monitoring the plate response. We observed the external combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kabeM8a3naaBaaaleaacaaIYaaabeaakiab% gUcaRiabeM8a3naaBaaaleaacaaI3aaabeaaaaa!45C9!\[\Omega \approx \omega _2 + \omega _7 \] in the quasi-isotropic plate and the external combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kaacIcacaaIXaGaai4laiaaikdacaGGPaGa% aiikaiabeM8a3naaBaaaleaacaaIYaaabeaakiabgUcaRiabeM8a3n% aaBaaaleaacaaI1aaabeaakiaacMcaaaa!4AAD!\[\Omega \approx (1/2)(\omega _2 + \omega _5 )\] and the internal combination resonance % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfM6axjabgIKi7kabeM8a3naaBaaaleaacaaI4aaabeaakiab% gIKi7kaacIcacaaIXaGaai4laiaaikdacaGGPaGaaiikaiabeM8a3n% aaBaaaleaacaaIYaaabeaakiabgUcaRiabeM8a3naaBaaaleaacaaI% XaGaaG4maaqabaGccaGGPaaaaa!4FDC!\[\Omega \approx \omega _8 \approx (1/2)(\omega _2 + \omega _{13} )\] in the ±75 plate, where the % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabeM8a3naaBaaaleaacaWGPbaabeaaaaa!3F16!\[\omega _i \] are the natural frequencies of the plate and is the excitation frequency. The results show that a low-amplitude high-frequency excitation can produce a high-amplitude low-frequency motion.     

Global solution to the incompressible viscous-multipolar material problem

In the paper we give a proof of the global existence of the weak solution to the initial-boundary-value problem describing an incompressible elasto-viscous-multipolar material in finite geometry. A brief introduction to the physical background of viscous-multipolar materials is given. We suggest the hypothesis% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaaxadabaGaeu4OdmfaleaacaWGPbGaaiilaGqaciaa-bcacaWG% QbGaa8hiaiabg2da9iaa-bcacaaIXaaabaGaaG4maaaakiaa-bcada% abdiqcaasaaOWaaSaaaKaaGeaacqGHciITcqqHOoqwcaGGOaGaamOr% aiaacYcacqaH4oqCcaGGPaaabaGaeyOaIyRaamOraOWaaSbaaSqaai% aadMgacaWGbbaabeaaaaqcaaIaa8hiaiaadAeakmaaBaaaleaacaWG% QbGaamyqaaqabaaajaaqcaGLhWUaayjcSdGaa8hiaiabgsMiJkaado% gakmaaBaaaleaacaWFVbaabeaakiaadwgacaGGOaGaamOraiaacYca% ieaacaGFGaGaeqiUdeNaaiykaiaa+bcacqGHRaWkcaGFGaGaam4yam% aaBaaaleaacaaIXaGaa4hiaiaacYcaaeqaaaaa!686E!\[\mathop \Sigma \limits_{i, j = 1}^3 \left| {\frac{{\partial \Psi (F,\theta )}}{{\partial F_{iA} }} F_{jA} } \right| \leqslant c_o e(F, \theta ) + c_{1 ,} \] which enables one to obtain a priori estimates.     

On Saint-Venant's principle in linear elastodynamics

We investigate the spatial behaviour of the steady state and transient elastic processes in an anisotropic elastic body subject to nonzero boundary conditions only on a plane end. For the transient elastic processes, it is shown that at distance x ₃ >ct from the loaded end, (c is a positive computable constant and t is the time), all the activity in the body vanishes. For x ₃ , an appropriate measure of the elastic process decays with the distance from the loaded end, the decay rate of end effects being controlled by the factor % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaKazaaiacaGGOaGccaaIXaGaaeiiaiabgkHiTiaabccadaWcaaqa% amXvP5wqonvsaeHbfv3ySLgzaGqbciab-Hha4naaBaaaleaacaqGZa% aabeaaaOqaaiaabogacaqG0baaaKazaakacaGGPaaaaa!4BB0!\[(1{\text{ }} - {\text{ }}\frac{{x_{\text{3}} }}{{{\text{ct}}}})\]. Next, it is shown that for isotropic materials, in the case of a steady state vibration, an analogue of the Phragmén-Lindelöf principle holds for an appropriate cross-sectional measure. One immediate consequence is that in the class of steady state vibrations for which a quasi-energy volume measure is bounded, this measure decays at least algebraically with the distance from the loaded end.     

Limit cycle analysis of a class of strongly nonlinear oscillation equations

The limit cycle of a class of strongly nonlinear oscillation equations of the form % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiqadwhagaWaaiabgUcaRmXvP5wqonvsaeHbbjxAHXgiofMCY92D% aGqbciab-DgaNjab-HcaOiaadwhacqWFPaqkcqWF9aqpcqaH1oqzca% WGMbGaaiikaiaadwhacaGGSaGabmyDayaacaGaaiykaaaa!50B8!\[\ddot u + g(u) = \varepsilon f(u,\dot u)\] is investigated by means of a modified version of the KBM method, where is a positive small parameter. The advantage of our method is its straightforwardness and effectiveness, which is suitable for the above equation, where g(u) need not be restricted to an odd function of u, provided that the reduced equation, corresponding to =0, has a periodic solution. A specific example is presented to demonstrate the validity and accuracy of our 09 method by comparing our results with numerical ones, which are in good agreement with each other even for relatively large .     

