Bifurcations of a Planar Pendulum Driven at a Tilted Angle

2005 ◽  
Author(s):  
Mike A. Koplow ◽  
Brian P. Mann
Keyword(s):  
Approximate Solution ◽  
High Frequency ◽  
Multiple Scales ◽  
Experimental Tests ◽  
Free Oscillations ◽  
Pendulum System ◽  
Small Deviations ◽  
Tilted Angle ◽  
Multiple Scales Analysis ◽  
Bifurcation Behavior

This paper examines the bifurcation behavior of a planar pendulum subjected to high-frequency parametric excitation along a tilted angle. Analytical and numerical results show that small deviations from either a perfectly vertical or horizontal excitation will result in symmetry breaking bifurcations. This behavior is confirmed through a series of experimental tests. Additionally, a multiple scales analysis is used to obtain an approximate solution for the free oscillations of a viscously damped planar pendulum. System identification is performed by comparing the approximate solution to transient experimental data.

Experimental and Measurement Uncertainty Associated With Characterizing Slurry Mixing Performance of Pulsating Jets at Multiple Scales

2015 ◽  
Author(s):  
Judith Ann Bamberger ◽  
Greg F. Piepel ◽  
Carl W. Enderlin ◽  
Brett G. Amidan ◽  
Alejandro Heredia-Langner
Keyword(s):  
Measurement Uncertainty ◽  
Multiple Scales ◽  
Experimental Tests ◽  
Experimental Results ◽  
Cloud Height ◽  
Solids Concentration ◽  
Relative Standard ◽  
Measurement Uncertainties ◽  
Standard Deviations ◽  
Pulsating Jets

Understanding how uncertainty manifests itself in complex experiments is important for developing the testing protocol and interpreting the experimental results. This paper describes experimental and measurement uncertainties, and how they can depend on the order of performing experimental tests. Experiments with pulse-jet mixers in tanks at three scales were conducted to characterize the performance of transient-developing periodic flows in Newtonian slurries. Other test parameters included the simulant, solids concentration, and nozzle exit velocity. Critical suspension velocity and cloud height were the metrics used to characterize Newtonian slurry flow associated with mobilization and mixing. During testing, near-replicate and near-repeat tests were conducted. The experimental results were used to quantify the combined experimental and measurement uncertainties using standard deviations and percent relative standard deviations (%RSD) The uncertainties in critical suspension velocity and cloud height tend to increase with the values of these responses. Hence, the %RSD values are the more appropriate summary measure of near-replicate testing and measurement uncertainty.

Fractal Basin Boundaries and Chaotic Motions in Spring-Pendulum System

2003 ◽  
Author(s):  
Aria Alasty ◽  
Rasool Shabani
Keyword(s):  
Numerical Simulations ◽  
Multiple Scales ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Global Behavior ◽  
Chaotic Motions ◽  
Pendulum System ◽  
Fractal Basin Boundaries ◽  
System Parameters ◽  
Basin Boundaries

This study investigates chaotic response in the spring-pendulum system. In this system beside of strange attractors, multiple regular attractors may coexist for some values of system parameters, where it is important to study the global behavior of the system using the basin boundaries of the attractors. Multiple scales method is used to distinguish the regions of stable and unstable attractors. In unstable regions, bifurcation diagram and poincare´ maps are used to study the existence of quasi-periodic and chaotic attractors. Results show that the jumping phenomena may occur when multiple regular attractors exist and for this case fractal basins of attraction are developed using numerical simulations.

The Small Scale Effect on Linear Vibration of Beam-Like Nanostructures

2012 ◽  
Vol 446-449 ◽  
pp. 3432-3435
Author(s):  
Cheng Li ◽  
Lin Quan Yao
Keyword(s):  
Axial Load ◽  
Vibration Mode ◽  
Multiple Scales ◽  
Separation Of Variables ◽  
Classical Mechanics ◽  
Small Scale ◽  
Transverse Displacement ◽  
Linear Vibration ◽  
Small Scale Effect ◽  
Multiple Scales Analysis

Transverse free dynamics of a beam-like nanostructure with axial load is investigated. The effects of a small size at nano-scale unavailable in classical mechanics are presented. Explicit solutions for natural frequency, vibration mode and transverse displacement are obtained by separation of variables and multiple scales analysis. Results by two methods are in close agreement.

scholarly journals An Evolutionary Analytic Method of Multi-DOF Nonlinear Coupling Dynamic Model for Controllable Close-Chain Linkage Mechanism System

2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Ru-Gui Wang ◽  
Gan-Wei Cai ◽  
Xiao-Rong Zhou
Keyword(s):  
Approximate Solution ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Dynamic Responses ◽  
Analytic Method ◽  
Nonlinear Coupling ◽  
Linkage Mechanism ◽  
First Order ◽  
Coupling System ◽  
Coupling Dynamic

The 2-DOF controllable close-chain linkage mechanism is investigated in this paper. Based on the characteristics of the multi-DOF nonlinear coupling dynamic equation of the system established by the finite element method, an analytic method of multiple-scales Newmark is presented after thinking about the method of perturbation and the method of numerical analysis. Firstly, the first-order approximate solution of the dynamic responses of the system at the time of t is calculated by the multiple scales method. Then, taken the first-order approximate solution as the initialization of the generalized coordinate of the system, the stable dynamic response of the system is obtained by the implicit Newmark method. The simulation and experimental results are given in the end. The studies indicate that the method of multiple-scales Newmark is correct and practicable to study the dynamic characteristics of such kind of multi-DOF nonlinear coupling system.

