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1、?n?Acta Phys.Sin.Vol.61,No.1(2012)010505?555?555?*?u 4kR(Hfi?U?,Hfi210016)(2010 c 7?16 F?;2011 c 4?15 F?Uv)?p“&?f$“&?-ye?5?LZVX?Duffing?fX?y?K?.)?(J?L,X?$“&?A?O?Cz?y?5X,?O?p“&?$“&?.?(J?L,?3;?y?Duffing X?,LN!?u?y?.?=?k?/?,?OrX?f$“&?A.?c:VX?,Duffing X?,?5?,?PACS:05.45.a,02.30.Ks,05.90.+m1?Cc5,?“&?-ye?
2、5X?AflK?0,&?f 1,&?“,=?&?O?$“f&?p“&?.3?(=0)?,?;?n?4,LN!?p“&?,X?A3“?y?y?,$“f&?3X?Or.?,Yang?Liu uy15,?&?C?,LN!?k?/?,l?$“f&?X?A.X?$“&?A?O5?,?Q=Q2s+Q2cf,(2)?Qs?QcO?X?3“?u?u Fourier?,Qs=2mTmT0 x(t)sin(t)dt,Qc=2mTmT0 x(t)cos(t)dt,(3)T=2/,m?.(2)“L?f$“&?LX?,?f?A3“?.2.1?ppp“&?-yyy?AAA?555KKK?k,?p“&?,=f=0?/,d?
3、(1)C?ddtx(t)=x3(t)+x(t )+F cos(t).(4)X?A?d?-&?,du?5?K?,b?4?)?x(t)=n=1Ansin(nt)+Bncos(nt),(5)?x(t)=n=1Cnsin(nt+n),(6)?Cn=A2n+B2n,n=atan(BnAn).(7)r?(5)?(4),?u An,Bn?X?|,?u A1,B1?|?3A31+3A1B21 4A1cos()4B1sin()4B1=0,(8)3B31+3A21B1 4B1cos()+4A1sin()+4A1 4F=0.(9)?5?|?(J?flK,?(8)?(9)?J?A1?A2?N)?“,?Juy A1?B
4、1?),F,sin(),cos()?.?C1?AT,F,sin(),cos()?,=C1=F1F,sin(),cos(),(10)?F1 L?P.3?(10),NuyC1(+2/)=F1F,sin(+2/),cos(+2/)=C1(),(11)dd?X?3“?5Cz?,?T?up“&?,=2/.L?y,X?3“n(n=2,.,)?5Cz,?2/(n).d,X?p“&?A?,?2/.fl?,3X?(4)?,?“83?,3?“?,?A?4?f.2.2?$“&?-yyy?AAA?555KKK?&?uX?,=F 6=0,f 6=0?/.?B$?,?X?A?kXe/010505-2?n?Acta Phy
5、s.Sin.Vol.61,No.1(2012)010505“?Cq)79x(t)=y(t)+sin(t+)(12)?y(t)?m”?TL=2/?A?$“?C,L?,I?(?.?m”?TL=2/?u TH=2/,d3 0TH?mS,?r y(t),y(t)?f cos(t)?n.?(12)“?(1)“,t 3 0TH?mS?1,z?ddty(t)y(t )+y3(t)+32/2y(t)=f cos(t).(13)aqu 2.1.!?(4)?,?(13)?Cq)?y(t)=n=1ansin(nt)+ancos(nt),(14)?y(t)=n=1cnsin(nt+n),(15)?cn=a2n+b2n
6、,n=atan(bn/an).(16)?(14)?(13),?u an,bn?X?|,?u a1,b1?|?3a31+3a1b21 4a1cos()4b1sin()+6a124b1+4f=0,(17)3b31+3a21b1+4a1sin()4b1cos()+6b12+4a1=0.(18)?(17)?(18)?b1,b2?)u,f,sin(),cos()?,c1?u,f,sin(),cos()?,P?c1=f1,f,sin(),cos().(19)?1=+2/?,f1,f,sin(1),cos(1)=f1,f,sin(),cos().(20)?y?c1?k5,=X?A3“?m?k5,?2/.2.
