On the Dynamics and Multiple Equilibria of an Inverted Flexible Pendulum With Tip Mass on a Cart

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Ojas Patil ◽  
Prasanna Gandhi
Keyword(s):  
Dynamic Model ◽  
Multiple Equilibria ◽  
Chaotic Behavior ◽  
Elastic Beam ◽  
Superior Performance ◽  
Flexible Link ◽  
Equilibrium Solutions ◽  
Lagrange Formulation ◽  
Compact Design ◽  
Tip Mass

Flexible link systems are increasingly becoming popular for advantages like superior performance in micro/nanopositioning, less weight, compact design, lower power requirements, and so on. The dynamics of distributed and lumped parameter flexible link systems, especially those in vertical planes are difficult to capture with ordinary differential equations (ODEs) and pose a challenge to control. A representative case, an inverted flexible pendulum with tip mass on a cart system, is considered in this paper. A dynamic model for this system from a control perspective is developed using an Euler Lagrange formulation. The major difference between the proposed method and several previous attempts is the use of length constraint, large deformations, and tip mass considered together. The proposed dynamic equations are demonstrated to display an odd number of multiple equilibria based on nondimensional quantity dependent on tip mass. Furthermore, the equilibrium solutions thus obtained are shown to compare fairly with static solutions obtained using elastica theory. The system is demonstrated to exhibit chaotic behavior similar to that previously observed for vibrating elastic beam without tip mass. Finally, the dynamic model is validated with experimental data for a couple of cases of beam excitation.

The Emergence of Multiple Equilibria in Electromechanical Systems

1994 ◽  
Vol 116 (4) ◽  
pp. 735-744 ◽  
Author(s):  
Neyram Hemati
Keyword(s):  
Multiple Equilibria ◽  
Chaotic Behavior ◽  
Multiple Equilibrium ◽  
Equilibrium Solutions ◽  
Multiple Steady State ◽  
Brushless Dc ◽  
Variable Reluctance ◽  
Stable Operating ◽  
The Stability ◽  
Steady State Solutions

The dynamic characteristics of brushless dc machines are considered. It is demonstrated that due to the inherent nonlinear dynamics of these systems, multiple steady-state solutions can exist. The presence of multiple equilibrium solutions is in turn used to provide an explanation for the loss of stability associated with a locally stable operating state. Numerical simulations are used to help verify the presence of multiple equilibria and their effect on the stability of the systems under investigation. Finally, an example is presented which demonstrates that the presence of multiple equilibria can lead to chaotic behavior in bldcm systems with variable reluctance.

On the Dynamics of an Inverted Flexible Pendulum With Tip Mass on a Cart

2012 ◽  
Author(s):  
Prasanna Gandhi ◽  
Ojas Patil
Keyword(s):  
Vertical Plane ◽  
Superior Performance ◽  
Positioning Systems ◽  
Proposed Model ◽  
Dynamics And Control ◽  
Stable Equilibria ◽  
Compact Design ◽  
And Control ◽  
Tip Mass ◽  
Method Approach

Flexible link systems are increasingly becoming popular for their superior performance in micro/nano positioning and several other advantages including less weight, compact design, lower power requirements and so on as compared to other precision micro-positioning systems. The dynamics and control of such systems is challenging, especially for cases where the system is in the vertical plane. A representative case, inverted flexible pendulum on cart system, is considered in this paper. A dynamic model for flexible pendulum with tip mass has been developed using the Euler Lagrange energy method with constraints for large deflection bending. The assumed modes method approach is used to represent significant contribution to dynamics by finite number of modes. For a lower tip mass, only one stable equilibrium in the center exists; however for higher mass this equilibrium becomes unstable and two stable equilibria on side emerge. An experimental setup of the system has also been developed and it is clear by measuring strain at the base of the pendulum that the nonlinear dynamics is captured well in the proposed model.

Dynamic Analysis of Mechanical Systems With Elastic Telescopic Members

1994 ◽  
Author(s):  
B. O. Al-Bedoor ◽  
Y. A. Khulief
Keyword(s):  
Dynamic Model ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Vibrational Motion ◽  
Dynamic Coupling ◽  
Gravitational Effects ◽  
Coupling Terms ◽  
Tip Mass ◽  
Stiffening Effect

Abstract A dynamic model for the vibrational motion of an elastic beam-like telescopic member is presented. In addition to translation, the elastic member is allowed to execute large reference rotation. The Lagrangian approach in conjunction with the assumed modes technique are employed in deriving the equations of motion. The developed model accounts for all the dynamic coupling terms, as well as the stiffening effect due to the beam reference rotation. The tip mass dynamics is included together with the associated dynamic coupling between the modal degrees of freedom. In addition, the devised dynamic model takes into account the gravitational effects, thus permitting motions in either vertical or horizontal planes. Numerical simulation of a mechanical system with an elastic telescopic member is presented.

