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.

Bifurcation Phenomena Caused by Two Nonlinear Dynamic Absorbers

2007 ◽  
Author(s):  
Takashi Ikeda
Keyword(s):  
Nonlinear Dynamic ◽  
Pitchfork Bifurcation ◽  
Harmonic Excitation ◽  
Hopf Bifurcations ◽  
Vibration Absorbers ◽  
Response Curves ◽  
Chaotic Vibration ◽  
Bifurcation Phenomena ◽  
Pitchfork Bifurcations ◽  
Steady State Solutions

The characteristics of two nonlinear vibration absorbers simultaneously attached to structures under harmonic excitation are investigated. The frequency response curves are theoretically determined using van der Pol’s method. It is found from the theoretical analysis that pitchfork bifurcations may appear on a part of the response curves which are stable in a system with one nonlinear dynamic absorber. Three steady-state solutions with different amplitudes appear just after the pitchfork bifurcation. After that, Hopf bifurcations may occur depending on the values of the system parameters, and amplitude- and phase-modulated motion including a chaotic vibration appears after the Hopf bifurcation. Lyapunov exponents are numerically calculated to prove the occurrence of a chaotic vibration. In addition, it is also found that only Hopf bifurcations, not pitchfork bifurcations, can occur even when the linear and nonlinear dynamic absorbers are combined.

Nonlinear Dynamic Responses of Elastic Structures With Two Rectangular Liquid Tanks Subjected to Horizontal Excitation

2010 ◽  
Vol 6 (2) ◽  
Author(s):  
Takashi Ikeda
Keyword(s):  
Frequency Response ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Pitchfork Bifurcation ◽  
Harmonic Excitation ◽  
Dynamic Responses ◽  
Response Curves ◽  
Chaotic Vibrations ◽  
Tuned Liquid Dampers ◽  
Theoretical Results

Nonlinear vibrations of an elastic structure with two partially filled liquid tanks subjected to horizontal harmonic excitation are investigated. The natural frequencies of the structure and sloshing satisfy the tuning condition 1:1:1 when tuned liquid dampers are used. The equations of motion for the structure and the modal equations of motion for the first, second, and third sloshing modes are derived by using Galerkin’s method, taking into account the nonlinearity of the sloshing. Then, van der Pol’s method is employed to determine the frequency response curves. It is found in calculating the frequency response curves that pitchfork bifurcation can occur followed by “localization phenomenon” for a specific excitation frequency range. During this range, sloshing occurs at different amplitudes in the two tanks, even if the dimensions of both tanks are identical. Furthermore, Hopf bifurcation may occur followed by amplitude- and phase-modulated motions including chaotic vibrations. In addition, Lyapunov exponents are calculated to prove the occurrence of both amplitude-modulated motions and chaotic vibrations. Bifurcation sets are also calculated to show the influence of the system parameters on the frequency response. Experiments were conducted to confirm the validity of the theoretical results. It was found that the theoretical results were in good agreement with the experimental data.

Nonlinear Vibrations of Elastic Structures Carrying Two Rectangular Liquid-Filled Tanks Under Horizontal Excitation

2005 ◽  
Author(s):  
Takashi Ikeda
Keyword(s):  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Natural Frequencies ◽  
Hydrodynamic Force ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Response Curves ◽  
System Parameters ◽  
Chaotic Vibrations ◽  
Multi Mode

Nonlinear vibrations of an elastic structure carrying two liquid-filled tanks under horizontal harmonic excitation are investigated. When a 1:1:1 ratio of internal resonance is satisfied among the natural frequencies of the structure and sloshing in the two liquid tanks, modal equations are derived by using Galerkin’s method, taking into account the nonlinearity of the hydrodynamic force. Then, frequency response curves are calculated by using Andronov and Witt’s method. Peculiar vibrations, referred to as ‘multi-mode vibrations’, sometimes may appear depending on the values of the system parameters. They never occur in a structure carrying only one liquid-filled tank. In other words, even if the dimensions of the two tanks are identical, the sloshing which occurs in each tank differs depending on the excitation frequency. The multi-mode vibrations include constant amplitude vibrations and amplitude modulated motion as well as chaotic vibrations.