7、3?(JJJ?(2)?“,?(1)?1?,?)?K?yX?$“&?A?O?k?5.?!?z 17?U?.4?Runge Kutta?,l?“?k1=x(n)3+x(n i/t)+f cos(nt)+F cos(nt),k2=x(n)+0.5k13+x(n i/t)+0.5k1+f cos(nt)+F cos(nt),k3=x(n)+0.5k23+x(n i/t)+0.5k2+f cos(nt)+F cos(nt),k4=x(n)+k33+x(n i/t)+k3+f cos(nt)+F cos(nt),x(n+1)=x(n)+16(k1+2k2+2k3+k4)t(21)(n=1,2,.,N,N=
8、tt,i=1,2,.,M,M=),?,t OL?m?.(21)“I?U?.4?Runge-Kutta?IO 4?Runge-Kutta?O?,?Bu?n,?z 18,19 L?!?Euler?1?(?.3?O?,?m=30.?“(2),X?p“&?A?O?R=R2s+R2c/F,(22)?Rs?RcO?X?3p“&?“?u?u Fourier?,Rs=2mTmT0 x(t)sin(t)dt,Rc=2mTmT0 x(t)cos(t)dt.(23)1?m(2/)S,X?$“&?A?O Q?m?X?.?X?Cz,Q y5Cz,?2/,?y?2.2.!)?(5.3z 15,du?,?uy5.3?Duf
9、fing?fX?K?V“&?-ye,O3?kV(V?)?(?)?Duffing?fX?5?,?d2dt2x(t)+ddtx(t)20 x(t )+x(t)3=f sin(t)+F sin(t),(24)d2dt2x(t)+ddtx(t)+20 x(t )+x(t)3=f sin(t)+F sin(t),(25)?0,ZX,?,0,?,?&?k?1 2!?&?A5.?!(2)“X?(22)?X?(23)?1?,?y?K?.3?L,?0=1,=30/2,=10,f=0.1,=0.1.?,=0?,3(a)?3(b)O?X?(22)?X?(23)$“&?A?O Q?p“&?F?m?X.3(a)LV;D
10、uffing X?3;?y?,?,?I F?.u =3,?F 1.3?,Q?F?O?/O?1.3 F 1.75?F?O?Q?N?.3(b)L,3?!?,LN!F?U?u?y?,Q?F?O?N?.OAu 3(a)?u)?u)?/,4?X?(22)?A?mS?.?u)?,X?3?(?,?y?m?B?,$“&?U?.?u)?,X?y?m?B?,$“&?k?/?.010505-4?n?Acta Phys.Sin.Vol.61,No.1(2012)010505 3?(=0)?,(a)V Duffing X?;?y?;(b)?Duffing X?y?y?4?(=0)?,(a)V Duffing X?;?y
11、?;(b)?Duffing X?y?y?5V Duffing X?Cz?X?$“&?A?O?5Cz =3 6V Duffing X?,F=1.0,=3?,(a)?u)?mS?,=0;(b)?u?mS?,=60.7010505-5?n?Acta Phys.Sin.Vol.61,No.1(2012)010505 7?Duffing X?Cz?X?$“&?A?O?5Cz =3 8?Duffing X?,F=1.0,=3?,(a)?u)?mS?,=0;(b)?u?mS?,=28.6?V“&?C?,5?$“&?A?O Q?K?.5L?X?O?,X?$“&?A?O?y?5,?O?&?.5(a)?3(a)?
12、uy,N!?U?X?r?.6?X?(22)?mS?K?.3 6(b),?u?,X?X?u)?mS?4?q,?l?m?q5.3(b)fiL,3?Duffing X?(23)?3;?y?.3 7,?C,LN!?,3X?(23)y?5?,?N?|?5,?3?3?X?pu?y?,?f$“&?.8?X?(23)?mS?K?,uy=3?X?,?u)?X?3/“?yaquVX?55?“B?”.k Duffing X?Cz?5?)?y,?1 2!?y?,?!?2K.4 o(?O?p“&?f$“&?-ye?LZV;X?Duffing X?y?.?3?5?,n?(J?LX?$“&?A?O?Cz?y?5,?O?u?
