Resolving Controversies in the Application of the Method of Multiple Scales and the Generalized Method of Averaging

I compare application of the method of multiple scales with reconstitution and the generalized method of averaging for determining higher-order approximations of three single-degree-of-freedom systems and a two-degree-of-freedom system. Three implementations of the method of multiple scales are considered, namely, application of the method to the system equations expressed as second-order equations, as first-order equations, and in complex-variable form. I show that all of these methods produce the same modulation equations.I address the problem of determining higher-order approximate solutions of the Duffing equation in the case of primary resonance. I show that the conclusions of Rahman and Burton that the method of multiple scales, the generalized method of averaging, and Lie series and transforms might lead to incorrect results, in that spurious solutions occur and the obtained frequency–response curves bear little resemblance to the actual response, is the result of their using parameter values for which the neglected terms are the same order as the retained terms. I show also that spurious solutions cannot be avoided, in general, in any consistent expansion and their presence does not constitute a limitation of the methods. In particular, I show that, for the Duffing equation, the second-order frequency–response equation does not possess spurious solutions for the case of hardening nonlinearity, but possesses spurious solutions for the case of softening nonlinearity. For sufficiently small nonlinearity, the spurious solutions are far removed from the actual response. But as the strength of the nonlinearity increases, these solutions move closer to the backbone and eventually distort it. This is not a drawback of the perturbation methods but an indication of an application of the analysis for parameter values outside the range of validity of the expansion.Also, I address the problem of obtaining non-Hamiltonian modulation equations in the application of the method of multiple scales to multi-degree-of-freedom Hamiltonian systems written as second-order equations in time and how this problem can be overcome by attacking the state-space form of the governing equations. Moreover, I show that application of a variation of the method of Rahman and Burton to multi-degree-of-freedom systems leads to results that do not agree with those obtained with the generalized method of averaging.Contributed by Prof. R.A. Ibrahim.     

Quasiperiodic solutions of the Navier-Stokes equations induced by the quasiperiodic shape of the boundaries of a two-dimensional flow domain

Steady quasiperiodic solutions of the Navier-Stokes equations in an infinite two-dimensional layer with quasiperiodic boundaries are obtained on the Reynolds number range 0 < Re* < 200. The calculations are performed using a spectral-difference method based on the representation of the quasiperiodic solutions in the form of convergent double Fourier series. The properties of these solutions and the distinctive features of their spectra are studied and their fundamental differences from periodic solutions are shown. The possibility of applying the quasiperiodic solutions for modeling flows in fractal layers is discussed.     

A note on the second problem of Stokes for Maxwell fluids

New exact solutions corresponding to the second problem of Stokes for Maxwell fluids have been established by means of Laplace transforms. For large times, these solutions reduce to the well-known steady-state solutions which are periodic in time and independent of the initial conditions. Furthermore, the transient solutions are in accordance with the previous solutions obtained using the Fourier sine transform. The required time to get the steady-state is determined by graphical illustrations. This time decreases if the frequency of the velocity increases. The effects of the material parameters on the decay of the transients in time are also investigated by graphs.     

Invariant solutions of the Westervelt model of nonlinear hydroacoustics without dissipation

We study three-dimensional Westervelt model of a nonlinear hydroacoustics without dissipation. We received all of its invariant submodels. We studied all invariant submodels described by the invariant solutions of rank 0 and 1. All invariant solutions of rank 0 and 1 are found either explicitly, or their search is reduced to the solution of the nonlinear integral equations. With a help of these invariant solutions we researched: (1) a propagation of the intensive acoustic waves (self-similar, axisymmetric, planar and one-dimensional) for which the acoustic pressure and a speed of its change, or the acoustic pressure and its derivative in the direction of one of the axes are specified at the initial moment of the time at a fixed point , (2) a spherically symmetric ultrasonic field for which the acoustic pressure and a speed of its change, or the acoustic pressure and its radial derivative are specified at the initial moment of the time at a fixed point. Solving of the boundary value problems describing these processes is reduced to the solving of nonlinear integral equations. We are established the existence and uniqueness of solutions of these boundary value problems under some additional conditions. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe nonlinear and diffraction effects in ultrasonic fields of a special kind, (2) these solutions can be used as a test solutions in the numerical calculations performed in studies of ultrasonic fields generated by powerful emitters. We found all the conservation laws of the first order for the Westerveld equation written in dimensionless variables.     

