矩形状平曲板と連結はりの非線形曲げ振動に関する研究

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(57) ( ) Chang , S., I., Bajaj, A., K., Krousgrill , C.,M., Non-linear vibrations and chaos in h armonically ex cited rect angular plates with one-to-one intern al resonan ce, Nonlinear Dyna mics, Vol.4 (1993 ), pp.433-460 . ( ) Murphy, K.D., Virgin, L.N. and Rizzi, S.A., Characterizing the Dynamic Respons e of a Th ermally Lo aded , Acoustical ly Excited Plate, Journal of Sound and Vibration, Vol. 196, (199 6), pp . 635-658. ( ) Nagai , K. and Yamaguchi , T. Chaotic Os cillations of a Shallo w Cylind rical Shell with Rect angular Bound ary und er Cyclic Ex citation. In: High pressu re technology, ASME, PVP Vol. 297, (1995), pp. 107-115. ( ) Yamaguchi , T. and. Nag ai, K.. Ch aotic Vibration of a Cylindrical. Shell-Panel with an In-Plan e Elastic-Suppo rt at Bound ary , Nonlinear Dynamics, Vol . 13 , (1997 ), pp. 259-277 . Amab ili, M., Non-lin ear vibrations of doubly curved shallow shells, Internatio nal Journal of Non-Linear Mechani cs, Vol. 40 (2005), pp . 6 83-710 . ( ) Nagai, K., Maruyama, S., Oya, M., Yamaguchi, T., Chaotic oscillations of a shallow cylindrical with a concentrated mas s under periodic excitation, Co mputers and Structures, Vol.82 (2004 ), pp.2 607-2619. ( ) Nagai, K., Maruyama, S., Murata, T., Yamaguchi, T., Experiments and analysis on chaotic vibrations o f a shallo w cy lindrical sh ell-pan el, Journal of Sound and Vibration, Vol . 305 (2007 ), pp . 4 92–520. ( ) Amabili , M., Non-linear vibrations of doubly curved shallow shells, International. Journal. of. Non-Linear. Mech anics,. Vol .. 40(2005 ),. pp.. 683-710. ( ) Amabili , M., Theory and experiments for l arge-amplitude vibrations of circul ar cylindrical p anels with geo metric imp erfections, Jo urnal of Sound and Vibration, Vol. 298 (2006 ), pp. 43-72 . ( ) Amabili , M. Pell egrini, M. To mmes ani, M., Ex peri ments on large-amplitude vibrations of a circular cylindri cal panel , Jou rnal of Sound and Vibration, Vol. 260, (2003) ,pp. 537-547.. 8.

(58) ( ) Maruyama, S., Nagai, K., Tsuruta, Y., Modal interaction in chaotic vibrations of a shallow double-curved shell-panel, Journal of So und and Vibration, Vol . 315 (2008 ), pp . 607–62 5. ( ) Kojima, Y., Load position error due to dimensional error in beam type load cell, Transactions of the Jap an society of mechanical engineers , Series C, Vol.52, No.483 (1986), pp.2869- 2874 . ( ) Kojima, Y., Sensitivity of beam type load cell to off-center loading (3rd report , shap es for reducing sensitivity), Transactions o f th e Jap an so ciety of mechanical engineers , Series C, Vol.52 , No.47 4 (1986 ), pp.746-749. ( ) Kojima, Y., Nonlinear characteristics of beam type load cell, Transactions of the Japan society of mechani cal engineers, Series C, Vol.57, No.540 (1991), pp .2764- 2769 . ( ) Yoshimura, Y., Amada, K. and Akasaka, T., Displacement characteristics of one piece parallel spring movement for balance (1st report) two symmetric leaf springs, Journal of th e Jap an soci ety of p recision engineering, Vol.53 , No.11, (1987), pp.1746-1750. ( ) Matsuda, T., Sato, M, Sato, K. and Kitagawa, T. Double beam type load cell for scales with stabilized pan-Analysis on strain characteristics and their improvement, Journal of the Japan society of precision engineering, Vol.52, No.2 (1990), pp.393 -398. ( ) Nayfeh, A.H., Zavodney, L.D.: Experimental observation of amplitude- and phase-modulated respons es of two internally coupled os cillators to a harmonic excitation. ASME J. Appl. Mech. 55 (1988), pp.706–710 ( ) Nayfeh,. A.H.,. Balachandran,. B.,. Colbert,. M.A.,. Nayfeh,. M.A.:. An. experi mental inv estigation of co mplicated responses o f a t wo-d egree-o ffreedo m stru cture. ASME J . Appl. Mech . 56, 9 60–967 (1989 ) ( ) Wang, F., Bajaj , A.K.: Nonlin ear no rmal mod es in multi mode models of an inertially coupled elastic structure. Nonlinear Dyn. 47, 25–47 (2007) ( ) Warminski, J., Cartmell, M.P., Boch enski, M., Iv anov , I.: An alytical and experi mental inv estigations o f an autoparametric b eam structu re. J . Sound Vib. 315, 486–508 (2008). 9.

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(94) p0>#'T@oPd-PY=Z p. Fig. 2.3 Configuration of the plate, nc =0.88. 15.

(95) Fig.2.4 Characteristics of the restoring force of the plate, the position of concentrated force (,)=(0.5,0.5) , nc =0.88. >*.O?h w w = 0 w = 1.5 Y4hL_=AbS i1M>*.O?hLU=. V g8<R. V iO. h = 0.4 h = 0.4 >*.O?h z `JC2]c;h : . Q ihf-7[.. Eh(\ = 0.2 h = 0.1 D: . 0Hhe;W9==P. i.

(96)

(97) HZ=6E@/B hNI*6E@/B 6E@/"#!. i h$FXZ. ^ 2.2 V. E 3Ih%I+I@/"#!. h .

(98) (3,1)ah(3,1)bh(3,1)c ^ i^ 2 h6E@/"#!XZhe;W ihHe;W9==Pi5@/"#!(2,1)(1,2) (3,1)a (3,1)c h

(99) "#!-hXZT& i6E@/ Bh-a,@G'. K h 3 21 2 12 d). 16. E i.

