Swing-Up Control of a Three-Link Underactuated Manipulator by High-Frequency Horizontal Excitation

2012 ◽  
Vol 8 (1) ◽  
Author(s):  
Kazuya Endo ◽  
Hiroshi Yabuno
Keyword(s):  
High Frequency ◽  
Multiple Scales ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Averaged Equations ◽  
Underactuated Manipulator ◽  
High Frequency Excitation ◽  
Pitchfork Bifurcations ◽  
The Stability

In the present paper, we consider a three-link underactuated manipulator, the first joint of which is active and the second and third joints of which exhibit passive motion, on a plane inclined at slight angle from horizontal the plane. We analytically investigate changes in the stability of equilibrium points of the free links connected to the passive joints using high-frequency horizontal excitation of the first link. We derive autonomous averaged equations from the dimensionless equations of motion using the method of multiple scales. We clarify that the two free links can be swung up through pitchfork bifurcations and stabilized at some configurations by producing nontrivial and stable equilibrium points due to the high-frequency excitation. Furthermore, it is experimentally verified that increasing the excitation frequency multiplies stable and nontrivial equilibrium points.

Motion Control of Three Link Underactuated Manipulator by Bifurcation Control

2014 ◽  
Author(s):  
Hiroshi Yabuno ◽  
Kazuya Endo
Keyword(s):  
Feedback Control ◽  
Motion Control ◽  
Multiple Scales ◽  
Equilibrium Points ◽  
Pitchfork Bifurcation ◽  
Upright Position ◽  
State Feedback Control ◽  
Vertical Excitation ◽  
Underactuated Manipulator ◽  
Pitchfork Bifurcations

Motion control of a three link underactuated manipulator, whose first joint has an actuator and a sensor and second and third joints do not have actuator or sensor, is theoretically proposed without feedback control with respect to the motion of the free links. By using high-frequency vertical excitation, so called Kapitza pendulum is stabilized at the upright position without state feedback control. The phenomenon can be regarded as a subcritical pitchfork bifurcation. On the other hand, it is known that the horizontal excitation causes supercritical pitchfork bifurcation in a pendulum. Also, the inclination of excitation from the horizontal and vertical directions produces the perturbation of the complete supercritical and subcritical pitchfork bifurcations, respectively. In this paper, we apply the method of multiple scales to obtain the averaged equations governing the motion of the free links. We perform the bifurcation analysis of the free links and clarify the equilibrium points in the free links and their stability. Then, we propose a strategy to swing up the free links and to stabilize them at the upright position by actuating the perturbation of the pitchfork bifurcations based on the change of the inclination of excitation.

scholarly journals Resonance in the Cart-Pendulum System—An Asymptotic Approach

2021 ◽  
Vol 11 (23) ◽  
pp. 11567
Author(s):  
Wael S. Amer ◽  
Tarek S. Amer ◽  
Roman Starosta ◽  
Mohamed A. Bek
Keyword(s):  
Multiple Scales ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Pendulum System ◽  
Damped System ◽  
Stability And Instability ◽  
Parametric Pendulum ◽  
The Stability ◽  
Fundamental Equations

The major objective of this research is to study the planar dynamical motion of 2DOF of an auto-parametric pendulum attached with a damped system. Using Lagrange’s equations in terms of generalized coordinates, the fundamental equations of motion (EOM) are derived. The method of multiple scales (MMS) is applied to obtain the approximate solutions of these equations up to the second order of approximation. Resonance cases are classified, in which the primary external and internal resonance are investigated simultaneously to establish both the solvability conditions and the modulation equations. In the context of the stability conditions of these solutions, the equilibrium points are obtained and graphically displayed to derive the probable steady-state solutions near the resonances. The temporal histories of the attained results, the amplitude, and the phases of the dynamical system are depicted in graphs to describe the motion of the system at any instance. The stability and instability zones of the system are explored, and it is discovered that the system’s performance is stable for a significant number of its variables.

Bifurcations of a High-Frequency Horizontally Excited Double Pendulum

2011 ◽  
Author(s):  
Hiroshi Yabuno ◽  
Kazuya Endo
Keyword(s):  
Bifurcation Point ◽  
High Frequency ◽  
Excitation Frequency ◽  
Pitchfork Bifurcation ◽  
Vertical Position ◽  
Double Pendulum ◽  
Bifurcation Phenomena ◽  
Underactuated Manipulator ◽  
High Frequency Excitation ◽  
Frequency Excitation

