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.

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.

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.

Vibration Control of 2DOF Structures Utilizing Sloshing in Square Tanks

2012 ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Shota Ninomiya
Keyword(s):  
Aspect Ratio ◽  
Vibration Control ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Tuned Liquid Damper ◽  
Response Curves ◽  
Chaotic Motions ◽  
Installation Angle ◽  
Theoretical Results

This paper investigates the vibration control of a tower-like structure utilizing a square tuned liquid damper (TLD) when the structure is subjected to horizontal, harmonic excitation. In the theoretical analysis, when the tuning condition, 1:1:1:1, is satisfied, the natural frequencies of the 2DOF structure and two predominate sloshing modes are nearly equal. 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 frequency response curves which are compared to the numerical simulation. 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 square TLDs work more efficiently than conventional rectangular TLDs, and its performance is further improved when the aspect ratio is slightly increased and the installation angle is equal to zero. Experiments were conducted in order to confirm the validity of the theoretical results.

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.

Nonlinear Responses of Dual Pendulum Dynamic Absorbers

2009 ◽  
Author(s):  
Takashi Ikeda
Keyword(s):  
Theoretical Analysis ◽  
Frequency Response ◽  
Pitchfork Bifurcation ◽  
Hopf Bifurcations ◽  
Response Curves ◽  
Pendulum System ◽  
Bifurcation Points ◽  
Chaotic Vibrations ◽  
Nonlinear Responses ◽  
Pitchfork Bifurcations

The nonlinear responses of a single-degree-of-freedom (SDOF) system with two pendulum tuned mass dampers (TMDs) 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 single- and dual-pendulum systems are shown. Pitchfork bifurcations occur followed by mode localization where both identical pendulums vibrate but at different amplitudes. Hopf bifurcations occur and then amplitude 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 non identical dual-pendulum system, perturbed pitchfork bifurcations occur and saddle-node bifurcation points appear instead of pitchfork bifurcation points. Hopf bifurcations and amplitude 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 pendulums were taken into consideration and the validity of the theoretical analysis was confirmed.

Bifurcation Modes of Periodic Solution in a Duffing System Under Constant Force as Well as Harmonic Excitation

2019 ◽  
Vol 29 (13) ◽  
pp. 1950173 ◽  
Author(s):  
Lei Hou ◽  
Xiaochao Su ◽  
Yushu Chen
Keyword(s):  
Frequency Response ◽  
Damping Ratio ◽  
Excitation Frequency ◽  
Harmonic Balance Method ◽  
Harmonic Excitation ◽  
Nonlinearity Parameter ◽  
Constant Force ◽  
Response Curves ◽  
Duffing System ◽  
Singularity Method

This paper focuses on the classification of the bifurcation modes of a Duffing system under the combined excitations of constant force and harmonic excitation. The Harmonic Balance method combined with the arc-length continuation is used to obtain the periodic solutions of the system, and the Floquet theory is employed to analyze the stability of the corresponding solutions. Accordingly, the frequency-response curves affected respectively by the constant force and the magnitude of the harmonic excitation are analyzed to show the basic dynamical properties of the system. Afterwards, the bifurcation investigations are carried out with the aid of the two-state variable singularity method. It is derived that there are a total of six different types of bifurcation modes due to the effects of the constant force and the magnitude of the harmonic excitation. At last, the effects of the nonlinearity parameter and the damping ratio on the bifurcation modes of the system are also discussed. The results obtained in this paper extend the findings in reference that the system can have markedly three types of frequency-response curves: with only one solution, or with maximum three or five solutions for a certain excitation frequency, and contribute to a better understanding of the significant influence of the constant force.

