CHAOTIC VIBRATIONS AND RESONANCES IN A FLEXIBLE-ARM ROBOT

Once flexibility is introduced into the arm of the robot, severe problems in the accuracy and stability are likely to occur which make control a critical issue. These problems can successfully be eliminated only if the nonlinear dynamics associated with the flexible–arm is properly accounted for. In this paper we study the behaviour of a two degree of freedom high speed robot with a flexible–arm, having quadratic nonlinearities with natural frequencies defined as ω1 and ω2, at ω1 ∝ 2ω2 internal resonance. We perform numerical simulations as well as analytical investigations on a simplified mathematical model of the system, subjected to periodic excitation. The two variable expansion perturbation method is used to show the existence of jump phenomena and ‘saturation’ when both forced resonance and internal resonance occur. Numerical studies indicate the existence of chaotic solutions in the resonance regions. The routes to chaos contain subharmonic bifurcations.

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Regulation of a Two-Degree-of-Freedom Structure Using Internal Resonance

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.2835186 ◽

1995 ◽

Vol 117 (2) ◽

pp. 247-251 ◽

Cited By ~ 7

Author(s):

Shafic S. Oueini ◽

Kevin L. Tuer ◽

M. Farid Golnaraghi

Keyword(s):

Internal Resonance ◽

Control Technique ◽

Control Law ◽

Nonlinear Feedback ◽

Flexible Arm ◽

Control Scheme ◽

Single Input ◽

Linear Differential ◽

Two Degree Of Freedom ◽

Input Torque

In this paper, we present a first attempt at using an energy based control technique to regulate the oscillations of a flexible joint, flexible arm device, through computer simulation. This technique takes advantage of the Internal Resonance (IR) phenomenon. The plant is governed by two coupled linear differential equations. The control scheme is implemented by introducing two software based controllers which are coupled dynamically with the plant through a nonlinear feedback control law. At Internal Resonance, the nonlinear coupling generates an energy link between the plant and the controllers. Thus, energy is transferred from the plant to the controllers where two active damping mechanisms subsequently dissipate it. Here the response of the structure is regulated with a single input torque applied to one plant coordinate. The theoretical analysis is based on the two-variable expansion perturbation method. Thereafter, the analytical findings are verified numerically. Simulation results indicate that the IR control strategy is able to effectively quench the oscillations of the plant.

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Bifurcations in Dynamical Systems With Internal Resonance

Journal of Applied Mechanics ◽

10.1115/1.3424438 ◽

1978 ◽

Vol 45 (4) ◽

pp. 895-902 ◽

Cited By ~ 35

Author(s):

P. R. Sethna ◽

A. K. Bajaj

Keyword(s):

Dynamical Systems ◽

Internal Resonance ◽

Excitation Frequency ◽

The Other ◽

Hopf Bifurcations ◽

System Of Equations ◽

Jump Phenomena ◽

Quadratic Nonlinearities

Dynamical systems with quadratic nonlinearities and exhibiting internal resonance under periodic excitations are studied. Two types of transition from stable to unstable motions are shown to occur. One kind are shown to be associated with jump phenomena while the other kind are shown to be associated with Hopf bifurcations of the averaged system of equations. In the case of the latter, the motions are shown to be amplitude modulated motions at the excitation frequency with the amplitude of modulation determined by the motion of a point on a torus.

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An Experimental Investigation of Complicated Responses of a Two-Degree-of-Freedom Structure

Journal of Applied Mechanics ◽

10.1115/1.3176197 ◽

1989 ◽

Vol 56 (4) ◽

pp. 960-967 ◽

Cited By ~ 37

Author(s):

A. H. Nayfeh ◽

B. Balachandran ◽

M. A. Colbert ◽

M. A. Nayfeh

Keyword(s):