The bifurcation phenomena produced in a double pendulum under high-frequency horizontal excitation are theoretically and experimentally examined. It has been well known as dynamic stabilization phenomenon that vertical high-frequency excitation can stabilize inverted pendulum. The phenomenon is produced through a sub-critical pitchfork bifurcation. On the other hand, under horizontal high-frequency excitation, the pendulum undergoes a supercritical pitchfork bifurcation and is swung up from the downward vertical position. There have so far been many researches on such dynamics of a single pendulum under the vertical and horizontal high-frequency excitations, but few investigations on multi-degrees-of-freedom system. Also, the utilization of these bifurcations phenomena under the high-frequency excitation is proposed for motion control of underactuated manipulators, but most researches on application is confined to a single pendulum to which a free of two-link underactuated manipulator corresponds. In this paper, toward the development of a three-link underactuated manipulator, we deal with a double pendulum to which two free links of the three-link underactuated manipulator correspond, and theoretically and experimentally investigate bifurcation phenomena in the two pendulums. First, we theoretically predict two pitchfork bifurcation points while increasing the excitation frequency by linear amplitude equations derived using the method of multiple scales. Furthermore, we experimentally examine the swing-up of the pendulums after the first pitchfork bifurcation point and observe that the system has the four types of stable configurations beyond the second pitchfork bifurcation point.

scholarly journals On the Stability of Collinear Points of the RTBP with Triaxial and Oblate Primaries and Relativistic Effects

2021 ◽  
pp. 56-73
Author(s):  
S. E. Abd El-Bar
Keyword(s):  
Characteristic Equation ◽  
Body Problem ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Relativistic Effects ◽  
Three Body Problem ◽  
Total Potential ◽  
The Mean ◽  
The Stability ◽  
Three Body

Under the influence of some different perturbations, we study the stability of collinear equilibrium points of the Restricted Three Body Problem. More precisely, the perturbations due to the triaxiality of the bigger primary and the oblateness of the smaller primary, in addition to the relativistic effects, are considered. Moreover, the total potential and the mean motion of the problem are obtained. The equations of motion are derived and linearized around the collinear points. For studying the stability of these points, the characteristic equation and its partial derivatives are derived. Two real and two imaginary roots of the characteristic equation are deduced from the plotted figures throughout the manuscript. In addition, the instability of the collinear points is stressed. Finally, we compute some selected roots corresponding to the eigenvalues which are based on some selected values of the perturbing parameters in the Tables 1, 2.

EFFECT OF ADDED TIP MASS ON THE NONLINEAR FLAPWISE AND CHORDWISE VIBRATION OF CANTILEVER COMPOSITE BEAM UNDER BASE EXCITATION

2012 ◽  
Vol 12 (02) ◽  
pp. 285-310 ◽  
Author(s):  
M. EFTEKHARI ◽  
M. MAHZOON ◽  
S. ZIAEI-RAD
Keyword(s):  
Cantilever Beam ◽  
Multiple Scales ◽  
Equilibrium Points ◽  
Laminated Composite ◽  
Base Excitation ◽  
System A ◽  
Autoparametric Resonance ◽  
The Stability ◽  
Tip Mass ◽  
Made In

In this paper, a comparative study is performed for a symmetrically laminated composite cantilever beam with and without a tip mass under harmonic base excitation. The base is subjected to both flapwise and chordwise excitations tuned to the primary resonances of the two directions and conditions of 2:1 autoparametric resonance. In the literature, the governing nonlinear equations of the same problem without tip mass have been derived using the extended Hamilton's principle. Extension is made in this study to include the effect of a tip mass on the response of the beam. The natural frequencies are obtained numerically using the diversity guided evolutionary algorithm (DGEA). Next, the multiple scales method is applied to determine the nonlinear response and stability of the system. A set of four first-order differential equations describing the modulation of the amplitudes and phases of interacting modes are derived for the perturbation analysis. For verification, the above equations are reduced to the special case of the cantilever beam without tip mass for comparison with existing results. Finally, the effect of the tip mass on the stability of the fixed points and on the amplitude of oscillation about the equilibrium points in both the frequency and force modulation responses is examined.

NONLINEAR LOW FREQUENCY WATER WAVES IN A CYLINDRICAL SHELL

2005 ◽  
Vol 19 (28n29) ◽  
pp. 1615-1618 ◽  
Author(s):  
H. W. PENG ◽  
D. J. WANG ◽  
C. B. LEE
Keyword(s):  
High Frequency ◽  
Water Waves ◽  
Water Wave ◽  
Excitation Frequency ◽  
Low Frequency ◽  
Viscous Effect ◽  
First Approximation ◽  
Quantitative Results ◽  
High Frequency Excitation ◽  
Frequency Excitation

The experiment was carried out to study the low frequency surface waves due to the horizontal high frequency excitation. The feature of the phenomenon was that the big amplitude axisymmetric surface wave frequency was typically about 1/50 of the excitation frequency. The viscous effect of water was neglected as a first approximation in the earlier papers on this subject. In contrast, we found the viscosity was important to achieve the low frequency water wave with the cooperation of hundreds of "finger" waves. Photographs were taken with stroboscopic lighting and thereafter relevant quantitative results were obtained based on the measurements with Polytec Scanning Vibrometer PSV 400.