Chaos in Inverted Flexible Pendulum With Tip Mass

2014 ◽  
Author(s):  
Prasanna Gandhi ◽  
Jaish Meena
Keyword(s):  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Excitation Amplitude ◽  
Harmonic Excitation ◽  
Distributed Parameter ◽  
Large Set ◽  
Flexible Link ◽  
Poincaré Maps ◽  
Poincare Maps ◽  
Tip Mass

Flexible link systems are increasingly being used in the robotic and other applications. The dynamics of distributed parameter single flexible link system, especially in the vertical planes, is known to demonstrate chaotic behavior upon harmonic excitation. However, to the best of authors knowledge, chaotic dynamics of ultra-large deflection flexible systems with distributed and lumped parameters considered together has not been considered in the literature so far. Dynamics of a representative case, an inverted flexible pendulum with tip mass on cart system, is analysed in this paper. Experimental results on a custom built system consisting of link having 1. ultra large deformation (300 times thickness) as compared to thickness, 2. a tip mass, and 3. base fixed to a cart, under harmonic excitation under several frequencies were obtained. Poincaré maps with large set of data show successive progression with a small cluster of points to start with splitting into two clusters finally leading to butterfly figure of chaotic vibrations. Effect of variation of the excitation amplitude is also explored leading to interesting change in the patterns of Poincaré maps observed.

scholarly journals Energy Harvesting in a Nonlinear Cantilever Piezoelastic Beam System Excited by Random Vertical Vibrations

2014 ◽  
Vol 14 (08) ◽  
pp. 1440018 ◽  
Author(s):  
Marek Borowiec ◽  
Grzegorz Litak ◽  
Michael I. Friswell ◽  
Sondipon Adhikari
Keyword(s):  
Power Generation ◽  
Energy Harvester ◽  
Elastic Beam ◽  
Stochastic Force ◽  
Beam System ◽  
Voltage Output ◽  
Single Well ◽  
Vertical Vibrations ◽  
Ambient Excitation ◽  
Tip Mass

The vertical elastic beam with vertical ambient excitation is proposed as an energy harvester. The beam has a tip mass and piezoelectric patches which transduce the bending strains induced by the stochastic force caused by vertical kinematic forcing into electrical charge. We focus on the region with a fairly large amplitude of voltage output where the beam overcomes the potential barrier. Increasing the noise level allows the transition from single well oscillations to inter-well stochastic jumps with more power generation.

On the Properties of a Dynamic Model of Flexible Robot Manipulators

1998 ◽  
Vol 120 (1) ◽  
pp. 8-14 ◽  
Author(s):  
Marco A. Arteaga
Keyword(s):  
Dynamic Model ◽  
Structural Properties ◽  
Robot Manipulators ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Control Design ◽  
Flexible Manipulator ◽  
Robot Dynamics ◽  
Flexible Link ◽  
Flexible Robot

Control design of flexible robot manipulators can take advantage of the structural properties of the model used to describe the robot dynamics. Many of these properties are physical characteristics of mechanical systems whereas others arise from the method employed to model the flexible manipulator. In this paper, the modeling of flexible-link robot manipulators on the basis of the Lagrange’s equations of motion combined with the assumed modes method is briefly discussed. Several notable properties of the dynamic model are presented and their impact on control design is underlined.

Analysis of High-Speed Electrical Generators for Utility Applications

2013 ◽  
Author(s):  
Miroslav P. Petrov
Keyword(s):  
Electricity Generation ◽  
High Speed ◽  
Renewable Energy Sources ◽  
Scale Up ◽  
Energy Utilization ◽  
Superior Performance ◽  
Electricity Production ◽  
Small Scale ◽  
Technical Challenges ◽  
Compact Design