(100) Table 2.2 Natural frequencies and vibration modes of the plate, nc =0.88. 6QDmK=YRMbTd. ? 2.5 ` oWi[X06QQ8. R nei[X0D7. wrms on?)(m ,n; j) mn n

(101) @UQ8%&$(m, n) j X1QMb[X0@UQ8R nPI2MbkO 2l1QMb. ` o[ ]. f oC[(m ,n; j)]n@UQ8%&. $(m, n) j X1QMb"!#Q8Mb ` oZ_JZhJ EM1QMb. Kon1QMbnQGV:E<aHA. ;FA;n

(102) CQGnFQG94o Z_JL05\NEM1QMbn = 18 j/nQ8 %&$(1,1) 3/2 X+3RXgY1QMbCQG. ` on = 31 j/. nQ8%&$(1,1)*1QMb] o = 40 j/nZ_JEM. Q8%&$(1,1) 2/3 X+3RXgY1QMbnQ8%&$(2,1)*1 QMbQ8%&$(1,2) 3/2 X+3RXgY1QMb. ,2l1Q. Mb] on = 90 j/nQ8%&$(2,1) 1/2 X3RXgY1 QZ_JMb. ` o'Sn = 44 ~ 49 c>nQ8%&$(1,1) 1/2. X3R XgY 1QM b(1,1;1/2)nZh- Z_J. ` o\nQGD7.. wrms = 0.5 ~ 1.2 nQ8%&$(1,1)^] Q8RnQGB6,( o. 17.

(103) Fig.2.5 Frequency response curves, nc =0.88 , pd = 0.38 ×10 3 , measured at = 0.60 , = 0.40. 5'4AH+ 5'6F- @BI5'4AH3 E&9 H! 8$;5'*H=D2%54A> .( =?2%54A>.( &GI H = 32.1 ~ 35.4 /515'4A(,. HC[(1,1;2/3),. (2,1;4/3)])H=?25'(1,1) 2/3 ;"&6;C<%54A>H 5'(2,1) 4/3 ;&6;C<%54A#I): H = 41.3 ~ 41.9 5'4A(,. HC[(1,1;1/2), (2,1;1), (1,2;3/2)])H=?25'(1,1). 1/2 ;&6;C<%54A>H5'(2,1)!%54A

(104) 5 '(1,2) 3/2 ;"&6;C<%54A#I7H = 49.6 ~ 50.1 0515'4A(,. HC[(1,1;1/2), (1,2;5/4)])H=D25'. (1,1) 1/2 ;&6;C<%54A>H5'(1,2) 5/4 ;"&6; C < % 5 4A#I = 78.6 ~ 81.0 5'4A ( ,. H C[(1,1;1/4),. (2,1;1/2), (1,2;3/4)])H=D25'(1,1) 1/4 ;&6;C<%54A >H5'(2,1) 1/2 ;&6;C<%54A

(105) 5'(1,2) 3/4. 18.

(106) h:D^hiB]Yu"<

(107) lsWl WB]Yup $#']HYu . . C[(1,1;2/3), (2,1;4/3)] C[(1,1;1/2), (1,2;5/4)]xDd" $#']HYuDdMi^Dd`S32.+-\^8[DDd {Kr!. . $#']HYu_iW Mi^Ddye" t

(108) O7(a)(b). . . O 2.6 O 2.7 . C[(1,1;2/3), (2,1;4/3)] C[(1,1;1/2),. (1,2;5/4)]UY

(109) _iWgG]Mb e "Qk Mb^"t

(110) Mi^DdyegmhADd]H^ sp "t }]V'0%)4 (&/4?"t

(111) O 2.6(a)(b)_iW_iWIz| 6ERH"t

(112) O 2.7(a)Mi^DdyeC[(1,1;2/3), (2,1;4/3)] G]]H[D Gh]H[D 2 3 ~]V'0%)4 , 5 % " t

(113) , 5% "t

(114) ; ] H[ D = h ]H [ D. (1 / 2) ~ (2 / 3) h]H[D (4 / 3) (5 / 2) ]H[D 3 @ 31c UY ]H[D,5%TP

(115) . ,5%"t

(116). ] H [ D J ] H 1 5 * q p

(117) ] H ^ U Y (2 / 3) 11 (4 / 3) 21 2 12 (5 / 2) 31a >"j

(118) ]H15* (1,1)qp

(119) ]HÛY

(120) ]H[D=]H[D o . . ]H]Vl WXAFnZ"t

(121) wNC. C[(1,1;2/3), (2,1;4/3)]$#']HYu]H15*(1,1) 2/3 h:D^ hiB]Yu]H15*(2,1) 4/3 h:D^hiB]Yu]H1 5*(1,2) 2 hiB]Yu9cv|"a

(122) ]H15*(3,1)a 5/2 h :D^hiB]Yu]H15*(3,1)c [ CB]Yu LfO 2.7(b) C[(1,1;1/2), (1,2;5/4)]G]]H[D G=h] H[D (1 / 2) h]H[D (3 / 2) 2 ~'0%)4,5% "t

(123) ,5%"t

(124) ;]H[D(5 / 6) (5 / 4) . 19.

(125) Fig. 2.6 Time responses at each region of chaotic responses measured at. =0.60 , =0.40 ;(a)C[(1,1;2/3),(2,1;4/3)], =33.1 ;(b) C[(1,1;1/2),(1,2;5/4)],. =49.9. Fig. 2.7 Fourier spectra at each region of chaotic responses measured at. =0.25 , =0.50 ;(a)C[(1,1;2/3),(2,1;4/3)], =33.1 ;(b) C[(1,1;1/2),(1,2;5/4)], )#(" 2 2 6 31c &')#(" %$7. . ) # ( " 6 (1 / 2) 11 6 (5 / 6) 21 6 (5 / 4) 12 6 (3 / 2) 31a 5-176C[(1,1;1/2), (1,2;5/4)] )#' / 6)#(1,1) 1/2 +"*+0, +"*+0,. )'/ 6)#(2,1) 5/6. )'/6)#(1,2) 5/4 +"*+0,. )'/

(126). )#(3,1)a 3/2 +"*+0, )'/)#(3,1)c 3( !4. )'/ 7. 20.

(127) B2@S C[(1,1;2/3), (2,1;4/3)] C[(1,1;1/2), (1,2;5/4)]#(' 1-5 5 2.8(a) (b)Q_#('1-5IWOJ.B= w

(128) ^TWOJ.Y> w,

(129) _5 C[(1,1;2/3), (2,1;4/3)] C[(1,1;1/2), (1,2;5/4)]#('1-5ÛDB2@SM60Z/B ?B2@S1-NC6PV_.