On the nonlinear slewing dynamics and control of the Space Station based Mobile Servicing System

A relatively general Lagrangian formulation for studying the nonlinear dynamics and control of space-craft with interconnected flexible members in a tree-type topology is developed. Versatility of the formulation is illustrated through a dynamical study of the Space Station based two-link Mobile Servicing System (MSS). The performance of the MSS undergoing inplane and out-of-plane slewing maneuvers is compared. Results indicate that, in absence of control, the maneuvers induce undesirable librational motion of the Space Station as well as vibration of the links. Nonlinear control, based on the Feedback Linearization Technique (FLT), appears promising. Quasi-Closed Loop Control (QCLC), a variation of the FLT, is applied to control the libration of the Space Station. Once the attitude of the Space Station is controlled, the performance of the MSS improves significantly. For a 5-minute maneuver of the MSS, the maximum control torque required is only 34.5 Nm.Nomenclature f _i ¹ , f _i,j ¹ fundamental frequency of bodies B _i and B _i,j, respectively - l _c, l _i, l _i,j length of bodies B _c, B _i, and B _i,j, respectively - m _c, m _i, m _i,j mass of bodies B _c, B _i, and B _i,j, respectively - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaerbhv2BYDwAHbacfiGab8xCayaaraqefavySfgDP52BGWuAU9gD% 5bxzaGGbciaa+zgacaWFSaGaa8hiaiqa-fhagaqeaiaa-jhaaaa!4B1F!\[\bar qf, \bar qr\] vector representing flexible and rigid generalized coordinates - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaerbhv2BYDwAHbacfiGaa8hkaiqa-fhagaqeaiaa-jhacaWFPaqe% favySfgDP52BGWuAU9gD5bxzaGGbciaa+rgaaaa!4A18!\[(\bar qr)d\] vector representing the desired rigid generalized coordinates - (I _xx)k, (I _yy)k, (I _zz)k principal inertia of body B _k about X _k, Y _k, and Z _k axes, respectively; ksc, i or i, j - K _p, K _v displacement and velocity gain matrices - N _q total number of generalized coordinates - % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaerbwvMCKfMBHbacfiGab8xuayaaraqefavySfgDP52BGWuAU9gD% 5bxzaGGbciaa+zgaieaacaqFSaGaa0hiaiqa-ffagaqeaGqaciaa8j% haaaa!4AEF!\[\bar Qf, \bar Qr\] control effort vectors for flexible and rigid coordinates, respectively - Q , Q , Q control effort for pitch, roll and yaw degree of freedom, respectively - _k ^y , _k ^z tip deflection of a beam type appendage (B _k) in the Y _k and Z _k directions, respectively.     

Radiation conditions in asymmetric elasticity

In the present paper the radiation conditions of the Sommerfield type for a linear homogeneous and isotropic micropolar elasticity are discussed. A regular solution (u, ) of the fundamental system of field equations in an infinite domain has been defined using the radiation conditions for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% WbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGWbGaaiykaaqabaGccaGGPaaaaa!4834!\[(u_{(p)} , + \varphi _{(p)} )\] and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% ZbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGPaaaaa!483A!\[(u_{(s)} , + \varphi _{(s)} )\], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaadwhacqGH9aqpcaWG1bWaaSbaaSqaaiaa% cIcacaWGWbGaaiykaaqabaGccqGHRaWkcaWG1bWaaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGSaGaaeiiaiabgEGirlabgEna0kaa% dwhadaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9iaaic% dacaGGSaGaaeiiaiabgEGirhrbmv3yPrwyGm0BUn3BSvgaiyGacaWF% 1bWaaSbaaSqaaGqaciaa+HcacaGFZbGaa4xkaaqabaGccqGH9aqpca% aIWaGaaiilaiaabccacqaHvpGAcqGH9aqpcqaHvpGAdaWgaaWcbaGa% aiikaiaadchacaGGPaaabeaakiabgUcaRiabew9aQnaaBaaaleaaca% GGOaGaam4CaiaacMcaaeqaaOGaaiilaiaabccacqGHhis0cqGHxdaT% cqaHvpGAdaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9i% aaicdacaGGSaGaaeiiaiabgEGirlabew9aQnaaBaaaleaacaqGOaGa% ae4CaiaabMcaaeqaaOGaeyypa0JaaGimaaaa!809B!\[u = u_{(p)} + u_{(s)} ,{\text{ }}\nabla \times u_{(p)} = 0,{\text{ }}\nabla u_{(s)} = 0,{\text{ }}\varphi = \varphi _{(p)} + \varphi _{(s)} ,{\text{ }}\nabla \times \varphi _{(p)} = 0,{\text{ }}\nabla \varphi _{{\text{(s)}}} = 0\], and formulae of the Betti type for an infinite domain with a cavity have been derived.

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Zusammenfassung Betrachtet werden die Ausstrahlungsbedingungen der Sommerfeldschen Art für lineare homogene mikropolare Elastizitätstheorie. Die reguläre Lösung (u, ) der grundlegenden Gleichungen für einen unendlichen Raum wird mit Hilfe der Bedingungen für Paare % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% WbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGWbGaaiykaaqabaGccaGGPaaaaa!4834!\[(u_{(p)} , + \varphi _{(p)} )\] und % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% ZbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGPaaaaa!483A!\[(u_{(s)} , + \varphi _{(s)} )\] definiert, wobei % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaadwhacqGH9aqpcaWG1bWaaSbaaSqaaiaa% cIcacaWGWbGaaiykaaqabaGccqGHRaWkcaWG1bWaaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGSaGaaeiiaiabgEGirlabgEna0kaa% dwhadaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9iaaic% dacaGGSaGaaeiiaiabgEGirhrbmv3yPrwyGm0BUn3BSvgaiyGacaWF% 1bWaaSbaaSqaaGqaciaa+HcacaGFZbGaa4xkaaqabaGccqGH9aqpca% aIWaGaaiilaiaabccacqaHvpGAcqGH9aqpcqaHvpGAdaWgaaWcbaGa% aiikaiaadchacaGGPaaabeaakiabgUcaRiabew9aQnaaBaaaleaaca% GGOaGaam4CaiaacMcaaeqaaOGaaiilaiaabccacqGHhis0cqGHxdaT% cqaHvpGAdaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9i% aaicdacaGGSaGaaeiiaiabgEGirlabew9aQnaaBaaaleaacaqGOaGa% ae4CaiaabMcaaeqaaOGaeyypa0JaaGimaaaa!809B!\[u = u_{(p)} + u_{(s)} ,{\text{ }}\nabla \times u_{(p)} = 0,{\text{ }}\nabla u_{(s)} = 0,{\text{ }}\varphi = \varphi _{(p)} + \varphi _{(s)} ,{\text{ }}\nabla \times \varphi _{(p)} = 0,{\text{ }}\nabla \varphi _{{\text{(s)}}} = 0\] ist. Es wird gezeigt, dass derartige reguläre Lösung mit Hilfe der Formel Bettischer Art für einem unendlichen Raum mit einem Hohlraum dargestellt werden kann.

