Planar Pendulum Research Articles

Effectiveness of Delayed Feedback Control Applied to a Small-Size Helicopter with a Suspended Load System

[abstFig src='/00280003/11.jpg' width=""250"" text='A small-size helicopter with a suspended load' ] In this study, a cable-suspended transportation system using a small-size helicopter was investigated. For a secure flight, it is necessary to suppress the swing of the suspended load. For this purpose, a cable angle feedback system was adopted because it is easy to mount the corresponding measuring device on a helicopter. Delayed feedback control was applied for efficient swing damping. The control parameters were obtained using a simple planar double pendulum model that takes into consideration the coupled dynamics of the helicopter and the load. To build the model, the system parameters were identified through frequency response flight tests. In this paper, an appropriate design strategy is reported for a trade-off relationship between vibration damping and piloted handling qualities; the robustness of the control method against disturbances and signal noise was verified by comparing the delayed feedback control method with the real-time angular velocity feedback control method. The damping effect was verified by performing flight tests with three cable lengths.

Dynamics of the N-link pendulum: a fractional perspective

ABSTRACTThis paper addresses the dynamics of an N-link planar pendulum in the perspective of fractional calculus. The proposed methodology leads to a novel point of view for signal propagation as a time-space wave within the system mechanical structure. Numerical simulations show the effectiveness of the approach both for signal visualisation and limit cycle detection.

The Foucault pendulum viewed as a spherical pendulum placed in a rotating reference system

For a pendulum of a given length, installed at any fixed latitude, the motion depends on three variables: the total mechanical energy, connected with the maximum angular distance of the bob from the vertical; the projection of the angular momentum along the vertical, connected with the shape of its orbit along the surface of a sphere, which may be described as a veering ellipse recalling the orbit of Mercury; and the angular velocity, Ω, of the rotating reference system given by that of the Earth. When the value of one of these variables is negligible, the ensuing motion has already been calculated in analytical grounds: a plummet, a Foucault pendulum, or a spherical pendulum including as a special case the so-called plane pendulum. This paper offers a Lagrangian approach in the case when Ω is the sidereal angular velocity of the Earth. To our knowledge, these are the first analytical solutions for any boundary conditions. When the boundary conditions are such that the bob passes through the vertical (i.e., the direction defined by the plummet), the behavior of the pendulum is as expected: the veering velocity of the plane equals the angular velocity of the Earth times the sine of the latitude. This state is reached when the projection of its angular momentum along the vertical vanishes. However, if that projection takes any other value, the motion is produced along a veering ellipse and the time rate of advance of the major axis is the sum of the previous result plus the spherical pendulum contribution. Our solutions include, as special cases, those of the spherical pendulum, or the ideal Foucault pendulum with finite amplitude.

A single mass dynamic vibration absorber producing linear damping, Coriolis damping and gyro-damping

To reduce the vibration of a planar pendulum, this paper proposes a single mass dynamic vibration absorber (DVA), which can simultaneously produce linear damping, Coriolis damping and gyro-damping. The proposed DVA contains a rotor inside a gimbal. The rotor has a constant angular speed and its axis is direct to the fulcrum. At the same time, the gimbal can move in the normal and tangential directions of the pendulum orbit. The translations in tangential and normal directions and the gimbal rotation result in the linear damping, Coriolis damping and gyro-damping, respectively. Because each type of damping has advantages and disadvantages, the combined DVA has the improved performance. A numerical demonstration is carried out to show the effectiveness of the proposed DVA.

The N-link pendulum: Embedding nonlinear dynamics into the multidimensional scaling method

This paper explores the use of multidimensional scaling for analyzing the dynamics of the N-link planar pendulum. Several integer- and fractional-order pendular systems are studied based on the time response and energy of the links. The dynamics get increasingly complex and chaotic with the number of degrees-of-freedom. The numerical experiments demonstrate the feasibility of the proposed methodology in analyzing the data and motivate its extension for other performance measures and dynamical systems.

Erratum to: Dynamics and control of a multi-body planar pendulum

Erratum to: Dynamics and control of a multi-body planar pendulum

The Dynamics of a Parametrically Driven Damped Pendulum

Ordered and chaotic states of a parametrically driven planar pendulum with viscous damping are numerically investigated. The damping makes the number of chaotic windows fewer but with larger width. Stroboscopic maps of the chaotic motion of the pendulum, driven either subharmonically or harmonically, show strange attractors with inversion symmetry in the phase plane.