(130) B2]8F[L?<G9&%" W7XJ. e ^TWG9&%". !AD5 2.9 Q_5I. !AD max

(131) _5 C[(1,1;2/3),. (2,1;4/3)] C[(1,1;1/2), (1,2;5/4)]

(132) ^ max K,4H_^ 3EB2@S^B2@S

(133) P:_^max 4 H\7XJ. e ^C[(1,1;2/3), (2,1;4/3)] C[(1,1;1/2), (1,2;5/4)] ê = 9 e = 10

(134) _^3EB2@S;* B2$)D+R> _. Fig.2.8 Poincaré projections of the responses at each region of chaotic responses, ; (a) C[(1,1;2/3),(2,1;4/3)], =33.1 ;(b) C[(1,1;1/2),(1,2;5/4)],. =49.9 , measured at =0.60 , =0.40. 21.

(135) Fig. 2.9 Maximum Lyapunov exponents related to embedding dimension, measured at =0.60 , =0.40. !8*6K/E,<D0 \5<XC42 \'7))? R]M@. 2.10 H ].OUJX#$"1&F]AU

(136) 1&F6 JX#$"Y3H ]. C[(1,1;2/3), (2,1;4/3)] \8*#$"(1,1)\(1,2)\(2,1)(>LN= 8*#$"(3,1)a\ (3,1)c 26 JX#$"5]C[(1,1;1/2), (1,2;5/4)]\8*#$"(1,1)\ (3,1)a (3,1)c \8*#$"(2,1)(1,2)PV#$"26 JX#$"5]5. !8*1& 'T8*#$". IZ\-C:)?M@%Q ]8*#$"(1,1)1&F\C[(1,1;2/3), (2,1;4/3)] C[(1,1;1/2), (1,2;5/4)]\

(137) 80% 89%] \+;. !8*6K\8*#$"(1,1)9WG]. \S:[B#$"(2,1)\(1,2)(3,1)a 1&F\C[(1,1;2/3), (2,1;4/3)]\20%H \C[(1,1;1/2), (1,2;5/4)]\10%H ]. 22.

(138) Fig. 2.10 Contribution ratio of each modes of vibration in the chaotic responses, :C[(1,1;2/3),(2,1;4/3)], =33.1 ; :C[(1,1;1/2),(1,2,5/4)],. =49.9

(139) TR58LLC pd %bR=8

(140) j'&(L9W "-?@h<%Hk C[(1,1;2/3), (2,1;4/3)]. C[(1,1;1/2), (1,2;5/4)]"-?@h<%##. ; 2.11(a)(b)Yk;QaTR58LL9M !jâTR58 LLC pd "k;/N]dj'&(L9h<%Yk;(a)! C[(1,1;2/3), (2,1;4/3)] L9Mh<D\"kTR58LLC pd = 0.30 × 103 pd = 0.38 × 103 =8"Gj'&(L9W "L9Mh<K>"k. j pd =81jC[(1,1;2/3), (2,1;4/3)]L9Mh<jiL9M4c Z"k,Oj;(b)!jC[(1,1;1/2), (1,2;5/4)]L9Mh<Uk. j. =81jC[(1,1;1/2), (1,2;5/4)]L9Mh<j$2L94cZ "k #jC[(1,1;2/3), (2,1;4/3)] C[(1,1;1/2),(1,2;5/4)]j##SXE %Y6LI[SÈ%Y6LI[Fg%: ""kj TR58LLC pd %bR=8 fj'&(L9I[B

(141) j0J77P %_k #!'&(L9A."L9*+)A.VjTR58L LC e3%HkC[(1,1;2/3), (2,1;4/3)] C[(1,1;1/2), (1,2;5/4)]7P. 23.

(142) 5-

(143) & 2.12(a):(b)4 ;&.70/"#**) pd :6 7%*$(14 ;&(a)4 C[(1,1;2/3), (2,1;4/3)] : pd '#. :9/*$(1,2)(1! :*$(1,1)+82. 3;,:&(b) C[(1,1;1/2), (1,2;5/4)] : pd . Fig. 2.11 Instability boundaries of the chaotic responses, (a) C[(1,1;2/3), (2,1;4/3)], (b) C[(1,1;1/2),(1,2,5/4)]. Fig. 2.12 Relation between the exciting amplitude pd and contribution ratios on each vibration mode (: mode (1,1), : mode (1,2), : mode (2,1)+(1,2)), (a) C[(1,1;2/3),(2,1;4/3)], (b) C[(1,1;1/2),(1,2,5/4)]. 24.

(144) QE. #W0wSRG!$ $C. C[(1,1;2/3),. (2,1;4/3)]PJfFhOfFh< @?fm7s $ " qfF+,* $# . # C[(1,1;1/2),. (1,2;5/4)]PJfFhO%#@?fm7s. $q+,*W0RG!$ . -XMU4XHge ${_v\nBl@N D! KlE]&:y (')fFV&@N D^Z X 88% $. "nt-t|_a>Duc&k @. ?fm7&s #(')fFbLfFhO} $ t|_. t|_Xb #?fb!x

(145) (')fFb. 5z2bAo&`!$p& # 6/. #. ( 1 ) j9q+,*q+,*@?fm7&s (')fFb $#Sf[(')fFbt|_Xb j9q+,* 2/3 q3Ahqr?fb=x

(146) #Yf[(')fFbt _Xb j9q+,* 1/2 qAhqr?fb=x

(147) #. ( 2 ) 1dAAop$(')fFbj9q+,*gz #2q.q!Lqq+,*. #W0w10% 20%. &I#. ( 3 ) Eff[QE8Sf[&~ (')fFfFhOt|_ ?fb;T fFh< #-iYf[&~ (')fF fFhOt_?fb;T 9fFh< #. 25.

