Experimental study of an inverted pendulum

Smith, H.; Blackburn, James A.

doi:10.1119/1.17012

Cited by 74 publications

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“…We shall investigate the stability of both fixed points and study their basins of attraction when they are (asymptotically) stable. We shall also investigate the existence of other attractors, and again determine the corresponding basins of attraction; analogous results for similar models can be found in literature (see Smith & Blackburn 1989, 1992Blackburn et al . 1992a, b;Capecchi & Bishop 1994).…”

Section: Introductionmentioning

confidence: 99%

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On the dynamics of a vertically driven damped planar pendulum

Bartuccelli

Gentile

Georgiou

2001

Proc. R. Soc. Lond. A

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Results on the dynamics of the planar pendulum with parametric vertical timeperiodic forcing are reviewed and extended. Numerical methods are employed to study the various dynamical features of the system about its equilibrium positions. Furthermore, the dynamics of the system far from its equilibrium points is systematically investigated by using phase portraits and Poincaré sections. The attractors and the associated basins of attraction are computed. We also calculate the Lyapunov exponents to show that for some parameter values the dynamics of the pendulum shows sensitivity to initial conditions.

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Section: Introductionmentioning

confidence: 99%

“…Note that for each basin there are regions in which the structure seems to be fractal (see Smith & Blackburn 1992); for instance, figure 8a reveals a dense core surrounded by fractal layers.…”

Section: Attractors and Basins Of Attractionmentioning

confidence: 99%

On the dynamics of a vertically driven damped planar pendulum

Bartuccelli

Gentile

Georgiou

2001

Proc. R. Soc. Lond. A

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“…The analysis of the dynamical behavior of a simple inverted pendulum has been studied in connection with stability problems, both from a theoretical and experimental viewpoint and with delay [1][2][3][4][5][6]. However, analytical solutions of the problem assuming oscillations in the suspension point are only considered under certain simplifications in the problem, as it appears in Ref [2].…”

Section: Introductionmentioning

confidence: 99%

“…1 shows the layout of the pendulum system as well as the notation used to deduce the Lagrangian of the system. The pendulum is modeled by a mass m hanging at the end of a rod of negligible mass and length l, which is fixed to a support O [4][5][6], [7][8][9][10][11][12][13][14][15][16]. Let O'XY be an inertial frame and …”

Section: Introductionmentioning

confidence: 99%

Stability and chaotic behavior of a PID controlled inverted pendulum subjected to harmonic base excitations by using the normal form theory

Pérez-Polo

Molina

Chica

et al. 2014

Applied Mathematics and Computation

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In this paper we investigate the stability and the onset of chaotic oscillations around the pointing-up position for a simple inverted pendulum that is driven by a control torque and is harmonically excited in the vertical and horizontal directions. The driven control torque is defined as a proportional plus integral plus derivative (PID) control of the deviation angle with respect to the pointing-down equilibrium position.The parameters of the PID controller are tuned by using the Routh criterion to obtain a stable weak focus around the pointing-up position, whose stability is investigated by using the normal form theory. The normal form theory is also used to deduce a simplified mathematical model that can be resolved analytically and compared with the numerical simulation of the complete mathematical model. From the harmonic prescribed motions for the pendulum base, necessary conditions for chaotic motion are deduced by means of the Melnikov function. When the pendulum is close to the unstable pointing-up position, the PID parameters are changed and the chaotic motion is destroyed, which is achieved by employing very small control signals even in the presence of random noise. The results of the analytical calculations are verified by full numerical simulations.

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“…This system has attracted great attention [12][13][14][15][16][17]. It presents a wide range of dynamical behaviour, such as: the stabilization of the hilltop saddle [18][19][20][21][22][23]; the occurrence of chaotic behaviour [24][25][26][27][28][29]; the observation of period-doubling cascades [30,31] and the existence of resonance regions [11,32]. Moreover, it can be used as qualitative analogue for more complex systems [31,33,34].…”

Section: Introductionmentioning

confidence: 99%

Tilted excitation implies odd periodic resonances

et al. 2016

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Our aim is to unveil how resonances of parametric systems are affected when symmetry is broken. We showed numerically and experimentally that odd resonances indeed come about when the pendulum is excited along a tilted direction. Applying the Melnikov subharmonic function, we not only determined analytically the loci of saddle-node bifurcations delimiting resonance regions in parameter space, but also explained these observations by demonstrating that, under the Melnikov method point of view, odd resonances arise due to an extra torque that appears in the asymmetric case.

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Experimental study of an inverted pendulum

Cited by 74 publications

References 0 publications

On the dynamics of a vertically driven damped planar pendulum

On the dynamics of a vertically driven damped planar pendulum

Stability and chaotic behavior of a PID controlled inverted pendulum subjected to harmonic base excitations by using the normal form theory

Tilted excitation implies odd periodic resonances

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