Autoparametric Resonances of Elastic Structures Coupled With Two Sloshing Modes in a Square Liquid Tank

2012 ◽  
Vol 8 (1) ◽  
Author(s):  
Takashi Ikeda ◽  
Masaki Takashima ◽  
Yuji Harata
Keyword(s):  
Equations Of Motion ◽  
Excitation Frequency ◽  
Excitation Amplitude ◽  
Liquid Level ◽  
Response Curves ◽  
Small Deviations ◽  
Tuned Liquid Dampers ◽  
Linear Damping ◽  
Liquid Dampers ◽  
Good Agreement

Nonlinear vibrations of an elastic structure coupled with liquid sloshing in a square tank subjected to vertical sinusoidal excitation are investigated. In the theoretical analysis, the ratios of the natural frequencies of the structure and two sloshing modes satisfy 2:1:1. The equations of motion for the structure and seven sloshing modes are derived using Galerkin’s method while considering the nonlinearity of sloshing. The linear damping terms are then incorporated into the modal equations to consider the damping effect of sloshing. The frequency response curves are determined using van der Pol’s method. The influences of the liquid level, the aspect ratio of the tank cross-section, the deviation of the tuning condition, and the excitation amplitude are investigated. When the liquid level is high, and depending on the excitation frequency, there are three patterns of sloshing: (i) both (1,0) and (0,1) sloshing modes appear at identical amplitudes; (ii) these two modes appear at different amplitudes; and (iii) either (1,0) or (0,1) mode appears. Small deviations of the tuning condition may cause Hopf bifurcations to occur followed by amplitude modulated motion including chaotic vibrations. Bifurcation sets are also calculated to illustrate the influence of the system parameters on the response of the system. It is found that for low liquid levels, square tuned liquid dampers (TLDs) work more effectively than rectangular TLDs. Experiments were also conducted in order to confirm the validity of the theoretical results and were in good agreement with the experimental data.

Autoparametric Resonances of Elastic Structures Coupled With Two Sloshing Modes in a Square Liquid Tank

2011 ◽  
Author(s):  
Takashi Ikeda ◽  
Masaki Takashima ◽  
Yuji Harata
Keyword(s):  
Vibration Suppression ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Liquid Level ◽  
Tuned Liquid Damper ◽  
Response Curves ◽  
Small Deviations ◽  
Sloshing Mode ◽  
Linear Damping ◽  
Good Agreement

Nonlinear vibrations of an elastic structure coupled with liquid sloshing in a square tank subjected to vertical sinusoidal excitation are investigated. Previous studies examined the vibrations of a structure coupled with only one sloshing mode in a rectangular tank. However, square tanks are expected to work more efficiently as a vibration suppression device (Tuned Liquid Damper, TLD) because two sloshing modes, (1,0) and (0,1) modes, simultaneously appear when the internal resonance ratio 2:1:1 is satisfied. In reality, it is impossible to build a perfectly square tank. Therefore, a nearly square liquid tank is also considered when the tuning condition is slightly deviated. In the theoretical analysis, the fluid in the tank is assumed to be perfect. The modal equations of motion for seven sloshing modes are derived using Galerkin’s method, considering the nonlinear terms. The linear damping terms are then incorporated into the modal equations to consider the damping effect of sloshing. The frequency response curves are determined using van der Pol’s method (based on the harmonic balance method). From these response curves, the influences of the liquid level, the aspect ratio of the tank cross section, and the deviation of the tuning condition are investigated. For a square tank it is found that (1,0) and (0,1) modes are nonlinearly coupled. When the liquid level is high, there are three patterns for sloshing: (I) both (1,0) and (0,1) sloshing modes appear at identical amplitudes; (II) these two modes appear at different amplitudes; and (III) either (1,0) or (0,1) mode appears. Compared with the performance of a rectangular TLD, a square TLD works more efficiently when the liquid level is low. Small deviations of the tuning condition may cause amplitude modulated motion to appear. Bifurcation sets are also calculated to illustrate the influence of the system parameters on the performance of the TLD. Experiments were also conducted in order to confirm the validity of the theoretical results. These results were in good agreement with the experimental data.

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 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.

scholarly journals Experimental Observation Of Nonlinear Vibrations Using A Closed-Loop Vibration System

2015 ◽  
Vol 22 (4) ◽  
pp. 559-564 ◽  
Author(s):  
Everaldo de Barros ◽  
Carlos d’Andrade Souto ◽  
Mauro Hugo Mathias
Keyword(s):  
Experimental Observation ◽  
Nonlinear Vibrations ◽  
Closed Loop ◽  
Response Spectrum ◽  
Excitation Frequency ◽  
Excitation Amplitude ◽  
Experimental Results ◽  
Vibration System ◽  
Response Curves ◽  
Flexural Mode

Abstract This paper presents experimental observation of nonlinear vibrations in the response of a flexible cantilever beam to transverse harmonic base excitations around its flexural mode frequencies. In the experimental setup, instead of manual control of the signal excitation frequency and amplitude, a closed-loop vibration system is used to keep the excitation amplitude constant during the frequency sweep and to increase confidence in the experimental results. The experimental results show the presence of the third mode in the response when varying the excitation frequency around the fourth mode. The frequency-response curves, response spectrum and Poincaré plots were used for characterization of nonlinear dynamic behaviour of the beam.

