scholarly journals Finite Element Simulation of Dynamic Stability of Plane Free-Surface of a Liquid under Vertical Excitation

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Siva Srinivas Kolukula ◽  
P. Chellapandi
Keyword(s):  
Finite Element ◽  
Free Surface ◽  
Parametric Resonance ◽  
Linear Equations ◽  
Parametric Instability ◽  
Initial Perturbation ◽  
Mathieu Equation ◽  
External Excitation ◽  
Nonlinear Potential Theory ◽  
Vertical Excitation

When partially filled liquid containers are excited vertically, the plane free-surface of the liquid can be stable or unstable depending on the amplitude and frequency of the external excitation. For some combinations of amplitude and frequency, the free-surface undergoes unbounded motion leading to instability called parametric instability or parametric resonance, and, for few other combinations, the free-surface undergoes bounded stable motion. In parametric resonance, a small initial perturbation on the free-surface can build up unboundedly even for small external excitation, if the excitation acts on the tank for sufficiently long time. In this paper, the stability of the plane free-surface is investigated by numerical simulation. Stability chart for the governing Mathieu equation is plotted analytically using linear equations. Applying fully nonlinear finite element method based on nonlinear potential theory, the response of the plane free-surface is simulated for various cases.

Parametric Instability of Tapered Beams by Finite Element Method

1982 ◽  
Vol 24 (4) ◽  
pp. 205-208 ◽  
Author(s):  
P. K. Datta ◽  
S. Chakraborty
Keyword(s):  
Finite Element ◽  
Dynamic Load ◽  
Parametric Instability ◽  
Mathieu Equation ◽  
Stability Diagram ◽  
Load Factor ◽  
Element Analysis ◽  
Load Factors ◽  
Principal Region ◽  
Dynamic Load Factor

The dynamic stability behaviour of a tapered beam has been studied using a finite element analysis. The instability zones of the parametric stability diagram have been discussed for the entire ranges of static and dynamic load factors. It has been observed that at high values of static load and beyond a particular value of the dynamic load factor, the periodic solution of the Mathieu equation does not exist in the principal region. This leads to unstable behaviour due to large displacement of the beam due to increasing values of static and dynamic load factors.

scholarly journals Parametric Instability in Mathieu Equation in Earthquake Dynamics

2017 ◽  
pp. 73-75
Author(s):  
H. Torres-Silva ◽  
J. López-Bonilla ◽  
A. Iturri-Hinojosa ◽  
D. Torres Cabezas
Keyword(s):  
Parametric Resonance ◽  
Parametric Instability ◽  
Mathieu Equation ◽  
Nonlinear Phenomena ◽  
Bifurcation Diagrams ◽  
Earthquake Dynamics ◽  
P Waves ◽  
S Waves ◽  
Theoretical Predictions ◽  
Good Agreement

We propose an study of parametric resonance between P-waves and S-waves, which can be used to describe various nonlinear phenomena qualitatively and to obtain bifurcation diagrams quantitatively. We have shown that it is a good simulation of parametric phenomena, and our results are in good agreement with theoretical predictions. In particular, it may be used to study the influence of pump P waves on the instability’s threshold and amplitude of S waves in earthquake phenomena that could be simulated with an electronic model.The Himalayan Physics Vol. 6 & 7, April 2017 (73-75)

scholarly journals Numerical Simulation of Sloshing in 2D Rectangular Tanks Based on the Prediction of Free Surface

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Haitao Zhang ◽  
Beibei Sun
Keyword(s):  
Numerical Simulation ◽  
Free Surface ◽  
Iterative Process ◽  
Numerical Solutions ◽  
Linear Equations ◽  
Standing Waves ◽  
External Excitation ◽  
Kinematic Condition ◽  
Aspect Ratios ◽  
The Relationship

A finite difference method for analyzing 2D nonlinear sloshing waves in a tank has been developed based on the potential flow theory. Afterσ-transformation, the free surface is predicted by the kinematic condition, and nonlinear terms are approximated; the governing equation and boundary conditions are discretized to linear equations in the iterative process of time. Simulations of standing waves and sloshing in horizontally excited tanks are presented. The results are compared with analytical and numerical solutions in other literatures, which demonstrate the effectiveness and accuracy of this numerical method. The beating phenomenon of sloshing in the tank with different aspect ratios is studied. The relationship between sloshing force and aspect ratio under the same external excitation is also discussed.

scholarly journals The Force Cone Method Applied to Explain Hidden Whirls in Tribology

Materials ◽  
2021 ◽  
Vol 14 (14) ◽  
pp. 3894
Author(s):  
Claus Mattheck ◽  
Christian Greiner ◽  
Klaus Bethge ◽  
Iwiza Tesari ◽  
Karlheinz Weber
Keyword(s):  
Finite Element ◽  
Free Surface ◽  
Simple Model ◽  
Alternative Explanation ◽  
Model Experiments ◽  
Complex Processes ◽  
Buried Interfaces ◽  
Internal Structures ◽  
Ideal Tool ◽  
Acting Force

