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1.
We prove, among other things, that if the acoustic tensor satisfies a suitable growth condition at infinity (the hyperbolicity condition) and the total initial energy is summable with a suitable weight, then the solution to the initial boundary value problem of linear elastodynamics in unbounded domains decays at infinity, at every instant, with a rate depending on the weight. Moreover, we show that the hyperbolicity condition is necessary and sufficient for the equipartition in mean of the total energy. 相似文献
2.
The paper considers the application of the method of direct separation of motions to the investigation of distributed systems.
An approach is proposed which allows one to apply the method directly to the initial equation of motion and to satisfy all
boundary conditions, arising for both slow and fast components of motion. The methodology is demonstrated by means of a classical
problem concerning the so-called Indian magic rope trick (Blekhman et al. in Selected topics in vibrational mechanics, vol.
11, pp. 139–149, [2004]; Champneys and Fraser in Proc. R. Soc. Lond. A 456:553–570, [2000]; in SIAM J. Appl. Math. 65(1):267–298, [2004]; Fraser and Champneys in Proc. R. Soc. Lond. A 458:1353–1373, [2002]; Galan et al. in J. Sound Vib. 280:359–377, [2005]), in which a wire with an unstable upper vertical position is stabilized due to vertical vibration of its bottom support
point. The wire is modeled as a heavy Bernoulli–Euler beam with a vertically vibrating lower end. As a result of the treatment,
an explicit formula is obtained for the vibrational correction to the critical flexural stiffness of the nonexcited system. 相似文献
3.
The exact linear three-dimensional equations for a elastically monoclinic (13 constant) plate of constant thickness are reduced
without approximation to a single 4th order differential equation for a thickness-weighted normal displacement plus two auxiliary equations for
weighted thickness integrals of a stress function and the normal strain. The 4th order equation is of the same form as in
classical (Kirchhoff) theory except the unknown is not the midsurface normal displacement. Assuming a solution of these plate
equations, we construct so-called modified Saint-Venant solutions—“modified” because they involve non-zero body and surface
loads. That is, solutions of the exact three-dimensional elasticity equations that exhibit no boundary layers and that are
subject to a special set of body and surface loads that leave the analogous plate loads arbitrary. 相似文献
4.
The response function of a network of springs and masses, an elastodynamic network, is the matrix valued function W(ω), depending on the frequency ω, mapping the displacements of some accessible or terminal nodes to the net forces at the terminals. We give necessary and
sufficient conditions for a given function W(ω) to be the response function of an elastodynamic network, assuming there is no damping. In particular we construct an elastodynamic
network that can mimic a suitable response in the frequency or time domain. Our characterization is valid for networks in
three dimensions and also for planar networks, which are networks where all the elements, displacements and forces are in
a plane. The network we design can fit within an arbitrarily small neighborhood of the convex hull of the terminal nodes,
provided the springs and masses occupy an arbitrarily small volume. Additionally, we prove stability of the network response
to small changes in the spring constants and/or addition of springs with small spring constants. 相似文献
5.
We analyze the quasiperiodic damped Mathieu equation
[(x)\ddot]+ g[(x)\dot]+ x ( 1 + d+ eq(t) )=0 ,\ddot{x}+ \gamma\dot{x}+ x \bigl( 1 + \delta+ \epsilon q(t) \bigr )=0 , 相似文献
6.
The present study is concerned with the out-of-plane vibrations of a rotating, internally damped (Kelvin-Voigt model) Bernoulli-Euler
beam carrying a tip mass. The centroid of the tip mass, possessing also a mass moment of inertia is offset from the free end
of the beam and is located along its extended axis. This system can be thought of as an extremely simplified model of a helicopter
rotor blade or a blade of an auto-cooling fan. The differential eigenvalue problem is solved by using Frobenius method of
solution in power series. The characteristic equation is then solved numerically. The simulation results are tabulated for
a variety of the nondimensional rotational speeds, tip mass, tip mass offset, mass moment of inertia and internal damping
parameters. These are compared with the results of a conventional finite element modeling as well, and excellent agreement
is obtained. Some numerical results are given in graphical form. The numerical results obtained, indicate clearly that the
tip mass offset and mass moment of inertia are important parameters on the eigencharacteristics of rotating beams so that
they have to be included in the modeling process. 相似文献
7.
