scholarly journals Near-resonant dynamics, period doubling and chaos of a 3-DOF vibro-impact system

2021 ◽  
Author(s):  
Pawel Fritzkowski ◽  
Jan Awrejcewicz
Keyword(s):  
Steady State ◽  
Multiple Scales ◽  
Period Doubling ◽  
Harmonic Excitation ◽  
Linear Oscillator ◽  
Model Parameters ◽  
Response Curves ◽  
Approximate Analytical Solutions ◽  
Resonant Dynamics ◽  
Periodic Steady State

Abstract A mechanical system composed of two weakly coupled oscillators under harmonic excitation is considered. Its main part is a vibro-impact unit composed of a linear oscillator with an internally colliding small block. This block is coupled with the secondary part being a damped linear oscillator. The mathematical model of the system has been presented in a non-dimensional form. The analytical studies are restricted to the case of a periodic steady-state motion with two symmetric impacts per cycle near 1:1 resonance. The multiple scales method combined with the sawtooth-function-based modelling of the non-smooth dynamics is employed. The approximate analytical solutions allow for stability analysis of the periodic motions. Moreover, the frequency-response curves and force-response curves with stable and unstable branches are determined, and the interplay between various model parameters is investigated. The theoretical predictions related to the motion amplitude and the range of stability of the periodic steady-state response is verified via a series of numerical experiments and computation of Lyapunov exponents.

scholarly journals Near-resonant dynamics, period doubling and chaos of a 3-DOF vibro-impact system

2021 ◽  
Vol 106 (1) ◽  
pp. 81-103
Author(s):  
Pawel Fritzkowski ◽  
Jan Awrejcewicz
Keyword(s):  
Steady State ◽  
Multiple Scales ◽  
Period Doubling ◽  
Harmonic Excitation ◽  
Linear Oscillator ◽  
Model Parameters ◽  
Response Curves ◽  
Resonant Dynamics ◽  
Periodic Steady State ◽  
The Stability

AbstractA mechanical system composed of two weakly coupled oscillators under harmonic excitation is considered. Its main part is a vibro-impact unit composed of a linear oscillator with an internally colliding small block. This block is coupled with the secondary part being a damped linear oscillator. The mathematical model of the system has been presented in a non-dimensional form. The analytical studies are restricted to the case of a periodic steady-state motion with two symmetric impacts per cycle near 1:1 resonance. The multiple scales method combined with the sawtooth-function-based modelling of the non-smooth dynamics is employed. A conception of the stability analysis of the periodic motions suited for this theoretical approach is presented. The frequency–response curves and force–response curves with stable and unstable branches are determined, and the interplay between various model parameters is investigated. The theoretical predictions related to the motion amplitude and the range of stability of the periodic steady-state response are verified via a series of numerical experiments and computation of Lyapunov exponents. Finally, the limitations and extensibility of the approach are discussed.

Nonlinear Vibrations of Viscoelastic Plane Truss Under Harmonic Excitation

2014 ◽  
Vol 14 (04) ◽  
pp. 1450009 ◽  
Author(s):  
Andrew Yee Tak Leung ◽  
Hong Xiang Yang ◽  
Ping Zhu
Keyword(s):  
Steady State ◽  
Fractional Derivatives ◽  
Harmonic Excitation ◽  
Homotopy Continuation ◽  
Response Curves ◽  
System A ◽  
Structural Responses ◽  
Voigt Model ◽  
Two Degree Of Freedom ◽  
Steady State Solutions

This paper is concerned with the steady state bifurcations of a harmonically excited two-member plane truss system. A two-degree-of-freedom Duffing system having nonlinear fractional derivatives is derived to govern the dynamic behaviors of the truss system. Viscoelastic properties are described by the fractional Kelvin–Voigt model based on the Caputo definition. The combined method of harmonic balance and polynomial homotopy continuation is adopted to obtain steady state solutions analytically. A parametric study is conducted with the help of amplitude-response curves. Despite its seeming simplicity, the mechanical system exhibits a wide variety of structural responses. The primary and sub-harmonic resonances and chaos are found in specific regions of system parameters. The dynamic snap-through phenomena are observed when the forcing amplitude exceeds some critical values. Moreover, it has been shown that, suppression of undesirable responses can be achieved via changing of viscosity of the system.

Free and Forced Vibrations of the Strongly Nonlinear Cubic-Quintic Duffing Oscillators

2017 ◽  
Vol 72 (1) ◽  
pp. 59-69 ◽  
Author(s):  
M.M. Fatih Karahan ◽  
Mehmet Pakdemirli
Keyword(s):  
Numerical Simulations ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Free And Forced Vibrations ◽  
Approximate Analytical Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

AbstractStrongly nonlinear cubic-quintic Duffing oscillatoris considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical solutions.