Theoretical and Experimental Studies on Reduction for Multi-Modal Seismic Responses of High-Rise Structures by Tuned Liquid Dampers

2004 ◽  
Vol 10 (7) ◽  
pp. 1041-1056 ◽  
Author(s):  
Hong-Nan Li ◽  
Ying Jia ◽  
Su-Yan Wang
Keyword(s):  
Passive Control ◽  
Equations Of Motion ◽  
Experimental Studies ◽  
Tall Buildings ◽  
Shaking Table ◽  
Earthquake Ground Motion ◽  
Seismic Responses ◽  
High Rise ◽  
Tuned Liquid Dampers ◽  
Liquid Dampers

This paper focuses on the use of multiple tuned liquid dampers (TLDs) as passive control devices to reduce the multi-modal responses of tall buildings and high-rise structures to earthquake ground motion excitation. A model of a three-story building with one and two TLDs was installed on a shaking-table. The system was subjected to three earthquake time histories. Then, the mechanical models and the equations of motion for the systems of tall buildings and high-rise structures with TLDs are established. Here, the solution of the dynamic liquid pressure is based on the method of the volume of fluid and the seismic responses are obtained by use of the state equation. The comparisons show that theoretical results are generally in good agreement with experiments. It is observed that the approach presented in this paper has proved to be quite effective both in the numerical example and in the seismic simulating tests.

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.

Experimental Verifications on Seismic Response Control of Tall Structures by Multiple Tuned Liquid Dampers

2004 ◽  
Author(s):  
Hong-Nan Li ◽  
Ying Jia ◽  
Su-Yan Wang
Keyword(s):  
Passive Control ◽  
Equations Of Motion ◽  
Tall Buildings ◽  
State Equation ◽  
Shaking Table ◽  
Earthquake Ground Motion ◽  
High Rise ◽  
Tuned Liquid Dampers ◽  
Tall Structures ◽  
Liquid Dampers

The focus of this paper is on the use of multiple Tuned Liquid Dampers (TLDs) as passive control devices to reduce the multi-modal responses of tall buildings and high-rise structures to earthquake ground motion excitation. A model of a 3-story building with one and two TLDs was installed on a shaking-table. The system was subjected to three earthquake time histories. Then the mechanical models and the equations of motion for the systems of tall buildings and high-rise structures with TLDs are established. Here, the solution of the dynamic liquid pressure is based on the method of the Volume of Fluid and the seismic responses are obtained by use of the state equation. The comparisons show that theoretical results are generally in good agreement with experiments. It is observed that the approach presented in this paper proved to be quite effective both in the numerical example and in the seismic simulating tests.

Nonlinear Transverse Vibrations and 3:1 Internal Resonances of a Beam With Multiple Supports

2008 ◽  
Vol 130 (2) ◽  
Author(s):  
E. Özkaya ◽  
S. M. Bağdatlı ◽  
H. R. Öz
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Mode Shapes ◽  
Response Curves ◽  
Internal Resonances ◽  
Transverse Vibrations ◽  
Second Mode ◽  
Euler Bernoulli Beam ◽  
Correction Terms

In this study, nonlinear transverse vibrations of an Euler–Bernoulli beam with multiple supports are considered. The beam is supported with immovable ends. The immovable end conditions cause stretching of neutral axis and introduce cubic nonlinear terms to the equations of motion. Forcing and damping effects are included in the problem. The general arbitrary number of support case is considered at first, and then 3-, 4-, and 5-support cases are investigated. The method of multiple scales is directly applied to the partial differential equations. Natural frequencies and mode shapes for the linear problem are found. The correction terms are obtained from the last order of expansion. Nonlinear frequencies are calculated and then amplitude and phase modulation figures are presented for different forcing and damping cases. The 3:1 internal resonances are investigated. External excitation frequency is applied to the first mode and responses are calculated for the first or second mode. Frequency-response and force-response curves are drawn.

Tuned Liquid Damper Experiment Using Shaking Table

2013 ◽  
Vol 421 ◽  
pp. 772-777 ◽  
Author(s):  
Ki Pyo You ◽  
Young Moon Kim ◽  
Jang Youl You
Keyword(s):  
Natural Frequency ◽  
Damping Ratio ◽  
Tall Buildings ◽  
Excitation Amplitude ◽  
Shaking Table ◽  
Tuned Liquid Damper ◽  
Tuned Liquid Dampers ◽  
Liquid Dampers ◽  
Water Tanks ◽  
Wind Induced Vibration

The present study examines the characteristics of rectangular and circular tuned liquid dampers, which control wind-induced vibration in tall buildings, according to the natural frequency. The tuned liquid dampers (TLD) were of frequencies: 0.44Hz, 0.55Hz, 0.64Hz and 0.73Hz. The tuning feature of TLD water tanks was better in circular water tanks than in rectangular water tanks. Excitation amplitude affected the damping ratio based on energy dissipation capacity. At low excitation (below 5mm) and low natural frequency (0.44Hz), circular water tanks were advantageous over rectangular water tanks. However, at high excitation (over 5mm) and high natural frequency (over 0.55Hz), rectangular water tanks were advantageous over circular water tanks.

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.

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