scholarly journals On the experimental identification of unstable static equilibria

2016 ◽  
Vol 472 (2190) ◽  
pp. 20160172 ◽  
Author(s):  
R. Wiebe ◽  
L. N. Virgin
Keyword(s):  
Energy Surface ◽  
Stable Equilibrium ◽  
Continuous System ◽  
Experimental System ◽  
Initial Energy ◽  
Unstable Equilibrium ◽  
Shallow Arch ◽  
Single Degree Of Freedom ◽  
Equilibrium Configurations ◽  
Unstable Configuration

This paper shows how the presence of unstable equilibrium configurations of elastic continua is reflected in the behaviour of transients induced by large perturbations. A beam that is axially loaded beyond its critical state typically exhibits two buckled stable equilibrium configurations, separated by one or more unstable equilibria. If the beam is then loaded laterally (effectively like a shallow arch) it may snap-through between these states, including the case in which the loading is applied dynamically and of short duration, i.e. an impact. Such impacts, if applied at random locations and of random strength, will generate an ensemble of transient trajectories that explore the phase space. Given sufficient variety, some of these trajectories will possess initial energy that is close to (just less than or just greater than) the energy required to cause snap-through and will have a tendency to slowdown as they pass close to an unstable configuration: a saddle point in a potential energy surface, for example. Although this close-encounter is relatively straightforward in a system characterized by a single degree of freedom, it is more challenging to identify in a higher order or continuous system, especially in a (necessarily) noisy experimental system. This paper will show how the identification of unstable equilibrium configurations can be achieved using transient dynamics.

Dynamics of an electrostatically charged elastic rod in fluid

2010 ◽  
Vol 467 (2126) ◽  
pp. 569-590 ◽  
Author(s):  
Sookkyung Lim ◽  
Yongsam Kim ◽  
David Swigon
Keyword(s):  
Immersed Boundary Method ◽  
Stable Equilibrium ◽  
Equations Of Motion ◽  
Initial Perturbation ◽  
Hard Core ◽  
Unstable Equilibrium ◽  
Immersed Boundary ◽  
Governing Equations ◽  
Elastic Rod ◽  
Equilibrium Configurations

We investigate the effects of electrostatic and steric repulsion on the dynamics of a pre-twisted charged elastic rod immersed in a viscous incompressible fluid. Equations of motion of the rod include the fluid–structure interaction, rod elasticity and a combination of two interactions that prevent self-contact, namely the electrostatic interaction and hard-core repulsion. The governing equations are solved using the generalized immersed-boundary method. We find that after perturbation, a pre-twisted minicircle collapses into a compact supercoiled configuration. The collapse proceeds along a complex trajectory that may pass near several unstable equilibrium configurations, before it settles in a locally stable equilibrium. The dwell time near an unstable equilibrium can be up to several microseconds. Both the final configuration and the transition path are sensitive to the initial excess link, ionic strength of the solvent and the initial perturbation.

STABILITY OF SPATIAL ELASTICA IN A GRAVITATIONAL FIELD

2012 ◽  
Vol 12 (02) ◽  
pp. 403-421 ◽  
Author(s):  
BOONCHAI PHUNGPAINGAM ◽  
LAWRENCE N. VIRGIN ◽  
SOMCHAI CHUCHEEPSAKUL
Keyword(s):  
Gravitational Field ◽  
Elastic Equilibrium ◽  
Experimental System ◽  
Primary Control ◽  
Arc Length ◽  
Euler Parameters ◽  
Reference Analysis ◽  
Horizontal Orientation ◽  
Equilibrium Configurations ◽  
Out Of Plane

