The development of wavepackets in unstable flows

1981 ◽  
Vol 373 (1755) ◽  
pp. 457-476 ◽  
Keyword(s):  
Dispersion Relation ◽  
Multiple Scales ◽  
Three Dimensional ◽  
Fourier Integral ◽  
Saddle Point Method ◽  
Alternative Derivation ◽  
Closed Form Solutions ◽  
Linear Evolution ◽  
Multiple Scales Technique ◽  
Complex Dispersion

By using a model form of the complex dispersion relation for unstable flows, the linear evolution of a localized three-dimensional wavepacket is determined. The disturbance is expressed as a double Fourier integral which is evaluated asymptotically by the saddle-point method. On making certain approximations, simple closed-form solutions are obtained, some of which resemble the curved wavepackets observed by Gaster & Grant (1975) and some the ‘ elliptic ’ packet found by Benjamin (1961). The range of validity of theories which lead to an elliptic packet is clarified. An alternative derivation of some of the results is given by using a multiple-scales technique. The relative merits of the two methods are thereby illuminated.

The linear growth of localized disturbances in unstable compressible flows

1984 ◽  
Vol 98 (3-4) ◽  
pp. 249-265
Author(s):  
S. Candler ◽  
A. D. D. Craik
Keyword(s):  
Dispersion Relation ◽  
Compressible Flows ◽  
Linear Growth ◽  
Incompressible Flows ◽  
Fourier Integral ◽  
Saddle Point Method ◽  
Parallel Flows ◽  
Viscous Incompressible Flows ◽  
Asymptotic Evaluation ◽  
Analytic Expression

SynopsisAccurate computation of the evolution of initially localized disturbances in compressible parallel flows is a tedious task requiring superposition of a large number of Fourier modes with differing temporal growth rates. An alternative approximate method, similar to that developed by Craik (1981, 1982) for viscous incompressible flows, is presented here. This involves asymptotic evaluation, by the saddle point method, of a double Fourier integral representation of the disturbance, with the actual dispersion relation replaced by a simpler analytic expression containing several parameters which may be adjusted to approximate the flow under investigation. Limiting cases yield informative results in simple closed form: these exemplify the possible shapes into which the disturbance may evolve. In particular, ‘splitting’ of the disturbance into two dominant regions is demonstrated.

A Three-Dimensional Finite Difference Solution for the Thermal Stresses in Railcar Wheels

1969 ◽  
Vol 91 (3) ◽  
pp. 891-896 ◽  
Author(s):  
G. E. Novak ◽  
B. J. Eck
Keyword(s):  
Computer Program ◽  
Thermal Stresses ◽  
Three Dimensional ◽  
Transient Temperature ◽  
Shear Stresses ◽  
Radial Temperature ◽  
Closed Form Solutions ◽  
Brake Application ◽  
History Of ◽  
Thin Disks

A numerical solution is presented for both the transient temperature and three-dimensional stress distribution in a railcar wheel resulting from a simulated emergency brake application. A computer program has been written for generating thermoelastic solutions applicable to wheels of arbitrary contour with temperature variations in both axial and radial directions. The results include the effect of shear stresses caused by the axial-radial temperature gradients and the high degree of boundary irregularity associated with this type of problem. The program has been validated by computing thermoelastic solutions for thin disks and long cylinders; the computed values being in good agreement with the closed form solutions. Currently, the computer program is being extended to general stress solutions corresponding to the transient temperature distributions obtained by simulated drag brake applications. When this work is completed, it will be possible to synthesize the thermal history of a railcar wheel and investigate the effects of wheel geometry in relation to thermal fatigue.

Solutions to the Vector Tetrahedron Equation

1965 ◽  
Vol 87 (2) ◽  
pp. 228-234 ◽  
Author(s):  
Milton A. Chace
Keyword(s):  
Closed Form ◽  
Digital Computer ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Spherical Coordinates ◽  
Tetrahedron Equation ◽  
Position Determination ◽  
Closed Form Solutions

A set of nine closed-form solutions are presented to the single, three-dimensional vector tetrahedron equation, sum of vectors equals zero. The set represents all possible combinations of unknown spherical coordinates among the vectors, assuming the coordinates are functionally independent. Optimum use is made of symmetry. The solutions are interpretable and can be evaluated reliably by digital computer in milliseconds. They have been successfully applied to position determination of many three-dimensional mechanisms.