High-speed alternators are believed to be well developed nowadays, following the improvement in performance and decrease of costs for electronic power converters and permanent magnet materials. Their compact design and their ability to vary the rotational speed in off-design conditions promise superior performance when compared to conventional generators. High-speed alternators are only available in limited sizes for small-scale applications, whereas improvements in efficiency and optimized part-load behavior are particularly important especially for small-scale electricity generation. Enhanced energy utilization for electricity production by small utility plants or by distributed units located at private homes or commercial buildings, based on thermodynamic cycles powered by natural gas or various renewable energy sources, is possible to be achieved through a wider application of grid-integrated high-speed technology. This study presents a critical review of previous research and demonstration work on high-speed electrical machines and a summary of the technical challenges limiting their performance and their expansion into larger sizes. Conclusions are drawn for finding appropriate solutions for practical high-speed electricity generation units and their readiness for a much wider deployment. Closer analysis is attempted on the thermal and mechanical integrity of high-speed alternators and the technical challenges that slow down their scale-up to MW-size units for utility applications. The necessary research and development work that needs to be done in the near future is outlined and discussed herein.

Dynamic Modeling of Planar Flexible Multi-Link Manipulators with Accounting for both Link Foreshortening and Link Material Damping

2011 ◽  
Vol 199-200 ◽  
pp. 19-24
Author(s):  
Jin Fu Zhang
Keyword(s):  
Dynamic Model ◽  
Dynamic Modeling ◽  
Dynamic Performance ◽  
Material Damping ◽  
Motion Responses ◽  
Case Simulation ◽  
Tip Mass ◽  
Lagrange Approach

In order to investigate dynamic performance of flexible multi-link manipulators more exactly, establishing the dynamic model with accounting for link foreshortening and link material damping is needed. In this paper, a new dynamic model for planar flexible multi-link manipulators is established by using Lagrange approach. Both link foreshortening and link material damping are accounted for in this model. As a case simulation, this model is applied to a planar flexible two-link manipulator with a tip mass, and the motion responses of the manipulator are obtained using Gear method.

Nonlinear Dynamic Modeling of Elastic Beam Fixed on a Moving Cart and Carrying Lumped Tip Mass Subjected to External Periodic Force

2009 ◽  
Author(s):  
Fadi A. Ghaith
Keyword(s):  
Linear Model ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Open Loop ◽  
Beam Deflection ◽  
Mode Shapes ◽  
Euler Beam ◽  
Periodic Force ◽  
Assumed Mode Method ◽  
Tip Mass

In the present work, a Bernoulli – Euler beam fixed on a moving cart and carrying lumped tip mass subjected to external periodic force is considered. Such a model could describe the motion of structures like forklift vehicles or ladder cars that carry heavy loads and military airplane wings with storage loads on their span. The nonlinear equations of motion which describe the global motion as well as the vibration motion were derived using Lagrangian approach under the inextensibility condition. In order to investigate the influence of the axial movement of the cart on the response of the system, unconstrained modal analysis has been carried out, and accurate mode shapes of the beam deflection were obtained. The assumed mode method was utilized for approximating the beam elastic deformation based on the single unconstrained mode shapes. Numerical simulation has been carried out to estimate the open-loop response of the nonlinear beam-mass-cart model as well as for the simplified linear model under the influence of the periodic excitation force. Also a comparison study between the responses of the linear and nonlinear models was established. It was shown that the maximum values of the beam tip deflection estimated from the nonlinear model are lower than the corresponding values obtained via the linear model, which reveals the importance of considering nonlinear hardening term in formulating the equations of motion for such system in order to come with more accurate and reliable model.

A Micro-Scale Swimmer Propelled by Traveling Surface Waves

2011 ◽  
Author(s):  
Izhak Bucher ◽  
Eyal Setter
Keyword(s):  
Travelling Wave ◽  
Superior Performance ◽  
Small Scale ◽  
Micro Scale ◽  
Viscous Forces ◽  
Macro Scale ◽  
Robotic Devices ◽  
Compact Design ◽  
Micro Organisms ◽  
Efficient Power

Micro-scale slender swimmers are frequently encountered in nature and recently in micro-robotic applications. The swimming mechanism examined in this article is based on small transverse axi-symmetrical travelling wave deformations of a cylindrical long shell. In very small scale, inertia forces become negligible and viscous forces dominate most propulsion mechanisms being used by micro-organisms and robotic devices. The present paper proposes a compact design principle that provides efficient power to propel and maneuver a micro-scale device. Shown in this paper is a numerical analysis which couples the MEMS structure to the surrounding fluid. Analytical results compare the proposed mechanism to commonly found tail (flagella) driven devices, and a parametric comparison is shown suggesting it has superior performance. Numerical studies are preformed to verify the analytical model. Finally, a macro-scale demonstrator swimming in an environment with similar Reynolds numbers to the ones found in small scale is shown and its behavior in the laboratory is compared to the theory.

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