COMPLEX DYNAMICS OF BILINEAR OSCILLATOR CLOSE TO GRAZING

2010 ◽  
Vol 20 (11) ◽  
pp. 3801-3817 ◽  
Author(s):  
EKATERINA PAVLOVSKAIA ◽  
JAMES ING ◽  
MARIAN WIERCIGROCH ◽  
SOUMITRO BANERJEE
Keyword(s):  
Complex Dynamics ◽  
Excitation Frequency ◽  
Excitation Amplitude ◽  
Basins Of Attraction ◽  
Experimental Investigations ◽  
Unstable Periodic Orbits ◽  
Chaotic Oscillations ◽  
Impact Oscillator ◽  
Parameter Values ◽  
Elastic Constraint

In this work the strange behavior of an impact oscillator with a one-sided elastic constraint discovered experimentally is compared with the predictions obtained using its mathematical model. Extensive experimental investigations undertaken on the rig developed at the Aberdeen University reveal different bifurcation scenarios under varying excitation frequency near grazing which were recorded for a number of values of the excitation amplitude. In the paper, particular attention is paid to the chaotic oscillations recorded near grazing frequency when a nonimpacting orbit becomes an impacting one under increasing excitation frequency. It was found that the evolution of the attractor is governed by a complex interplay between smooth and nonsmooth bifurcations, and the interactions between a number of coexisting orbits. The occurrence of coexisting attractors is manifested in the experimental results through discontinuous transitions from one orbit to another via boundary crisis. In some cases, the basins of attraction have a fractal structure. Detailed numerical exploration also revealed coexisting unstable periodic orbits. These stable and unstable coexisting orbits are often born far from the parameter values at which they influence the system dynamics. The very rich dynamics of the bilinear oscillator close to grazing is demonstrated and typical mechanisms of the attractors' appearance and disappearance are explained using stability analysis.

Experimental study of impact oscillator with one-sided elastic constraint

2007 ◽  
Vol 366 (1866) ◽  
pp. 679-705 ◽  
Author(s):  
James Ing ◽  
Ekaterina Pavlovskaia ◽  
Marian Wiercigroch ◽  
Soumitro Banerjee
Keyword(s):  
Excitation Frequency ◽  
Excitation Amplitude ◽  
Experimental Investigations ◽  
Coexisting Attractors ◽  
Impact Oscillator ◽  
Good Correspondence ◽  
Experimental Procedure ◽  
Acceleration Signal ◽  
The Stability ◽  
Elastic Constraint

In this paper, extensive experimental investigations of an impact oscillator with a one-sided elastic constraint are presented. Different bifurcation scenarios under varying the excitation frequency near grazing are shown for a number of values of the excitation amplitude. The mass acceleration signal is used to effectively detect contacts with the secondary spring. The most typical recorded scenario is when a non-impacting periodic orbit bifurcates into an impacting one via grazing mechanism. The resulting orbit can be stable, but in many cases it loses stability through grazing. Following such an event, the evolution of the attractor is governed by a complex interplay between smooth and non-smooth bifurcations. In some cases, the occurrence of coexisting attractors is manifested through discontinuous transition from one orbit to another through boundary crisis. The stability of non-impacting and impacting period-1 orbits is then studied using a newly proposed experimental procedure. The results are compared with the predictions obtained from standard theoretical stability analysis and a good correspondence between them is shown for different stiffness ratios. A mathematical model of a damped impact oscillator with one-sided elastic constraint is used in the theoretical studies.

Experimental Investigations Into Nonlinear Vibro-Acoustics for Detection of Delaminations in a Composite Laminate

2018 ◽  
Vol 2 (1) ◽  
Author(s):  
Ashish Kumar Singh ◽  
Vincent B. C. Tan ◽  
Tong Earn Tay ◽  
Heow Pueh Lee
Keyword(s):  
Frequency Dependence ◽  
Composite Laminate ◽  
Excitation Frequency ◽  
Excitation Amplitude ◽  
Experimental Investigations ◽  
Frequency Sweep ◽  
Ultrasonic Methods ◽  
Acoustic Methods ◽  
Different Depths ◽  
Excitation Frequencies

