Interaction Between Coexisting Orbits in Impact Oscillators

2017 ◽  
Vol 12 (6) ◽  
Author(s):  
Narasimha Suda ◽  
Soumitro Banerjee
Keyword(s):  
Abrupt Onset ◽  
Exponential Dependence ◽  
Square Root ◽  
Stiffness Ratio ◽  
Chaotic Orbit ◽  
Practical Applications ◽  
Impact Oscillators ◽  
Chaotic Vibration ◽  
Onset Of Chaos ◽  
Square Root Singularity

Impact oscillators exhibit an abrupt onset of chaos close to grazing due to the square-root singularity in their discrete time maps. In practical applications, this large-amplitude chaotic vibration needs to be avoided. It has been shown that this can be achieved if the ratio of the natural frequency of the oscillator ω0 and the forcing frequency is an even integer. But, in practice, it is difficult to set a parameter at such a precise value. We show that in systems with square-root singularity (prestressed impacting surface), there exists a range of ω0 around the theoretical value over which the chaotic orbit does not occur, and that this is due to an interplay between the main attractor and coexisting orbits. We show that this range of forcing frequency has exponential dependence on the amount of prestress as well as on the stiffness ratio of the springs.

scholarly journals Unconventional singularities and energy balance in frictional rupture

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Efim A. Brener ◽  
Eran Bouchbinder
Keyword(s):  
Energy Balance ◽  
Square Root ◽  
Earthquake Physics ◽  
Macroscopic Theory ◽  
Scale Separation ◽  
Rate Dependent ◽  
Theoretical Predictions ◽  
Spatially Extended ◽  
Singularity Order ◽  
Square Root Singularity

AbstractA widespread framework for understanding frictional rupture, such as earthquakes along geological faults, invokes an analogy to ordinary cracks. A distinct feature of ordinary cracks is that their near edge fields are characterized by a square root singularity, which is intimately related to the existence of strict dissipation-related lengthscale separation and edge-localized energy balance. Yet, the interrelations between the singularity order, lengthscale separation and edge-localized energy balance in frictional rupture are not fully understood, even in physical situations in which the conventional square root singularity remains approximately valid. Here we develop a macroscopic theory that shows that the generic rate-dependent nature of friction leads to deviations from the conventional singularity, and that even if this deviation is small, significant non-edge-localized rupture-related dissipation emerges. The physical origin of the latter, which is predicted to vanish identically in the crack analogy, is the breakdown of scale separation that leads an accumulated spatially-extended dissipation, involving macroscopic scales. The non-edge-localized rupture-related dissipation is also predicted to be position dependent. The theoretical predictions are quantitatively supported by available numerical results, and their possible implications for earthquake physics are discussed.

Influence of a square-root singularity on the behaviour of piecewise smooth maps

Nonlinearity ◽  
2010 ◽  
Vol 23 (2) ◽  
pp. 445-463 ◽  
Author(s):  
Viktor Avrutin ◽  
Partha Sharathi Dutta ◽  
Michael Schanz ◽  
Soumitro Banerjee
Keyword(s):  
Piecewise Smooth ◽  
Square Root ◽  
Smooth Maps ◽  
Piecewise Smooth Maps ◽  
Square Root Singularity

Extended Stokes series: laminar flow through a loosely coiled pipe

1978 ◽  
Vol 86 (1) ◽  
pp. 129-145 ◽  
Author(s):  
Milton Van Dyke
Keyword(s):  
Laminar Flow ◽  
Flow Speed ◽  
Square Root ◽  
Mean Flow ◽  
Dean Number ◽  
Curvature Ratio ◽  
Wide Range ◽  
Square Root Singularity ◽  
Flow Through ◽  
The One

Dean's series for steady fully developed laminar flow through a toroidal pipe of small curvature ratio has been extended by computer to 24 terms. Analysis suggests that convergence is limited by a square-root singularity on the negative axis of the square of the Dean number. An Euler transformation and extraction of the leading and secondary singularities at infinity render the series accurate for all Dean numbers. For curvature ratios no greater than$\frac{1}{250} $, experimental measurements of the laminar friction factor agree with the theory over a wide range of Dean numbers. In particular, they confirm our conclusion that the friction in a loosely coiled pipe grows asymptotically as the one-quarter power of the Dean number based on mean flow speed. This contradicts a number of incomplete boundary-layer analyses in the literature, which predict a square-root variation.

Some Boundary Value Problems of Elastic Media Containing Rigid Inclusions Under Compressive Mechanical Loads

2009 ◽  
Author(s):  
Vahagn Makaryan ◽  
Michael Sutton ◽  
Gurgen Chlingaryan ◽  
Davresh Hasanyan ◽  
Xiaomin Deng
Keyword(s):  
Boundary Value Problems ◽  
Elastic Medium ◽  
Integral Transformation ◽  
Logarithmic Singularity ◽  
Boundary Value ◽  
Compressive Loading ◽  
Geometrical Shape ◽  
Square Root ◽  
End Points ◽  
Square Root Singularity

In this work we discuss some 2D boundary-value problems related to an elastic medium containing a thin rigid inclusion with general geometrical shape located in the interface between two separate elastic half-planes and subjected to compressive loading. Assuming perfect bonding between the inclusion and elastic medium, Fourier and Henkel integral transformation techniques are used to obtain the exact solution for the problem. Explicit forms are presented for arbitrary forms of thin inclusions, demonstrating that the tangent shear stress at the end-points of the inclusion has a square-root singularity. It is also shown that the normal stress has a logarithmic singularity when the end-points of the inclusion are approached from the inside of the inclusion and a square-root singularity when the end-points of the inclusion are approached from the outside of the inclusion. For special, extreme cases the solutions for anti-cracks are also presented.

