On the phenomenon of parametric resonance of a nonlinear vibrator under the action of electromagnetic force

[...]... inistyczny “ klasycznego” ujẹcia m echaniki kwantowej C r o x a c n m e c K H Ỉí H AeTep.MHHiiCTHMecKKH acneK Tbi “ lcriaccM M ecKoro” noaxoaa K KBaHTOBoft MCXAHHKC S K a l isk i, Averaged e q u a tio n s o f c u m u la tiv e laser heating of a plasm a focus w ith c o n sid e ratio n o f th e h e a t o f n u c le a r fu sion U srednione rồvvnania kum ulacyjno-laserow ego nagrzew ania "fo c u sa ” plazm y z... e theory o f non-linear oscillations, Moscow 1963 5 N g u y e n v a n D a o , Fundamental m ethods o f the theory o f non-linear oscil/ationsy H a n o i 1971 Streszczcnic ZJA W ISK O PA R A M ETR Y C ZN EG O REZONANSU N IELIN IO W EG O O SCYLA TO RA P O D DZIALAN1EM ELEK TR O M A G N ETY C ZN EJ SILY w pracy rozpatrzono zjawisko parametrycznego rezonansu nieliniowego sprẹzystego oscylatora pobudzanego... ig 4 a W v /V W V W W V W F i g 5 a b /W ÍW W V W W W V W W W \A Fio 6 [ 289 ] Nguyen van Dao 290 Increasing from zero the frequency of external force V it was observed that, at a value V - 19Hz the mass m vibrates s t r o n g l y with doubled frequency V (Fig 5b) This is the principal resonance Figure 5a shows photographs of that oscillation (curve I) For con­ venience of comparison, in Fie 5a is... elektromagnetycznie (rys 1) Ustalono, ze w rozpatrywanym ukladzie oprócz znanego podstaw ow ego rezonansu, gdy czẹstoấé drgart w elektrycznym obw odzic je s t dw a razy m niejsza ođ czẹstoấci On the phenomenon o f param etric resonance o f a nonlinear vibrator w ta s n e j o s c y la t o r a , w y s tẹ p u jc r o w n ie z p r z e d z i a l c z ẹ s to s c i, w e le k trv c z n e g o o b u o d u R e z u lta... represented the electrical sicnal with frequency 2v = 38 Hz (curve 0) F ig 7 For values V from 36 Hz to 40Hz, there exist parametric oscillations of the mass m with frequency V (F is 6 b) The curve 7 in Fig 6 acorresponds to oscillations of the mass m inside th e r e so n a n c e (v = 3 0 H z ) ; curve 2 for the value V = 3 7 H z, a n d curve 3 f o r the value V = 39Hz (paramctric resonance) For comparison,... cucmeM 3.ưKm a H biM 6036yncdenuH iÌỊ ripHO anuũ poM wnH U M ốopocTpoeH H e, 2 , 1 9 6 0 2 A E H echokob, K tneopuu u pa cu em y 3AeK»ipoMcuHwiiHOĩo eu6pa m opa3 3/ieKTpH'iecTBO, 1 2 , 1961 3 D D K a n a , Parametric coupling in a nonlinear eìectro-mechanicơl system , Transactions o f the ASME\ Proc of the Vibrations Conference, 2, 1967 4 N N B o g o liu b o v an d Yu A M itr o p o ls k i, A sym ptotic... P a c n p o c T p a u c H ite ruiocK oft V A apiioft B o.iH bi Harp>'3KK B ỖH/iiiHeHHOM y n p y ro M CTCp>KHC S t ì NJ a n d a l , D iíĩraction o f elastic waves by a r i g i d half-plane D yfrakcja fal sprẹzystych na sztywnej pốỉplaszczyznie H Ộ U yiipynix B JIH na /K K M noJiynpocTpaH H paK itH O ecT O CTBe S K a lis k i a n d E W l o d a r c z y k , Shock w a ve p ro d u c e d in a solid by m eans... y z uwzglẹdnieniem ciepla syntezy jạdrow ej ycpcAneHifbic ypaBiieiW KyM H \v!«unoHHc-.ia3cpnoro HarpcBa “ộoKyca” n.ia3M c y C O bi MT M T en.ia H A epnoro cHHTC 3a E W L O D A R C Z Y ^ P r o p a g a t i o n o f a p l a n e l o a d i n g s h o c k w a v e in a b i l i n e a r e l a s t i c b a r P r o p a g a c j a p la s k i e j u d e r z c n i o w e j fall o b c i ^ z e n i a w b i l i n i c w y m... a. 1113 E PAR TMENT OF MATHEMATICS AND PHYSICS POLYTECHNI C INSTITUTE, H a n o i Received Xovem bcr 6, 1971 p C 3 0 H a n c) The next number of Proceedings o f Vibration Problems will contain the following papers: S K a l is k i, T h e s to c h a s tic a n d d e te r m in is tic a s p e c ts o f t h e c la ssic a l a p p r o a c h t o p r o b le m s o f q u a n t u m m echanics A spekty stochastyczny... obody riapa.M Tpim K e K ieỐ ỉia C T Ml c 3B M C CC H H C oốoA C eC H O aH ỈỈC C b V H T n H M B kí A W ie l g u s, A n a ly s is o f self-cxcited a n d fo r c c d v ib r a tio n s o f a r e c ta n g u la r p la te o n m a n y s u p p o r t s in supersonic flow A naliza sam ow zbudnvch i w ym uszonych drgan p r o s t o k ^ t n c j ptyty na wielu podp o rach w oplywie naddzv\iẹko\vym A H a; iH 3 ca.M0B03Õy>KAaK>mHxcfl . of a cantilever beam and a block of mass /?/, which is made of magnetic material. The elastic force of the beam is assumed to have a nonlinear char acteristic where A is the coordinate of the. the parametric oscillatiòn. On the phenomenon o f parametric resonance o f a nonlinear vibrator 283 It is to be noted that the linear oscillation of the mass m in a similar electromechanical system. that the quantity B tends asymptotically to zero. Consequently, On the phenomenon o f parametric resonance o f a nonlinear vibrator 285 below we shall be interested in the last two equations of