     

A two dimensional crack problem for an elastic layer bonded to an infinite elastic medium

In this paper we consider the problem of determining the distribution of stress in the neighbourhood of a crack in an infinitely long strip bonded to semi-infinite elastic planes on either side. By the use of Fourier transforms we reduce the problem to solving a single Fredholm integral equation of the second kind. Analytical expressions up to the order of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabes7aKnaaCaaaleqabaGaeyOeI0IaaGym% aiaaicdaaaaaaa!41AF!\[\delta ^{ - 10} \], where 2 is the thickness of the strip for 1 are derived for the shape of the deformed crack and for the crack energy. Some numerical results have been displayed graphically.

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Zusammenfassung In dieser Arbeit betrachten wir das Problem der Spannungsverteilung in der Nachbarschaft eines Sprunges auf ethem unendlich langen Band welches an beiden Seiten an halbseitig-unendliche elastische Platten aufgeheftet ist. Mit Hilfe von Fourier-Transformationen reduzieren wir das Problem zu einer einzelnen Fredholm Integralgleichung der zweiten Art. Für die Sprung-Energie und die Gestalt des deformierten Sprunges leiten wir analytische Ausdrücke bis zur Ordnung % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabes7aKnaaCaaaleqabaGaeyOeI0IaaGym% aiaaicdaaaaaaa!41AF!\[\delta ^{ - 10} \] her, wobei 2 für 1 die Dicke des Bandes ist. Einige numerische Resultate haben wir graphisch veranschaulicht.

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This work was supported by National Research Council of Canada through NRC-Grant No. A4177. This work was completed while the author was visiting the University of Glasgow.     

Chaotic motions of a rigid rotor in short journal bearings

In the present paper the conditions that give rise to chaotic motions in a rigid rotor on short journal bearings are investigated and determined. A suitable symmetry was given to the rotor, to the supporting system, to the acting system of forces and to the system of initial conditions, in order to restrict the motions of the rotor to translatory whirl. For an assigned distance between the supports, the ratio between the transverse and the polar mass moments of the rotor was selected conveniently small, with the aim of avoiding conical instability. Since the theoretical analysis of a system's chaotic motions can only be carried out by means of numerical investigation, the procedure here adopted by the authors consists of numerical integration of the rotor's equations of motion, with trial and error regarding the three parameters that characterise the theoretical model of the system: m, the half non-dimensional mass of the rotor, , the modified Sommerfeld number relating to the lubricated bearings, and , the dimensionless value of rotor unbalance. In the rotor's equations of motion, the forces due to the lubricating film are written under the assumption of isothermal and laminar flow in short bearings. The number of numerical trials needed to find the system's chaotic responses has been greatly reduced by recognition of the fact that chaotic motions become possible when the value of the dimensionless static eccentricity % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabew7aLnaaBaaaleaacaWGZbaabeaaaaa!4046!\[\varepsilon _s \] is greater than 0.4. In these conditions, non-periodic motions can be obtained even when rotor unbalance values are not particularly high (=0.05), whereas higher values (>0.4) make the rotor motion periodic and synchronous with the driving rotation. The present investigation has also identified the route that leads an assigned rotor to chaos when its angular speed is varied with prefixed values of the dimensionless unbalance . The theoretical results obtained have then been compared with experimental data. Both the theoretical and the experimental data have pointed out that in the circumstances investigated chaotic motions deserve more attention, from a technical point of view, than is normally ascribed to behaviours of this sort. This is mainly because such behaviours are usually considered of scarce practical significance owing to the typically bounded nature of chaotic evolution. The present analysis has shown that when the rotor exhibits chaotic motions, the centres of the journals describe orbits that alternate between small and large in an unpredictable and disordered manner. In these conditions the thickness of the lubricating film can assume values that are extremely low and such as to compromise the efficiency of the bearings, whereas the rotor is affected by inertia forces that are so high as to determine severe vibrations of the supports.Nomenclature C radial clearance of bearing (m) - D diameter of bearing (m) - e dimensional eccentricity of journal (m) - e _s value of e corresponding to the static position of the journal - E dimensional static unbalance of rotor (m) - f _x, f _y =F _x/(P), F _y/(P): non-dimensional components of fluid film force - F _x, F _y dimensional components of fluid film force (N) - g acceleration of gravity (m/s²) - L axial length of bearing (m) - m % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiabg2da9maalaaabaGaeqyYdC3aaWbaaSqa% beaacaaIYaaaaaGcbaGaeqyYdC3aa0baaSqaaGabciaa-bdaaeaaca% WFYaaaaaaakiabg2da9maalaaabaGaeqyYdC3aaWbaaSqabeaacaaI% YaaaaOGaam4qaaqaaiabeo8aZjaadEgaaaaaaa!4C14!\[ = \frac{{\omega ^2 }}{{\omega _0^2 }} = \frac{{\omega ^2 C}}{{\sigma g}}\]: half non-dimensional mass of rotor - M half mass of rotor (kg) - n angular speed of rotor (in r.p.m.=60/2) - t time     