(148) () Amabili, M., Non-linear vibrations of doubly curved shallow shells, International Journal of Non-Linear Mechanics, Vol. 40(2005), pp. 683-710. () Amabili, M., Theory and experiments for large-amplitude vibrations of circular cylindrical panels with geometric imperfections, Journal of Sound and Vibration, Vol. 298 (2006), pp. 43-72. () Nagai, K., Maruyama, S., Oya, M., Yamaguchi, T., Chaotic oscillations of a shallow cylindrical with a concentrated mass under periodic excitation, Computers and Structures, Vol.82 (2004), pp.2607-2619. () Nagai, K., Maruyama, S., Murata, T., Yamaguchi, T., Experiments and analysis on chaotic vibrations of a shallow cylindrical shell-panel, Journal of Sound and Vibration, Vol. 305 (2007), pp. 492–520. () Maruyama, S., Nagai, K., Tsuruta, Y., Modal interaction in chaotic vibrations of a shallow double-curved shell-panel, Journal of Sound and Vibration, Vol. 315 (2008), pp. 607–625. () Maruyama, S., Onozato, N., Nagai, K., Yamaguchi, T., Experiments on nonlinear vibrations of a cylindrical shallow shell-panel with clamped edges under an in-plane elastic constraint, Transactions of the Japan society of mechanical engineers, Series C, Vol.74, No.743 (2008),pp.1696-1701. () Chang, S., I., Bajaj, A., K., Krousgrill, C.,M., Non-linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance, Nonlinear Dynamics, Vol.4 (1993), pp.433-460. ( ) Yamaguchi, T., Nagai, K., Maruyama, S., Identification of spatial modes in chaotic vibration involving dynamic snap-through using KL method, Transactions of the Japan Society of Mechanical Engineers, Series C, Vol.69, No.687 (2003), pp.2937-2942. ( ) Azeez, M. F. A. ,Vakakis, A. F. , Proper orthogonal decomposition of a class of vibroimpact oscillations, Journal of Sound and Vibration, Vol. 240 (2001), pp. 859-889. () Wolf, A. , Determining Lyapunov exponents from a time series, Physica , Vol. 16D (1985), pp. 285-317.. 26.

(149) !. . (*

(150) ")$+ # '. £\´¡¤zv´ ©=³9e=|"%$µ´ [t. 9e³nt´x¦ R %$µntIp. ¨/ PDnt<d´8p`. Qs.a°' C. $´& iZ®è'

(151) µ §Ańti<¬

(152) $. U²OH(1)~(5) %. $µ ´ %"-2I¢?jhntOH#´!# U KTQs.'GU²OHXµ qU²´5lFQ'KT´lFQ'?jh

(153) $5 nt®ì<U²'µ´KT+¢¯4_egr µ 8p¯4L:!#^Y ntW Ńqe ´8p&´ c1:e"Koi<kKoi<)*(¬0'|µ "ń tylFIp;¥]',´®ì<dI}kdn'B« µ SiZd. ®Ipd'

(154) i<k±Mńt<d'I}k. 6u

(155) $3´m7}`'REm~T µ %!#´-f66u' ´ki<)*( %$®ì<dí<)*(V ,'T µ. " &%. nt!jhw'J 3.1

(156) µnt´+¢ a = b = 140 mm { l`Q'h´@ h = 0.20 mm ń>b R = 4.8 × 103 mm ªt$µ. 27.

(157) inzN ineG_58%PA YqD

(158) inJ h = 0.24 mm ^

(159) in} 27$w E = 102 GPa UY = 7.47 × 103 kg/m 3 . '7t = 0.33 inOym@ ChKVMS{hKVI gc MS9B[`bl MS9Si wH] R = 4.8 × 103 mm k i\vFX

(160) ,6(#ind -4) ra BhKGMSTx

(161) 9hMS<on? :&3"*,6(# &3"*,6(#~G inB [`bl; 9hI gcOym@inOygcpJ 0.06 mm ER|+!4/ ^+!4/inOyu >=f| in

(162) i1807)jWra ZsL} BI gcu x MSu y SQhK z S. A. A Shell-panel. y. b =140mm. x adhesive film. a=140mm spring plate. z. slide block. h'=0.24mm. W(x,y,t). Cross Section of A-A. 3. R=4.8×10mm. Fig.3.1 Fixture of the shell-panel and the elastic-constraint 28.

(163) . µL_P*U 'miqbpg

(164) ¸ "`x

(165) ¸tJ Px%^yR}^yR:>7°F*"¹¸miq [Nj¡®*A'Qyf·*¢¹¸µL_P¸ { (!->1³!¦%"¹ tJPx*"'f·¸~Z±B¡®)!°F*tJPx

(166) ¹±B¡®¸ *¡®¦$&¸BdQ¹ )!¸<>0bC¦e

(167) ¹ ^yR}e¸uh¶_*A¸QyyR}*zn

(168) ¸ RbC*e

(169) ¹^yR}¸RbCKy$wac* yR}

(170) ¹¸^yR:>7¸yR:>7C*¨ ' e

(171) ¹¸mi¡® N cr = 31 N ¸µL_P*«bS

(172) ¸ D^yR}h'I%|e

(173) ¹ yRf·¸Qy\$&¸~Z[Q¬l*E ' ¸´pyRv*©ª '¹] 3.2 yRf·£¥*¹ yRf·£¸Qy£¸v*M'£%N£% '¹ Qy£¸Q¬lyk?e[Q¬lA%('¹. ¸Q. yOs£ 1 $&oHWy ('#¸QyyR}zn ('¹y (HW¸ak\ 2 $&ak (¸²Qy\ 3 $' [QyV '¹Q¬l2=/ 4 ¸E'Q¬lyk *M

(174) ¸Q¬lHW*QyOs£ 1 9+>785,'¹ ($&¸Q yBQ¬lyk?eG('¹Qy$&©ª (Rv M¸¤}<>0bC¦ 5 bC¦Os£ 6 $&¢)('¹¸< >0bC¦¸@R£ 7 $&¸@Xe§'¹ @XRv¸Ye (¸613;<.>4 8 T¯ ('¹ Rvg'N$¢¹. ¸QyyR}*z. n

(175) ¸vyk*e

(176) ¸[}v*r'¹QyyR}¸[. 29.

(177) e]vfJ 9 fO. 35'N:vV

(178) CmXp}Lv 10 . PB<[SN\. . He]Xp_rA[[C]. _a

(179) !PB<"&4-05) 11 w{ c_aCm XpHe]?bFFT #+1$' 12 s.#4%3>=K

(180) !zT"A[H`D{ V. CmXp;E"W?. J 13 zT;EN\ :nv 14 ,2(l[J 15 A[H` G`

(181) 9Y:nu,2(;E"jZ ,2(;EM_ a

(182) !zT" FFT #+1$' 12 w{ ~rUXpQ7. [C/5*"hO

(183) G^fO.