This paper considers the behavior of a spatial elastica in a gravitational field. The slenderness of the system considered is such that the weight becomes an important consideration in determining elastic equilibrium configurations. Both ends of the elastica are clamped in an initially (planar) horizontal orientation at a fixed distance apart. However, one of the ends allows an increase in arc-length, that is, it is a sleeve joint. Thus, the total arc-length is the primary control parameter. This kind of elastica typically loses stability, resulting in out-of-plane deflections, when the total arc-length is increased beyond a critical value. A small mid-length torque can used to perturb a planar equilibrium configuration in order to test for stability. The aim of this study is to assess the effect of self-weight of the elastica (which is typically ignored) on promoting or delaying the loss of stability. To this end, it is useful to compare and contrast the results of orientation, that is, the system is configured in both an initial "upright" orientation and then in an "upside-down" orientation to highlight the influence of gravity. The results of the weightless elastica are used as a reference. Analysis is based on Kirchhoff's rod theory and Euler parameters, and the resulting set of governing differential equations are solved using a shooting method. The results from an experimental system using a slender superelastic wire made from Nitinol (Nickel Titanium Naval Ordnance Laboratory) exhibit close agreement with the analytical results.

scholarly journals Curvature Dependent Electrostatic Field in the Deformable MEMS Device: Stability and Optimal Control

2020 ◽  
Vol 11 (1) ◽  
pp. 35-54
Author(s):  
Paolo Di Barba ◽  
Luisa Fattorusso ◽  
Mario Versaci
Keyword(s):  
Optimal Control ◽  
Electrostatic Field ◽  
Stable Equilibrium ◽  
Micro Electro Mechanical System ◽  
Operating Conditions ◽  
Electric Voltage ◽  
External Voltage ◽  
Device Stability ◽  
Equilibrium Configurations ◽  
Mems Device

AbstractThe recovery of the membrane profile of an electrostatic micro-electro-mechanical system (MEMS) device is an important issue because, when applying an external voltage, the membrane deforms with the consequent risk of touching the upper plate of the device (a condition that should be avoided). Then, during the deformation of the membrane, it is useful to know if this movement admits stable equilibrium configurations. In such a context, our present work analyze the behavior of an electrostatic 1D membrane MEMS device when an external electric voltage is applied. In particular, starting from a well-known second-order elliptical semi-linear di erential model, obtained considering the electrostatic field inside the device proportional to the curvature of the membrane, the only possible equilibrium position is obtained, and its stability is analyzed. Moreover, considering that the membrane has an inertia in moving and taking into account that it must not touch the upper plate of the device, the range of possible values of the applied external voltage is obtained, which accounted for these two particular operating conditions. Finally, some calculations about the variation of potential energy have identified optimal control conditions.

Pressure-Impulse Diagrams for Blast Loaded Continuous Beams Based on Dimensional Analysis

2013 ◽  
Vol 80 (5) ◽  
Author(s):  
A. S. Fallah ◽  
E. Nwankwo ◽  
L. A. Louca
Keyword(s):  
Structural Model ◽  
Continuous System ◽  
Degree Of Freedom ◽  
Pulse Loading ◽  
Single Degree Of Freedom ◽  
Pressure Impulse ◽  
Elastic Plastic ◽  
Single Degree ◽  
Continuous Beams ◽  
Shape Effects