Internal Resonance ◽

Degree Of Freedom ◽

Time Dependent ◽

Theoretical Studies ◽

Chaotic Motions ◽

Single Degree Of Freedom ◽

Single Degree ◽

Second Mode ◽

Quadratic Nonlinearities ◽

Two Degree Of Freedom

Recent theoretical studies indicate that whereas large excitation amplitudes are needed to produce chaotic motions in single-degree-of-freedom systems, extremely small excitation levels can produce chaotic motions in multi-degree-of-freedom systems if they possess autoparametric resonances. To verify these results, we conducted an experimental study of the response of a two-degree-of-freedom structure with quadratic nonlinearities and a two-to-one internal resonance to a primary resonant excitation of the second mode. The responses were analyzed using hardware and software developed for performing time-dependent modal decomposition. We observed periodic, quasi-periodic, and chaotic responses, as predicted by theory. Conditions were found under which extremely small excitation levels produced chaotic motions.

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Internal resonance in a parametrically excited two-degree-of-freedom system with nonlinear spring characteristics. (2nd report, On the combination type)

TRANSACTIONS OF THE JAPAN SOCIETY OF MECHANICAL ENGINEERS Series C ◽

10.1299/kikaic.52.657 ◽

1986 ◽

Vol 52 (474) ◽

pp. 657-663

Author(s):

Yukio ISHIDA ◽

Toshio YAMAMOTO

Keyword(s):

Internal Resonance ◽

Degree Of Freedom ◽

Parametrically Excited ◽

Nonlinear Spring ◽

Two Degree Of Freedom

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Rotor-Dynamics of Different Shaft Configurations for a 6 kW Micro Gas Turbine for Concentrated Solar Power

Volume 8: Microturbines, Turbochargers and Small Turbomachines; Steam Turbines ◽

10.1115/gt2016-56479 ◽

2016 ◽

Cited By ~ 2

Author(s):

A. Arroyo ◽

M. McLorn ◽

M. Fabian ◽

M. White ◽

A. I. Sayma

Keyword(s):

Gas Turbines ◽

High Speed ◽

Dynamic Models ◽

Solar Power ◽

Critical Issue ◽

Rotor Dynamics ◽

Mode Shapes ◽

Concentrated Solar Power ◽

Element Analysis ◽

Rotor Dynamic

Rotor-dynamics of Micro Gas Turbines (MGTs) under 30 kW have been a critical issue for the successful development of reliable engines during the last decades. Especially, no consensus has been reached on a reliable MGT arrangement under 10 kW with rotational speeds above 100,000 rpm, making the understanding of the rotor-dynamics of these high speed systems an important research area. This paper presents a linear rotor-dynamic analysis and comparison of three mechanical arrangements of a 6 kW MGT intended for utilising Concentrated Solar Power (CSP) using a parabolic dish concentrator. This application differs from the usual fuel burning MGT in that it is required to operate at a wider operating speed range. The objective is to find an arrangement that allows reliable mechanical operation through better understanding of the rotor dynamics for a number of alternative shaft-bearings arrangements. Finite Element Analysis (FEA) was used to produce Campbell diagrams and to determine the critical speeds and mode shapes. Experimental hammer tests using a new approach based on optical sensing technology were used to validate the rotor-dynamic models. The FEA simulation results for the natural frequencies of a shaft arrangement were within 5% of the measurements, while the deviation for the shaft-bearings arrangement increased up to 16%.

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A Study on the Energy-Harvesting Device with a Magnetic Spring for Improved Durability in High-Speed Trains

Micromachines ◽

10.3390/mi12070830 ◽

2021 ◽

Vol 12 (7) ◽

pp. 830

Author(s):

Jaehoon Kim

Keyword(s):

Energy Harvesting ◽

High Speed ◽

Permanent Magnets ◽

Critical Issue ◽

Sensor Nodes ◽

High Speed Train ◽

Vibration Energy Harvesting ◽

Magnetic Spring ◽

High Speed Trains ◽

Energy Harvesting Devices

Durability is a critical issue concerning energy-harvesting devices. Despite the energy-harvesting device’s excellent performance, moving components, such as the metal spring, can be damaged during operation. To solve the durability problem of the metal spring in a vibration-energy-harvesting (VEH) device, this study applied a non-contact magnetic spring to a VEH device using the repulsive force of permanent magnets. A laboratory experiment was conducted to determine the potential energy-harvesting power using the magnetic spring VEH device. In addition, the characteristics of the generated power were studied using the magnetic spring VEH device in a high-speed train traveling at 300 km/h. Through the high-speed train experiment, the power generated by both the metal spring VEH device and magnetic spring VEH device was measured, and the performance characteristics required for a power source for wireless sensor nodes in high-speed trains are discussed.