The proof of Fermat's Last Theorem

ＴＨＥＰＲＯＯＦＯＦＦＥＲＭＡＴ＇ＳＬＡＳＴＴＨＥＯＲＥＭＷｏｎｇＣｈｉａｈｏ（汪家訸）（ＲｅｃｅｉｖｅｄＡｐｒｉｌ１０,１９９５）Ａｂｓｔｒａｃｔ：（ｉ）Ｉｎｓｔｅａｄｏｆｘ￣ｎ＋ｙ￣ｎ＝ｚ￣ｎ,ｗｅｕｓｅａｓｔｈｅｇｅｎｅｒａｌｅｑｕａｔｉｏｎｏ...     

Boundary conditions at the edge of a thin or thick plate bonded to an elastic support

At the clamped edge of a thin plate, the interior transverse deflection ω(x ₁, x₂) of the mid-plane x ₃=0 is required to satisfy the boundary conditions ω=?ω/?n=0. But suppose that the plate is not held fixed at the edge but is supported by being bonded to another elastic body; what now are the boundary conditions which should be applied to the interior solution in the plate? For the case in which the plate and its support are in two-dimensional plane strain, we show that the correct boundary conditions for ω must always have the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakqaabeqaaiaabEhacaqGTaWa% aSaaaeaacaGG0aGaae4vamaaCaaaleqabaGaamOqaaaaaOqaaiaaco% dadaqadaqaaiaacgdacqGHsislcaqG2baacaGLOaGaayzkaaaaaiaa% bIgadaahaaWcbeqaaiaackdaaaGcdaWcaaqaaiaabsgadaahaaWcbe% qaaiaackdaaaGccaqG3baabaGaaeizaiaabIhafaqabeGabaaajaaq% baqcLbkacaGGYaaajaaybaqcLbkacaGGXaaaaaaakiabgUcaRmaala% aabaGaaiinaiaabEfadaahaaWcbeqaaiaadAeaaaaakeaacaGGZaWa% aeWaaeaacaGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGOb% WaaWbaaSqabeaacaGGZaaaaOWaaSaaaeaacaqGKbWaaWbaaSqabeaa% caGGZaaaaOGaae4DaaqaaiaabsgacaqG4bqcaaubaeqabiqaaaqcaa% saaiaacodaaKaaafaajugGaiaacgdaaaaaaOGaeyypa0Jaaiimaiaa% cYcaaeaadaWcaaqaaiaabsgacaqG3baabaGaaeizaiaabIhaliaacg% daaaGccqGHsisldaWcaaqaaiaacsdacqqHyoqudaahaaWcbeqaaiaa% bkeaaaaakeaacaGGZaWaaeWaaeaacaGGXaGaeyOeI0IaaeODaaGaay% jkaiaawMcaaaaacaqGObWaaSaaaeaacaqGKbWaaWbaaSqabeaacaGG% YaaaaOGaae4DaaqaaiaabsgacaqG4bqbaeqabiqaaaqcaauaaKqzGc% GaaiOmaaqcaawaaKqzGcGaaiymaaaaaaGccqGHRaWkdaWcaaqaaiaa% csdacqqHyoqudaahaaWcbeqaaiaabAeaaaaakeaacaGGZaWaaeWaae% aacaGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGObWaaWba% aSqabeaacaGGYaaaaOWaaSaaaeaacaqGKbWaaWbaaSqabeaacaGGZa% aaaOGaae4DaaqaaiaabsgacaqG4bqcaaubaeqabiqaaaqcaasaaiaa% codaaKaaafaajugGaiaacgdaaaaaaOGaeyypa0JaaiimaiaacYcaaa% aa!993A!\[\begin{gathered}{\text{w - }}\frac{{4{\text{W}}^B }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^2 \frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} + \frac{{4{\text{W}}^F }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^3 \frac{{{\text{d}}^3 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}3 \\1 \\\end{array} }} = 0, \hfill \\\frac{{{\text{dw}}}}{{{\text{dx}}1}} - \frac{{4\Theta ^{\text{B}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}\frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} + \frac{{4\Theta ^{\text{F}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}^2 \frac{{{\text{d}}^3 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}3 \\1 \\\end{array} }} = 0, \hfill \\\end{gathered}\]with exponentially small error as L/h→∞, where 2h is the plate thickness and L is the length scale of ω in the x ₁-direction. The four coefficients W ^B, W^F, Θ ^B , Θ ^F are computable constants which depend upon the geometry of the support and the elastic properties of the support and the plate, but are independent of the length of the plate and the loading applied to it. The leading terms in these boundary conditions as L/h→∞ (with all elastic moduli remaining fixed) are the same as those for a thin plate with a clamped edge. However by obtaining asymptotic formulae and general inequalities for Θ ^B , W ^F, we prove that these constants take large values when the support is ‘soft’ and so may still have a strong influence even when h/L is small. The coefficient W ^F is also shown to become large as the size of the support becomes large but this effect is unlikely to be significant except for very thick plates. When h/L is small, the first order corrected boundary conditions are w=0,% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaadaWcaaqaaiaabsgacaqG% 3baabaGaaeizaiaabIhaliaacgdaaaGccqGHsisldaWcaaqaaiaacs% dacqqHyoqudaahaaWcbeqaaiaabkeaaaaakeaacaGGZaWaaeWaaeaa% caGGXaGaeyOeI0IaaeODaaGaayjkaiaawMcaaaaacaqGObWaaSaaae% aacaqGKbWaaWbaaSqabeaacaGGYaaaaOGaae4DaaqaaiaabsgacaqG% 4bqbaeqabiqaaaqcaauaaKqzGcGaaiOmaaqcaawaaKqzGcGaaiymaa% aaaaGccqGH9aqpcaGGWaGaaiilaaaa!5DD4!\[\frac{{{\text{dw}}}}{{{\text{dx}}1}} - \frac{{4\Theta ^{\text{B}} }}{{3\left( {1 - {\text{v}}} \right)}}{\text{h}}\frac{{{\text{d}}^2 {\text{w}}}}{{{\text{dx}}\begin{array}{*{20}c}2 \\1 \\\end{array} }} = 0,\]which correspond to a hinged edge with a restoring couple proportional to the angular deflection of the plate at the edge.     