Intrinsic Localized Modes of Principal Parametric Resonances in Pendulum Arrays Subjected to Vertical Excitation

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Chongyue Shi ◽  
Keisuke Nishimura
Keyword(s):  
Experimental Data ◽  
Theoretical Analysis ◽  
Frequency Response ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Localized Modes ◽  
Response Curves ◽  
Chaotic Motions ◽  
Intrinsic Localized Modes ◽  
Spring Constants

Intrinsic localized modes (ILMs) are investigated in an N-pendulum array subjected to vertical harmonic excitation. The pendula behave nonlinearly and are coupled with each other because they are connected by torsional, weak, linear springs. In the theoretical analysis, van der Pol's method is employed to determine the expressions for frequency response curves for the principal parametric resonance, considering the nonlinear restoring moment of the pendula. In the numerical results, frequency response curves for N = 2 and 3 are shown to examine the patterns of ILMs, and demonstrate the influences of the connecting spring constants and the imperfections of the pendula. Bifurcation sets are also calculated to show the excitation frequency range and the conditions for the occurrence of ILMs. Increasing the connecting spring constants results in the appearance of Hopf bifurcations. The numerical simulations reveal the occurrence of ILMs with amplitude modulated motions (AMMs), including chaotic motions. ILMs were observed in experiments, and the experimental data were compared with the theoretical results. The validity of the theoretical analysis was confirmed by the experimental data.

Nonlinear liquid sloshing in a square tank subjected to obliquely horizontal excitation

2012 ◽  
Vol 700 ◽  
pp. 304-328 ◽  
Author(s):  
Takashi Ikeda ◽  
Raouf A. Ibrahim ◽  
Yuji Harata ◽  
Tasuku Kuriyama
Keyword(s):  
Theoretical Analysis ◽  
Frequency Response ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Frequency Range ◽  
Response Curves ◽  
Chaotic Motions ◽  
Nonlinear Responses ◽  
Good Agreement

AbstractNonlinear responses of surface waves in rigid square and nearly square tanks partially filled with liquid subjected to obliquely horizontal, sinusoidal excitation are investigated theoretically and experimentally. Two predominant modes of sloshing are significantly coupled nonlinearly because their natural frequencies are nearly identical resulting in 1:1 internal resonance. Therefore, if only one of these modes is directly excited, the other mode is indirectly excited due to the nonlinear coupling. In the nonlinear theoretical analysis, the modal equations of motion are derived for the two predominant sloshing modes as well as five higher sloshing modes. The linear viscous terms are incorporated in order to consider the damping effect of sloshing. The expressions for the frequency response curves are determined using van der Pol’s method. The influences of the excitation direction and the aspect ratio of the tank cross-section on the frequency response curves are numerically examined. Planar and swirl motions of sloshing, and Hopf bifurcations followed by amplitude modulated motions including chaotic motions, are predicted when the excitation frequency is close to one of the natural frequencies of the two predominant sloshing modes. Lyapunov exponents are calculated and reveal the excitation frequency range over which liquid chaotic motions occur. In addition, bifurcation sets are shown to clarify the influences of the parameters on the change in the structural stability. The theoretically predicted results are in good agreement with the measured data, thus the theoretical analysis was experimentally validated.

Modal Analysis to Interpret Localization Phenomena of Harmonic Oscillations in Nonlinear Oscillator Arrays

2019 ◽  
Author(s):  
Yuji Harata ◽  
Takashi Ikeda
Keyword(s):  
Coordinate System ◽  
Frequency Response ◽  
Equations Of Motion ◽  
Harmonic Excitation ◽  
Vibrational Modes ◽  
Response Curves ◽  
Internal Resonances ◽  
Harmonic Oscillations ◽  
Oscillator Arrays ◽  
Phase Angles

Abstract This paper investigates localization phenomena in a nonlinear array with N Duffing oscillators connected by weak, linear springs when the array is subjected to harmonic excitation. In the theoretical analysis, the equations of motion are derived for: (1) the physical coordinate system, and (2) modal coordinate system. The modal equations of motion form an autoparametric system, i.e., the excitation acts directly on the first mode of vibration, and the other modes are indirectly excited because they are nonlinearly coupled with the first mode. Van der Pol’s method is employed to obtain the solutions of the harmonic oscillations, and then the expressions of the frequency response curves are given. In the numerical calculations, the frequency response curves of the amplitudes and phase angles in the cases of N = 2 and 3 are presented. The frequency response curves, obtained in the modal coordinate system, demonstrate that localization phenomena occur in the physical coordinate system when multiple vibrational modes simultaneously appear. When imperfections exist in the N Duffing oscillators, the modal equations of motion do not form an autoparametric system because the external excitation directly acts on all modes. Instead, internal resonances may occur in such systems.

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