Approximate solution for high-frequency Q-switched lasers

2016 ◽  
Vol 55 (16) ◽  
pp. 4282
Author(s):  
Antonio Agnesi
Keyword(s):  
Approximate Solution ◽  
High Frequency

Approximate Solution of High-Frequency-Factor Vibrations of Rigid Bodies on Elastic Media

1971 ◽  
Vol 38 (1) ◽  
pp. 111-117 ◽  
Author(s):  
A. O. Awojobi
Keyword(s):  
Approximate Solution ◽  
Elastic Medium ◽  
Half Space ◽  
High Frequency ◽  
Low Frequency ◽  
Frequency Factor ◽  
Rigid Bodies ◽  
Elastic Media ◽  
Stress Distributions ◽  
Rectangular Body

The mixed boundary-value problems of the vibrations of rigid bodies on elastic media are generally considered in the low-frequency-factor range. It is first established that, quite apart from a consideration of resonance, the usual assumption that this range predominates in practice is erroneous. The present work, therefore, is concerned with vibrations at frequency factors which are much greater than unity. Five cases have been considered: torsional vibration of a rigid circular body on a semi-infinite elastic medium and on an infinitely wide elastic stratum on a rigid bed; vertical vibration of a rigid circular body and of an infinitely long rectangular body on a semi-infinite elastic medium; rocking of a long rectangular body on a semi-infinite elastic medium. An estimate of both the unknown dynamic stress distribution under the rigid bodies and their amplitude responses has been obtained by finding an approximate solution to the exact governing dual integral equations. It is shown that at high-frequency factors, stress distributions are approximately constant for vertical vibrations and vary linearly from the center for rotational vibrations as in a Winkler model of theoretical soil statics contrary to increasing stresses with infinite edge stresses for low-frequency and static stress distributions of rigid bodies on elastic half space. We also obtain the important conclusion for amplitude response that it is predominantly governed by the inertia of the bodies because the contribution due to the dispersion of waves in the elastic medium is generally of a lower order of frequency factor than the inertia term except for an incompressible medium which has been analyzed separately and found to be of the same order leading to expressions for equivalent inertia of the vibrating medium. The theoretical results are used to derive the “tails” of resonance curves for both half space and stratum cases where experimental results are available. The agreement is fair and improves with increasing frequency factor.

Nonlinear Stability of Thermoelastic Sliding Contact

2005 ◽  
Vol 74 (3) ◽  
pp. 595-598
Author(s):  
Jason D. Miller ◽  
D. Dane Quinn
Keyword(s):  
Multiple Scales ◽  
Dynamical Behavior ◽  
Stable State ◽  
Nonlinear Behavior ◽  
Sliding Contact ◽  
Stability Criteria ◽  
Multiple Scales Analysis ◽  
Two Equilibria

A model for sliding contact of a thermoelastic rod is considered and is subjected to a multiple scales analysis to uncover its nonlinear behavior near a neutrally stable state. The analysis reveals a combination of the contact resistance and frictional intensity that describes the generic unfolding of this critical state and its associated bifurcations. In particular, the system can describe how two equilibria coalesce in a saddle-node bifurcation and generalizes stability criteria that have been presented previously in the literature for this model. Moreover, this analysis describes the role of the initial deformation of the rod on its long-term dynamical behavior.

scholarly journals Excitation device for high frequency vibration analysis: Design and test results

2017 ◽  
Vol 24 (19) ◽  
pp. 4620-4629 ◽  
Author(s):  
Paolo Neri
Keyword(s):  
High Frequency ◽  
Test Bench ◽  
Experimental Tests ◽  
Wide Frequency Range ◽  
Response Analysis ◽  
Design And Optimization ◽  
Vibrational Response ◽  
High Frequency Excitation ◽  
Frequency Excitation ◽  
Harmonic Response Analysis

In the present paper, the design and optimization of a high frequency excitation source is presented. The device was developed for a harmonic response analysis test bench, aimed at dynamic characterization and resonance prediction of mechanical structures. A wide frequency range must be covered, depending on the analyzed structure: the range 1–10 kHz was considered in the present work. The device was designed for a test bench aimed at investigating the vibrational response of centrifugal compressor bladed wheels. A really compact solution was needed since the final test bench provides one exciter for each blade (up to 20 devices on the circumference hoop). Both contact and contactless solutions were considered, but only the contact solution was found to fulfill all the specifications. Finally, different stinger solutions were proposed and compared in the paper. The investigated solutions were: a beam stinger (diameter 1 mm); a wire stinger (diameter 0.2 mm); and a ball stinger (diameter 3 mm) with two different support solutions. Experimental tests performed on a device prototype allowed to verify the specifications fulfillment and to choose the best stinger solution for the application.

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