Modal Analysis to Interpret Localization Phenomena in Two Nonlinear Tuned Mass Dampers

2021 ◽  
Author(s):  
Yuji Harata ◽  
Takashi Ikeda
Keyword(s):  
Steady State ◽  
Modal Analysis ◽  
Equations Of Motion ◽  
Harmonic Excitation ◽  
Mode Shapes ◽  
Response Curves ◽  
Tuned Mass Dampers ◽  
Second Mode ◽  
Steady State Solutions ◽  
Modal Coordinates

Abstract This study investigates localization phenomena in two identical nonlinear tuned mass dampers (TMDs) installed on an elastic structure, which is subjected to external, harmonic excitation. In the theoretical analysis, the mode shapes of the system are determined, and the modal equations of motion are derived using modal analysis. These equations are demonstrated as forming an autoparametric system in which external excitation directly acts on the first and third vibration modes, whereas the second vibration mode is indirectly excited due to the nonlinear coupling with the other modes. Van der Pol’s method is employed to obtain the frequency response curves for both physical and modal coordinates. The two TMDs vibrate in phase for the first and third modes, but vibrate out of phase for the second mode. Consequently, when all modes appear, the two TMDs may vibrate at different amplitudes, i.e., localization phenomena may occur because the TMD motions are expressed by the summation of motions for all modes. The numerical calculations clarify that the localization phenomena may occur in the two TMDs when all three modes appear simultaneously. Moreover, there are two steady-state solutions of the harmonic oscillations for the second mode with identical amplitudes; however, their phases differ by π. Hence, which TMD vibrates at higher amplitudes depends on which of these two steady-state solutions for the phase.

Estimation of Cumene Cracking Reaction-Diffusion Model Parameters: Uniqueness and Parametric Sensitivity Analysis

2003 ◽  
Vol 1 (1) ◽  
Author(s):  
Peter Schwan ◽  
Klaus P Möller
Keyword(s):  
Sensitivity Analysis ◽  
Steady State ◽  
Parameter Estimates ◽  
Physical Parameters ◽  
Model Parameters ◽  
Response Curves ◽  
Cumene Cracking ◽  
Irreversible Reaction ◽  
Reaction Diffusion Model ◽  
Order Of Magnitude

The pulse response of cumene cracking over ZSM5 extrudates has been measured using a Jetloop recycle reactor. A model assuming first order irreversible reaction with constant macro-pore diffusivity and linear adsorption was used to describe the response curves of the reactants and products. The model parameters adsorption, diffusion and reaction rate are in general highly correlated. Relationships for regions of parameter insensitivity and correlation functions between dependent parameters are given. With the aid of independent measurement of adsorption, a sensitivity analysis and a similarity analysis between equations, it was possible to reduce the 7 parameter model into a 2 parameter model for conditions of strong diffusion limitation observed in these experiments. Although good model fits could be achieved, a high degree of uncertainty in the parameter estimates remained, which reflects the high correlation of the physical parameters. Comparison with steady state results shows that the transient diffusivity for cumene is approximately equal to the Knudsen diffusivity, but an order of magnitude lower than the steady state diffusivity. The transient Thiele modulus for cumene was an order of magnitude higher than the steady state value.

A Broadband Internally Resonant Vibratory Energy Harvester

2016 ◽  
Vol 138 (6) ◽  
Author(s):  
Li-Qun Chen ◽  
Wen-An Jiang ◽  
Meghashyam Panyam ◽  
Mohammed F. Daqaq
Keyword(s):  
Steady State ◽  
Internal Resonance ◽  
Frequency Tuning ◽  
Resonance Energy ◽  
Harmonic Excitation ◽  
Design Parameters ◽  
Modal Interactions ◽  
Response Curves ◽  
Modal Frequency ◽  
Experimental Findings

The objective of this paper is twofold: first to illustrate that nonlinear modal interactions, namely, a two-to-one internal resonance energy pump, can be exploited to improve the steady-state bandwidth of vibratory energy harvesters; and, second, to investigate the influence of key system’s parameters on the steady-state bandwidth in the presence of the internal resonance. To achieve this objective, an L-shaped piezoelectric cantilevered harvester augmented with frequency tuning magnets is considered. The distance between the magnets is adjusted such that the second modal frequency of the structure is nearly twice its first modal frequency. This facilitates a nonlinear energy exchange between these two commensurate modes resulting in large-amplitude responses over a wider range of frequencies. The harvester is then subjected to a harmonic excitation with a frequency close to the first modal frequency, and the voltage–frequency response curves are generated. Results clearly illustrate an improved bandwidth and output voltage over a case which does not involve an internal resonance. A nonlinear model of the harvester is developed and validated against experimental findings. An approximate analytical solution of the model is obtained using perturbation methods and utilized to draw several conclusions regarding the influence of key design parameters on the harvester’s bandwidth.