Dynamics and control of a multi-body planar pendulum

The explicit equations of motion for a gen- eral n-body planar pendulum are derived in a simple and concise manner. A new and novel approach for obtaining these equations using mathematical induc- tion on the number bodies in the pendulum system is used. Assuming that the parameters of the system are precisely known, a simple method for its control that is inspired by analytical dynamics is developed. The control methodology provides closed-form nonlinear control and makes no approximations/linearizations of the nonlinear system. No a priori structure is imposed on the controller. Globally, asymptotic Lyapunov sta- bilityisachievedalongwiththeminimizationofauser- provided control cost at each instant of time. This con- trolmethodologyisthenextendedtoincludeuncertain- tiesintheparametersofthesystemthroughtheuseofan

Rotary motion of the parametric and planar pendulum under stochastic wave excitation

In this paper a rotary motion of a pendulum subjected to a parametric and planar excitation of its pivot mimicking random nature of sea waves has been studied. The vertical motion of the sea surface has been modelled and simulated as a stochastic process, based on the Shinozuka approach and using the spectral representation of the sea state proposed by Pierson–Moskowitz model. It has been investigated how the number of wave frequency components used in the simulation can be reduced without the loss of accuracy and how the model relates to the real data. The generated stochastic wave has been used as an excitation to the pendulum system in numerical and experimental studies. For the first time, the rotary response of a pendulum under stochastic wave excitation has been studied. The rotational number has been used for statistical analysis of the results in the numerical and experimental studies. It has been demonstrated how the forcing arrangement affects the probability of rotation of the parametric pendulum.

Resonance rotations of a pendulum with a vibrating suspension

Resonance rotations of the form s:n (where s and n are integers), for which the pendulum executes exactly s rotations during n oscillations of the suspension point are investigated for a plane pendulum with a vibrating suspension point, when there is external and internal damping. It is shown analytically that asymptotically stable resonance rotations of the form 1:2, 1:3, 2:1, 2:3 and 3:1 exist. Many other resonance rotations of higher orders are found from the results of a numerical experiment.

Stabilizing Planar Inverted Pendulum System Based on Fuzzy Nine-point Controller

In order to stabilize planar inverted pendulum, after analyzing the physical characteristics of the planar inverted pendulum system, a pendulum nine-point controller and a car nine-point controller for X-axis and Y-axis were designed respectively. Then a fuzzy coordinator was designed using the fuzzy control theory based on the priority of those two controllers, and the priority level of the pendulum is higher than the car. Thus, the control tasks of each controller in each axis were harmonized successfully. The designed control strategy did not depend on mathematical model of the system, it depended on the control experience of people or the control experts. The compared experiments showed that the control strategy was easy and effective, what was’s more, it had a very good robust feature. DOI : http://dx.doi.org/10.11591/telkomnika.v12i1.4146

On the Damping-Induced Self-Recovery Phenomenon in Mechanical Systems with Several Unactuated Cyclic Variables

The damping-induced self-recovery phenomenon refers to the fundamental property of underactuated mechanical systems: if an unactuated cyclic variable is under a viscous damping-like force and the system starts from rest, then the cyclic variable will always move back to its initial condition as the actuated variables come to stop. The regular momentum conservation phenomenon can be viewed as the limit of the damping-induced self-recovery phenomenon in the sense that the self-recovery phenomenon disappears as the damping goes to zero. This paper generalizes the past result on damping-induced self-recovery for the case of a single unactuated cyclic variable to the case of multiple unactuated cyclic variables. We characterize a class of external forces that induce new conserved quantities, which we call the damping-induced momenta. The damping-induced momenta yield first-order asymptotically stable dynamics for the unactuated cyclic variables under some conditions, thereby inducing the self-recovery phenomenon. It is also shown that the viscous damping-like forces impose bounds on the range of trajectories of the unactuated cyclic variables. Two examples are presented to demonstrate the analytical discoveries: the planar pendulum with gimbal actuators and the three-link planar manipulator on a horizontal plane.

Pendulum with a square-wave modulated length

Parametric excitation of a rigid planar pendulum caused by a square-wave modulation of its length is investigated both analytically and with the help of computer simulations. The threshold and other characteristics of parametric resonance are found and discussed in detail. The role of non-linear properties of the pendulum in restricting the resonant swinging is emphasized. The boundaries of parametric instability are determined as functions of the modulation depth and the quality factor. Stationary oscillations at these boundaries and at the threshold conditions are investigated. The feedback providing active optimal control of pumping and damping is analyzed. Phase locking between the drive and the pendulum at large amplitudes and the phenomenon of parametric autoresonance are discussed.

