Equilibrium, Stability, and Dynamics of Rectangular Liquid-Filled Vessels

2011 ◽  
Vol 6 (4) ◽  
Author(s):  
Russell Trahan ◽  
Tamás Kalmár-Nagy
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Mathieu Equation ◽  
Liquid Volume ◽  
Position Vector ◽  
Equilibrium Stability ◽  
Bifurcation Diagrams ◽  
Equilibrium Configurations ◽  
Theoretical Predictions ◽  
The Stability

Here we focus on the stability and dynamic characteristics of a rectangular, liquid-filled vessel. The position vector of the center of gravity of the liquid volume is derived and used to express the equilibrium angles of the vessel. Analysis of the potential function determines the stability of these equilibria, and bifurcation diagrams are constructed to demonstrate the co-existence of several equilibrium configurations of the vessel. To validate the results, a vessel of rectangular cross section was built. The results of the experiments agree well with the theoretical predictions of stability. The dynamics of the unforced and forced systems with a threshold constraint is discussed in the context of the nonlinear Mathieu equation.

Rectangular Cross Section Beams Calculation on the Stability of a Flat Bending Shape Taking into Account the Initial Imperfections

2019 ◽  
Vol 974 ◽  
pp. 551-555 ◽  
Author(s):  
I.M. Zotov ◽  
Anastasia P. Lapina ◽  
Anton S. Chepurnenko ◽  
B.M. Yazyev
Keyword(s):  
Cross Section ◽  
Basic Equation ◽  
Applied Load ◽  
Rectangular Cross Section ◽  
Twist Angle ◽  
Stability Loss ◽  
Lateral Buckling ◽  
Initial Imperfections ◽  
Beam Stability ◽  
The Stability

The article presents the derivation of the resolving equation for the calculation of lateral buckling of rectangular beams. When deriving the basic equation, the initial imperfections of the beam are taken into account, which are specified in the form of the eccentricity of the applied load, the initial deflection in the plane of least stiffness and the initial twist angle. The influence of initial imperfections on the process of beam stability loss is investigated.

scholarly journals Calculation of wooden beams on the stability of a flat bending shape enhancement

2018 ◽  
Vol 196 ◽  
pp. 01003 ◽  
Author(s):  
Anton Chepurnenko ◽  
Vera Ulianskaya ◽  
Serdar Yazyev ◽  
Ivan Zotov
Keyword(s):  
Differential Equation ◽  
Cross Section ◽  
Stability Problem ◽  
Secular Equation ◽  
Rectangular Cross Section ◽  
Center Of Gravity ◽  
Critical Force ◽  
Distributed Load ◽  
Matlab Package ◽  
The Stability

Flat bending stability problem of constant rectangular cross section wooden beam, loaded by a distributed load is considered. Differential equation is provided for the cases when load is located not in the center of gravity. The solution of the equation is performed numerically by the method of finite differences. For the case of applying a load at the center of gravity, the problem reduces to a generalized secular equation. In other cases, the iterative algorithm developed by the authors is implemented, in the Matlab package. A relationship between the value of the critical force and the position of the load application point is obtained. A linear approximating function is selected for this dependence.

Recommendations for Designing Wooden Arches on Metal-toothed Plates

2021 ◽  
Author(s):  
Bohdan Demchyna ◽  
◽  
Yaroslav Shydlovskyi
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Laboratory Testing ◽  
Horizontal Plane ◽  
Thrust Force ◽  
Rectangular Cross Section ◽  
Rectangular Section ◽  
Pilot Studies ◽  
Different Types ◽  
The Stability

This paper presents the findings of the pilot studies and recommendations for designing of two-hinged wooden arches. The prototype models of wooden arches with the span of 6mand the rise of 1m were designed. The models had a rectangular cross-section of 180x40mm and a T-section of 180x40mm with a plywood plate with the thickness of 6 mm and the width of 500mm. The main objective of the T-section was to ensure the stability of the arch. Each arch was composed of six segments –boards joined by clamping plates. The bowstring truss including two inclined tie bars enables carrying asymmetric loads and provides in-plane stability of the arch. A methodology for laboratory testing of the prototype models of wooden arches subjected to different types of loads was developed. Two prototypes of wooden arches were tested with rectangular cross-sections and two T-section ones subjected to the loading across the span, and two prototypes subjected to the half-span loading. In total, eight arches were tested. Deflections of arches, cross-section deformations and arch thrust force were recorded. The arches were tested until failure. The results of testing revealed insufficient stability of the arches with rectangular cross-section in the horizontal plane. For the arches with T-section the whole arch rib was damaged, the in-plane stability was ensured by the T-section. The collapsing force of the T-section arch was about 1.3 times greater than the collapsing force of the rectangular section arches.