Electrostatic Vibration Energy Harvesters with Linear and Nonlinear Resonators

2014 ◽  
Vol 24 (11) ◽  
pp. 1430030 ◽  
Author(s):  
Peter Harte ◽  
Elena Blokhina ◽  
Orla Feely ◽  
Danièle Fournier-Prunaret ◽  
Dimitri Galayko
Keyword(s):  
Steady State ◽  
Multiple Scales ◽  
Perturbation Technique ◽  
Period Doubling ◽  
Vibration Energy ◽  
Energy Harvesters ◽  
Transition To Chaos ◽  
Nonlinear Resonator ◽  
Linear And Nonlinear ◽  
Filippov Type

This paper discusses the time-dependent dynamics of electrostatic vibration energy harvesters (eVEHs) with linear and nonlinear mechanical resonators. These eVEHs are fundamentally nonlinear regardless of whether a linear or nonlinear resonator is being used. The model of the system under investigation has the form of a piecewise-smooth dynamical system of a Filippov type that has a specific discontinuity in the form of a hold-on term. We use a perturbation technique called the multiple scales method to develop a theory to analyze the steady-state dynamics of the system, be it with a linear or a nonlinear resonator. We then analyze the stability of the steady-state orbit to determine when the first doubling bifurcation occurs in the system. This gives an upper bound on the region of steady-state oscillations which allows us to determine a theoretical limit on the power convertible by the eVEH. We then turn our discussion to the nonlinear behavior we see in the system's transition to chaos. Since the eVEH studied here is a Filippov type system, sliding modes and sliding bifurcations are possible in the system. We discuss the evolution of the sliding region and give particular examples of sliding phenomena and sliding bifurcations. An understanding of sliding phenomena is required for analyzing the transition to chaos since segments of sliding motion appear on trajectories that undergo period-doubling bifurcations. The transition to chaos is explained in detail by the example of the system with a linear resonator, however we discuss examples of the system with mechanical nonlinearities and discuss the difference between the linear and nonlinear cases.

Bifurcation Phenomena Caused by Multiple Nonlinear Vibration Absorbers

2010 ◽  
Vol 5 (2) ◽  
Author(s):  
Takashi Ikeda
Keyword(s):  
Steady State ◽  
Nonlinear Vibration ◽  
Excitation Frequency ◽  
Pitchfork Bifurcation ◽  
Harmonic Excitation ◽  
Vibration Absorbers ◽  
Response Curves ◽  
System Parameters ◽  
Chaotic Vibrations ◽  
Steady State Solutions

The characteristics of two, three, and four nonlinear vibration absorbers or nonlinear tuned mass dampers (NTMDs) attached to a structure under harmonic excitation are investigated. The frequency response curves are theoretically determined using van der Pol’s method. When the parameters of the absorbers are equal, it is found from the theoretical analysis that pitchfork bifurcations may occur on the part of the response curves, which are unstable in the multi-absorber systems, but are stable in a system with one NTMD. Multivalued steady-state solutions, such as three steady-state solutions for a dual-absorber system with different amplitudes, five steady-state solutions for a triple-absorber system, and seven steady-state solutions for a quadruple-absorber system, appear near bifurcation points. The NTMDs behave in that one of them vibrates at high amplitudes while the others vibrate at low amplitudes, even if the dimensions of the NTMDs are identical. Namely, “localization phenomenon” or “mode localization” occurs. After the pitchfork bifurcation, Hopf bifurcations may occur depending on the values of the system parameters, and amplitude- and phase-modulated motions, including chaotic vibrations, appear after the Hopf bifurcation when the excitation frequency decreases. Lyapunov exponents are numerically calculated to prove the occurrence of chaotic vibrations. Bifurcation sets are also calculated to investigate the influence of the system parameters on the response of the systems.

A Nonlinear Oscillator Lanchester Damper

1987 ◽  
Vol 109 (4) ◽  
pp. 343-347 ◽  
Author(s):  
K. R. Asfar ◽  
A. H. Nayfeh ◽  
K. A. Barrash
Keyword(s):  
Steady State ◽  
Nonlinear Oscillator ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Harmonic Excitation ◽  
Second Order ◽  
Numerical Results ◽  
Nonlinear Spring ◽  
Uniform Expansion ◽  
Steady State Solutions

The method of multiple scales is used to investigate the effect of a nonlinear spring in the main system on the performance of Lanchester-type absorbers. A second-order uniform expansion is obtained for the response of the system to a harmonic excitation. Numerical results for steady-state solutions illustrating the influence of the nonlinearity and damping factors on the response are presented. A softening-type effective nonlinearity dominates the system and considerably improves its damping.

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