In recent years, nonlinear vibro-acoustic methods have shown potential to identify defects which are difficult to detect using linear ultrasonic methods. However, these methods come with their own challenges such as frequency dependence, requirement for a high excitation amplitude, and difficulties in distinguishing nonlinearity from defect with nonlinearity from other sources to name a few. This paper aims to study the dependence of nonlinear vibro-acoustic methods for detection of delaminations inside a composite laminate, on the excitation methods and excitation frequencies. It is shown that nonlinear vibro-acoustic methods are highly frequency dependent and commonly used excitation signals which utilize particular values of excitation frequencies might not always lead to a clear distinction between intact and delaminated regions of the specimen. To overcome the frequency dependence, signals based on frequency sweep are used. Interpretation of output response to sweep signals to identify damage is demonstrated using an earlier available approach, and a simpler approach is proposed. It is demonstrated that the damage detection with sweep signal excitations is relatively less dependent on excitation frequency than the conventional excitation methods. The proposed interpretation technique is then applied to specimens with delamination of varying sizes and with delaminations at different depths inside the laminate to demonstrate its effectiveness.

Modelling and analysis of ship roll oscillations interacting with stationary icebergs

2008 ◽  
Vol 222 (10) ◽  
pp. 1873-1884 ◽  
Author(s):  
I F Grace ◽  
R A Ibrahim
Keyword(s):  
Initial Conditions ◽  
Excitation Frequency ◽  
Excitation Amplitude ◽  
Basins Of Attraction ◽  
Analytical Models ◽  
Safe Operation ◽  
Rigid Barrier ◽  
Ship Roll ◽  
Different Response ◽  
Impact Motion

Impact dynamic interaction of ships with solid ice or stationary rigid structures is a serious problem that affects the safe operation and navigation in arctic regions. The purpose of this study is to present two analytical models of impact interaction between ship roll dynamics and one-side rigid barrier. These models are the phenomenological modelling represented by a power law in stiffness and damping forces, and Zhuravlev non-smooth coordinate transformation. Extensive numerical simulations are carried out for all initial conditions covered by the ship grazing orbit for different values of excitation amplitude and frequencies of external wave roll moment. The basins of attraction of safe operation are obtained and reveal the coexistence of different response regimes such as non-impact periodic oscillations, modulation impact motion, period-added impact oscillations, chaotic impact motion, and unbounded rotational motion. The results are summarized in the bifurcation diagram in terms of response-excitation amplitudes plane. The stability fraction index is obtained for different values of excitation frequency based on the ratio of the area of bounded roll oscillations to the total area of the grazing orbit.

scholarly journals Complex response analysis of a non-smooth oscillator under harmonic and random excitations

2021 ◽  
Vol 42 (5) ◽  
pp. 641-648
Author(s):  
Shichao Ma ◽  
Xin Ning ◽  
Liang Wang ◽  
Wantao Jia ◽  
Wei Xu
Keyword(s):  
Excitation Frequency ◽  
System Stability ◽  
Response Analysis ◽  
External Excitation ◽  
Stochastic Response ◽  
Impact Oscillator ◽  
Nonlinear Parameters ◽  
Bifurcation Phenomena ◽  
Mc Simulation ◽  
Smooth System

AbstractIt is well-known that practical vibro-impact systems are often influenced by random perturbations and external excitation forces, making it challenging to carry out the research of this category of complex systems with non-smooth characteristics. To address this problem, by adequately utilizing the stochastic response analysis approach and performing the stochastic response for the considered non-smooth system with the external excitation force and white noise excitation, a modified conducting process has proposed. Taking the multiple nonlinear parameters, the non-smooth parameters, and the external excitation frequency into consideration, the steady-state stochastic P-bifurcation phenomena of an elastic impact oscillator are discussed. It can be found that the system parameters can make the system stability topology change. The effectiveness of the proposed method is verified and demonstrated by the Monte Carlo (MC) simulation. Consequently, the conclusions show that the process can be applied to stochastic non-autonomous and non-smooth systems.

