Enhanced nonlinear 3D Euler–Bernoulli beam with flying support

Using Hamilton’s principle the coupled nonlinear partial differential motion equations of a flying 3D Euler–Bernoulli beam are derived. Stress is treated three dimensionally regardless of in-plane and out-of-plane warpings of cross-section. Tension, compression, twisting, and spatial deflections are nonlinearly coupled to each other. The flying support of the beam has three translational and three rotational degrees of freedom. The beam is made of a linearly elastic isotropic material and is dynamically modeled much more accurately than a nonlinear 3D Euler–Bernoulli beam. The accuracy is caused by two new elastic terms that are lost in the conventional nonlinear 3D Euler–Bernoulli beam theory by differentiation from the approximated strain field regarding negligible elastic orientation of cross-sectional frame. In this paper, the exact strain field concerning considerable elastic orientation of cross-sectional frame is used as a source in differentiations although the orientation of cross-section is negligible.     

Existence and Non-linear Stability of Rotating Star Solutions of the Compressible Euler–Poisson Equations

We prove the existence of rotating star solutions which are steady-state solutions of the compressible isentropic Euler–Poisson (Euler–Poisson) equations in three spatial dimensions with prescribed angular momentum and total mass. This problem can be formulated as a variational problem of finding a minimizer of an energy functional in a broader class of functions having less symmetry than those functions considered in the classical Auchmuty–Beals paper. We prove the non-linear dynamical stability of these solutions with perturbations having the same total mass and symmetry as the rotating star solution. We also prove finite time stability of solutions where the perturbations are entropy-weak solutions of the Euler–Poisson equations. Finally, we give a uniform (in time) a priori estimate for entropy-weak solutions of the Euler–Poisson equations.     

A dynamical model for nonlinear viscoelastic corrugated circular plates with shallow sinusoidal corrugations

An integrated mathematic model and an efficient algorithm on the dynamical behavior of homogeneous viscoelastic corrugated circular plates with shallow sinusoidal corrugations are suggested. Based on the nonlinear bending theory of thin shallow shells, a set of integro-partial differential equations governing the motion of the plates is established from extended Hamilton’s principle. The material behavior is given in terms of the Boltzmann superposition principle. The variational method is applied following an assumed spatial mode to simplify the governing equations to a nonlinear integro-differential variation of the Duffing equation in the temporal domain, which is further reduced to an autonomic system with four coupled first-order ordinary differential equation by introducing an auxiliary variable. These measurements make the numerical simulation performs easily. The classical tools of nonlinear dynamics, such as Poincaré map, phase portrait, Lyapunov exponent, and bifurcation diagrams, are illustrated. The influences of geometric and physical parameters of the plate on its dynamic characteristics are examined. The present mathematic model can easily be used to the similar problems related to other dynamical system for viscoelastic thin plates and shallow shells.     

Canonical forms of Nielsen's and Cenov's dynamical equations

In this paper, with Poincaré's formalism, and an indirect method, the canonical forms of the generalized equations of motion due to Nielsen and Cenov of a holonomic dynamical system in the velocity-phase space and the acceleration-phase space are obtained in terms of the Poincaré parameters This paper was presented at the International Congress of Mathematicians (ICM), 21–29 August, 1990, Kyoto University, Japan.     

Derivation of the Zakharov Equations

This article continues the study, initiated in [27, 7], of the validity of the Zakharov model which describes Langmuir turbulence. We give an existence theorem for a class of singular quasilinear equations. This theorem is valid for prepared initial data. We apply this result to the Euler–Maxwell equations which describes laser-plasma interactions, to obtain, in a high-frequency limit, an asymptotic estimate that describes solutions of the Euler–Maxwell equations in terms of WKB approximate solutions, the leading terms of which are solutions of the Zakharov equations. Due to the transparency properties of the Euler–Maxwell equations evidenced in [27], this study is carried out in a supercritical (highly nonlinear) regime. In such a regime, resonances between plasma waves, electromagnetric waves and acoustic waves could create instabilities in small time. The key of this work is the control of these resonances. The proof involves the techniques of geometric optics of JOLY, MéTIVIER and RAUCH [12, 13]; recent results by LANNES on norms of pseudodifferential operators [14]; and a semiclassical paradifferential calculus.     

Complicated nonlinear responses of a simply supported FGM rectangular plate under combined parametric and external excitations

In this paper, we use the asymptotic perturbation method based on the Fourier expansion and the temporal rescaling to investigate the nonlinear oscillations and chaotic dynamics of a simply supported rectangular plate made of functionally graded materials (FGMs) subjected to a through-thickness temperature field together with parametric and external excitations. Material properties are assumed to be temperature-dependent. Based on the Reddy’s third-order plate theory, the governing equations of motion for the plate are derived using the Hamilton’s principle. The Galerkin procedure is employed to obtain a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms. The resonant case considered here is 1:2 internal resonance, principal parametric resonance-1/2 subharmonic resonance. Based on the averaged equation in polar coordinate form, the stability of steady state solutions is analyzed. The phase portrait, waveform and Poincaré map are used to analyze the periodic and chaotic motions of the FGM rectangular plate. It is found that the FGM rectangular plate exhibits the chaotic motions under certain circumstances. It is seen that the nonlinear dynamic responses of the FGM rectangular plate are more sensitive to transverse excitation. The excitation force can be used as a controlling factor which can change the response of the FGM rectangular plate from periodic motion to the chaotic motion.     