The semi-infinite strip x≥0, −1≤y≤1; completeness of the Papkovich-Fadle eigenfunctions when Φxx(0,y), Φyy(0,y) are prescribed

For the problem of bending of a semi-infinite strip x0, –1y1, with the sides y=±1 clamped, we give a proof that the end-data% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaarmWu51MyVXgaiuGacqWFgpGzdaWgaaWcbaGaaeiEaiaabIha% aeqaaGqbaOGae4hiaaIaaiikaiaaicdacaGGSaGae4hiaaIaamyEai% aacMcacqGFGaaicqGH9aqpcqGFGaaicaWGMbGaaiikaiaadMhacaGG% PaGaaiilaaqaaiab-z8aMnaaBaaaleaacaqG5bGaaeyEaaqabaGccq% GFGaaicaGGOaGaaGimaiaacYcacqGFGaaicaWG5bGaaiykaiab+bca% Giabg2da9iab+bcaGiaadAgacaGGOaGaamyEaiaacMcacaGGSaaaaa% a!5D6D!\[\begin{array}{l} \phi _{{\rm{xx}}} (0, y) = f(y), \\ \phi _{{\rm{yy}}} (0, y) = f(y), \\ \end{array}\] where f(y), g(y) are arbitrary independent functions prescribed on (–1,1), may be expanded as a series of the bi-orthogonal Papkovich-Fadle eigenfunctions for the strip. This represents an advance on the standard work of R. T. C. Smith [6], who proved such an expansion, but under conditions which are often not satisfied in practice. In particular we are able to solve this bi-harmonic boundary value problem when f, g do not satisfy the side conditions% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaacaWGMbGaaiikaiabgglaXkaaigdacaGGPaqedmvETj2BSbac% faGae8hiaaIaeyypa0Jae8hiaaIaamOzamaaCaaaleqabaGaai4jaa% aakiab-bcaGiaacIcacqGHXcqScaaIXaGaaiykaiab-bcaGiabg2da% 9iab-bcaGiaaicdacaGGSaaabaGaam4zaiaacIcacqGHXcqScaaIXa% Gaaiykaiab-bcaGiabg2da9iab-bcaGiaadEgadaahaaWcbeqaaiaa% cEcaaaGccqWFGaaicaGGOaGaeyySaeRaaGymaiaacMcacqWFGaaicq% GH9aqpcqWFGaaicaaIWaGaaiilaaaaaa!6222!\[\begin{array}{l} f( \pm 1) = f^' ( \pm 1) = 0, \\ g( \pm 1) = g^' ( \pm 1) = 0, \\ \end{array}\]and when the conditions of consistency% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa8qmaeaacaWGNbGaaiikaiaadMhacaGGPaqedmvETj2BSbacfaGa% e8hiaaIaamizaiaadMhacqWFGaaicqWF9aqpcqWFGaaidaWdXaqaai% aadMhacaWGNbGaaiikaiaadMhacaGGPaGae8hiaaIaamizaiaadMha% cqWFGaaicqGH9aqpcqWFGaaicaaIWaaaleaacqWFsislcqWFXaqmae% aacqWFXaqma0Gaey4kIipaaSqaaiabgkHiTiaaigdaaeaacaaIXaaa% niabgUIiYdaaaa!5A1B!\[\int_{ - 1}^1 {g(y) dy = \int_{ - 1}^1 {yg(y) dy = 0} } \]are not satisfied.The present completeness proof thus answers questions raised recently (in the mathematically equivalent context of Stokes flow) by Joseph [3], and Joseph and Sturges [5], who showed that if the side conditions (A), (B) are relaxed then the corresponding eigenfunction series may still converge; but they left open the more difficult question of whether these series still converge to the data.The method of proof used here also succeeds in proving a corresponding completeness theorem for the Williams eigenfunctions for the wedge with the data.% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcea% qabeaadaabciqaamaalaaabaGaeyOaIylabaGaeyOaIyRaamOCaaaa% daqadiqaamaalaaabaGaaGymaaqaaiaadkhaaaqedmvETj2BSbacfi% Gae8NXdygacaGLOaGaayzkaaaacaGLiWoadaWgaaWcbaGaamOCaiab% g2da9iaaigdaaeqaaGqbaOGae4hiaaIaeyypa0Jae4hiaaIaamOzai% aacIcacqaH4oqCcaGGPaGaaiilaaqaamaaeiGabaWaaSaaaeaacqGH% ciITdaahaaWcbeqaaiaaikdaaaGccqaHgpGzaeaacqGHciITcqaH4o% qCdaahaaWcbeqaaiaaikdaaaaaaOWaaeWaceaadaWcaaqaaiaaigda% aeaacaWGYbaaaiab-z8aMbGaayjkaiaawMcaaaGaayjcSdWaaSbaaS% qaaiaadkhacqGH9aqpcaaIXaaabeaakiab+bcaGiabg2da9iab+bca% GiaadEgacaGGOaGaeqiUdeNaaiykaiaacYcaaaaa!6B9C!\[\begin{array}{l} \left. {\frac{\partial }{{\partial r}}\left( {\frac{1}{r}\phi } \right)} \right|_{r = 1} = f(\theta ), \\ \left. {\frac{{\partial ^2 \phi }}{{\partial \theta ^2 }}\left( {\frac{1}{r}\phi } \right)} \right|_{r = 1} = g(\theta ), \\ \end{array}\]prescribed on –<<, (where 2 is the wedge angle).Department of Mathematics, University of ManchesterOn leave of absence at the University of British Columbia, Vancouver, B.C. Canada, during 1977–79. This work was supported in part by N.R.C. grants Nos. A 9259 and A9117.     