Generalized projective synchronization of the fractional-order Chen hyperchaotic system 总被引:2,自引:0,他引:2
In this paper, we numerically investigate the hyperchaotic behaviors in the fractional-order Chen hyperchaotic systems. By utilizing the fractional calculus techniques, we find that hyperchaos exists in the fractional-order Chen hyperchaotic system with the order less than 4. We found that the lowest order for hyperchaos to have in this system is 3.72. Our results are validated by the existence of two positive Lyapunov exponents. The generalized projective synchronization method is also presented for synchronizing the fractional-order Chen hyperchaotic systems. The present technique is based on the Laplace transform theory. This simple and theoretically rigorous synchronization approach enables synchronization of fractional-order hyperchaotic systems to be achieved and does not require the computation of the conditional Lyapunov exponents. Numerical simulations are performed to verify the effectiveness of the proposed synchronization scheme. 相似文献
8.
Kadry Zakaria 《Nonlinear dynamics》2011,66(4):457-477
The aim of this work is to study the instability of interacting waves between two immiscible magnetic liquids. The effects
of gravitation and a uniform normal magnetic field are taken into account. The method of multiple scales is used to determine
the stability criteria of the considered problem. The various stability criteria are discussed both analytically and graphically.
According to the numerical examples, we have remarked that the increase of the ratio of the permeability of the liquids appears
to be the destabilizing effect of the magnetic field. The short waves below the critical wavenumbers are stable whereas a
number of long waves are unstable. The viscosity effect on the stability criteria is a dual-role one, depending on the strength
of the applied magnetic field. 相似文献
9.
To overcome the shortcomings of extreme time-consuming in solving the Reynolds equation, two efficient calculation methods,
based on the free boundary theory and variational principles for the unsteady nonlinear Reynolds equation in the condition
of Reynolds boundary, are presented in the paper. By employing the two mentioned methods, the nonlinear dynamic forces as
well as their Jacobians of the journal bearing can be calculated saving time but with the same accuracy. Of these two methods,
the one is called a Ritz model which manipulates the cavitation region by simply introducing a parameter to match the free
boundary condition and, as a result, a very simple approximate formulae of oil-film pressure is being obtained. The other
one is a one-dimensional FEM method which reduces the two-dimensional variational inequality to the one-dimensional algebraic
complementary equations, and then a direct method is being used to solve these complementary equations, without the need of
iterations, and the free boundary condition can be automatically satisfied. Meanwhile, a new order reduction method is contributed
to reduce the degrees of freedom of a complex rotor-bearing system. Thus the nonlinear behavior analysis of the rotor-bearing
system can be studied time-sparingly. The results in the paper show the high efficiency of the two methods as well as the
abundant nonlinear phenomenon of the system, compared with the results obtained by the usual numerical solution of the Reynolds
equation. 相似文献
10.
Yury M. Vetyukov 《Journal of Elasticity》2010,98(2):141-158
A new asymptotic approach to the theory of thin-walled rods of open profile is suggested. For the problem of linear static
deformation of a noncircular cylindrical shell we consider solutions, which are slowly varying along the axial coordinate.
A small parameter is introduced in the equations of the modern theory of shells. Conditions of compatibility for the shell
strain measures are employed. The principal terms of the series expansion of the solution are determined from the conditions
of solvability for the minor terms. We conclude the procedure with the subsequent solution for the field of displacements.
The analysis shows that the known equations of thin-walled rods, which were previously obtained with some approximate methods
using hypotheses and approximations of displacements, are asymptotically exact. The presented semi-numerical analysis of the
shell equations allows us to estimate the accuracy of the obtained solution. The results of the paper constitute a sound basis
to the equations of the theory of thin-walled rods and provide trustworthy information concerning the distribution of stresses
in the cross-section. 相似文献
11.