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A New Shaft Coupling Design for a High-Speed Rotor Wheel

Volume 2: 11th Biennial Conference on Reliability, Stress Analysis, and Failure Prevention; 7th International Conference on Design Theory and Methodology; JSME Symposium on Design and Production; Mechanical Design Education and History; Computer-Integrated Concurrent Design Conference ◽

10.1115/detc1995-0136 ◽

1995 ◽

Author(s):

Tibor Kiss ◽

Wing-Fai Ng ◽

Larry D. Mitchell

Keyword(s):

High Speed ◽

High Stress ◽

Critical Issue ◽

Working Environment ◽

Outer Diameter ◽

Stress Concentrations ◽

Coupling Region ◽

Rotor Wheel ◽

High Stress Concentration ◽

Safety Margins

Abstract A high-speed rotor wheel for a wind-tunnel experiment has been designed. The rotor wheel was similar to one in an axial turbine, except that slender bars replaced the blades. The main parameters of the rotor wheel were an outer diameter of 10“, a maximum rotational speed of 24,000 RPM and a maximum transferred torque of 64 lb-ft. Due to the working environment, the rotor had to be designed with high safety margins. The coupling of the rotor wheel with the shaft was found to be the most critical issue, because of the high stress concentration factors associated with the conventional coupling methods. The efforts to reduce the stress concentrations resulted in an advanced coupling design which is the main subject of the present paper. This new design was a special key coupling in which six dowel pins were used for keys. The key slots, now pin-grooves, were placed in bosses on the inner surface of the hub. The hub of the rotor wheel was relatively long, which allowed for applying the coupling near the end faces of the hub, that is, away from the highly loaded centerplane. The long hub resulted in low radial expansion in the coupling region. Therefore, solid contact between the shaft and the hub could be maintained for all working conditions. To develop and verify the design ideas, stress and deformation analyses were carried out using quasi-two-dimensional finite element models. An overall safety factor of 3.7 resulted. The rotor has been built and successfully accelerated over the design speed in a spin test pit.

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Chaotic Amplitude Dynamics for Parametrically Excited Systems With Quadratic Nonlinearities

Volume 3A: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration of Nonlinear, Random, and Time-Varying Systems ◽

10.1115/detc1995-0284 ◽

1995 ◽

Author(s):

Bappaditya Banerjee ◽

Anil K. Bajaj

Keyword(s):

Harmonic Balance ◽

Chaotic Dynamics ◽

Degrees Of Freedom ◽

Order Approximation ◽

Amplitude Equations ◽

Analytical Technique ◽

First Order ◽

Numerical Studies ◽

Quadratic Nonlinearities ◽

First Order Approximation

Abstract Dynamical systems with two degrees-of-freedom, with quadratic nonlinearities and parametric excitations are studied in this analysis. The 1:2 superharmonic internal resonance case is analyzed. The method of harmonic balance is used to obtain a set of four first-order amplitude equations that govern the dynamics of the first-order approximation of the response. An analytical technique, based on Melnikov’s method is used to predict the parameter range for which chaotic dynamics exist in the undamped averaged system. Numerical studies show that chaotic responses are quite common in these quadratic systems and chaotic responses occur even in presence of damping.

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Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4041771 ◽

2018 ◽

Vol 14 (1) ◽

Cited By ~ 9

Author(s):

Feras K. Alfosail ◽

Amal Z. Hajjaj ◽

Mohammad I. Younis

Keyword(s):

Internal Resonance ◽

Nonlinear Response ◽

Multiple Scales ◽

Hopf Bifurcations ◽

Chaotic Motions ◽

Different Types ◽

Quadratic Nonlinearities ◽

Multiple Scales Perturbation Method ◽

Good Agreement ◽

Perturbation Solutions

We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.

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