(184) _a6Fg^q@eU. R 8Z??b"s

(185) 8Z??bKarhunen-Loéve d(6) Proper Orthogonal Decomposition(7)I 7ihOk. . [CQ7 o|/5*/5*Q. t][C/5*G^xy. 8. 6. w. f. 9. 11 wrms. 10. 7. 12 5. wrms f. 4. 13 14. w,ωτ. w. 15. τ dB ωsp. 3. 1. w,ωτ w. 2. λmax e. Fig. 3.2 Vibration test apparatus of the shell-panel. 30. ~.

(186) bkCfY kCX5

(187) $7kC/2+VF$® ©JtkCfY sT10.,-jn'{)(*kC'V ® W ¬ v ' o

(188) $ ` (2.1) y = ¢ ! y = r . = a 2 / (Rh) 'Z> ® nc ª?O¡ N c '_\¡ N cr = 31N ¥ < O¡z

(189) $®W¬ruy=r = 20 $®O ¡z nc = 1.6 " nc = 1.9 ®BkW¬y=Bkk]' pd = 0.75 × 103 pd = 0.87 × 103 ®NV'Iru|ê. [SC!#ª?a pªASD.

(190) ®W¬. 6ruJ|^' 22.5 ± 0.5°C L?i . " . . ! . qHª?OA'B¦ru&b'M 3.3

(191) ®M 3.3(a) ª?OA: QGru = 0.5 ¤ Yb $®M 3.3(b)_\cSb $O¡z nc = 1.6 QGz qH &ru6U8%ru = 0.5 ¤ ©Y& b$® "M 3.3(c)ª?OA'RB nc = 1.9 QGru = 0.5 ¤ w©Y&b$®%_\c Sb& @tb4omh$K

(192) $®.

(193) u&¡¤;#d=Ag'M 3.4

(194) ®x9 = 0.4 . = 0.6 $ruy=& w ' y=x§6¡ q '

(195) ®&E~ru¨£#G9®M!#O¡z. nc = 1.6 QGd=Ag}-}b'

(196) ®}bg'

(197) &«P w = 0 " w = 1^$®3q nc = 1.9 QG}bg'

(198) &«P w = 0 " w = 4 ^#}bl $®. 31.

(199) (a) η =1. η =0.5. η. (b). η =0.5. η =1. ξ. η =1. ξ. ξ =1 Nc. ξ =1. (c). η. η =0.5. η. ξ. 0.87mm. ξ =1 Nc. Fig. 3.3 Deflection of the shell-panel under each compressive in-plane force, (a) nc = 0 , (b) nc = 1.6 , (c) nc = 1.9.

(200)

(201) 3RS\J nc = 1.6 nc = 1.9 a;9>CFLI&2DA,B.2D A,. !HXU 3.1 P

(202) bU 3.1(a)anc = 1.6 "a2DA,B. ]%a 3 11 31 (['AE#K

(203) b 8 aU 3.1(b)P

(204) _(3R )7 nc = 1.9 "a2DA,B]%a 2 11 12 a 3 11 31 a 2 21 31 WB(['AE#K

(205) b_(3R)5*$. 2DA,B6- a3RS\J nc = 1.6 nc = 1.9 57 4/a. (1,1) a(1, 2) (3,1) A, !8?

(206) 2DA,B 5* bMa (1, 2). !2DA,B 12 5*:`Tb a1 3.3(c) . =Na = 0.5 Z8 a^8Q6==N@ a (1, 2) A,=a^8Q6=V<+GO Y0

(207) b. 32. !.

(208) : n c =1.6 : n c =1.9. 500. q. 0 -500 -1000 -4. -3. -2. -1. w. 0. 1. Fig.3.4 Characteristics of restoring force of the shell-panel, concentrated force is loaded on =0.5 , =0.5 and the deflection is measured at =0.4 , =0.6 Table 3.2 Natural frequencies and vibration modes of the shell-panel (a) nc = 1.6 (m,n). (1,1). (1,2). (2,1). (1, 3). (3,1). (2,2). fmn [H z] 74.9 ω mn 40.8. 103 55.9. 130 70.7. 187 102. 223 121. 232 126. η. ξ. (b) nc = 1.9 (m,n). (1,1). (2,1). (1,2). (3, 1). fmn [H z] 88.3 ω mn 48.0. 127 69.2. 172 93.0. 263 142. η. ξ. 33.

(209) .TU[G nc = 1.6 nc = 1.9 ]538ADE@*+C&Z4]&= =(?>6](N\R7;QI0 ^9 +H?;QAR, 3.5 P ÊXKF"&==(? ]SXKF" 1'. wrms . ^ (m, n) j F#=;Q (m, n; j) ]-B=(? mn )

(210) , P ^, 3.5(a)P nc = 1.6 /) ] = 40 Y! (1,1) # =;QM]JW-JO7:"%L<2; #=;QP ^ = 75 Y ! ](1,1) F$?FVH#=;QM ^] = 85 ~ 90. :measured at (0.6,0.4), pd=0.75 103. wrms. 3. (1,1;1) (1,1;1/2). 2 11. 12. 1. A1. 13. 22. 31. 21. 40. 60. 80. 100. 120. 140. (a) nc = 1.6 :measured at (0.6,0.4), pd=0.87103. 3. wrms. (1,1;1). B1. 2 1. 21. 11. 30. 40. 50. 60. 70. (b) nc = 1.9. Fig.3.5 Frequency response curves under each compressive in-plane force. 34.

(211) (:-gA1 e<)gC4MIg 12 + 31 2 d/R

(212) [8`Q3GDY?GADYX

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(215) h g (1,1) *,&;KG7H 11 c1(:-gB1 e<)g #G7DY W

(216) h

(217) : 3.5(a) A1 e<[8`Q3GDYg: 3.5(b) B1 e<. #G7. DY@ 4M^

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(219) 9LH. / e g\bUP2

(220) w h9QH4M[NObUP24 MG7H sp X g\bGA#)!%+$"(+0X h9QH4M[ Ng6GG7HF4 g 12 g 31 @D G7F4g. w. f]#)!%+',!_hUP26GG7H G7F4 12 g 31. 1.0 0.5 0.0 -0.5 -1.0 -1.5 50. = 89.4 60. 70. 80. 90. 100. e. 20. 2. A[dB]. 0 -20 -40 -60 10. = 89.4 2. 11 3. 4. 12 5. 31. 6 7 8 9. 100. 2. sp. Fig. 3.6 Time response and Fourier spectrum at = 89.4 , measured at =0.6 , =0.4 , nc = 1.6 35.