Y. A. Bogan 《Journal of Elasticity》2011,103(2):269-280
By definition, the principal problem of the two-dimensional theory of elasticity consists in solving the equation for the
Airy’s stress function in a region with its first order derivatives assigned at a boundary. In this paper, an indirect formulation
of this problem based on integral equations with weakly singular kernels is proposed. In a bounded region with a Lyapunov
boundary it is reduced to the solution of weakly singular integral equations. Differential properties of its solution are
investigated. 相似文献
12.
This paper presents a detailed analysis on the dynamics of a ring network with small world connection. On the basis of Lyapunov stability approach, the asymptotic stability of the trivial equilibrium is first investigated and the delay-dependent criteria ensuring global stability are obtained. The existence of Hopf bifurcation and the stability of periodic solutions bifurcating from the trivial equilibrium are then analyzed. Further studies are paid to the effects of small world connection on the stability interval and the stability of periodic solution. In particular, some complex dynamical phenomena due to short-cut strength are observed numerically, such as: period-doubling bifurcation and torus breaking to chaos, the coexistence of multiple periodic solutions, multiple quasi-periodic solutions, and multiple chaotic attractors. The studies show that small world connection may be used as a simple but efficient “switch” to control the dynamics of a system. 相似文献
13.
The interior transmission problem (ITP), which plays a fundamental role in inverse scattering theories involving penetrable
defects, is investigated within the framework of mechanical waves scattered by piecewise-homogeneous, elastic or viscoelastic
obstacles in a likewise heterogeneous background solid. For generality, the obstacle is allowed to be multiply connected,
having both penetrable components (inclusions) and impenetrable parts (cavities). A variational formulation is employed to
establish sufficient conditions for the existence and uniqueness of a solution to the ITP, provided that the excitation frequency does not belong
to (at most) countable spectrum of transmission eigenvalues. The featured sufficient conditions, expressed in terms of the
mass density and elasticity parameters of the problem, represent an advancement over earlier works on the subject in that
(i) they pose a precise, previously unavailable provision for the well-posedness of the ITP in situations when both the obstacle
and the background solid are heterogeneous, and (ii) they are dimensionally consistent, i.e., invariant under the choice of
physical units. For the case of a viscoelastic scatterer in an elastic solid it is further shown, consistent with earlier
studies in acoustics, electromagnetism, and elasticity that the uniqueness of a solution to the ITP is maintained irrespective
of the vibration frequency. When applied to the situation where both the scatterer and the background medium are viscoelastic, i.e., dissipative, on the other hand, the same type of analysis
shows that the analogous claim of uniqueness does not hold. Physically, such anomalous behavior of the “viscoelastic-viscoelastic”
case (that has eluded previous studies) has its origins in a lesser known fact that the homogeneous ITP is not mechanically
insulated from its surroundings—a feature that is particularly cloaked in situations when either the background medium or
the scatterer are dissipative. A set of numerical results, computed for ITP configurations that meet the sufficient conditions
for the existence of a solution, is included to illustrate the problem. Consistent with the preceding analysis, the results
indicate that the set of transmission values is indeed empty in the “elastic-viscoelastic” case, and countable for “elastic-elastic”
and “viscoelastic-viscoelastic” configurations. 相似文献
14.
A visco-elastoplastic model for the impact between a compact body and a composite target is presented. The model is a combination
of a nonlinear contact law that includes energy loss due to plastic deformation and a viscous element that accounts for energy
losses due to wave propagation and/or damping. The governing nonlinear equations are solved numerically to obtain the response.
A piecewise linear version of the model is also presented, which facilitates analytical solution. The model predictions are
compared to those of the well-known and commonly used Hunt–Crossley model. The effects of the various impact parameters, such
as impactor mass, velocity, plasticity, and damping, on the impact response and coefficient of restitution are investigated.
The model appears to be suitable for a wide range of impact situations, with parameters that are well defined and easily calculated
or measured. Furthermore, the resulting coefficient of restitution is shown to be a function of impact velocity and damping,
as confirmed by published experimental data. 相似文献
15.
