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1.
Lyapunov’s first method, extended by Kozlov to nonlinear mechanical systems, is applied to study the instability of the equilibrium
position of a mechanical system moving in the field of conservative and dissipative forces. The cases with a tensor of inertia
or a matrix of coefficients of the Rayleigh dissipative function are analyzed singularly in the equilibrium position. This
fact renders the impossible application of Lyapunov’s approach in the analysis of the stability because, in the equilibrium
position, the conditions of the existence and uniqueness of the solutions to the differential equations of motion are not
fulfilled. It is shown that Kozlov’s generalization of Lyapunov’s first method can also be applied in the mentioned cases
on the conditions that, besides the known algebraic expression, more are fulfilled. Three theorems on the instability of the
equilibrium position are formulated. The results are illustrated by an example. 相似文献
2.
For a generalized Hamiltonian system, stability for the manifolds of equilibrium states is presented based on Lyapunov’s stability
theories. Equilibrium equations, perturbation equations and first approximate equations of the system are given. A theorem
for the stability of manifolds of equilibrium states of general autonomous system is used to the generalized Hamiltonian system,
and three propositions on the stability of manifolds of equilibrium states of the system are obtained. Two examples are given
to illustrate application of the method and results. 相似文献
3.
4.
The vibration of a ship pitch-roll motion described by a non-linear spring pendulum system (two degrees of freedom) subjected
to multi external and parametric excitations can be reduced using a longitudinal absorber. The method of multiple scale perturbation
technique (MSPT) is applied to analyze the response of this system near the simultaneous primary, sub-harmonic and internal
resonance. The steady state solution near this resonance case is determined and studied applying Lyapunov’s first method.
The stability of the system is investigated using frequency response equations. Numerical simulations are extensive investigations
to illustrate the effects of the absorber and some system parameters at selected values on the vibrating system. The simulation
results are achieved using MATLAB 7.0 programs. Results are compared to previously published work. 相似文献
5.
The two-dimensional nonlinear problem of steady gravity waves on water of finite depth is considered. The Benjamin–Lighthill
conjecture is proved for these waves provided Bernoulli’s constant attains near-critical values. In fact this is a consequence
of the following more general results. If Bernoulli’s constant is near-critical, then all corresponding waves have sufficiently
small heights and slopes. Moreover, for every near-critical value of Bernoulli’s constant, there exist only the following
waves: a solitary wave and the family of Stokes waves having their crests strictly below the crest of this solitary wave;
this family is parametrised by wave heights which increase from zero to the height of the solitary wave. All these waves are
unique up to horizontal translations. Most of these results were proved in our previous paper (Kozlov and Kuznetsov in Arch
Rational Mech Anal 197, 433–488, 2010), in which it was supposed that wave slopes are bounded a priori. Here we show that the latter condition is superfluous by
proving the following theorem. If any steady wave has the free-surface profile of a sufficiently small height, then the slope
of this wave is also small. 相似文献
6.
Nonlocal generalizations of Burgers’ equation were derived in earlier work by Hunter (Contemp Math, vol 100, pp 185–202. AMS, 1989), and more recently by Benzoni-Gavage and Rosini (Comput Math Appl 57(3–4):1463–1484, 2009), as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage (Differ Integr Equ 22(3–4):303–320, 2009) under an appropriate stability condition originally pointed out by Hunter. The same stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov (Benzoni-Gavage et al. in Adv Math 227(6):2220–2240, 2011). In this article, we show how the verification of Hunter’s stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. We also show that the resulting amplitude equation has a Hamiltonian structure when the original boundary value problem has a variational origin. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity. 相似文献
7.
The Cauchy’s theorem for balance laws is proved in a general context using a simpler and more natural method in comparison
to the one recently presented in Segev (Arch. Ration. Mech. Anal. 154:183–198, 2000). By “generality” we mean that the ambient space is considered to be an orientable smooth manifold, and not only the Euclidean
space. 相似文献
8.
A rotor- active magnetic bearing (AMB) system with a periodically time-varying stiffness subjected to multi- external, -parametric
and -tuned excitations is studied and solved. The method of multiple scales is applied to analyze the response of the two
modes of the system near the simultaneous sub-harmonic, super-harmonic and combined resonance case. The stability of the steady
state solution near this resonance case is determined and studied applying Lyapunov’s first method. Also, the system exhibits
many typical nonlinear behaviors including multi-valued solutions, jump phenomenon, softening nonlinearities. The effects
of the different parameters on the steady state solutions are investigated and discussed. Simulation results are achieved
using MATLAB 7.0 program. 相似文献
9.
S. V. Babenko 《International Applied Mechanics》2011,47(1):86-96
Linear systems of dynamic equations with periodic coefficients and structural perturbations on time scale are analyzed for
Lyapunov stability. Sufficient conditions for the asymptotic stability of the equations are established based on the matrix-value
concept of Lyapunov’s direct method for all values of the structural matrix from the structural set. A system of two dynamic
equations on time scale is considered as an example of applying the theoretical results obtained 相似文献
10.