(221) 2 12 + 31 2. d (1, 2) (3,1) ),&|~ . tA}b5XUq n sp = 34 0(1,1) ) ,&X<W8MU ',$JF (1,1) ),&E[X<Y 11 = 41 1X<Y',$ktA}b5XUqXOeR T4:hV. o rC6(!+#*73D. D 3.7 o . (!+#*73D`ga4XO w wga4Q w, (!+#*73D

(222) Ni_n{ tA}b5XUq W88]. HfcK SZs9bRM /. D 3.8 o DX<L. p),&. yt^. PM L.jo L.j (3,1) (1, 2) . (1,1) @E[X<),&MU p),& ,&L.jI 8 XUql

(223) . S (3,1) (1, 2) ). ? (3,1) (1, 2) ),&tA}b5 o 8.7%L.j. o (1,1) ),&Bb. Y8]hK X<W8-x .

(224) nc = 1.9 ;XX<Y = 47.2 #"%X<UqZs9bRBbY8]t^ D 3.9 o DZs9bR=uvB\mG>zB. w,. bY8]t^;XX<YW8 aX<W8 2 3 . 1.4 1.2 1.0 0.8 0.6 0.4 -0.8. 89.4 == 89.4. -0.6. -0.4. w. -0.2. Fig.3.7 Poincaré projection of the response at = 89.4 , measured at =0.6 , =0.4 , nc = 1.6 36.

(225) Contribution ratio. 1. 8 7 6 5 4. (3,1): 65.8%. = 89.4. (1,2): 22.4%. 3 2. (1,1): 8.7%. 0.1. 8. 1. 2. 3. Order of eigenvalues. Fig.3.8 Principal components in the combination response at = 89.4 , nc = 1.6 4 5 wYQ(/')3,4'MI "w,4'$o Y. CV@ Ejd=H_YC\PU 11 2 12 3 31 :$ i . 21 |< (3 / 2) (/')3,4'MI. " #! (1,1) 04*8>YUr (1, 2) 04* 2 dyh>YUr (3,1) 04* 3 dyh>YUrB (2,1) 04* 3/2 d9@\dy h>YUr}V ?~>YS&%(YCUrl

(226) " &%(YCUr$o ]sAhSP Wolf Wg(8)K ^L21.+-X\$ftb$G 3.10 o GczJ{d= uz^L21.+-X\ max "G! max 0.4 pRD` e; $" #! T #". #]Ur &%(YCUr" nN. e = 6 |<D` " . J{d= e = 5 . &%(. YCUrO7 "8YC04*\6" &%(YCUr]sAhSP 8V@@a$xtb$G 3.11 o O7k (1,1) (1, 2) (2,1) 04*T. # # Fh\@atb. " J{d=D`; ". (3,1) 04*PU "q ZN YCV@5v. ZN #"O704*\5v. (1,1) 04*O7k 89%$o . ?~>YS&%(YCUr"". 37. (1,1)04*[m.

(227) 2. w. 1 0 -1. = 47.2. 260. 280. 300. 320. 340. A[dB]. e. 20 0 -20 -40 -60 -80 10. 4. 3. 2. 5. 3/2 = 47.2. 11. 2. 3. 4. 21 5. 12. 31. 6 7 8 9. 2. 100. sp. Fig. 3.9 Time response and Fourier spectrum at = 47.2 , measured at =0.7 , =0.7 , nc = 1.9 . 2. max. 10 4 2. 1 4 2. 0.1 2. 4. 6. e. 8. 10. 12. Fig. 3.10 Maximum Lyapunov exponents related to embedding dimension, = 47.2 , measured at =0.7 , =0.7 , nc = 1.9 38.

(228) Contribution ratio. 1. 8 6 4. (1,1): 88.5%. 2. (1,1): (2,1): 3.7%. 0.1. 8 6 4. (1,2): (1,1): 6.0%. 2. 0.01. = 47.2. 1. (1,1): 1.6% (3,1):. 2. 3. 4. Order of eigenvalues. Fig.3.11 Principal components in the chaotic response at = 47.2 , nc = 1.9 . FCH0lVZ9X1Ds{+b . Q4Lm8|qGQ. 4P4@eU 0lT7=c:?fqT7=c5 oRO

(229) =cY(&j}/;r2:?k3|qG Q4Lm. Q4Lm1[uIp\vn

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(232) . ( 2 )}/;r2><6J-2`M_xGRze`Ke . Q4LmW*]"#!%.QLmy,a . ( 3 ) %N11[p\p6w^.QLmAiQ4"#! |Ai. 39.

(233) '!4 3+#1. $-. 4)+. $5(4'!&/

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(235) () Amabili, M., Non-linear vibrations of doubly curved shallow shells, International Journal of Non-Linear Mechanics, Vol. 40(2005), pp. 683-710. () Amabili, M., Theory and experiments for large-amplitude vibrations of circular cylindrical panels with geometric imperfections, Journal of Sound and Vibration, Vol. 298 (2006), pp. 43-72. () Nagai, K., Maruyama, S., Oya, M., Yamaguchi, T., Chaotic oscillations of a shallow cylindrical with a concentrated mass under periodic excitation, Computers and Structures, Vol.82 (2004), pp.2607-2619. () Nagai, K., Maruyama, S., Murata, T., Yamaguchi, T., Experiments and analysis on chaotic vibrations of a shallow cylindrical shell-panel, Journal of Sound and Vibration, Vol. 305 (2007), pp. 492–520. () Maruyama, S., Nagai, K., Tsuruta, Y., Modal interaction in chaotic vibrations of a shallow double-curved shell-panel, Journal of Sound and Vibration, Vol. 315 (2008), pp. 607–625. () Maruyama, S., Onozato, N., Nagai, K., Yamaguchi, T., Experiments on nonlinear vibrations of a cylindrical shallow shell-panel with clamped edges under an in-plane elastic constraint, Transactions of the Japan society of mechanical engineers, Series C, Vol.74, No.743 (2008),pp.1696-1701. () Chang, S., I., Bajaj, A., K., Krousgrill, C.,M., Non-linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance, Nonlinear Dynamics, Vol.4 (1993), pp.433-460. ( ) Yamaguchi, T., Nagai, K., Maruyama, S., Identification of spatial modes in chaotic vibration involving dynamic snap-through using KL method, Transactions of the Japan Society of Mechanical Engineers, Series C, Vol.69, No.687 (2003), pp.2937-2942. ( ) Azeez, M. F. A. ,Vakakis, A. F. , Proper orthogonal decomposition of a class of vibroimpact oscillations, Journal of Sound and Vibration, Vol. 240 (2001), pp. 859-889. () Wolf, A. , Determining Lyapunov exponents from a time series, Physica , Vol. 16D (1985), pp. 285-317.. 41.