Using quaternion matrices to describe the kinematics and nonlinear dynamics of an asymmetric rigid body

A set of four quaternion matrices is used to represent the equations of finite rotation theory and to describe the kinematics and nonlinear dynamics of an asymmetric rigid body in space. The results obtained are tested in setting up direction-cosine matrices, calculating three-index symbols, establishing the relationship between the components of angular velocity in body-fixed and space-fixed frames of reference, and using a set of three independent rotations. Euler–Lagrange equations and a set of four quaternion matrices are used to construct a block-matrix model describing the nonlinear dynamics of a free asymmetric rigid body in three-dimensional space. The model gives the matrix Euler’s equations of motion and other special cases. Algorithms adapted to use in a numerical experiment are developed Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 133–143, February 2009.     

Investigation of the size effects in Timoshenko beams based on the couple stress theory

In this paper, a size-dependent Timoshenko beam is developed on the basis of the couple stress theory. The couple stress theory is a non-classic continuum theory capable of capturing the small-scale size effects on the mechanical behavior of structures, while the classical continuum theory is unable to predict the mechanical behavior accurately when the characteristic size of structures is close to the material length scale parameter. The governing differential equations of motion are derived for the couple-stress Timoshenko beam using the principles of linear and angular momentum. Then, the general form of boundary conditions and generally valid closed-form analytical solutions are obtained for the axial deformation, bending deflection, and the rotation angle of cross sections in the static cases. As an example, the closed-form analytical results are obtained for the response of a cantilever beam subjected to a static loading with a concentrated force at its free end. The results indicate that modeling on the basis of the couple stress theory causes more stiffness than modeling by the classical beam theory. In addition, the results indicate that the differences between the results of the proposed model and those based on the classical Euler–Bernoulli and classical Timoshenko beam theories are significant when the beam thickness is comparable to its material length scale parameter.     

On the motion of a gyrostat similar to Lagrange’s gyroscope under the influence of a gyrostatic moment vector

In this paper, the problem of the motion of a gyrostat fixed at one point under the action of a gyrostatic moment vector whose components are ℓ _i (i=1,2,3) about the axes of rotation, similar to a Lagrange gyroscope is investigated. We assume that the center of mass G of this gyrostat is displaced by a small quantity relative to the axis of symmetry, and that quantity is used to obtain the small parameter ε (Elfimov in PMM, 42(2):251–258, [1978]). The equations of motion will be studied under certain initial conditions of motion. The Poincaré small parameter method (Malkin in USAEC, Technical Information Service, ABC. Tr-3766, [1959]; Nayfeh in Perturbation methods, Wiley-Interscience, New York, [1973]) is applied to obtain the periodic solutions of motion. The periodic solutions for the case of irrational frequencies ratio are given. The periodic solutions are analyzed geometrically using Euler’s angles to describe the orientation of the body at any instant t of time. These solutions are performed by our computer programs to get their graphical representations.     

Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system

Thermo-mechanical vibrations of a simply supported spring-mass-beam system are investigated analytically in this paper. Taking into account the thermal effects, the nonlinear equations of motion and internal/external boundary conditions are derived through Hamilton’s principle and constitutive relations. Under quasi-static assumptions, the equations governing the longitudinal motion are transformed into functions of transverse displacements, which results in three integro-partial differential equations with coupling terms. These are solved using the direct multiple-scale method, leading to closed-form solutions for the mode functions, nonlinear natural frequencies and frequency–response curves of the system. The influence of system parameters on the linear and nonlinear natural frequencies, mode functions, and frequency–response curves is studied through numerical parametric analysis. It is shown that the vibration characteristics depend on the mid-plane stretching, intra-span spring, point mass, and temperature change.     

Gas-dynamic analogy for vortex free-boundary flows

The classical shallow-water equations describing the propagation of long waves in flow without a shear of the horizontal velocity along the vertical coincide with the equations describing the isentropic motion of a polytropic gas for a polytropic exponent γ = 2 (in the theory of fluid wave motion, this fact is called the gas-dynamic analogy). A new mathematical model of long-wave theory is derived that describes shear free-boundary fluid flows. It is shown that in the case of one-dimensional motion, the equations of the new model coincide with the equations describing nonisentropic gas motion with a special choice of the equation of state, and in the multidimensional case, the new system of long-wave equations differs significantly from the gas motion model. In the general case, it is established that the system of equations derived is a hyperbolic system. The velocities of propagation of wave perturbations are found. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 8–15, May–June, 2007.     