13、p“&?$“&?.u?3;?Duffing X?,LN!?,?pu?y?.|N!?=?k?/?5X?y?,?OrX?f$“&?A.?(J3?+?&?9?f$“&?u?flKJ?g.8c,k?3?“&?-y?5X?A?u?,?/.k,?/“?5?,u?5?010505-6?n?Acta Phys.Sin.Vol.61,No.1(2012)010505K?E,?,?3?,3?/q?X?k?u?.?g,?X?/,?LZ?/?1?.ujZ?/,X?AK?U?yb,u?,?y?.?,?2?3?aX?,X?mX?,p?X?,$?E,?X?K?k?u?.u?n,?X?|?5Jp&?3?fS?D?,$?k?U
14、?)kX?“&?A?JK.1Knoblauch A,Palm G 2005 BioSystems 79 832Su D,Chiu M,Chen C 1996 J.Soc.Precis.Eng.18 1613Maksimov A 1997 Ultrasonics 35 794Landa P S,McClintock P V E 2000 J.Phys.A 33 L4335Gammaitoni L,H anggi P,Jung P,Marchesoni F 1998 Rev.Mod.Phys.70 2236Gitterman M 2001 J.Phys.A 34 L3557Jeyakumari S
15、,Chinnathambi V,Rajasekar S,Sanjuan M A F 2009Chaos 19 0431288Jeyakumari S,Chinnathambi V,Rajasekar S,Sanju an M A F 2009Phys.Rev.E 80 0466089Lin M,Huang Y M 2007 Acta Phys.Sin.56 6173(in Chinese)?fl,?Xr 2007?n?56 617310 Lin M,Meng Y 2010 Acta Phys.Sin.59 3627(in Chinese)?fl,?C 2010?n?59 362711 Balt
16、an as J P,L opez L,Blechman I I,Landa P,Zaikin A,Kurths J,Sanju an M A F 2003 Phys.Rev.E 67 06611912 Ullner E,Zaikin A,Garc a-Ojalvo J,B ascones R,Kurths J 2003Phys.Lett.A 312 34813 Deng B,Wang J,Wei X 2009 Chaos 19 01311714 Deng B,Wang J,Wei X,Tsang K M,Chan W L 2010 Chaos 2001311315 Yang J H,Liu X
17、 B 2010 J.Phys.A 43 12200116 Yao C,Zhan M 2010 Phys.Rev.E 81 06112917 Yang D X,Hu N Q 2003 J.Natl.Univ.Def.Technol.25 91(inChinese)?#,?)?2003 I?E?25 9118 Yang J H,Liu X B 2010 Chaos 20 03312419 Yang J H,Liu X B 2010 Phys.Scr.82 025006010505-7?n?Acta Phys.Sin.Vol.61,No.1(2012)010505Analysis of period
18、ic vibrational resonance induced bylinear time delay feedbackYang Jian-Hua Liu Xian-Bin(Institute of Vibration Engineering Research,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China)(Received 16 July 2010;revised manuscript received 15 April 2011)AbstractUnder the excitations o
19、f the high-frequency and weak low-frequency signals,the effects of linear time delay feedback on thevibrational resonance in overdamped bistable system and Duffing systems are investigated respectively.Both the analytical and the nu-merical results show that the response amplitude of the system to t
20、he low-frequency signal varies periodically with the delay parametersimultaneously(with two different periods,i.e.,the periods of the two exciting signals).Numerical results also indicate that the delayfeedback can induce vibrational resonance in the monostable Duffing system in which there exists n
21、o traditional vibrational resonance.By adjusting the delay parameter,not only the vibrational resonance can be effectively controlled,but also the response of the systemto the weak low-frequency signal can be further improved.Keywords:bistable system,Duffing system,linear time delay feedback,vibrational resonancePACS:05.45.a,02.30.Ks,05.90.+m*Project supported by the National Natural Science Foundation of China(Grant No.11072107),and the Specialized Research Fund for theDoctoral Program of Higher Education of China(Grant No.20093218110003).E-mail:010505-8