Nonlinear Vibrations of an Elastic Structure Subjected to Vertical Excitation and Coupled With Liquid Sloshing in a Cylindrical Tank: Resonance With an Axisymmetric Mode

1999 ◽  
Author(s):  
Takashi Ikeda ◽  
Shin Murakami
Keyword(s):  
Natural Frequency ◽  
Resonance Curve ◽  
Liquid Sloshing ◽  
Elastic Structure ◽  
Cylindrical Tank ◽  
Axisymmetric Mode ◽  
Vertical Excitation ◽  
Structure Changes ◽  
Flat Shape ◽  
Resonance Curves

Abstract The nonlinear coupled vibrations of an elastic structure and liquid sloshing in a cylindrical tank are investigated. When the structure is vertically subjected to a sinusoidal excitation, and when the natural frequency of the structure is equal to twice the natural frequency of the first axisymmetric mode of sloshing, modal equations governing the coupled motions are derived. Then, the theoretical resonance curves are presented by using the method of harmonic balance and an FFT analysis. As a result, it is demonstrated that the resonance curve for the structure changes from a shape with a peak to a flat shape as the liquid level decreases. It is also clarified that amplitude-modulated motions appear when the tuning condition is deviated. In experiments, theoretical resonance curves were qualitatively in good agreement with the experimental results.

Nonlinear Vibrations in an Elastic Structure Subjected to Vertical Excitation and Coupled With Liquid Sloshing in a Rectangular Tank

1997 ◽  
Author(s):  
Takashi Ikeda
Keyword(s):  
Theoretical Analysis ◽  
Natural Frequency ◽  
Liquid Surface ◽  
Excitation Frequency ◽  
Harmonic Balance Method ◽  
Liquid Sloshing ◽  
Elastic Structure ◽  
Vertical Excitation ◽  
Rectangular Tank ◽  
Resonance Curves

Abstract The nonlinear coupled vibrations of an elastic structure and liquid sloshing in a rectangular tank, partially filled with liquid, are investigated. The structure containing the tank is vertically subjected to a sinusoidal excitation. In the theoretical analysis, the resonance curves for the responses of the structure and liquid surface are presented by the harmonic balance method, when the natural frequency of the structure is equal to twice the natural frequency of one of the sloshing modes. From the theoretical analysis, the following predictions have been obtained: (a) Due to the nonlinearity of the fluid force, harmonic oscillations appear in the structure, while subharmonic oscillations occur on the liquid surface, (b) the shapes of the resonance curves markedly change depending on the liquid depth, and (c) when the detuning condition is slightly deviated, almost periodic oscillations and chaotic oscillations appear at certain intervals of the excitation frequency. These were qualitatively in good agreement with the experimental results.

Nonlinear Vibrations in a Parametrically Excited Fluid-Structure Interaction System With One-to-One Internal Resonance

2003 ◽  
Author(s):  
Takashi Ikeda
Keyword(s):  
Natural Frequency ◽  
Internal Resonance ◽  
Fluid Structure Interaction ◽  
Liquid Sloshing ◽  
Fluid Structure ◽  
Inertia Effects ◽  
Structure Interaction ◽  
Interaction System ◽  
Resonance Curves ◽  
One To One

Theoretical resonance curves prove that a structure’s resonance can facilitate liquid sloshing even when the internal resonance ratio is one-to-one. An investigation of nonlinear sloshing liquid vibrations in a rectangular tank supported by an elastic structure that is subjected to a vertical and sinusoidal excitation reveals that liquid sloshing occurs when the structure’s natural frequency is approximately equal to the natural frequency of sloshing, that is, in the state of one-to-one internal resonance, and that amplitude-modulated motions appear when the condition of the internal resonance deviates to some extent. A special consideration of the nonlinear inertia effects of liquid force and the use of Galerkin’s method help derive the differential (modal) equations governing the dynamic behaviors of the fluid-structure interaction system, while van der Pol’s method helps express the theoretical resonance curves. These theoretical results are in quantitative agreement with the experimental data.