In tribologically loaded materials, folding instabilities and vortices lead to the formation of complex internal structures. This is true for geological as well as nanoscopic contacts. Classically, these structures have been described by Kelvin–Helmholtz instabilities or shear localization. We here introduce an alternative explanation based on an intuitive approach referred to as the force cone method. It is considered how whirls are situated near forces acting on a free surface of an elastic or elastoplastic solid. The force cone results are supplemented by finite element simulations. Depending on the direction of the acting force, one or two whirls are predicted by the simplified force cone method. In 3D, there is always a ring shaped whirl present. These modelling findings were tested in simple model experiments. The results qualitatively match the predictions and whirl formation was found. The force cone method and the experiments may seem trivial, but they are an ideal tool to intuitively understand the presence of whirls within a solid under a tribological load. The position of these whirls was found at the predicted places and the force cone method allows a direct approach to understand the complex processes in the otherwise buried interfaces of tribologically loaded materials.

Eulerian FEA With Volume of Solid (VOS) Application in Metal Forming

2010 ◽  
Author(s):  
K. S. Al-Athel ◽  
M. S. Gadala
Keyword(s):  
Finite Element ◽  
Free Surface ◽  
Process Improvement ◽  
Metal Forming ◽  
Material Flow ◽  
Solid Mechanics ◽  
Uniform Mesh ◽  
Element Formulation ◽  
Volume Of Fluid Method ◽  
Whole Process

The adaptation of the volume of fluid method (VOF) to solid mechanics (VOS) is presented in this work with the focus on metal forming applications. The method is discussed for a general non-uniform mesh with Eulerian finite element formulation. The implementation of the VOS method in metal forming applications is presented by focusing on topics such as the contact between the tool and the workpiece, tracking of the free surface of the material flow and the connectivity of the free surface during the whole process. Improvement on the connectivity of the free surface and the representation of curves is achieved by considering the mechanics of different metal forming processes. Different applications are simulated and discussed to highlight the capability of the VOS method.

Nonlinear Vibration of Sheet Metal Plates Under Interacting Parametric and External Excitation During Manufacturing

2005 ◽  
Vol 127 (1) ◽  
pp. 36-43 ◽  
Author(s):  
Chung Hwan Kim ◽  
Chong-Won Lee ◽  
N. C. Perkins
Keyword(s):  
Sheet Metal ◽  
Parametric Resonance ◽  
Parametric Excitation ◽  
Synergistic Effects ◽  
Broad Band ◽  
Metal Plate ◽  
Time Dependent ◽  
Single Frequency ◽  
Cubic Nonlinearity ◽  
External Excitation

This study is motivated by the vibrations that plague coating processes used in the manufacturing of coated sheet metal. These vibrations arise from time-dependent tension fluctuations within the sheet metal plate as well as from the eccentricity of the rollers used to transport the plate. The time-dependent tension is observed to be rather broad-band and creates multi-frequency parametric excitation. By contrast, the roller eccentricity is largely single-frequency (synchronized with the roller speed) and creates single-frequency external excitation. The plate and excitation sources are studied herein using a single-degree-of-freedom model with a cubic nonlinearity, subject to combined parametric and external excitation. In our study, we investigate the resonances that arise from the synergistic effects of multi-frequency parametric excitation and single-frequency external excitation. For the simpler case of single-frequency parametric excitation, we observe both sum and difference combination resonances in addition to principal parametric resonance. For the case of multi-frequency parametric excitation, we observe a frequency shift for the parametric resonance that derives from the cubic nonlinearity and external excitation. Moreover, the phase relationships of the external and each parametric excitation source have a significant effect on the resulting response amplitude. We use these analyses to explain the resonance mechanisms observed in experiments conducted on an example sheet metal coating process.

Influence of Axial Excitations and of Boundary Conditions on the Parametric Instability of a Beam

1995 ◽  
Author(s):  
Régis Dufour ◽  
Alain Berlioz ◽  
Thomas Streule
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Parametric Instability ◽  
Experimental Tests ◽  
Time Varying ◽  
Periodic Force ◽  
Modal Reduction ◽  
Time Varying Parameter ◽  
The Stability

Abstract In this paper the stability of the lateral dynamic behavior of a pinned-pinned, clamped-pinned and clamped-clamped beam under axial periodic force or torque is studied. The time-varying parameter equations are derived using the Rayleigh-Ritz method. The stability analysis of the solution is based on Floquet’s theory and investigated in detail. The Rayleigh-Ritz results are compared to those of a finite element modal reduction. It shows that the lateral instabilities of the beam depend on the forcing frequency, the type of excitation and the boundary conditions. Several experimental tests enable the validation of the numerical results.

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