Nonlinear Coupled Transverse and Axial Vibration of Variable Cross-Section Beam Flexures Interconnecting Rigid Body

2016 ◽  
Author(s):  
Hasan Malaeke ◽  
Hamid Moeenfard ◽  
Amir H. Ghasemi
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Cross Section ◽  
Multiple Scales ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Closed Form Solutions ◽  
Dynamic Performances

The objective of this paper is to analytically study the nonlinear behavior of variable cross-section beam flexures interconnecting an eccentric rigid body. Hamilton’s principle is utilized to obtain the partial differential equations governing the nonlinear vibration of the system as well as the corresponding boundary conditions. Using a single mode approximation, the governing equations are reduced to a set of two nonlinear ordinary differential equations in terms of end displacement components of the beam which are coupled due to the presence of the transverse eccentricity. The method of multiple scales are employed to obtain parametric closed-form solutions. The obtained analytical results are compared with the numerical ones and excellent agreement is observed. These analytical expressions provide design insights for modeling and optimization of more complex flexure mechanisms for improved dynamic performances.

Spring constants of filament-wound composite circular rings

2001 ◽  
Vol 215 (2) ◽  
pp. 211-226 ◽  
Author(s):  
P C Tse ◽  
S R Reid ◽  
S P Ng
Keyword(s):  
Three Dimensional ◽  
Element Analysis ◽  
Closed Form Solutions ◽  
Composite Ring ◽  
Filament Wound ◽  
Dimensional Finite Element ◽  
Filament Wound Composite ◽  
Circular Rings ◽  
Three Dimensional Finite Element ◽  
Wound Composite

Closed-form solutions from complementary strain energy are derived for the spring stiffnesses of mid-surface symmetric, filament-wound, composite circular rings under unidirectional loading. A three-dimensional finite element analysis (FEA) including the effects of transverse shear has also been applied to study the problem. Four > 45° and four > 75° E-glass/epoxy composite rings of odd numbers of covers were tested. Comparisons of the results obtained from the two methods with experimental data are made and the results are found to be in good agreement. The FEA prediction of stiffness is always higher than the theoretical result. The relationships between the spring stiffnesses and the winding angles and geometry of the filament-wound composite ring are considered and discussed.

scholarly journals Reconstruction of three-dimensional maps based on closed-form solutions of the variational problem of multisensor data registration

2019 ◽  
Vol 484 (6) ◽  
pp. 672-677
Author(s):  
A. V. Vokhmintcev ◽  
A. V. Melnikov ◽  
K. V. Mironov ◽  
V. V. Burlutskiy
Keyword(s):  
Closed Form ◽  
Large Scale ◽  
Three Dimensional ◽  
Dynamic Environment ◽  
Closed Form Solution ◽  
Point Clouds ◽  
Accurate Method ◽  
Form Solution ◽  
Closed Form Solutions ◽  
Reconstruction Methods

A closed-form solution is proposed for the problem of minimizing a functional consisting of two terms measuring mean-square distances for visually associated characteristic points on an image and meansquare distances for point clouds in terms of a point-to-plane metric. An accurate method for reconstructing three-dimensional dynamic environment is presented, and the properties of closed-form solutions are described. The proposed approach improves the accuracy and convergence of reconstruction methods for complex and large-scale scenes.

Analyses of Full-Scale Tank Car Shell Impact Tests

2007 ◽  
Author(s):  
Y. H. Tang ◽  
H. Yu ◽  
J. E. Gordon ◽  
M. Priante ◽  
D. Y. Jeong ◽  
...  
Keyword(s):  
Finite Element ◽  
Test Data ◽  
Three Dimensional ◽  
Full Scale ◽  
Collision Dynamics ◽  
Dynamics Modeling ◽  
Dynamics Model ◽  
Rigid Plastic ◽  
Element Analysis ◽  
Closed Form Solutions

This paper describes analyses of a railroad tank car impacted at its side by a ram car with a rigid punch. This generalized collision, referred to as a shell impact, is examined using nonlinear (i.e., elastic-plastic) finite element analysis (FEA) and three-dimensional (3-D) collision dynamics modeling. Moreover, the analysis results are compared to full-scale test data to validate the models. Commercial software packages are used to carry out the nonlinear FEA (ABAQUS and LS-DYNA) and the 3-D collision dynamics analysis (ADAMS). Model results from the two finite element codes are compared to verify the analysis methodology. Results from static, nonlinear FEA are compared to closed-form solutions based on rigid-plastic collapse for additional verification of the analysis. Results from dynamic, nonlinear FEA are compared to data obtained from full-scale tests to validate the analysis. The collision dynamics model is calibrated using test data. While the nonlinear FEA requires high computational times, the collision dynamics model calculates gross behavior of the colliding cars in times that are several orders of magnitude less than the FEA models.