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.

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.

On the Dynamics of Vibration Absorbers With Motion-Limiting Stops

1998 ◽  
Vol 65 (1) ◽  
pp. 223-233 ◽  
Author(s):  
H. Luo ◽  
S. Hanagud
Keyword(s):  
Steady State ◽  
Mechanical Model ◽  
Piecewise Linear ◽  
Nonlinear Effects ◽  
Vibration Absorbers ◽  
Chaotic Vibrations ◽  
Piecewise Linear System ◽  
Period 2 ◽  
Steady State Solutions ◽  
Theoretical Results

The dynamics of a class of vibration absorbers with elastic stops is discussed in this paper. The mechanical model proposed in previously published papers are modified to explain certain nonlinear effects, chaotic vibrations, and lower damping observed in our studies. Refined contact-noncontact criteria are presented. Exact steady-state solutions are obtained for a piecewise linear system by using the proposed contact-noncontact criteria. Numerical simulations are presented and compared with the results of the previous work. Significant differences that have been found include some chaotic responses of the system. Experiments are conducted to validate the theoretical results. Chaotic and period-2 responses are also detected experimentally.

Vibration Control of Horizontally Excited Structures Utilizing Internal Resonance of Liquid Sloshing in Nearly Square Tanks

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata
Keyword(s):  
Frequency Response ◽  
Internal Resonance ◽  
Passive Control ◽  
Harmonic Excitation ◽  
Liquid Sloshing ◽  
Maximum Efficiency ◽  
System Response ◽  
Response Curves ◽  
System Parameters ◽  
Theoretical Results

Passive control of vibrations in an elastic structure subjected to horizontal, harmonic excitation by utilizing a nearly square liquid tank is investigated. When the natural frequency ratio 1:1:1 is satisfied among the natural frequencies of the structure and the two predominant sloshing modes (1,0) and (0,1), the performance of a nearly square tank as a tuned liquid damper (TLD) is expected to be superior to rectangular TLDs due to internal resonance. In the theoretical analysis, Galerkin's method is used to determine the modal equations of motion for liquid sloshing considering the nonlinearity of sloshing. Then, van der Pol's method is used to obtain the expressions for the frequency response curves for the structure and sloshing modes. Frequency response curves and bifurcation set diagrams are shown to investigate the influences of the aspect ratio of the tank cross section and the tank installation angle on the system response. From the theoretical results, the optimal values of the system parameters can be determined in order to achieve maximum efficiency of vibration suppression for the structure. Hopf bifurcations occur and amplitude modulated motions (AMMs) may appear depending on the values of the system parameters. Experiments were also conducted, and the theoretical results agreed well with the experimental data.

Nonlinear Responses of Dual-Pendulum Dynamic Absorbers

2010 ◽  
Vol 6 (1) ◽  
Author(s):  
Takashi Ikeda
Keyword(s):  
Theoretical Analysis ◽  
Frequency Response ◽  
Pitchfork Bifurcation ◽  
Hopf Bifurcations ◽  
Response Curves ◽  
Pendulum System ◽  
Mode Localization ◽  
Bifurcation Points ◽  
Chaotic Vibrations ◽  
Nonlinear Responses

The nonlinear responses of a single-degree-of-freedom system with two pendulum tuned mass dampers under horizontal sinusoidal excitation are investigated. In the theoretical analysis, van der Pol’s method is applied to determine the expressions for the frequency response curves. In the numerical results, the differences between the responses in single- and dual-pendulum systems are shown. A pitchfork bifurcation occurs followed by mode localization where both identical pendula vibrate at constant but different amplitudes. Hopf bifurcations occur, and then amplitude- and phase-modulated motions including chaotic vibrations appear in the identical dual-pendulum system. The Lyapunov exponents are calculated to prove the occurrence of chaotic vibrations. In a nonidentical dual-pendulum system, a perturbed pitchfork bifurcation occurs and saddle-node bifurcation points appear instead of pitchfork bifurcation points. Hopf bifurcations and amplitude- and phase-modulated motions also appear. The deviation of the tuning condition is also investigated by showing the frequency response curves and bifurcation sets. The numerical simulations are shown to be in good agreement with the theoretical results. In experiments, the imperfections of the two pendula were taken into consideration, and the validity of the theoretical analysis was confirmed.