Pressure-impulse diagrams are commonly used in preliminary blast resistant design to assess the maxima of damage related parameter(s) in different types of structures as a function of pulse loading parameters. It is well established that plastic dynamic response of elastic-plastic structures is profoundly influenced by the temporal shape of applied pulse loading (Youngdahl, 1970, “Correlation Parameters for Eliminating the Effect of Pulse Shape on Dynamic Plastic Deformation,” ASME, J. Appl. Mech., 37, pp. 744–752; Jones, Structural Impact (Cambridge University Press, Cambridge, England, 1989); Li, and Meng, 2002, “Pulse Loading Shape Effects on Pressure–Impulse Diagram of an Elastic–Plastic, Single-Degree-of-Freedom Structural Model,” Int. J. Mech. Sci., 44(9), pp. 1985–1998). This paper studies pulse loading shape effects on the dynamic response of continuous beams. The beam is modeled as a single span with symmetrical semirigid support conditions. The rotational spring can assume different stiffness values ranging from 0 (simply supported) to ∞ (fully clamped). An analytical solution for evaluating displacement time histories of the semirigidly supported (continuous) beam subjected to pulse loads, which can be extendable to very high frequency pulses, is presented in this paper. With the maximum structural deflection, being generally the controlling criterion for damage, pressure-impulse diagrams for the continuous system are developed. This work presents a straightforward preliminary assessment tool for structures such as blast walls utilized on offshore platforms. For this type of structures with semirigid supports, simplifying the whole system as a single-degree-of-freedom (SDOF) discrete-parameter model and applying the procedure presented by Li and Meng (Li and Meng, 2002, “Pulse Loading Shape Effects on Pressure–Impulse Diagram of an Elastic–Plastic, Single-Degree-of-Freedom Structural Model,” Int. J. Mech. Sci., 44(9), pp. 1985–1998; Li and Meng, 2002, “Pressure-Impulse Diagram for Blast Loads Based on Dimensional Analysis and Single-Degree-of-Freedom Model,” J. Eng. Mech., 128(1), pp. 87–92) to eliminate pulse loading shape effects on pressure-impulse diagrams would be very conservative and cumbersome considering the support conditions. It is well known that an SDOF model is a very conservative simplification of a continuous system. Dimensionless parameters are introduced to develop a unique pulse-shape-independent pressure-impulse diagram for elastic and elastic-plastic responses of continuous beams.

scholarly journals A non-linear mathematical model for the X-ray variability classes of the microquasar GRS 1915+105 – II. Transition and swaying classes

2020 ◽  
Vol 496 (2) ◽  
pp. 1697-1705 ◽  
Author(s):  
E Massaro ◽  
F Capitanio ◽  
M Feroci ◽  
T Mineo ◽  
A Ardito ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Input Function ◽  
Unstable Equilibrium ◽  
Light Curves ◽  
Extended Model ◽  
X Ray ◽  
Non Linear ◽  
Non Linear System

ABSTRACT The complex time evolution in the X-ray light curves of the peculiar black hole binary GRS 1915+105 can be obtained as solutions of a non-linear system of ordinary differential equations derived from the Hindmarsh–Rose model and modified introducing an input function depending on time. In the first paper, assuming a constant input with a superposed white noise, we reproduced light curves of the classes ρ, χ, and δ. We use this mathematical model to reproduce light curves, including some interesting details, of other eight GRS 1915+105 variability classes either considering a variable input function or with small changes of the equation parameters. On the basis of this extended model and its equilibrium states, we can arrange most of the classes in three main types: (i) stable equilibrium patterns (classes ϕ, χ, α″, θ, ξ, and ω) whose light curve modulation follows the same time-scale of the input function, because changes occur around stable equilibrium points; (ii) unstable equilibrium patterns characterized by series of spikes (class ρ) originated by a limit cycle around an unstable equilibrium point; and (iii) transition pattern (classes δ, γ, λ, κ, and α′), in which random changes of the input function induce transitions from stable to unstable regions originating either slow changes or spiking, and the occurrence of dips and red noise. We present a possible physical interpretation of the model based on the similarity between an equilibrium curve and literature results obtained by numerical integrations of slim disc equations.

THE STABILITY OF SOLUTE CRYSTALLIZATION EQUILIBRIUM

2005 ◽  
Vol 12 (04) ◽  
pp. 623-630
Author(s):  
SHIMIN ZHANG
Keyword(s):  
Stable Equilibrium ◽  
Unstable Equilibrium ◽  
Energy Peak ◽  
The Stability ◽  
Solute Crystallization

Solute crystallization equilibrium contains a stable equilibrium and an unstable equilibrium. The system must get across an energy peak of unstable equilibrium first, and then get into an energy valley of stable equilibrium during the solute crystallization. When solution is diluted, the energy peak becomes high, and the energy valley becomes shallow, the solute crystallization becomes difficult and the dissolution of the crystallized product becomes easy; when solution is diluted to a certain extent, the energy peak and the energy valley combine into one, and crystallization of the solute becomes impossible.

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