A－HIGH－ORDER ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING THE EQUATION OF TWO－DIMENSIONAL PARABOLIC TYPE

Ａ－ＨＩＧＨ－ＯＲＤＥＲＡＣＣＵＲＡＣＹＥＸＰＬＩＣＩＴＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＦＯＲＳＯＬＶＩＮＧＴＨＥＥＱＵＡＴＩＯＮＯＦＴＷＯ－ＤＩＭＥＮＳＩＯＮＡＬＰＡＲＡＢＯＬＩＣＴＹＰＥＭａＭｉｎｇｓｈｕ（马明书）（ＲｅｃｅｉｖｅｄＪｕｎｅ２,１...     

Existence of solutions of three-point boundary value problems for nonlinear fourth order differential equation

In this paper, the author uses the methods in [1, 2] to study the existence of solutions of three point boundary value problems for nonlinear fourth order differential equation.% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaCa% aaleqabaGaaiikaiaaisdacaGGPaaaaOGaeyypa0JaaGOKbiaacIca% caWG0bGaaiilaiaadMhacaGGSaGabmyEayaafaGaaiilaiqadMhaga% GbaiaacYcaceWG5bGbaibacaGGPaaaaa!4497!\[y^{(4)} = f(t,y,y',y',y')\] with the boundary conditions% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiGaaqaabe% qaaiaadEgacaGGOaGaamyEaiaacIcacaWGHbGaaiykaiaacYcaceWG% 5bGbauaacaGGOaGaamyyaiaacMcacaGGSaGabmyEayaagaGaaiikai% aadggacaGGPaGaaiilaiqadMhagaGeaiaacIcacaWGHbGaaiykaiaa% cMcacqGH9aqpcaaIWaGaaiilaiaadIgacaGGOaGaamyEaiaacIcaca% WGIbGaaiykaiaacYcaceWG5bGbayaacaGGOaGaamOyaiaacMcacaGG% PaGaeyypa0JaaGimaaqaaiqadMhagaqbaiaacIcacaWGIbGaaiykai% abg2da9iaadkgadaWgaaWcbaGaaGymaaqabaGccaGGSaGaam4Aaiaa% cIcacaWG5bGaaiikaiaadogacaGGPaGaaiilaiqadMhagaqbaiaacI% cacaWGJbGaaiykaiaacYcaceWG5bGbayaacaGGOaGaam4yaiaacMca% caGGSaGabmyEayaasaGaaiikaiaadogacaGGPaGaaiykaiabg2da9i% aaicdaaaGaayzFaaaaaa!7059!\[\left. \begin{gathered} g(y(a),y'(a),y'(a),y'(a)) = 0,h(y(b),y'(b)) = 0 \hfill \\ y'(b) = b_1 ,k(y(c),y'(c),y'(c),y'(c)) = 0 \hfill \\ \end{gathered} \right\}\] For the boundary value problems of nonlinear fourth order differential equation% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaCa% aaleqabaGaaiikaiaaisdacaGGPaaaaOGaeyypa0JaaGOKbiaacIca% caWG0bGaaiilaiaadMhacaGGSaGabmyEayaafaGaaiilaiqadMhaga% GbaiaacYcaceWG5bGbaibacaGGPaaaaa!4497!\[y^{(4)} = f(t,y,y',y',y')\] many results have been given at the present time. But the existence of solutions of boundary value problem (*). (**) studied in this paper has not been involved by the above researches. Morcover, the corollary of the important theorem in this paper, i. e. existence of solutions of the boundary value problem of equation (*) with the following boundary conditions.% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGHb% WaaSbaaSqaaiaaicdaaeqaaOGaamyEaiaacIcacaWGHbGaaiykaiab% gUcaRiaadggadaWgaaWcbaGaaGymaaqabaGcceWG5bGbauaacaGGOa% GaamyyaiaacMcacqGHRaWkcaWGHbWaaSbaaSqaaiaaikdaaeqaaOGa% bmyEayaagaGaaiikaiaadggacaGGPaGaey4kaSIaamyyamaaBaaale% aacaaIZaaabeaakiqadMhagaGeaiaacIcacaWGHbGaaiykaiabg2da% 9iaadMhadaWgaaWcbaGaaGimaaqabaGccaGGSaGaamOyamaaBaaale% aacaaIWaaabeaakiaadMhacaGGOaGaamOyaiaacMcacqGHRaWkcaWG% IbWaaSbaaSqaaiaaikdaaeqaaOGabmyEayaagaGaaiikaiaadkgaca% GGPaGaeyypa0JaamyEamaaBaaaleaacaaIXaaabeaaaOqaaiqadMha% gaqbaiaacIcacaWGIbGaaiykaiabg2da9iaadMhadaWgaaWcbaGaaG% OmaaqabaGccaGGSaGaam4yamaaBaaaleaacaaIWaaabeaakiaadMha% caGGOaGaam4yaiaacMcacqGHRaWkcaWGJbWaaSbaaSqaaiaaigdaae% qaaOGabmyEayaafaGaaiikaiaadogacaGGPaGaey4kaSIaam4yamaa% BaaaleaacaaIYaaabeaakiqadMhagaGbaiaacIcacaWGJbGaaiykai% abgUcaRiqadogagaGeaiaacIcacaWGJbGaaiykaiabg2da9iaadMha% daWgaaWcbaGaaG4maaqabaaaaaa!7DF7!\[\begin{gathered} a_0 y(a) + a_1 y'(a) + a_2 y'(a) + a_3 y'(a) = y_0 ,b_0 y(b) + b_2 y'(b) = y_1 \hfill \\ y'(b) = y_2 ,c_0 y(c) + c_1 y'(c) + c_2 y'(c) + c'(c) = y_3 \hfill \\ \end{gathered} \] has not been dealt with in previous works.     

