Inverted oscillations of a driven pendulum

被引:22
作者
Clifford, MJ
Bishop, SR
机构
[1] Univ Nottingham, Div Theoret Mech, Nottingham NG7 2RD, England
[2] Univ London Univ Coll, Ctr Nonlinear Dynam & Applicat, London WC1E 6BT, England
来源
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES | 1998年 / 454卷 / 1979期
关键词
parametrically forced pendulum; nodding oscillations; subharmonic oscillations; manifolds; horseshoe; braids;
D O I
10.1098/rspa.1998.0282
中图分类号
O [数理科学和化学]; P [天文学、地球科学]; Q [生物科学]; N [自然科学总论];
学科分类号
07 ; 0710 ; 09 ;
摘要
Inverted oscillations of a parametrically driven planar pendulum are considered, together with their relationship to the inverted solution. In particular, a horse shoe structure of the associated manifolds is identified which explains the similarity between the bifurcations of the inverted position and the hanging position. This allows us to apply a large body of existing knowledge to the dynamics enabling a lower bound on the forcing required to achieve inverted oscillations to be established.
引用
收藏
页码:2811 / 2817
页数:7
相关论文
共 23 条
[1]   A PENDULUM THEOREM [J].
ACHESON, DJ .
PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 1993, 443 (1917) :239-245
[2]   UPSIDE-DOWN PENDULUMS [J].
ACHESON, DJ ;
MULLIN, T .
NATURE, 1993, 366 (6452) :215-216
[3]   MULTIPLE-NODDING OSCILLATIONS OF A DRIVEN INVERTED PENDULUM [J].
ACHESON, DJ .
PROCEEDINGS OF THE ROYAL SOCIETY-MATHEMATICAL AND PHYSICAL SCIENCES, 1995, 448 (1932) :89-95
[4]  
[Anonymous], 2012, Practical numerical algorithms for chaotic systems
[5]  
[Anonymous], 1993, NATURE CHAOS
[6]  
Birman J. S., 1974, ANN MATH STUDY, V82
[7]   The use of manifold tangencies to predict orbits, bifurcations and estimate escape in driven systems [J].
Bishop, SR ;
Clifford, MJ .
CHAOS SOLITONS & FRACTALS, 1996, 7 (10) :1537-1553
[8]   Zones of chaotic behaviour in the parametrically excited pendulum [J].
Bishop, SR ;
Clifford, MJ .
JOURNAL OF SOUND AND VIBRATION, 1996, 189 (01) :142-147
[9]  
BISHOP SR, 1994, EUR J MECH A-SOLID, V13, P581
[10]   ON A PERIODICALLY FORCED, WEAKLY DAMPED PENDULUM .3. VERTICAL FORCING [J].
BRYANT, PJ ;
MILES, JW .
JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY SERIES B-APPLIED MATHEMATICS, 1990, 32 :42-60