Effects of Acoustic Excitation on a Swirling Diffusion Flame

2010 ◽  
Vol 132 (12) ◽  
Author(s):  
Michael E. Loretero ◽  
Rong F. Huang
Keyword(s):  
Flow Field ◽  
Bluff Body ◽  
Excitation Frequency ◽  
Excitation Amplitude ◽  
Mie Scattering ◽  
Flame Length ◽  
Laser Doppler Velocimeter ◽  
Scattering Method ◽  
Acoustic Excitation ◽  
Characteristic Modes

A swirling double concentric jet is commonly used for nonpremixed gas burner application for safety reasons and to improve the combustion performance. Fuel is generally spurted at the central jet while the annular coflowing air is swirled. They are normally separated by a blockage disk where the bluff-body effects further enhance the recirculation of hot gas at the reaction zone. This paper aims to experimentally investigate the behavior of flame and flow in a double concentric jet combustor when the fuel supply is acoustically driven. Laser-light sheet assisted Mie scattering method has been used to visualize the flow, while the flame lengths were measured by a conventional photography technique. The fluctuating velocity at the jet exit was measured by a two-component laser Doppler velocimeter. Flammability and stability at first fuel tube resonant frequency are reported and discussed. The evolution of flame profile with excitation level is presented and discussed, together with the reduction in flame length. The flame in the unforced reacting axisymmetric wake is classified into three characteristic modes, which are weak swirling flame, lifted flame, and transitional reattached flame. These terms reflect their primary features of flame appearances, and when the acoustic excitation is applied, the flame behaviors change with the excitation frequency and amplitude. Four additional characteristic modes are identified; e.g., at low excitation amplitudes, wrinkling flame with a blue annular film is observed because the excitation induces vortices in the central fuel jet and hence gives rise to the wrinkling of flame. The central jet vortices become larger with the increase in excitation amplitude and thus lead to a wider and shorter flame. If the excitation amplitude is increased above a certain value, the central jet vortices change the rotation direction and pacing with the annular jet vortices. These changes in the flow field induce large turbulent intensity and mixing and therefore make the flame looks blue and short. Further increase in the excitation amplitude would lift the flame because the flow field would be dramatically modified.

Modeling and Experimental Validations of a Piezoelectric Energy Harvester Under Combined Galloping and Base Excitations

2014 ◽  
Author(s):  
Amin Bibo ◽  
Abdessattar Abdelkefi ◽  
Mohammed F. Daqaq
Keyword(s):  
Bluff Body ◽  
Energy Harvester ◽  
Constitutive Relations ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Beam Theory ◽  
Excitation Amplitude ◽  
The Body ◽  
Base Excitation ◽  
Piezoelectric Energy

This paper develops an experimentally validated model of a piezoelectric energy harvester under combined aeroelastic-galloping and base excitations. To that end, an energy harvester consisting of a thin piezoelectric cantilever beam subjected to vibratory base excitation is considered. To permit galloping excitation, a bluff body is rigidly attached at the free end such that a net aerodynamic lift is generated as the incoming airflow separates on both sides of the body giving rise to limit cycle oscillations when the flow velocity exceeds a critical value. A nonlinear electromechanical distributed-parameter model of the harvester under the combined excitation is derived using the energy approach and by adopting the nonlinear Euler-Bernoulli beam theory, linear constitutive relations for the piezoelectric transduction, and the quasi-steady assumption for the aerodynamic loading. The partial differential equations of the system are discretized and a reduced-order-model is obtained. The mathematical model is validated by conducting a series of experiments with different loading conditions represented by wind speed, base excitation amplitude, and excitation frequency around the primary resonance.

Two Groups of Chaotic Vibrations in a Single-Degree-of-Freedom Maglev System

1997 ◽  
Author(s):  
Zhixiang Xu ◽  
Hideyuki Tamura
Keyword(s):  
Magnetic Levitation ◽  
Excitation Frequency ◽  
Power Spectra ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Moving Base ◽  
Period Doubling Bifurcations

Abstract In this paper, a single-degree-of-freedom magnetic levitation dynamic system, whose spring is composed of a magnetic repulsive force, is numerically analyzed. The numerical results indicate that a body levitated by magnetic force shows many kinds of vibrations upon adjusting the system parameters (viz., damping, excitation amplitude and excitation frequency) when the system is excited by the harmonically moving base. For a suitable combination of parameters, an aperiodic vibration occurs after a sequence of period-doubling bifurcations. Typical aperiodic vibrations that occurred after period-doubling bifurcations from several initial states are identified as chaotic vibration and classified into two groups by examining their power spectra, Poincare maps, fractal dimension analyses, etc.