The analysis of eigenstrains outside of an ellipsoidal inclusion

The stress and displacement fields due to an eigenstrain outside of an ellipsoid are analyzed both inside and outside the ellipsoid and are given in terms of nonsingular surface integrals. Constant and inverse power eigenstrains are discussed as examples.Phase transformation strains, plastic strains, thermal strains, stress-free strains, and other inelastic strains are collectively labeled eigenstrains because the mathematical analysis generated from elasticity theory can, to a large extent, be determined independently of the physical source of the strain. The eigenstrain when added to the elastic strain gives the total strain. The stress field corresponding to this eigenstrain is called an eigenstress.In this paper we will assume that the eigenstrain is zero inside an ellipsoid and generally nonzero outside the ellipsoid in D- where D denotes the whole body. We will assume D to be a three-dimensional, infinitely extended, anisotropic material. The stress and displacement fields will be found in terms of surface integrals. These can be analyzed exactly for some simple cases and numerically on a computer for more complex cases.Many authors have studied eigenstrains when they are nonzero inside . Eshelby [1] solved the case where the eigenstrain is a constant and discussed the problem of an ellipsoidal inhomogeneity in an isotropic material subjected to constant stresses. This was done by using an eigenstrain to simulate the effect of the inhomogeneity. This solution has been generalized to solve problems involving anisotropic materials by Kinoshita and Mura [2], polynomial eigenstrains by Asaro and Barnett [3] and periodic eigenstrains by Mura, Mori and Kato [4]. Many interesting solutions have been found, including ones for the two inclusion problem, martensitic transformation, and thermal inclusions among others (see [4]-[7]). Mura and Cheng [8] found a general solution when the eigenstrain is defined inside . In this paper we will examine the case where the eigenstrain is defined in D- and zero inside . Nobody appears to have published solutions for this case before in spite of the fact that it leads to some exact solutions. Solutions of this kind should be useful in solving problems dealing with plastic fields located outside an ellipsoid, precipitate formation, and multiple inclusion problems among others. We will specifically examine the case where the eigenstrain is (1) constant and (2) of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaacqaHYoGyliaabMgacaqG% QbGccaqGHbWaaWbaaSqabeaacaqGqbaaaOGaaiikaiaabIhajaaifa% qabeGabaaabaGaaiOmaaqaaiaacgdaaaGaey4kaSIccaqG4bqcaasb% aeqabiqaaaqaaiaackdaaeaacaGGYaaaaiabgUcaROGaaeiEaKaaGu% aabeqaceaaaeaacaGGYaaabaGaai4maaaakiaacMcadaahaaWcbeqa% aiabgkHiTiaadchacaGGVaGaaiOmaaaaaaa!57D6!\[\beta {\text{ija}}^{\text{P}} ({\text{x}}\begin{array}{*{20}c}2 \\1 \\\end{array} + {\text{x}}\begin{array}{*{20}c}2 \\2 \\\end{array} + {\text{x}}\begin{array}{*{20}c}2 \\3 \\\end{array} )^{ - p/2}\]where p > 0, is a sphere of radius a, _ij are constants, and the material is isotropic.This research was supported by US Army Research Grant DAAG29-81-K-0090.     

Alloy separation of a binary mixture in a stressed elastic sphere

We consider the static state of a spherical isotropic binary elastic solid mixture whose boundary is given a uniform radial displacement. The elastic volumetric strain energy is given by the classical quadratic form from linear elasticity theory,% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaacaqGxbGaeyypa0ZaaSaa% aeaacaGGXaaabaGaaiOmaaaacaGG7bGaaCOUdiaacIcacaqGJbGaam% ykaiaadIcacaWG0bGaamOCaiaadwgacaWGPaWaaWbaaSqabeaacaGG% YaaaaOGaey4kaSIaaiOmaiabeY7aTjaacIcacaqGJbGaaiykaiaacY% hacaWGLbGaeyOeI0IaaiiiamaalaaabaGaamymaaqaaiaadodaaaGa% aiikaiaadshacaWGYbGaamyzaiaacMcacaWGXaGaaiiFamaaCaaale% qabaGaaiOmaaaakiaac2hacaGGUaaaaa!63E0!\[{\text{W}} = \frac{1}{2}\{ {\mathbf{\kappa }}({\text{c}})(tre)^2 + 2\mu ({\text{c}})|e - \frac{1}{3}(tre)1|^2 \} .\]Here, e is the infinitesimal strain tensor, c[0, 1] is the volumetric concentration of the mixture, and (·) and (·) are the (positive) bulk and shear material moduli, respectively, which are given functions of the concentration. As a function of c and e, the strain energy function is generally nonconvex. Thus, we consider the nonconvex problem of minimizing the potential energy of the body, among all spatial concentration and displacement fields, subject to a given boundary displacement and a fixed amount of component materials. Assuming spherical symmetry, we find that the two component materials must be separated in the optimal state of minimum potential energy. The harder material forms the central core of the sphere, and the softer material is segregated into a surrounding shell. This behavior is remindful of a general notion in metallurgy that in the casting of materials the harder material tends to migrate toward the center.Partial support of the NSF under grant MSS-9024637 and Alliant Techsystems Inc. is gratefully acknowledged.Professor R. Bartelletti of the Università di Pisa is gratefully acknowledged.     