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 CRITERIA OF MINIMUM MATERIALS CONSUMPTION FOR BARS WITH RECTANGULAR CROSS-SECTION AND RESTRICTIONS ON STABILITY OR LIMITATIONS ON THE VALUE OF THE FIRST NATURAL FREQUENCY

2017 ◽  
Vol 13 (1) ◽  
pp. 9-22 ◽  
Author(s):  
Leonid S. Lyakhovich ◽  
Pavel A. Akimov ◽  
Boris A. Tukhfatullin
Keyword(s):  
Mathematical Programming ◽  
Natural Frequency ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Optimal Solutions ◽  
Material Consumption ◽  
Variable Parameters ◽  
Special Cases ◽  
The Stability ◽  
Minimal Consumption

Apparatus of mathematical programming is normally used in the most part of research works, dealing with structural optimization. However, the special properties of optimal systems have been identified in several studies. Besides, corresponding criteria, which have been formulated as well, can be used for assessments of proximity of optimal solutions to minimal material consumption. Particularly relevant criteria for bars with rectangular cross-section and restrictions on the stability or limitations on the value of the first natural frequency have been formulated. However, not all the features of some of the criteria have been observed. In addition it seems appropriate to identify relevant criteria for special cases set variable parameters. The distinctive paper contains additional property proximity criterion of optimal solutions to minimal consumption of materials for the bars with a rectangular cross-section and limitations on the value of the first natural frequency, modification of one of the previously proposed criteria and formulation of appropriate criterion for the case where one of the parameters of variable rectangular cross-section is constant along the length of the bar.

scholarly journals Almost sure stability of the thin-walled beam subjected to end moments

2003 ◽  
pp. 193-208 ◽  
Author(s):  
Ratko Pavlovic ◽  
Predrag Kozic
Keyword(s):  
Cross Section ◽  
Rectangular Cross Section ◽  
Damping Coefficient ◽  
Physical Parameters ◽  
Almost Sure Stability ◽  
Liapunov Method ◽  
Process Variance ◽  
Thin Walled ◽  
The Stability ◽  
Asymptotic Stability Condition

The dynamic stability problem of the thin-walled beams subjected to end moments is studied. Each moment consists of constant part and time-dependent stochastic non-white function. Closed form analytical solutions are obtained for simply supported boundary conditions. By using the direct Liapunov method almost sure asymptotic stability condition is obtained as function of stochastic process variance, damping coefficient, geometric and physical parameters of the beam. The stability regions for I-cross section and narrow rectangular cross section are shown in variance - damping coefficient plane when stochastic part of moment is Gaussian zero-mean process with variance ?2 and harmonic process with amplitude A.

Pressure Surge Propagation in Thick-Walled Conduits of Rectangular Cross Section

1976 ◽  
Vol 98 (3) ◽  
pp. 455-459 ◽  
Author(s):  
A. R. D. Thorley ◽  
C. Guymer
Keyword(s):  
Experimental Data ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Side Wall ◽  
Shear Forces ◽  
Theoretical Predictions ◽  
Pressure Surge ◽  
Surge Phenomena

This paper discusses the propagation of pressure surge phenomena in liquids contained in thick-walled pipes and ducts of rectangular cross section. It is shown that the effects of shear forces and side wall elongation should be considered in calculating the velocity of surge propagation in such conduits. Experimental data from three pipelines are included and compared with the theoretical predictions.

Parametric Torsional Stability of a Bar Under Axial Excitation

1968 ◽  
Vol 35 (1) ◽  
pp. 13-19 ◽  
Author(s):  
W. K. Tso
Keyword(s):  
Steady State ◽  
Rectangular Cross Section ◽  
Mathieu Equation ◽  
Axial Loading ◽  
Response Curves ◽  
Vibrational Response ◽  
State Response ◽  
The Stability ◽  
Torsional Stability ◽  
Boundary Of Stability

A study is made on the parametric torsional stability of an elastic cantilever of rectangular cross section under dynamic axial loading. The coupling between the longitudinal and torsional motions exists due to the “shortening effect.” The problem is so formulated that the stability of torsional vibrations is represented by a Mathieu equation, the stability of which is well known. The effect of longitudinal vibrations on the torsional stability is investigated. The steady-state torsional-vibrational response curves are given analytically, and the effect of longitudinal damping on the boundary of stability and the steady-state response curves is also determined.

scholarly journals Stability of a viscous liquid contained between two rotating cylinders

1923 ◽  
Vol 102 (718) ◽  
pp. 541-542 ◽  
Keyword(s):  
Viscous Fluid ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Outer Cylinder ◽  
Steady Motion ◽  
Annular Space ◽  
Rotating Cylinders ◽  
High Speeds ◽  
Approximate Form ◽  
The Stability

Part I .—The stability for symmetrical disturbances of a viscous fluid in steady motion between concentric rotating cylinders is investigated mathematically. It is shown that at slow speeds the motion is always stable, but that at high speeds the motion is only stable when the ratio of the speed of the outer cylinder to that of the inner one exceeds a certain value. When the ratio is less than this or when it is negative the motion becomes unstable at high speeds. The “criterion” for stability is found, and in cases suitable for experimental verification an approximate form for the “criterion” is developed which is useful for numerical computation. The type of instability which may be expected to appear when the speed of the cylinders is slowly increased is shown to consist of symmetrical ring-shaped vortices spaced at regular intervals along the length of the cylinders. These vortices rotate alternately in opposite directions. Their dimensions are calculated and it is shown that they are contained in partitions of rectangular cross-section. In the case when the instability arises while both cylinders are rotating in the same direction, these rectangles are squares, so that the vortices are spaced at distances apart equal to the thickness of the annular space between the two cylinders. In the case when the cylinders rotate in opposite directions the spacing, or distance between the centres of neighbouring vortices, is smaller than this; and at the same time two systems of vortices develop—an inner system which is similar to the system which appears when the two cylinders rotate in the same direction, and an outer system, which is much less vigorous and rotates in the opposite direction to the adjacent members of the inner system.

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