Exact solutions of the system of the equations of thermal motion of gas in the rarefied space

We study the model describing thermal motion of gas in the rarefied space. This model can be used, in particular, in the study of the state of the medium behind the front of shock wave after very strong blast, in the study of the processes taking place inside of tornado, in the study of the motion of the gas in outer space. For any given initial distribution of the pressure a specific selection of mass Lagrange variables leads to reduction of the system of differential equations describing this motion to the system, for which the number of independent variables is less on the unit. For the obtained system we found all nontrivial conservation laws of the first order. In addition to the classical conservation laws the system has other conservation laws, which generalizes the energy conservation law. We obtained the exact solutions of this system. These solutions describe a variety of different physical processes taking place in the rarefied medium. Using the symmetry properties of the system we got the generating formulas for the receipt of the new solutions using already found earlier solutions of the system.     

Starting solutions for the oscillating motion of a Maxwell fluid in cylindrical domains

The exact solutions for the motion of a Maxwell fluid due to longitudinal and torsional oscillations of an infinite circular cylinder are determined by means of the Laplace transform. These solutions are presented as sum of the steady-state and transient solutions and describe the motion of the fluid for some time after its initiation. After that time, when the transients disappear, the motion is described by the steady-state solution which is periodic in time and independent of the initial conditions. Finally, by means of graphical illustrations, the required times to reach the steady-state are determined for sine, cosine and combined oscillations of the boundary.     

Invariant submodels of the Westervelt model with dissipation

We study three-dimensional Westervelt model of nonlinear hydroacoustics with dissipation. We received all its invariant submodels. With the help of invariant solutions, we explored some wave processes, specifying their physical meaning. The boundary value problems describing these processes are reduced to the nonlinear integro-differential equations. We established the existence and uniqueness of the solutions of these boundary value problems under some additional conditions. Also we considered the invariant solutions of rank 2 and 3. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe nonlinear and diffraction effects in ultrasonic fields of a special kind, (2) these solutions can be used as a test solutions in the numerical calculations performed in studies of ultrasonic fields generated by powerful emitters.     

基于第二类四边形面积坐标的弹性力学纯弯解及MacNeal局限定理的破解

为了提高有限元的性能,弹性力学的解析解(齐次方程的通解)常常可用作有限元的试探函数。然而单元自由度数与完备的直角坐标解析解个数并不匹配,不完备的试函数会导致单元有方向依赖性。利用新型局部自然坐标——第二类四边形面积坐标QACM-II(S,T),给出了平面问题对应任意方向纯弯曲状态的应力函数解析解,即S3和T3的线性组合,并推导出了这两组应力函数对应的应力、应变和位移解析解。之后,利用QACM-II表示的解析解构造了非对称的平面4节点8自由度单元USQ4,该单元可以同时通过常应力/应变分片检验和纯弯测试,从而破解了MacNeal局限定理对平面低阶单元的限制。     

APPLICATION OF THE MODIFIED METHOD OF MULTIPLE SCALES TO THE BENDING PROBLEMS FOR CIRCULAR THIN PLATE AT VERY LARGE DEFLECTION AND THE ASYMPTOTICS OF SOLUTIONS(Ⅱ)

This paper is a continuation of part (Ⅰ), on the asymptotics behaviors of the series solutions investigated in (Ⅰ). The remainder terms of the series solutions are estimated by the maximum norm.     

An explicit tensor expression for the fundamental solutions of a bimaterial space problem

I.IntroductionWiththedevelopmentofmaterialscience,muchattentiontothestudyonelasticmechanicsproblemsofcompositematerialsisbeingpaidbymoreandmoreresearchersengaginginsolidmechanics.Inmechanicsmodel,manyelasticproblemsformulti-phasematerialarebasedontheonesf…     

Studying the Complex Oscillations of a Star in the Field of a Galaxy

The oscillations of a star in the filed of a galaxy are studied. Initial energy conditions are established under which regular and complex oscillations occur. The latter are different from regular oscillations by the presence of saddle solutions in the trajectory. Unstable oscillations are due to only saddle solutions, with the trajectory tending to infinity     

Classification of the Spatial Equilibria of the Clamped Elastica: Symmetries and Zoology of Solutions

We investigate the configurations of twisted elastic rods under applied end loads and clamped boundary conditions. We classify all the possible equilibrium states of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods. We show that all solutions of the clamped boundary value problem exhibit a π-flip symmetry. The Kirchhoff equations which describe the equilibria of these rods are integrated in a formal way which enable us to describe the boundary conditions in terms of 2 closed form equations involving 4 free parameters. We show that the flip symmetry property is equivalent to a reversibility property of the solutions of the Kirchhoff differential equations. We sort these solutions according to their period in the phase plane. We show how planar untwisted configurations as well as circularly closed configurations play an important role in the classification. This revised version was published online in June 2006 with corrections to the Cover Date.     