(236)

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(242) " "sflEHlcH z y# GWBH!. . ;TI7mX[`WBCZ;{L#3vx"T<Qn #\8 !;TItr]u 2.3 oA^!. b=140 mm. y. a=140 mm. x. z. shell-panel. w(x,y,t). rigid block 3. spring plates. R=4.8 10 mm. slide block. strain gauge. Fig.4.1 Fixture of the shell-panel 43.

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(258) i z ]H[@.Z `6 iIP:L=>_O jie%89)iAE. @.6 (=2*.

(259) j. η η = 0.98. η = 0.02 ξ = 0.02 ξ. η. η = 0.02 ξ = 0.02. η = 0.98. w = 5.2. ξ. ξ = 0.98. (a) nc =0 (b) nc =-0.78 Fig. 4.2. Configurations of the panel under each in-plane force.. 45. w = 4.2 ξ = 0.98.

(260)

(261)

(262) + 4.1 ., nc = 0

(263) , nc = 0.78 *! %$!. f mn .. mn

(264) ! #&'/.!. ( ." )*!.$. 600. - (1,3)a.. : nc = 0 : nc = -0.78. 400 200. q. 0 -200 -400 -600 -800 -2.0. -1.0. 0.0. 1.0. w Fig.4.3 Characteristics of the restoring force of the panel, measured at = 0.6 , = 0.4 , the point of concentrated load =0.5 , =0.5 . Table 4.1 Natural frequencies and vibration modes of the panel. nc. i (m,n) F. 0. C. η. 1 (1,2). 2 (1,1). Cξ. F. f m,n Hz 53.5 68.0 ω m,n 30.4 38.7 (m,n) (1,3)a (1,2) F C. 3 4 5 6 (1,3)a (1,3)b (1,3)c (1,4). 114 64.7 (1,1). 117 147 176 66.7 83.6 100 (1,3)b (1,3)c (1,3)d. 71.2 40.4. 110 62.6. Cξ. -0.78 η F f m,n Hz 49.0 ω m,n 27.9. 62.0 35.2. 46. 116 65.8. 153 87.0.

(265) (1,3)b(1,3)c (1,3)d

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(267) . ` âIr

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(269) \?8TF:

(270)

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(274) "!#X<St{u P@gZ7b_G*)'%&WZ.U77b] B 4.4(a)69 nc = 0 E=X<ZD = 29 38 GXK" !#X<St C[(1,1;1),(1,2;1)]X<(+$(1,1)h-hoOjT. q. .5XStl "!#X<St C[(1,1;1),(1,2;1)]X<(+$(1,1) 5XStX<(+$(1,2)5XSt65XSt = 70 3 "!#X<St C[(1,1;1/2)].X<(+$(1,1) 1/2 e7Ze}g5XSt >cB(b)6LN9 nc = 0.78 E= = 26 46 "!#X< St C[(1,2;1),(1,1;1)] C[(1,1;1), (1,2;4/5)]GXK. q j = 26 41. "!#X<St C[(1,2;1),(1,1;1)]hoOjT. q X<(+$. (1,2).5XStl "!#X<St C[(1,2;1),(1,1;1)]X<(+$ (1,1)5XStX<(+$(1,2)5XSt65XSt = 56 3"!#X<St C[(1,1;2/3), (1,2;3/5)]X<(+$(1,1) 2/3 e/7Ze}. 47.

(275) 9%1.<1("(1,2) 3/5 8$'28@9%1.< &B%1 :E.

(276) 5C D6)A,1-1(.< C[(1,1;1), (1,2;1)]DC[(2,1;1), (1,1;1)]

(277) C[(1,1;1), (1,2;4/5)] '7?E

(278) D1(. <'7D+92.<3>D+92'7D4,!. 02#/'. '7=*; ?E. 2.5. nc= 0. C[(1,1;1),(1,2;1)]. w rms. 2.0 [(1,4;1), (1,3b;2/3)]. 1.5 (1,1;1). [(1,4;1), (1,1;1/3)]. 1.0. [(1,2;1/3),(1,3c;1)]. C[(1,1;1/2)]. 0.5. (a) 0.0. 20. 12. 40. 11. 60. 80 13b. 13a. 13c. 100. 14. 2.5. n c = -0.78. C[(1,2;1),(1,1;1)] C[(1,1;1),(1,2;4/5)] 2.0. w rms. (1,1; 2) 1.5. C[(1,1;2/3),(1,2;3/5)] 1.0. [(1,2;1/3),(1,3b;3/5)] [(1,3a;1/2),(1,3b;1)]. 0.5 0.0. 12 20. 13a. 40. 11. (b). 13b 60. 13c. 80. 100 13d. 14. Fig.4.4 Frequency response curves, measured at = 0.6 , = 0.4 , pd = 0.32 ×10 3 . 48.