Stanisław Sieniutycz 《Transport in Porous Media》2007,69(2):239-257
A Fermat-like principle of minimum time is formulated for nonlinear steady paths of fluid flow in inhomogeneous isotropic
porous media where fluid streamlines are curved by a location dependent hydraulic conductivity. The principle describes an
optimal nature of nonlinear paths in steady Darcy’s flows of fluids. An expression for the total path resistance leads to
a basic analytical formula for an optimal shape of a steady trajectory. In the physical space an optimal curved path ensures
the maximum flux or shortest transition time of the fluid through the porous medium. A sort of “law of bending” holds for
the frictional fluid flux in Lagrange coordinates. This law shows that—by minimizing the total resistance—a ray spanned between
two given points takes the shape assuring that a relatively large part of it resides in the region of lower flow resistance
(a ‘rarer’ region of the medium). 相似文献
11.
We study the rheological response of monodomain ellipsoidal biaxial liquid crystal polymers (BLCP) as well as bent-core or
V-shaped liquid crystal polymers (VLCP) subject to steady and time-dependent small amplitude oscillatory shear in selected
regions of the model as well as flow parameter space. We adopt the two newly developed hydrodynamical kinetic theories for
ellipsoidal BLCPs and VLCPs, respectively (Sircar and Wang, PRE 78:061702, 2008, J Rheol 53:819–858, 2009; Sircar et al., Comm Math Sci (in press), 2010), in which a generalized Straley’s potential is used to represent the pairwise mean-field interaction of the mesoscopic system
in biaxial phases. Transient shear stresses and normal stress differences corresponding to steady and small amplitude oscillatory
shear are investigated; their variations with respect to the strength of the intermolecular potential, types of biaxial interaction,
and changes in the aspect ratios for ellipsoidal BLCPs and the bent angle for VLCPs are explored. 相似文献
12.
Marius-F. Danca 《Nonlinear dynamics》2010,60(4):525-534
In this paper, the chaos persistence in a class of discontinuous dynamical systems of fractional-order is analyzed. To that
end, the initial value problem is first transformed, by using the Filippov regularization (Filippov in Differential Equations
with Discontinuous Right-Hand Sides, 1988), into a set-valued problem of fractional-order, then by Cellina’s approximate selection theorem (Aubin and Cellina in Differential
Inclusions Set-valued Maps and Viability Theory, 1984; Aubin and Frankowska in Set-valued Analysis, 1990). The problem is approximated into a single-valued fractional-order problem, which is numerically solved by using a numerical
scheme proposed by Diethelm et al. (Nonlinear Dyn. 29:3–22, 2002). Two typical examples of systems belonging to this class are analyzed and simulated. 相似文献
13.
Yu. N. Shevchenko R. G. Terekhov N. N. Tormakhov 《International Applied Mechanics》2006,42(4):421-430
Equations relating the components of the stress and strain tensors (constitutive equations) are formulated in terms of Euler
coordinates. The equations describe the finite elastoplastic deformation of an isotropic body along paths of small curvature.
It is assumed that the stress deviator is coaxial with the plastic-strain differential deviator. The relationships between
the first and second invariants of the stress and strain tensors in the case of complex elastoplastic deformation of the body’s
elements are determined from base tests on tubular specimens loaded along rectilinear paths for several values of the stress
mode angle. Methods for specification of these relationships are proposed. The assumptions adopted to derive the constitutive
equations are validated experimentally
__________
Translated from Prikladnaya Mekhanika, Vol. 42, No. 4, pp. 62–72, April 2006. 相似文献
14.
In this paper we consider the problem of adhesive frictionless contact of an elastic half-space by an axi-symmetric punch.
We obtain integral equations that define the tractions and displacements normal to the surface of the half-space, as well
as the size of the contact regions, for the cases of circular and annular contact regions. The novelty of our approach resides
in the use of Betti’s reciprocity theorem to impose equilibrium, and of Abel transforms to either solve or substantially simplify
the resulting integral equations. Additionally, the radii that define the annular or circular contact region are defined as
local minimizers of the function obtained by evaluating the potential energy at the equilibrium solutions for each pair of
radii. With this approach, we rather easily recover Sneddon’s formulas (Sneddon, Int. J. Eng. Sci., 3(1):47–57, 1965) for circular contact regions. For the annular contact region, we obtain a new integral equation that defines the inverse
Abel transform of the surface normal displacement. We solve this equation numerically for two particular punches: a flat annular
punch, and a concave punch. 相似文献
15.
In an earlier paper in this journal (Mocikat et al. in Exp Fluids 34:442–448, 2003) LDV measurements in a simple geometry but for a complex flow have been provided as a database for CFD evaluation purposes.
With special inflow devices swirl could now be added to the flow. By changing the exit position of the test section in order
to get a non-symmetric flow field, a steady swirling flow without instability induced precessing motions could be established.