SINGULAR PERTURBATION FOR A NONLINEAR BOUNDARY VALUE PROBLEM OF FIRST ORDER SYSTEM

ＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＦＯＲＡＮＯＮＬＩＮＥＡＲＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＯＦＦＩＲＳＴＯＲＤＥＲＳＹＳＴＥＭＣｈｅｎＳｏｎｇｌｉｎ（陈松林）（ＲｅｃｅｉｖｅｄＡｐｒｉｌ８,１９８４;ＲｅｖｉｓｅｄＡｐｒｉｌ１５,１９...     

Uniqueness theorems for displacement fields with locally finite energy in linear elastostatics

Uniquencess theorems are proved for the fundamental boundary value problems of linear elastostatics in bodies of arbitrary shape. The displacement fields are required to have finite strain energy in bounded portions of the bodies and satisfy the principle of virtual work. For bounded bodies, the total strain energy is finite and uniquencess is proved without additional hypotheses. In particular, no restrictions other than the energy condition are placed on the field singularities that may occur at sharp edges and corners. For unbounded bodies, uniqueness can be proved as in the bounded case if the total strain energy is finite. Sufficient conditions for this are shown to be the finiteness of the strain energy in bounded portions of the body together with the growth restriction % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaadaWdraqaaiaabwhadaWg% aaWcbaGaaeyAaaqabaGccaGGOaGaaeiEaiaacMcacaqG1bWaaSbaaS% qaaiaabMgaaeqaaOGaaiikaiaabIhacaGGPaGaaeizaiaabIhacaqG% 9aGaaGimaiaacIcacaqGYbGaaiykaiaacYcacaqGYbGaeyOKH4Qaey% OhIukaleaacqGHPoWvdaWgaaadbaGaaeOCaiaacYcacqaH0oazaeqa% aaWcbeqdcqGHRiI8aaaa!5E73!\[\int_{\Omega _{{\text{r}},\delta } } {{\text{u}}_{\text{i}} ({\text{x}}){\text{u}}_{\text{i}} ({\text{x}}){\text{dx = }}0({\text{r}}),{\text{r}} \to \infty } \] on the displacement fieldu _i , where _r, is the portion of the body that lies between concentric spheres with radiir andr+ and >0.This research was supported by the Air Force Office of Scientific Research. Reproduction in whole or part is permitted for any purpose of the United States Government.Prepared under Contract No. F 49620-77-C-0053 for Air Force Office of Scientific Research.     

On an inversion theorem for conical regions in elasticity theory

Summary Two-dimensional stress singularities in wedges have already drawn attention since a long time. An inverse square-root stress singularity (in a 360° wedge) plays an important role in fracture mechanics.Recently some similar three-dimensional singularities in conical regions have been investigated, from which one may be also important in fracture mechanics.Spherical coordinates are r, , . The conical region occupied by the elastic homogeneous body (and possible anisotropic) has its vertex at r=0. The mantle of the cone is described by an arbitrary function f(, )=0. The displacement components be u. For special values of (eigenvalues) there exist states of displacements (eigenstates) % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakabbaaa6daaahjxzL5gapeqa% aiaadwhadaWgaaWcbaGaeqOVdGhabeaakiabg2da9iaadkhadaahaa% WcbeqaaiabeU7aSbaakiaadAgadaWgaaWcbaGaeqOVdGhabeaakiaa% cIcacqaH7oaBcaGGSaGaeqiUdeNaaiilaiabfA6agjaacMcaaaa!582B!\[u_\xi = r^\lambda f_\xi (\lambda ,\theta ,\Phi )\],which may satisfy rather arbitrary homogeneous boundary conditions along the generators.The paper brings a theorem which expresses that if is an eigenvalue, then also-1- is an eigenvalue. Though the theorem is related to a known theorem in Potential Theory (Kelvin's theorem), the proof has to be given along quite another line.

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Zusammenfassung Zwei-dimensionale Spannungssingularitäten in keilförmigen Gebieten sind schon längere Zeit untersucht worden und neuerdings auch ähnliche drei-dimensionale Singularitäten in konischen Gebieten.Kugelkoordinaten sind r, , . Das konische Gebiet hat seine Spitze in r=0. Der Mantel des Kegels lässt sich beschreiben mittels einer willkürlichen Funktion f(, )=0. Die Verschiebungskomponenten seien u. Für spezielle Werte von (Eigenwerte) bestehen Verschiebunszustände % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakabbaaa6daaahjxzL5gapeqa% aiaadwhadaWgaaWcbaGaeqOVdGhabeaakiabg2da9iaadkhadaahaa% WcbeqaaiabeU7aSbaakiaadAgadaWgaaWcbaGaeqOVdGhabeaakiaa% cIcacqaH7oaBcaGGSaGaeqiUdeNaaiilaiabfA6agjaacMcaaaa!582B!\[u_\xi = r^\lambda f_\xi (\lambda ,\theta ,\Phi )\],welche homogene Randwerte der Beschreibenden des Kegels entlang genügen.Das Bericht bringt ein Theorem, welches aussagt, das und =–1– beide Eigenwerte sind.