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.

Intrinsic Localized Modes of Harmonic Oscillations in Pendulum-Arrays Subjected to Horizontal Excitation

2013 ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Keisuke Nishimura
Keyword(s):  
Theoretical Analysis ◽  
Frequency Response ◽  
Excitation Frequency ◽  
System Response ◽  
Stable Steady State ◽  
Frequency Range ◽  
Localized Modes ◽  
Response Curves ◽  
Harmonic Oscillations ◽  
Intrinsic Localized Modes

The behavior of intrinsic localized modes (ILMs) is investigated for an array with N pendula which are connected with each other by weak, linear springs when the array is subjected to horizontal, sinusoidal excitation. In the theoretical analysis, van der Pol’s method is employed to determine the expressions for the frequency response curves for fundamental harmonic oscillations. In the numerical calculations, the frequency response curves are presented for N = 2 and 3 and compared with the results of the numerical simulations. Patterns of oscillations are classified according to the stable steady-state solutions of the response curves, and the patterns in which ILMs appear are discussed in detail. The influence of the connecting springs of the pendula on the appearance of ILMs is examined. Increasing the values of the connecting spring constants may affect the excitation frequency range of ILMs and cause Hopf bifurcation to occur, followed by amplitude modulated motions (AMMs) including chaotic vibrations. The influence of the imperfections of the pendula on the system response is also investigated. Bifurcation sets are calculated to examine the influence of the system parameters on the excitation frequency range of ILMs and determine the threshold value for the connecting spring constant after which ILMs do not appear. Experiments were conducted for N = 2, and the data were compared with the theoretical results in order to confirm the validity of the theoretical analysis.

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.

Intrinsic Localized Modes of Harmonic Oscillations in Pendulum Arrays Subjected to Horizontal Excitation

2015 ◽  
Vol 10 (2) ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Keisuke Nishimura
Keyword(s):  
Theoretical Analysis ◽  
Frequency Response ◽  
Excitation Frequency ◽  
System Response ◽  
Stable Steady State ◽  
Frequency Range ◽  
Localized Modes ◽  
Response Curves ◽  
Harmonic Oscillations ◽  
Intrinsic Localized Modes

The behavior of intrinsic localized modes (ILMs) is investigated for an array with N pendula which are connected with each other by weak, linear springs when the array is subjected to horizontal, sinusoidal excitation. In the theoretical analysis, van der Pol's method is employed to determine the expressions for the frequency response curves for fundamental harmonic oscillations. In the numerical calculations, the frequency response curves are presented for N = 2 and 3 and compared with the results of the numerical simulations. Patterns of oscillations are classified according to the stable steady-state solutions of the response curves, and the patterns in which ILMs appear are discussed in detail. The influence of the connecting springs of the pendula on the appearance of ILMs is examined. Increasing the values of the connecting spring constants may affect the excitation frequency range of ILMs and cause Hopf bifurcation to occur, followed by amplitude modulated motions (AMMs) including chaotic vibrations. The influence of the imperfections of the pendula on the system response is also investigated. Bifurcation sets are calculated to examine the influence of the system parameters on the excitation frequency range of ILMs and determine the threshold value for the connecting spring constant above which ILMs do not appear. Experiments were conducted for N = 2, and the data were compared with the theoretical results in order to confirm the validity of the theoretical analysis.

Localization Phenomena in Pendulum Arrays Subjected to Vertical Excitation

2014 ◽  
Author(s):  
Takashi Ikeda ◽  
Yuji Harata ◽  
Chongyue Shi ◽  
Keisuke Nishimura
Keyword(s):  
Experimental Data ◽  
Theoretical Analysis ◽  
Frequency Response ◽  
Excitation Frequency ◽  
Harmonic Excitation ◽  
Response Curves ◽  
Chaotic Motions ◽  
Vertical Excitation ◽  
Parametric Resonances ◽  
Spring Constants

Localization phenomena, also referred to as intrinsic localized modes (ILMs), are investigated in an N-pendulum array subjected to vertical harmonic excitation. The pendula behave nonlinearly and are connected with each other by 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 resonances, 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 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 constant results in the appearance of Hopf bifurcation. 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 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.

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.

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 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.

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.

Frequency Response Curves and Dynamic Characteristics of a Ginkgo Tree in Different Growth Periods

2020 ◽  
Vol 63 (6) ◽  
pp. 1673-1684
Author(s):  
Jie Zhou ◽  
Linyun Xu ◽  
Guanhua Liu ◽  
Yan Xuan ◽  
Hongping Zhou ◽  
...  
Keyword(s):  
Frequency Response ◽  
Frequency Domain ◽  
Dynamic Characteristics ◽  
Natural Frequencies ◽  
Excitation Frequency ◽  
Fruit Trees ◽  
Modal Parameters ◽  
Response Curves ◽  
Resonance Frequencies ◽  
Modal Parameters Identification

HighlightsThe frequency domain modal parameters identification method was applied to a ginkgo tree.Dynamic characteristics of the ginkgo tree were tested during five phenological periods.Almost all resonance frequencies were near the peaks of the frequency response curves.Leaves caused the number of natural frequencies of the ginkgo tree to be greatly reduced.Abstract. Understanding the dynamic characteristics of fruit trees is the premise of effective mechanized harvesting. This study performed a tracking test on a ginkgo tree in five phenological periods from the dormant period to the leaf-unfolding period. The frequency domain modal parameters identification method was applied to the ginkgo tree, and the relationship between the natural frequencies and resonance frequencies of the ginkgo tree was obtained. The main factors affecting the fundamental frequency and damping ratio of the ginkgo tree were not the elastic modulus and moisture content but rather the growth of the leaves. The growth of leaves caused the number of natural frequencies in the low-frequency band to be greatly reduced, and the value of the natural frequencies exhibited a slightly decreasing tendency. The damping caused by leaves had a significant weakening effect on the transmission of vibrational energy on the lateral branches. The resonance frequencies that caused strong response of the ginkgo tree were mostly near the peak frequencies of the frequency response curves (natural frequencies), but eccentric motor excitation could not effectively stimulate all the natural frequencies of the ginkgo tree to reach resonance. In the frequency response curves of the ginkgo tree, the main natural frequency with the maximum energy might not cause the maximum vibration response of the ginkgo tree, even if this excitation frequency could induce resonance. Resonance could be used to maximize the transfer of excitation energy, but each position of the tree had its own independent frequency spectrum characteristics. A single excitation frequency could not cause all positions of the ginkgo tree to resonate simultaneously. Changing the excitation frequency of harvesting equipment within a small frequency range could achieve the maximum resonance response of most positions on fruit trees. Keywords: Dynamic characteristics, Growth periods, Leaves, Natural frequencies, Resonance.

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