Random Excitation of a Structure Interacting With Liquid Sloshing Dynamics

2002 ◽  
Author(s):  
Takashi Ikeda ◽  
Raouf A. Ibrahim
Keyword(s):  
Free Surface ◽  
Natural Frequency ◽  
Internal Resonance ◽  
Gaussian White Noise ◽  
Liquid Sloshing ◽  
Elastic Structure ◽  
Center Frequency ◽  
System Response ◽  
Liquid Free Surface ◽  
Sloshing Dynamics

The nonlinear random interaction of an elastic structure with liquid sloshing dynamics in a cylindrical tank is investigated in the neighborhood of 1:2 internal resonance. Such internal resonance takes place when the natural frequency of the elastic structure is close to twice the natural frequency of the antisymmetric sloshing mode (1,1). The excitation is generated from the response of a linear shaping filter subjected to a Gaussian white noise. The analytical model involves three sloshing modes; (1,1), (0,1) and (2,1). The system response statistics and stability boundaries are numerically estimated using Monte Carlo simulation. The influence of the excitation center frequency, its bandwidth, and the liquid level on the system responses is studied. It is found that there is an irregular energy exchange between the structure and the liquid free surface motion when the center frequency is close to the structure natural frequency. Depending on the excitation power spectral density, the liquid free surface experiences zero motion, uncertain motion (intermittency), partially developed motion, and fully developed random motion. The structure response probability density function is almost Gaussian, while the liquid elevation deviates from normality. The unstable region, where the liquid motion occurs, becomes wider as the excitation intensity increases or as the bandwidth decreases. As the liquid depth decreases, the region of nonlinear interaction shrinks which is associated with a shift of the peak of the structure mean square response toward the left side of the frequency axis.

Nonlinear Parametric Vibrations of an Elastic Structure With a Cylindrical Liquid Container (in the Case of an Anti-Symmetric Sloshing Mode)

2001 ◽  
Author(s):  
Takashi Ikeda ◽  
Shin Murakami
Keyword(s):  
Natural Frequency ◽  
Harmonic Balance ◽  
Small Deviation ◽  
Harmonic Balance Method ◽  
Mathieu Equation ◽  
Elastic Structure ◽  
Coupled Vibration ◽  
Sloshing Mode ◽  
Resonance Curves ◽  
Fft Analysis

Abstract The nonlinear-coupled vibrations of an elastic structure and liquid sloshing in a cylindrical container are investigated. Since the structure is vertically subjected to a sinusoidal excitation, the behavior of the liquid surface is governed by a kind of the Mathieu equation. Modal equations governing the coupled motions are derived, when the natural frequency of the structure is equal to twice the natural frequency of an anti-symmetric mode of sloshing. The theoretical resonance curves are also presented by using an FFT analysis and the improved harmonic balance method. The influences of a liquid level and a detuning parameter on the theoretical resonance curves are shown. A small deviation of the tuning condition can cause amplitude-modulated motions and separate the occurrence region of the coupled vibration into two regions. In the experiments, the theoretical resonance curves were qualitatively in agreement with the experimental data. In addition, amplitude-modulated motions were observed.

scholarly journals Prediction of Isolated Resonance Curves Using Nonlinear Normal Modes

2015 ◽  
Author(s):  
R. J. Kuether ◽  
L. Renson ◽  
T. Detroux ◽  
C. Grappasonni ◽  
G. Kerschen ◽  
...  
Keyword(s):  
Normal Modes ◽  
Resonance Curve ◽  
Element Model ◽  
Nonlinear Normal Modes ◽  
Initial Guess ◽  
Forced Response ◽  
Numerical Continuation ◽  
Continuation Algorithm ◽  
Resonance Curves ◽  
Nonlinear Normal

Isolated resonance curves are separate from the main nonlinear forced-response branch, so they can easily be missed by a continuation algorithm and the resonant response might be underpredicted. The present work explores the connection between these isolated resonances and the nonlinear normal modes of the system and adapts an energy balance criterion to connect the two. This approach provides new insights into the occurrence of isolated resonances as well as a method to find an initial guess to compute the isolated resonance curve using numerical continuation. The concepts are illustrated on a finite element model of a cantilever beam with a nonlinear spring at its tip. This system presents jumps in both frequency and amplitude in its response to a swept sinusoidal excitation. The jumps are found to be the result of a modal interaction that creates an isolated resonance curve that eventually merges with the main resonance branch as the excitation force increases. Excellent insight into the observed dynamics is provided with the NNM theory, which supports that NNMs can also be a useful tool for predicting isolated resonance curves and other behaviors in the damped, forced response.