M. S. Matbuly 《Meccanica》2009,44(5):547-554
The present work concerns with the multiple crack propagation along the interface of two bonded dissimilar strips. The crack
faces are subjected to anti-plane shear traction. Galilean transformation is employed to reduce the problem to a quasi-static
one. Then, using Fourier transforms and asymptotic analysis, the quasi-static problem is reduced to a pair of singular integral
equations. That are solved numerically, using Gauss-Chebyshev integration formulae. The values of the dynamic stress intensity
factors are obtained and compared with the previous similar works. Further, a parametric study is introduced to investigate
the effect of crack growth rate, geometric and elastic characteristics of the composite on the values of dynamic stress intensity
factors. 相似文献
16.
17.
Determination of the global responses characteristics of a piecewise smooth dynamical system with contact 总被引:2,自引:0,他引:2
Jun Jiang 《Nonlinear dynamics》2009,57(3):351-361
In this paper the global response characteristics of a piecewise smooth dynamical system with contact, which is specifically
used to describe the rotor/stator rubbing systems, is studied analytically. A method to derive the global response characteristics
of the model is proposed by studying each piece of the equations corresponding to different phases of the rotor motion, i.e.,
the phase without rubbing, the phase with rubbing and the phase of self-excited backward whirl. After solving the typical
responses in each phase and deriving the corresponding existence boundaries in the parameter space, an overall picture of
the global response characteristics of the model is obtained. As is shown, five types of the coexistences of the different
rotor responses and deep insights into the interactive effect of parameters on the dynamic behavior of the model are gained. 相似文献
18.
In this paper the physically-based approach to non-local elasticity theory is introduced. It is formulated by reverting the
continuum to an ensemble of interacting volume elements. Interactions between adjacent elements are classical contact forces
while long-range interactions between non-adjacent elements are modelled as distance-decaying central body forces. The latter
are proportional to the relative displacements rather than to the strain field as in the Eringen model and subsequent developments.
At the limit the displacement field is found to be governed by an integro-differential equation, solved by a simple discretization
procedure suggested by the underlying mechanical model itself, with corresponding static boundary conditions enforced in a
quite simple form. It is then shown that the constitutive law of the proposed model coalesces with the Eringen constitutive
law for an unbounded domain under suitable assumptions, whereas it remains substantially different for a bounded domain. Thermodynamic
consistency of the model also has been investigated in detail and some numerical applications are presented for different
parameters and different functional forms for the decay of the long range forces. For simplicity, the problem is formulated
for a 1D continuum while the general formulation for a 3D elastic solid has been reported in the appendix. 相似文献
19.
E. R. Viana R. M. Rubinger H. A. Albuquerque F. O. Dias A. G. de Oliveira G. M. Ribeiro 《Nonlinear dynamics》2012,67(1):385-392
We report a Periodicity-Detection algorithm, implemented in a LabVIEW routine for real-time data analysis on experimental chaos, to evaluate the periodicity P of experimental time series. The Periodicity-Detector (PD) algorithm was applied to the forced Chua’s circuit with the aim
to build the Periodicity-parameter-space (P-parameter-space). As results of the P-parameter-space, we could observe very complex dynamical behaviors, as regions of periodic structures, a new sequence of
accumulation boundary, and the periodic structures organizing themselves in a period-adding bifurcation cascade. Those results
agree with the maximal Lyapunov exponent and the bifurcation diagram analysis, presented in a previous work. 相似文献
20.
The propagation of plane waves in fibre-reinforced, rotating thermoelastic half-space proposed by Lord-Shulman is discussed.
The problem has been solved numerically using a finite element method. Numerical results for the temperature distribution,
the displacement components and the thermal stress are given and illustrated graphically. Comparisons are made with the results
predicted by the coupled theory and the theory of generalized thermoelasticity with one relaxation time in the presence and
absence of rotation and reinforcement. It is found that the rotation has a significant effect and the reinforcement has great
effect on the distribution of field quantities when the rotation is considered. 相似文献
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