An approximate analytical solution of the cylindrical detonation wave generated by the linear explosion

The solution of the cylindrical detonation wave generated by the linear explosion was obtained by numerical method in ref.[1].In this paper,when the ratio of specific heatγ》1 by using the enlargement coordinate method,the first-order analytical solutions are obtained.The perturbation parameter isε=1/γ~2 .The correction of these solutions is checked at the end of this paper.     

Verification of Euler/Navier–Stokes codes using the method of manufactured solutions

The method of manufactured solutions is used to verify the order of accuracy of two finite‐volume Euler and Navier–Stokes codes. The Premo code employs a node‐centred approach using unstructured meshes, while the Wind code employs a similar scheme on structured meshes. Both codes use Roe's upwind method with MUSCL extrapolation for the convective terms and central differences for the diffusion terms, thus yielding a numerical scheme that is formally second‐order accurate. The method of manufactured solutions is employed to generate exact solutions to the governing Euler and Navier–Stokes equations in two dimensions along with additional source terms. These exact solutions are then used to accurately evaluate the discretization error in the numerical solutions. Through global discretization error analyses, the spatial order of accuracy is observed to be second order for both codes, thus giving a high degree of confidence that the two codes are free from coding mistakes in the options exercised. Examples of coding mistakes discovered using the method are also given. Copyright © 2004 John Wiley & Sons, Ltd.     

The analytical solutions for the wave propagation in a stretched string with a moving mass

This paper derives the analytical solutions for a stretched string subjected to a concentrated mass moving at a constant velocity. From the derived analytical solutions of the contact force between the string and the mass, the displacement responses of the string can be easily obtained. The solutions cover an infinite, semi-infinite or finite string subjected to a moving mass at subsonic, sonic or supersonic velocities. For the semi-infinite or finite strings, the solutions for different types of boundary conditions are presented in both a unified form and in the form of a series of exponential and polynomial functions. The formula derived is shown to be correct by comparison with the semi-analytical method.     

DISCUSSION ON THE PERIODIC SOLUTIONS FOR HIGHER-ORDER LINEAR EQUATION OF NEU

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Two-dimensional Hierarchical Models for Prismatic Shells with Thickness Vanishing at the Boundary

We construct variational hierarchical two-dimensional models for elastic, prismatic shells of variable thickness vanishing at boundary. With the help of variational methods, existence and uniqueness theorems for the corresponding two-dimensional boundary value problems are proved in appropriate weighted functional spaces. By means of the solutions of these two-dimensional boundary value problems, a sequence of approximate solutions in the corresponding three-dimensional region is constructed. We establish that this sequence converges in the Sobolev space H¹ to the solution of the original three-dimensional boundary value problem. Mathematics Subject Classifications (2000) 74K20, 74K25.     

Spectrum distribution of the sedond order generalized distributed parameter systems

IntroductionGeneralizeddistributedparametersystemsuchastemperaturedistributioninacompositeheatconductor,signalpropagationinasystemofelectricalcables,voltagedistributioninelectromagneticallycoupledsuperconductivecircuits,wavetransmissionandsoonismoreextensivesystemthandistributedparametersystem[1～3].Itisessentiallydifferentfromdistributedparametersystem .Thestudyofgeneralizeddistributedparametersystemshasonlylastedfortenyears.Thecentresofstudyaremodelling ,solutionfinding ,stability ,poleassig…     

Influence of the Locality of Loading on the Stress Distribution in Anisotropic Shells of Revolution

The class of stress problems for orthotropic shells of revolution loaded along narrow ring zones or by forces concentrated in the meridional direction is analyzed on the basis of a refined model. It is established that the solutions of these two problems for essentially anisotropic shells do not fully agree     

Reproduction of solutions of bidimensional ideal plasticity

The symmetries of a system of differential equations allowed the transformation of its solutions to a solution of this system. New analytical exact solutions of a system of two-dimensional ideal plasticity equations were constructed from two well-known solutions, that for a circular cavity stressed by normal pressure, and Prandtl's solution for a block compressed between perfectly rough plates, for the case where the thickness of the block was rather small. A mechanical sense of new solutions was discussed.