A New Hybrid SPH–FEM Model to Evaluate Seismic Response of TSD Equipped-Structures

2019 ◽  
Vol 13 (02) ◽  
pp. 1950007 ◽  
Author(s):  
Amir M. Halabian ◽  
Amin Karamnasab ◽  
Mohammad R. Chamani
Keyword(s):  
Seismic Response ◽  
Natural Frequency ◽  
Nonlinear Behavior ◽  
Numerical Data ◽  
Liquid Sloshing ◽  
Wind Loading ◽  
Weakly Compressible ◽  
Highly Nonlinear ◽  
Smoothed Particle ◽  
Higher Modes Effect

Tuned Sloshing Dampers (TSD) are passive devices, working based on shallow liquid sloshing in a rigid tank to suppress the horizontal structural vibrations induced by wind loading or earthquake excitations. The key parameters in design of a TSD could be referred to the natural frequency of the liquid sloshing motion and the inherent damping of the TSD during the excitation. Due to the highly nonlinear behavior of the liquid free-surface occurring in TSDs, accurate prediction of the TSD-structure’s behavior during strong excitations is highly desirable. In the current paper, Weakly Compressible form of Smoothed Particle Hydrodynamic (SPH) method is used to simulate the flow within rectangular TSDs during large movements. Characteristics of the flow such as wave height and sloshing forces acting on the container’s walls are calculated and compared with the existing experimental and numerical data. A hybrid SPH-Finite Element Method (FEM) was developed to investigate the seismic response of MDOF structures equipped with multiple TSDs. The proposed model was employed to evaluate the dynamic response of MDOF structures under severe seismic excitations with different frequency contents. The results showed that depending on the frequency content of the ground motion, having the TSDs tuned to a frequency close to the natural frequency of the structure could significantly alter the seismic response of the structures. The effectiveness of TSD is also related to the higher modes effect for MDOF structures and location of TSDs placed on the structural floors.

On Investigation of Liquid Sloshing in Cylindrical Tanks With Single and Multiply Connected Domains Using Isogeometric Boundary Element Method

2020 ◽  
Vol 143 (2) ◽  
Author(s):  
Quansheng Zang ◽  
Jun Liu ◽  
Yang Zhou ◽  
Gao Lin
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Basis Functions ◽  
Liquid Sloshing ◽  
Multiply Connected Domains ◽  
Cylindrical Tank ◽  
Multiply Connected ◽  
Computer Aided ◽  
Cylindrical Tanks ◽  
Lagrange Basis

Abstract This paper explores an isogeometric boundary element method (IGA-BEM) for sloshing problems in cylindrical tanks with single and multiply connected domains. Instead of the Lagrange basis functions used in the standard BEM, the nonuniform rational B-splines (NURBS) basis functions are introduced to approximate the geometries of the problem boundaries and the unknown variables. Compared with the Lagrange basis functions, NURBS basis functions can accurately reconstruct the geometric boundary of analysis domain with almost no error, and all the data information for NURBS basis functions can be directly obtained from the computer-aided design (cad) or computer-aided engineering (cae) commercial software, which implies the modeling process of IGA-BEM is more simple than that of the standard BEM. NURBS makes it possible for the IGA-BEM to realize the seamless connection between cad and cae software with relative higher calculation accuracy than the standard BEM. Based on the weighted residual method as well as the divergence theorem, the IGA-BEM is developed for the single and multiply connected domains, whose boundaries are separately defined in the parameter space by different knot vectors. The natural sloshing frequencies of the liquid sloshing in a circular cylindrical tank with a coaxial or an off-center circular pipe, an elliptical cylindrical tank with an elliptical pipe, a circular cylindrical tank with multiple pipes are estimated with the introduced method by assuming an ideal (inviscid and incompressible) liquid, irrotational small-amplitude sloshing, and the linear free-surface condition. The comparison between the results obtained by the proposed method and those in the existing literatures shows very good agreements, which verifies the proposed model well. Meanwhile, the effects of radius ratio, liquid depth, number, and location of internal pipe (pipes) on the natural sloshing frequency and sloshing mode are analyzed carefully, and some conclusions are outlined finally.

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