This flow can be interpreted as a superposition of a swirling motion to an otherwise swirl-free flow by introducing “swirl
influence factors” for various aspects of the flow field. With a modified inflow device a periodically unsteady flow with
swirl emerged. The turbulence features of this flow are distinctively different from the steady flow case with swirl. For
all flows under consideration the three time-averaged components of the velocity vector and all components of the Reynolds
stress tensor are measured in selected cross sections and provided as a data base for CFD calculations. 相似文献
16.
In the present paper we investigate conservation and balance laws in the framework of linear elastodynamics considering the
strain energy density depending on the gradients of the displacement up to the third order, as originally proposed by Mindlin
(Int. J. Solids Struct. 1, 417–438, 1965). The conservation and balance laws that correspond to the symmetries of translation, rotation, scaling and addition of solutions
are derived using Noether’s theorem. Also, the formulas of the dynamical J,L and M-integrals are presented for the problem under study. Moreover, the balance law of addition of solutions gives rise to explore
the dynamical reciprocal theorem as well as the restrictions under which it is valid.
相似文献
17.
Hot-wire and oil-film interferometry measurements are taken for 3D rough wall boundary layers at very high Reynolds numbers
(61,000 < Re θ < 120,000) with low blockage ratios, 10 < δ/H < 135, and high roughness, 100 < H
+ < 4,900. The results cover flows over both rough walls and over obstacles and are compared with and provide extension to
recent lower Reynolds number results. The validity of the Townsend ‘wall similarity hypothesis’ in relation to consistently
increasing 3D roughness is interrogated. In agreement with recent work, Schultz and Flack (J Fluid Mech 580:381–405, 2007) and Castro (J Fluid Mech 585:469–485, 2007) found that, for relatively low roughness, Townsend’s hypothesis holds for the mean velocity field. With increasing roughness,
the equilibrium layer diminishes and gradually vanishes. The viscous component of the wall shear stress decreases, while the
turbulent component increases as the roughness effects extend across the boundary layer. 相似文献
18.
Thomas J. Waters 《Nonlinear dynamics》2010,60(3):341-356
In the following we consider a 2-dimensional system of ODEs containing quasiperiodic terms. The system is proposed as an extension
of Mathieu-type equations to higher dimensions, with emphasis on how resonance between the internal frequencies leads to a
loss of stability. The 2-d system has two ‘natural’ frequencies when the time-dependent terms are switched off, and it is
internally driven by quasiperiodic terms in the same frequencies. Stability charts in the parameter space are generated first
using numerical simulations and Floquet theory. While some instability regions are easy to anticipate, there are some surprises:
within instability zones, small islands of stability develop, and unusual ‘arcs’ of instability arise also. The transition
curves are analyzed using the method of harmonic balance, and we find we can use this method to easily predict the ‘resonance
curves’ from which bands of instability emanate. In addition, the method of multiple scales is used to examine the islands
of stability near the 1:1 resonance. 相似文献
19.
V. L. Berdichevsky 《Continuum Mechanics and Thermodynamics》2008,20(4):219-229
Ideal incompressible fluid is a Hamiltonian system which possesses an infinite number of integrals, the circulations of velocity
over closed fluid contours. This allows one to split all the degrees of freedom into the driving ones and the “slave” ones,
the latter to be determined by the integrals of motions. The “slave” degrees of freedom correspond to “potential part” of
motion, which is driven by vorticity. Elimination of the “slave” degrees of freedom from equations of ideal incompressible
fluid yields a closed system of equations for dynamics of vortex lines. This system is also Hamiltonian. The variational principle
for this system was found recently (Berdichevsky in Thermodynamics of chaos and order, Addison-Wesly-Longman, Reading, 1997;
Kuznetsov and Ruban in JETP Lett 67, 1076–1081, 1998). It looks striking, however. In particular, the fluid motion is set
to be compressible, while in the least action principle of fluid mechanics the incompressibility of motion is a built-in property.
This striking feature is explained in the paper, and a link between the variational principle of vortex line dynamics and
the least action principle is established. Other points made in this paper are concerned with steady motions. Two new variational
principles are proposed for steady vortex flows. Their relation to Arnold’s variational principle of steady vortex motion
is discussed.
相似文献
20.
External heat transfer prediction is performed in two-dimensional turbine blade cascades using the Reynolds-averaged Navier–Stokes
equations. For this purpose, six different turbulence models including the algebraic Baldwin–Lomax (AIAA paper 78-257, 1978), three low-Re
k−ɛ models (Chien in AIAA J 20:33–38, 1982; Launder and Sharma in Lett Heat Mass Transf 1(2):131–138, 1974; Biswas and Fukuyama in J Turbomach 116:765–773, 1994), and two k−ω models (Wilcox in AIAA J 32(2):247–255, 1994) are taken into account. The computer code developed employs a finite volume method to solve governing equations based on
an explicit time marching approach with capability to simulate subsonic, transonic and supersonic flows. The Roe method is
used to decompose the inviscid fluxes and the gradient theorem to decompose viscous fluxes. The performance of different turbulence
models in prediction of heat transfer is examined. To do so, the effect of Reynolds and Mach numbers along with the turbulent
intensity are taken into account, and the numerical results obtained are compared with the experimental data available. 相似文献