An energy perturbation approach to limit cycle analysis in legged locomotion systems

1990 ◽  
Author(s):  
Z. Li ◽  
J. He
Keyword(s):  
Limit Cycle ◽  
Legged Locomotion ◽  
Perturbation Approach ◽  
Energy Perturbation

A Free Energy Perturbation Approach to Estimate the Intrinsic Solubilities of Drug-like Small Molecules

2019 ◽  
Author(s):  
Sayan Mondal ◽  
Gary Tresadern ◽  
Jeremy Greenwood ◽  
Byungchan Kim ◽  
Joe Kaus ◽  
...  
Keyword(s):  
Solid State ◽  
Small Molecules ◽  
Three Dimensional ◽  
Aqueous Solubility ◽  
Computational Approach ◽  
Free Energy Perturbation ◽  
Perturbation Approach ◽  
Energy Perturbation ◽  
State Form ◽  
Good Agreement

<p>Optimizing the solubility of small molecules is important in a wide variety of contexts, including in drug discovery where the optimization of aqueous solubility is often crucial to achieve oral bioavailability. In such a context, solubility optimization cannot be successfully pursued by indiscriminate increases in polarity, which would likely reduce permeability and potency. Moreover, increasing polarity may not even improve solubility itself in many cases, if it stabilizes the solid-state form. Here we present a novel physics-based approach to predict the solubility of small molecules, that takes into account three-dimensional solid-state characteristics in addition to polarity. The calculated solubilities are in good agreement with experimental solubilities taken both from the literature as well as from several active pharmaceutical discovery projects. This computational approach enables strategies to optimize solubility by disrupting the three-dimensional solid-state packing of novel chemical matter, illustrated here for an active medicinal chemistry campaign.</p>

Analysis of the non-periodic oscillations of a self-excited friction-damped system with closely-spaced modes

2021 ◽  
Author(s):  
Lukas Woiwode ◽  
Alexander F. Vakakis ◽  
Malte Krack
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Dry Friction ◽  
The Self ◽  
Perturbation Approach ◽  
Friction Damping ◽  
Periodic Oscillations ◽  
Analytical Results ◽  
Fold Bifurcation ◽  
Phase Motion

Abstract It is widely known that dry friction damping can bound the self-excited vibrations induced by negative damping. The vibrations typically take the form of (periodic) limit cycle oscillations. However, when the intensity of the self-excitation reaches a condition of maximum friction damping, the limit cycle loses stability via a fold bifurcation. The behavior may become even more complicated in the presence of any internal resonance conditions. In this work, we consider a two-degree-of-freedom system with an elastic dry friction element (Jenkins element) having closely spaced natural frequencies. The symmetric in-phase motion is subjected to self-excitation by negative (viscous) damping, while the symmetric out-of-phase motion is positively damped. In a previous work, we showed that the limit cycle loses stability via a secondary Hopf bifurcation, giving rise to quasi-periodic oscillations. A further increase of the self-excitation intensity may lead to chaos and finally divergence, long before reaching the fold bifurcation point of the limit cycle. In this work, we use the method of Complexification-Averaging to obtain the slow flow in the neighborhood of the limit cycle. This way, we show that chaos is reached via a cascade of period doubling bifurcations on invariant tori. Using perturbation calculus, we establish analytical conditions for the emergence of the secondary Hopf bifurcation and approximate analytically its location. In particular, we show that non-periodic oscillations are the typical case for prominent nonlinearity, mild coupling (controlling the proximity of the modes) and sufficiently light damping. The range of validity of the analytical results presented herein is thoroughly assessed numerically. To the authors' knowledge, this is the first work that shows how the challenging Jenkins element can be treated formally within a consistent perturbation approach in order to derive closed-form analytical results for limit cycles and their bifurcations.

Limit Cycles of Spinning Thermoacoustic Modes in Annular Combustors: A Bloch-Wave and Adjoint-Perturbation Approach

2017 ◽  
Author(s):  
Georg A. Mensah ◽  
Jonas P. Moeck
Keyword(s):  
Helmholtz Equation ◽  
Limit Cycle ◽  
Rotational Symmetry ◽  
Wave Theory ◽  
Bloch Wave ◽  
Generic Model ◽  
Perturbation Approach ◽  
Dependent Manner ◽  
Combustion System ◽  
Linear Character