Stability and Three-Wave Interaction of Disturbances in a Supersonic Boundary Layer With Cooling

2010 ◽  
Vol 5 (3) ◽  
pp. 52-62
Author(s):  
Sergey A. Gaponov ◽  
Natalya M. Terekhova
Keyword(s):  
Boundary Layer ◽  
Three Dimensional ◽  
Wave Interaction ◽  
Considerable Change ◽  
Supersonic Boundary Layer ◽  
Surface Cooling ◽  
Weakly Nonlinear ◽  
Linear Evolution ◽  
Weakly Nonlinear Theory ◽  
Compressed Gas

In linear and nonlinear approach (weakly nonlinear theory of stability) interaction of disturbances on a boundary layer of compressed gas is considered at surface cooling. The regimes of moderate (Max number М = 2) and high (М = 5.35) are considered at supersonic speeds. It is established that the surface cooling leads to considerable change of linear evolution of disturbances: the vortical disturbances of the first mode are stabilised, and the acoustic disturbances of the second mode are destabilised, the change degree is defined by the degree of change of the temperature factor. The nonlinear interaction in three-wave systems on high (М = 5.35) supersonic regimes on a boundary layer of compressed gas is carried out between waves of the different nature (acoustic and vortical) in a regime of a parametrical resonance. As a rating wave the flat acoustic wave which raises three-dimensional subharmonic components of the vortical modes. However, the similar interactions for vortical waves at М = 2 considerably weaken. It is possible to expect that surface cooling will lead to delay of a laminar regime at М = 2 and to accelerate of turbulization at М = 5.35

New techniques for imaging, digitization and analysis of three-dimensional neural morphology on multiple scales

Neuroscience ◽  
2005 ◽  
Vol 136 (3) ◽  
pp. 661-680 ◽  
Author(s):  
S.L. Wearne ◽  
A. Rodriguez ◽  
D.B. Ehlenberger ◽  
A.B. Rocher ◽  
S.C. Henderson ◽  
...  
Keyword(s):  
Multiple Scales ◽  
Three Dimensional ◽  
New Techniques

scholarly journals Forced Vibration in Cutting Process considering the Nonlinear Curvature and Inertia of a Rotating Composite Cutter Bar

2020 ◽  
Vol 2020 ◽  
pp. 1-19
Author(s):  
Yongsheng Ren ◽  
Donghui Yao
Keyword(s):  
Cutting Force ◽  
Forced Vibration ◽  
Multiple Scales ◽  
Damping Ratio ◽  
Three Dimensional ◽  
Forced Response ◽  
Cutting Process ◽  
Response Curves ◽  
Rotating Speed ◽  
Cutting System

Forced vibration of the cutting system with a three-dimensional composite cutter bar is investigated. The composite cutter bar is simplified as a rotating cantilever shaft which is subjected to a cutting force including regenerative delay effects and harmonic exciting items. The nonlinear curvature and inertia of the cutter bar are taken into account based on inextensible assumption. The effects of the moment of inertia, gyroscopic effect, and internal and external damping are also considered, but shear deformation is neglected. Equation of motion is derived based on Hamiltonʼs extended principle and discretized by the Galerkin method. The analytical solutions of the steady-state response of the cutting system are constructed by using the method of multiple scales. The response of the cutting system is studied for primary and superharmonic resonances. The effects of length-to-diameter ratio, damping ratio, cutting force coefficients, ply angle, rotating speed, and internal and external damping are investigated. The results show that nonlinear curvature and inertia imposed a significant effect on the dynamic behavior of the cutting process. The equivalent nonlinearity of the cutting system shows hard spring characteristics. Multiple solutions and jumping phenomenon of typical Duffing system are found in forced response curves.

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