(279)

(280)

(281) lSo (, ) = (0.6,0.4) (, ) = (0.25,0.10) "'&*\DY{` kX%I 4.5(a),(b)(c)x I 4.5 hB\Hc e %Mm Hc ] / e " pi< $ w " I!' &*\ DY{ C[(1,1;1),(1,2;1)]C[(1,2;1), (1,1;1)] C[(1,1;1), (1,2;4/5)]`kX6 AE}~ND%x . ad7Q9lSo (, ) = (0.6,0.4) . "`kX$P j ^GFyW"4^sz ; (, ) = (0.25,0.10) ^G$!\UNDP #(, ) = (0.25,0.10) "adv$j^G?" " Hk]@e|f%I 4.6(a),(b),(c)x I 4.6 hpi<@e\D ] sp %x \U*0(,2+)/2:%x IHk]@e|f !\U*0(,2.3(FyWP V\D]LRK " #!#'&*\DY{O\D[@%b " "q\D[@ I 4.6(a)B\\D[@ "B\\ D[@ \D13-(1,1)\D13-(1,2)C\\D]TY " # !'&*\DY{ C[(1,1;1),(1,2;1)]\D13-(1,1)8=\Y{. . \D13-(1,2)8=\Y{[ >=\!r

(282) " \D1 3-(1,1) (1,2) #"\D[@~XJb\D]t" # \D13-(1,1)ur "C\\D]\D\U!n-nwX qZx "Fg\D13-(1,1)ur "C\\D]nw XqZ!5_ " I 4.6(b)B\\D[@ (9 / 4) \D[@"B\\D [ @ \ D 1 3 - (1,2) (1,1) C \ \ D ] T Y " # ! C[(1,2;1),(1,1;1)]\D13-(1,2)8=\Y{\D13-(1,1)8=\ Y{">=\X'&*\DY{" I 4.6(c)B\\D[@ (4 / 5) \D[@"\D[@ (4 / 5) ##\D13-(1,1)\D13-(2,1)C\\D]T Y " #!C[(1,1;1),(1,2;4/5)]\D13-(1,1)8=\Y{\. 49.

(283)

(284) (1,2) 4/5 . 4. ( ξ , η) =(0.6,0.4). w. 2. (0.25,0.1). 0 -2 -4 -6 (a) C[(1,1;1),(1,2;1)] 4. (0.6,0.4). w. 2. (0.25,0.1). 0 -2 -4 -6 (b) C[(1,2;1),(1,1;1)] 4 (0.6,0.4). w. 2. (0.25,0.1). 0 -2 -4 -6. 50. 60. 70. 80 ε. 90. 100 50. 60. (c) C[(1,1;1),(1,2;4/5)]. 70. 80 ε. 90. 100. Fig. 4.5 Time responses at each region of chaotic responses, (a) nc =0 , = 35.4 ;(b) nc =-0.78 , = 40.4 (c) nc =-0.78 , = 44.9 . 50.

(285) 20. A [dB]. (a) C[(1,1;1),(1,2;1)] 0 -20 -40. 12. -60 2. 11. 3. 13 a. 4. 5. 6. 13 c. 13 b 7. 8. 14. 9. 2. 100. A [dB]. 20. (b) C[(1,2;1),(1,1;1)]. 0. 9/4. -20 -40 -60 2. 3. 13 b. 11. 12. 13 a. 4. 5. 6. 13 c 7. 13 d. 8. 9. 2. 100. A [dB]. 20. 4/5. (c) C[(1,1;1),(1,2;4/5)]. 0 -20 -40 13 a. -60 2. 3. 12. 13 b. 11 4. 5. 6. 13 c 7. 8. 13 d 9. 2. 100 sp. Fig. 4.6 Fourier spectra at each region of chaotic responses, measured at = 0.25 , = 0.90 ;(a) nc =0 , = 35.4 ;(b) nc =-0.78 , = 40.4 (c) nc =-0.78 , = 44.9 .

(286) 95 -&,7 C[(1,1;1), (1,2;1)] C[(1,1;1), (1,2;4/5)] %!(( 4.7(a) (b)

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(289) ?/ %!3>'6A. 51.

(290)

(291) (. &4 C[(1,1;1), (1,2;1)]:C[(1,2;1), (1,1;1)] C[(1,1;1), (1,2;4/5)]. *92%+$')1;5-" 4.8 3;" .7

(292) # 8/ e :67

(293) +$') max ;":(. &4. C[(1,1;1), (1,2;1)]: C[(1,2;1), (1,1;1)] C[(1,1;1), (1,2;4/5)] +$ ') max

(294) : max = 1.5 :max = 0.98 max = 2.1 0 !,;. 6. 3. (a). (b) 2. 4. 1 0. w,. w,. 2 0. -1 -2. -2. -3 -4 -4. -2. 0. 2. -4 -3. 4. -2. -1. w. 0. 1. 2. w. Fig.4.7 Poincaré projections of the response at each region of chaotic responses,(a)C[(1,1;1),(1,2;1)], = 35.4 ;(b)C[(1,1;1),(1,2;4/5)], = 44.9 measured at = 0.6 , = 0.4 .. max. 10. : C[(1,1;1),(1,2;1)], ω = 35.4 : C[(1,2;1),(1,1;1)], ω = 40.4 : C[(1,1;1),(1,2;4/5)], ω = 44.9. 8 6 4 2. 1. 8. 5. 10. e. 15. 20. Fig. 4.8 Maximum Lyapunov exponents related to embedding dimension, measured at =0.60 , =0.40. 52.

(295)

(296) #!s#MpWE'&(K7"

(297) `@ #"t. s. O?/.,*+JL 8P"q<mU5 e s#'&(K7G ds9 10 "t s9X@YMpWEBsO?/.,*+JL max %Vs max 8P3%FteS%; 4.9 at;TlX@Y%asflO?/ .,*+JL8P3"t;!smax 8P3sCI_^"

(298) k. #"tZsNQE[$ ?"g]cn4X@Y. P3 (, ) = (0.25,0.10) s max 8P3 rh"t

(299) #s; 4.5 Y. (, ) = (0.25,0.10) "MpWE s? KD=7%a

(300) "t '&(K7Gd C[(1,1;1),(1,2;1)]sC[(1,2;1), (1,1;1)] C[(1,1;1), (1,2;4/5)] % >Y:M8oMpWEBs2H66R%iteS%; 4.10 at ;Tl2jbp-0)sflbp-0)A1\"t. max. 3.5. F. o. P2 P1. P4. Convergence values of. ξ. P3. 3.0 C. 2.5. η. 2.0. P5. C. P6 P7. F. 1.5 1.0 0.5. P1. P2. P3. P4. P5. P6. P7. Fig. 4.9 Convergence values of maximum Lyapunov exponents, measured at each point, :C[(1,1;1), (1,2;1)], = 35.4 ; : C[(1,2;1), (1,1;1)],. = 40.4 C[(1,1;1), (1,2;4/5)], = 44.9 .. 53.

(301) )0,>%& 2'1: C[(1,1;1), (1,2;1)]@?*#68 2'. !