A Combined Steady-State/Transient Approach to Study Very Large Amplitude Ship Rolling

1995 ◽  
Author(s):  
J. Falzarano ◽  
R. Kota ◽  
I. Esparza
Keyword(s):  
Steady State ◽  
Large Amplitude ◽  
Wave Amplitude ◽  
Practical Importance ◽  
Rolling Motion ◽  
Frequency Range ◽  
Response Curves ◽  
Type Of Behavior ◽  
Steady State Solutions ◽  
Ship Rolling

Abstract For ships, rolling motion is the most critical due to the possibility of capsizing. In a regular (periodic) sea, if no bounded steady state solutions exist, then capsizing may be imminent. Determining for exactly which wave amplitude and frequency the steady-state solutions disappear or become unstable is of great practical importance. In previous works (Falzarano, Esparza, and Taz Ul Mulk, 1994) and abstracted presentations (Falzarano, 1993), the global transient dynamics of large amplitude ship rolling motion was studied. The effect on the steady-state solutions of changing wave frequency for a fixed wave amplitudes was studied. It was shown how the in-phase and out-of-phase solutions evolve as the frequency passes through the linear natural frequency. For small wave amplitudes (external forcing) there exists a single steady-state throughout the frequency range, for moderate wave amplitudes there exists a frequency range where multiple steady state harmonic solutions exists. As the wave amplitude was increased further there existed a frequency range where no steady-state harmonic solution existed. In the present work, the very large amplitude ship rolling motion in the region where no steady-state solutions exist will be studied in more detail. Moreover, the mechanisms (bifurcations) that cause this type of behavior to evolve from more simple behavior will be studied using a combination of both frequency response curves and Poincaré maps. It is expected that global chaotic bifurcations such as those previously described (e.g., Thompson and Stewart, 1989) will be identified.

Vibration Control of Two-Degree-of-Freedom Structures Utilizing Sloshing in Nearly Square Tanks

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Shota Ninomiya
Keyword(s):  
Vibration Control ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Degree Of Freedom ◽  
Response Curves ◽  
Chaotic Motions ◽  
Installation Angle ◽  
Two Degree Of Freedom ◽  
Theoretical Results

This paper investigates the vibration control of a towerlike structure with degrees of freedom utilizing a square or nearly square tuned liquid damper (TLD) when the structure is subjected to horizontal, harmonic excitation. In the theoretical analysis, when the two natural frequencies of the two-degree-of-freedom (2DOF) structure nearly equal those of the two predominant sloshing modes, the tuning condition, 1:1:1:1, is nearly satisfied. Galerkin's method is used to derive the modal equations of motion for sloshing. The nonlinearity of the hydrodynamic force due to sloshing is considered in the equations of motion for the 2DOF structure. Linear viscous damping terms are incorporated into the modal equations to consider the damping effect of sloshing. Van der Pol's method is employed to determine the expressions for the frequency response curves. The influences of the excitation frequency, the tank installation angle, and the aspect ratio of the tank cross section on the response curves are examined. The theoretical results show that whirling motions and amplitude-modulated motions (AMMs), including chaotic motions, may occur in the structure because swirl motions and Hopf bifurcations, followed by AMMs, appear in the tank. It is also found that a square TLD works more effectively than a conventional rectangular TLD, and its performance is further improved when the tank width is slightly increased and the installation angle is equal to zero. Experiments were conducted in order to confirm the validity of the theoretical results.

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