scholarly journals A CONDITIONAL STOCHASTIC PROJECTION METHOD APPLIED TO A PARAMETRIC VIBRATIONS PROBLEM

2014 ◽  
Vol 20 (6) ◽  
pp. 810-818 ◽  
Author(s):  
Wlodzimierz Brzakala ◽  
Aneta Herbut
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Polynomial Chaos ◽  
Excitation Frequency ◽  
Finite Sequence ◽  
Excitation Amplitude ◽  
Random Excitation ◽  
Parametric Vibrations ◽  
Representative Points ◽  
Cable Stayed Bridges

Parametric vibrations can be observed in cable-stayed bridges due to periodic excitations caused by a deck or a pylon. The vibrations are described by an ordinary differential equation with periodic coefficients. The paper focuses on random excitations, i.e. on the excitation amplitude and the excitation frequency which are two random variables. The excitation frequency ωL is discretized to a finite sequence of representative points, ωL,i Therefore, the problem is (conditionally) formulated and solved as a one-dimensional polynomial chaos expansion generated by the random excitation amplitude. The presented numerical analysis is focused on a real situation for which the problem of parametric resonance was observed (a cable of the Ben-Ahin bridge). The results obtained by the use of the conditional polynomial chaos approximations are compared with the ones based on the Monte Carlo simulation (truly two-dimensional, not conditional one). The convergence of both methods is discussed. It is found that the conditional polynomial chaos can yield a better convergence then the Monte Carlo simulation, especially if resonant vibrations are probable.

Investigation on a Type of Flow Control to Weaken Unsteady Separated Flows by Unsteady Excitation in Axial Flow Compressors

2004 ◽  
Author(s):  
Xin-Qian Zheng ◽  
Xiao-Bo Zhou ◽  
Sheng Zhou
Keyword(s):  
Wide Spectrum ◽  
Excitation Frequency ◽  
Axial Flow ◽  
Maximum Velocity ◽  
Excitation Amplitude ◽  
High Order Scheme ◽  
Wide Range ◽  
Optimal Excitation ◽  
Suction And Blowing ◽  
Axial Flow Compressors

By solving unsteady Reynolds-averaged 2-D N-S equations discretized by a high-order scheme, the results showed that the disordered unsteady separated flow could be effectively controlled by periodic suction and blowing in a wide range of incidence, resulting in enhancement of time-averaged aerodynamic performances. The effects of unsteady excitation frequency, amplitude and excitation location were investigated in detail. The effective excitation frequency spans a wide spectrum and there is an optimal excitation frequency that is nearly equal to the Characteristic frequency of vortex shedding. Excitation amplitude exhibits a threshold value (nearly 10% in term of the ratio of maximum velocity of periodic suction and blowing to the velocity of free flow) and an optimal value (nearly 35%). The optimal excitation location is just upstream of the separation point. We also explored feasible unsteady actuators by utilizing upstream wake for constraining unsteady separation in axial flow compressors.

scholarly journals Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Chengwei Dong ◽  
Lian Jia ◽  
Qi Jie ◽  
Hantao Li
Keyword(s):  
Periodic Orbits ◽  
Nonlinear Dynamical Systems ◽  
Unstable Periodic Orbits ◽  
Nonlinear Dynamical ◽  
System A ◽  
Return Maps ◽  
Phase Portrait Analysis ◽  
Encoding Method ◽  
First Return Maps ◽  
Parameter Values

To describe and analyze the unstable periodic orbits of the Rucklidge system, a so-called symbolic encoding method is introduced, which has been proven to be an efficient tool to explore the topological properties concealed in these periodic orbits. In this work, the unstable periodic orbits up to a certain topological length in the Rucklidge system are systematically investigated via a proposed variational method. The dynamics in the Rucklidge system are explored by using phase portrait analysis, Lyapunov exponents, and Poincaré first return maps. Symbolic encodings of the periodic orbits with two and four letters based on the trajectory topology in the phase space are implemented under two sets of parameter values. Meanwhile, the bifurcations of the periodic orbits are explored, significantly improving the understanding of the dynamics of the Rucklidge system. The multiple-letter symbolic encoding method could also be applicable to other nonlinear dynamical systems.

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