The most straightforward way to assess the thermoacoustic stability of a combustion system is based on modal approaches. The modes are typically computed from linearized equations in the frequency domain, such as the Helmholtz equation. Due to the linear character, nonlinear saturation effects cannot be computed with such models. Flame describing functions have been suggested to fill this gap. They describe the flame response in an amplitude-dependent manner and have been successfully used in recent work for the prediction of limit-cycle amplitudes in single-burner systems and annular combustors. This paper presents a more efficient approach of computing limit-cycle amplitudes of spinning thermoacoustic modes in an annular combustion chamber. As one important feature, adjoint perturbation theory is utilized for the solution of the thermoacoustic Helmholtz equation associated with a flame describing function. This avoids iterations over different amplitude levels to find the limit cycle amplitude, i.e., the amplitude level at which the modal growth rate is zero, as required in previous approaches. Moreover, based on the discrete rotational symmetry of the system, the computation is also accelerated by means of Bloch-wave theory, which reduces computations for annular combustors to a single burner/flame segment. Results for a generic model and a laboratory-scale annular combustion system are presented and discussed.

A Perturbation Approach to Bifurcation Analysis and Limit-Cycle Amplitudes for Brake Squeal Problems

2010 ◽  
Author(s):  
Hartmut Hetzler ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
Limit Cycle ◽  
Bifurcation Analysis ◽  
Perturbation Approach ◽  
Brake Squeal

scholarly journals A little damping goes a long way: a simulation study of how damping influences task-level stability in running

2020 ◽  
Vol 16 (9) ◽  
pp. 20200467
Author(s):  
Steve Heim ◽  
Matthew Millard ◽  
Charlotte Le Mouel ◽  
Alexander Badri-Spröwitz
Keyword(s):  
Limit Cycle ◽  
Simulation Study ◽  
Functional Role ◽  
Binary Classification ◽  
Legged Locomotion ◽  
Model Parameters ◽  
Damping Term ◽  
The Face ◽  
Task Level

It is currently unclear if damping plays a functional role in legged locomotion, and simple models often do not include damping terms. We present a new model with a damping term that is isolated from other parameters: that is, the damping term can be adjusted without retuning other model parameters for nominal motion. We systematically compare how increased damping affects stability in the face of unexpected ground-height perturbations. Unlike most studies, we focus on task-level stability: instead of observing whether trajectories converge towards a nominal limit-cycle, we quantify the ability to avoid falls using a recently developed mathematical measure. This measure allows trajectories to be compared quantitatively instead of only being separated into a binary classification of ‘stable' or ‘unstable'. Our simulation study shows that increased damping contributes significantly to task-level stability; however, this benefit quickly plateaus after only a small amount of damping. These results suggest that the low intrinsic damping values observed experimentally may have stability benefits and are not simply minimized for energetic reasons. All Python code and data needed to generate our results are available open source.

scholarly journals Analysis of the non-periodic oscillations of a self-excited friction-damped system with closely spaced modes

2021 ◽  
Author(s):  
Lukas Woiwode ◽  
Alexander F. Vakakis ◽  
Malte Krack
Keyword(s):  
Hopf Bifurcation ◽  
Limit Cycle ◽  
Dry Friction ◽  
The Self ◽  
Perturbation Approach ◽  
Friction Damping ◽  
Periodic Oscillations ◽  
Analytical Results ◽  
Fold Bifurcation ◽  
Phase Motion

AbstractIt is widely known that dry friction damping can bound the self-excited vibrations induced by negative damping. The vibrations typically take the form of (periodic) limit cycle oscillations. However, when the intensity of the self-excitation reaches a condition of maximum friction damping, the limit cycle loses stability via a fold bifurcation. The behavior may become even more complicated in the presence of any internal resonance conditions. In this work, we consider a two-degree-of-freedom system with an elastic dry friction element (Jenkins element) having closely spaced natural frequencies. The symmetric in-phase motion is subjected to self-excitation by negative (viscous) damping, while the symmetric out-of-phase motion is positively damped. In a previous work, we showed that the limit cycle loses stability via a secondary Hopf bifurcation, giving rise to quasi-periodic oscillations. A further increase in the self-excitation intensity may lead to chaos and finally divergence, long before reaching the fold bifurcation point of the limit cycle. In this work, we use the method of complexification-averaging to obtain the slow flow in the neighborhood of the limit cycle. This way, we show that chaos is reached via a cascade of period-doubling bifurcations on invariant tori. Using perturbation calculus, we establish analytical conditions for the emergence of the secondary Hopf bifurcation and approximate analytically its location. In particular, we show that non-periodic oscillations are the typical case for prominent nonlinearity, mild coupling (controlling the proximity of the modes), and sufficiently light damping. The range of validity of the analytical results presented herein is thoroughly assessed numerically. To the authors’ knowledge, this is the first work that shows how the challenging Jenkins element can be treated formally within a consistent perturbation approach in order to derive closed-form analytical results for limit cycles and their bifurcations.

Sign in / Sign up

Export Citation Format

Share Document