Design of Parameter Adaptive Fuzzy Controller for the Planar Double Inverted Pendulum

In this paper, The mathematical model is established using analytical dynamics method for planar double inverted pendulum, based on optimal control theory, parameter adaptive fuzzy controller is designed by use of variable fusion technology, thereby it can reduce the dimension of the input variables of the fuzzy controller, greatly decrease the number of the fuzzy rules, and the impact of the quantization factor on the control effectiveness is studied. The designed controller for system is applied in simulation experiments, the results show that the fuzzy controller can guarantee good control precision, fast response, control planar inverted pendulum stability.

Jacobi zeta function and action-angle coordinates for the pendulum

The Jacobi elliptic functions and integrals play a defining role in analytically describing the motion of the planar pendulum. In the present paper, the Jacobi zeta function is given the physical interpretation as the generating function of the canonical transformation from the pendulum coordinates ϑ and p≡∂ϑ/∂t to the action-angle coordinates (J,ζ) for both the librating pendulum and the rotating pendulum.

Dynamics Modeling of the Planar Double Inverted Pendulum

planar double inverted pendulum is a strong coupling, uncertain and complex nonlinear system, and the dynamics model of which is the basis of control, simulation and analysis. In the paper coordinate systems of the planar double inverted pendulum were first defined, and then the dynamics model of which was built up based on screw theory and the Lagrange principle. The modeling method used being systematic and standardized, it is easy to extend to dynamics modeling of higher order planar inverted pendulums or other multi-body systems.

The planarity of the stickface motion in the field hockey hit

The field hockey hit is an important but poorly understood stroke. In this study, we investigated the planarity of the stickface motion during the downswing to better characterize the kinematics and to assess the suitability of planar pendulum models for simulating the hit. Thirteen experienced female field hockey players were filmed executing hits with a single approach step, and the kinematics of the centre of the stickface were measured. A method was developed for identifying how far back from impact the stickface motion was planar. Orthogonal regression was used to fit least-squares planes to the stickface path during sections of the downswing of varying length, with each section ending at impact. A section was considered planar if the root mean square residual between the stickface path and the fitted plane was less than 0.25% of the distance travelled by the stickface during that period. On average, the stickface motion was planar for the last 83 ± 12% of its downswing path, with the length of the planar section ranging from 1.85 m to 2.70 m. The suitability of a planar model for the stickface motion was supported, but further investigation of the stick and arm kinematics is warranted.

LQR self-adjusting based control for the planar double inverted pendulum

Abstract Firstly, the mathematical model of planar double inverted pendulum was established by means of analytical dynamics method. Based on the linear quadratic optimal theory, LQR self-adjusting controller was presented with optimize factor. Further the output of LQR controller is refined through optimize factor which is the function of the states of planar pendulum, and on account of that, control action exerted on the pendulum is improved. Simulation results together with pilot scale experiment verify the efficacy of the suggested scheme. The results show that the controller designed is simple and real-time is good in the lab. Moreover it can ensure fast response, good stability and robustness in the different operating conditions.

Geometric integration of Hamiltonian systems perturbed by Rayleigh damping

Explicit and semi-explicit geometric integration schemes for dissipative perturbations of Hamiltonian systems are analyzed. The dissipation is characterized by a small parameter ε, and the schemes under study preserve the symplectic structure in the case ε=0. In the case 0<ε≪1 the energy dissipation rate is shown to be asymptotically correct by backward error analysis. Theoretical results on monotone decrease of the modified Hamiltonian function for small enough step sizes are given. Further, an analysis proving near conservation of relative equilibria for small enough step sizes is conducted. Numerical examples, verifying the analyses, are given for a planar pendulum and an elastic 3D pendulum. The results are superior in comparison with a conventional explicit Runge-Kutta method of the same order.

Exponential stability and spectral analysis of the pendulum system under position and delayed position feedbacks

This article addresses the exponential stability and spectral analysis of the linearised pendulum system with both position and delayed position (PDP for short) feedbacks. The semigroup approach is adopted in investigation. The asymptotic spectral expression of the system is presented. It is shown that the spectrum of the system is located in the left half complex plane and its real part goes to −∞ at the high frequency end when the feedback gains and the delay satisfy some additional conditions. Therefore, exponential stability of the system follows from the spectrum determined growth property. Finally, some numerical simulations of a planar pendulum are presented.