An Alternative Decomposition of the Strain Gradient Tensor

2001 ◽  
Vol 69 (2) ◽  
pp. 139-141 ◽  
Author(s):  
H. Jiang ◽  
Y. Huang ◽  
T. F. Guo ◽  
K. C. Hwang
Keyword(s):  
Strain Gradient ◽  
Strain Field ◽  
Numerical Study ◽  
Volumetric Strain ◽  
Crack Tip ◽  
Square Root ◽  
Crack Tip Field ◽  
Gradient Tensor ◽  
Classical Plasticity ◽  
Square Root Singularity

An alternative decomposition of the strain gradient tensor is proposed in this paper in order to ensure that the deviatoric strain gradient vanishes for an arbitrary volumetric strain field, which is consistent with the physical picture of plastic deformation. The theory of mechanism-based strain gradient (MSG) plasticity is then modified accordingly based on this new decomposition. The numerical study of the crack-tip field based on the new theory shows that the crack tip in MSG plasticity has the square-root singularity, and the stress level is much higher than the HRR field in classical plasticity.

scholarly journals Square root singularity in the viscosity of neutral colloidal suspensions at large frequencies

1997 ◽  
Vol 87 (5-6) ◽  
pp. 1037-1049 ◽  
Author(s):  
R. Verberg ◽  
I. M. de Schepper ◽  
M. J. Feigenbaum ◽  
E. G. D. Cohen
Keyword(s):  
Colloidal Suspensions ◽  
Square Root ◽  
Square Root Singularity

Flame characteristics of wind-driven surface fires

1986 ◽  
Vol 16 (6) ◽  
pp. 1293-1300 ◽  
Author(s):  
Ralph M. Nelson Jr. ◽  
Carl W. Adkins
Keyword(s):  
Boundary Layer ◽  
Tilt Angle ◽  
Fire Behavior ◽  
Flame Length ◽  
Fire Intensity ◽  
Square Root ◽  
Pine Needle ◽  
Weather Variables ◽  
Practical Applications ◽  
Surface Fires

Twenty-two fires in a laboratory wind tunnel and 8 field fires were studied with video techniques to determine relationships between their flame characteristics and fire behavior. The laboratory fires were in pine needle fuel beds with and without an overlying stratum of live vegetation. These fuels simulated 2-year roughs in southeastern fuel types. The field bums were in 1- and 2-year roughs in similar fuels. Byram's fire intensity ranged from 98 to 590 kW/m in the laboratory, and from 355 to 2755 kW/m in the field. Flame lengths were proportional to the square root of fire intensity when fuel consumption exceeded 0.5 kg/m2, in agreement with predictions from buoyant flame theory. However, for burns in the needle layer (consumption approximately 0.5 kg/m2), flame lengths were constant at about 0.5 m, regardless of intensity. Similar values were observed on two of the field fires. It is speculated that flame length is limited by a boundary layer pattern for the overall flow, even though the flames themselves did not exhibit boundary layer characteristics. Also, laboratory correlations of flame tilt angle and fire intensity with other fire and weather variables depart from buoyant flame theory. Further study under field conditions is needed before relationships involving flame tilt angle, fire intensity, and wind speed should be used in practical applications.

scholarly journals Flow-induced forces arising during the impact of two circular cylinders

2008 ◽  
Vol 616 ◽  
pp. 205-234 ◽  
Author(s):  
N. BAMPALAS ◽  
J. M. R. GRAHAM
Keyword(s):  
Relative Motion ◽  
Conformal Transformation ◽  
Steady Motion ◽  
Inviscid Flow ◽  
Circular Cylinders ◽  
Square Root ◽  
Two Dimensional ◽  
Incident Flow ◽  
Square Root Singularity ◽  
The Impact

This paper presents numerical simulations of two-dimensional incompressible flow around two circular cylinders in relative motion, which may result in impact. Viscous flow computations are carried out using a streamfunction–vorticity method for two equal-diameter cylinders undergoing a two-dimensional impact in otherwise stationary fluid and for cases of similar impact of two cylinders in a steady incident flow. These results are supported by potential flow calculations carried out using a Möbius conformal transformation and infinite arrays of image singularities. The inviscid flow results are compared with other published work and show that the inviscid forces induced on the cylinders have an inverse square root singularity with respect to the time to impact. All impacts considered in this paper result from steady motion of the cylinders along the line joining their centres.

scholarly journals Square root singularity in boundary reflection matrix

1996 ◽  
Vol 388 (3) ◽  
pp. 550-556 ◽  
Author(s):  
J.D. Kim ◽  
I.G. Koh
Keyword(s):  
Square Root ◽  
Reflection Matrix ◽  
Square Root Singularity
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