scholarly journals Dynamics of Satellites Around Asteroids in Presence of Solar Radiation Pressure

2015 ◽  
Vol 10 (S318) ◽  
pp. 259-264
Author(s):  
Xiaosheng Xin ◽  
Daniel J. Scheeres ◽  
Xiyun Hou ◽  
Lin Liu
Keyword(s):  
Solar Radiation ◽  
Radiation Pressure ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Solar Radiation Pressure ◽  
Approximate Solutions ◽  
Triaxial Ellipsoid ◽  
Stability Maps ◽  
The Stability ◽  
Near Earth Asteroids

AbstractDue to the close distance to the Sun, solar radiation pressure (SRP) plays an important role in the dynamics of satellites around near-Earth asteroids (NEAs). In this paper, we focus on the equilibrium points of a satellite orbiting around an asteroid in presence of SRP in the asteroid rotating frame. The asteroid is modelled as a uniformly rotating triaxial ellipsoid. When SRP comes into play, the equilibrium points transformed into periodic orbits termed as``dynamical substitutes". We obtain the analytical approximate solutions of the dynamical substitutes from the linearised equations of motion. The analytical solutions are then used as initial guesses and are numerically corrected to compute the accurate orbits of the dynamical substitutes. The stability of the dynamical substitutes is analysed and the stability maps are obtained by varying parameters of the ellipsoid model as well as the magnitude of SRP.

Nonlinear free vibrations analysis of overhung rotors under the influence of gravity

2019 ◽  
Vol 234 (2) ◽  
pp. 575-588
Author(s):  
M Moradi Tiaki ◽  
SAA Hosseini ◽  
H Shaban Ali Nezhad
Keyword(s):  
Natural Frequency ◽  
High Frequency ◽  
Multiple Scales ◽  
Perturbation Technique ◽  
Equations Of Motion ◽  
Beat Frequency ◽  
Dynamical Behavior ◽  
Free Vibrations ◽  
Beam Model ◽  
Rayleigh Beam

In this paper, nonlinear free vibration of a cantilever flexible shaft carrying a rigid disk at its free end (overhung rotor) is investigated. The Rayleigh beam model is used and the rotor has large amplitude vibrations. With the assumption of inextensibility, the effect of nonlinear curvature and inertia is considered. The effect of disk mass on the dynamical behavior of the system is studied in the presence and absence of gravity (horizontal and vertical rotors). By using perturbation technique (method of multiple scales), the main focus is on the influence of gravity on equations of motion and on quantities such as amplitude and damped natural frequency. Here, a different behavior is observed due to the rotor weight. Indeed, the combination effects of gyroscopic term, nonlinearity and gravity are studied on the modal behavior of the system. It is shown that the static deflection creates second order nonlinear terms and changes the nonlinear damped natural frequency. With considering of gravity, both beat and high frequency in beat phenomenon increase. With increasing of the rotor weight, the minimum value of amplitude is extremely amplified in the direction of gravity but in the other transverse direction, amplitude of vibrations decreases. In addition, it is found that the weight has directly influence on beat frequency, while the mass ratio between disk and beam affects the high frequency.

Performance Enhancement of an Autoparametric Vibration Absorber Using Multiple Pendulums

2000 ◽  
Author(s):  
Ashwin Vyas ◽  
Anil K. Bajaj ◽  
Arvind Raman
Keyword(s):  
Performance Enhancement ◽  
Excitation Frequency ◽  
Single Mode ◽  
Frequency Interval ◽  
Vibration Absorbers ◽  
Coupled Mode ◽  
Averaged Equations ◽  
Stability And Bifurcations ◽  
The Stability ◽  
Mode Response

Abstract This paper analyzes the dynamics of a resonantly excited single degree-of-freedom linear system coupled to an array of nonlinear autoparametric vibration absorbers (pendulums). The case of a 1:1:…:2 internal resonance between pendulums and the primary oscillator is studied. The method of averaging is used to obtain first-order approximations to the nonlinear response of the system. The stability and bifurcations of the equilibria of the averaged equations are computed. It is shown that the frequency interval of the unstable single-mode response, or the absorber bandwidth, can be enlarged substantially compared to that of a single pendulum absorber by adjusting individually the internal mistunings of the pendulums. Use of multiple pendulums is also shown to engender degenerate bifurcations as the coupled-mode response “switches” from one pendulum to the other with changing external excitation frequency. This results in a significant enhancement of the performance of autoparametric vibration absorbers.

Sign in / Sign up

Export Citation Format

Share Document