Fractal First-Order Partial Differential Equations

The present paper is concerned with semi-linear partial differential equations involving a particular pseudo-differential operator. It investigates both fractal conservation laws and non-local Hamilton–Jacobi equations. The idea is to combine an integral representation of the operator and Duhamel's formula to prove, on the one hand, the key a priori estimates for the scalar conservation law and the Hamilton–Jacobi equation and, on the other hand, the smoothing effect of the operator. As far as Hamilton–Jacobi equations are concerned, a non-local vanishing viscosity method is used to construct a (viscosity) solution when existence of regular solutions fails, and a rate of convergence is provided. Turning to conservation laws, global-in-time existence and uniqueness are established. We also show that our formula allows us to obtain entropy inequalities for the non-local conservation law, and thus to prove the convergence of the solution, as the non-local term vanishes, toward the entropy solution of the pure conservation law.     

Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method

The literature regarding the free vibration analysis of Bernoulli–Euler and Timoshenko beams under various supporting conditions is plenty, but the free vibration analysis of Reddy–Bickford beams with variable cross-section on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded and semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton’s principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed.     

Damped vibration of a graphene sheet using a higher-order nonlocal strain-gradient Kirchhoff plate model

A higher-order nonlocal strain-gradient model is presented for the damped vibration analysis of single-layer graphene sheets (SLGSs) in hygrothermal environment. Based on Kirchhoff plate theory in conjunction with a higher-order (bi-Helmholtz) nonlocal strain gradient theory, the equations of motion are obtained using Hamilton's principle. The higher-order nonlocal strain gradient theory has lower- and higher-order nonlocal parameters and a material characteristic parameter. The presented model can reasonably interpret the softening effects of the SLGS, and indicates a reasonably good match with the experimental flexural frequencies. Finally, the roles of viscous and structural damping coefficients, small-scale parameters, hygrothermal environment and elastic foundation on the vibrational responses of SLGSs are studied in detail.     

Peculiarities of wave dynamics in media with oscillating inclusions

The article is concerned with mathematical models for media with oscillating inclusions. These models consist of mutually connected equations, one of which is the wave equation for carrying medium and others are equations of motion for partial oscillators. To close these models, we use cubic and nonlocal equations of state for the carrying medium. Travelling wave solutions to these models are studied in detail. Using qualitative analysis methods, the phase space is shown to contain periodic, homo- and heteroclinic trajectories. Moreover, in the case of nonlocal models we observe the creation of quasiperiodic and chaotic regimes. Bifurcations of localized regimes are studied via the Poincaré section technique.     

On the motion of incompressible-medium particles in a domain with a periodically varying boundary

The motion of incompressible-medium particles in a cavity bounded by a plane bottom, vertical lateral walls, and a top boundary deformed in accordance with an arbitrary periodic law is investigated. The problem is reduced to solving the Hamilton equations with a time-periodic Hamiltonian. To study the system, the Hamilton system averaging method and the KAM (Kolmogorov, Arnold, Mozer) theory are used. A novel modification of the averaging procedure using the Poincaré point map is proposed. Correct to an exponentially small cavity boundary deformation amplitude, the Poincaré map points lie on the closed integral curves of an averaged autonomous Hamiltonian system. An asymptotic expansion of the averaged Hamiltonian in the amplitude is written down. The method is applied to the solution of the following problems: (i) the Stokes problem of mass transfer by a progressive wave on the surface of a heavy finite-depth fluid and (ii) the problems of particle motion in a thin layer of viscous or viscoplastic medium with a deformable boundary. For the case of finite amplitudes, qualitative agreement between the results of the asymptotic theory and the numerical calculations is obtained. The reasons for the appearance of a stochastic regime are discussed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 12–19, July–August, 2000. The work received financial support from the Russian Foundation for Basic Research (project 99-01-00250).     

A fractional gradient representation of the Poincaré equations

A gradient representation and a fractional gradient representation of the Poincaré equations are studied. Firstly, the condition presented here for the Poincaré equation can be considered as a gradient system. Then, a condition under which the Poincaré equation can be considered as a fractional gradient system is obtained. Finally, two examples are given to illustrate applications of the result.     

Convex Sobolev Inequalities Derived from Entropy Dissipation

We study families of convex Sobolev inequalities, which arise as entropy–dissipation relations for certain linear Fokker–Planck equations. Extending the ideas recently developed by the first two authors, a refinement of the Bakry–émery method is established, which allows us to prove non-trivial inequalities even in situations where the classical Bakry–émery criterion fails. The main application of our theory concerns the linearized fast diffusion equation in dimensions d ≧ 1, which admits a Poincaré, but no logarithmic Sobolev inequality. We calculate bounds on the constants in the interpolating convex Sobolev inequalities, and prove that these bounds are sharp on a specified range. In dimension d = 1, our estimates improve the corresponding results that can be obtained by the measure-theoretic techniques of Barthe and Roberto. As a by-product, we give a short and elementary alternative proof of the sharp spectral gap inequality first obtained by Denzler and McCann. In further applications of our method, we prove convex Sobolev inequalities for a mean field model for the redistribution of wealth in a simple market economy, and the Lasota model for blood cell production.     

The governing equations for moiré interferometry and their identity to equations of geometrical moiré

The equations prescribing the gradient and inclination of fringes in moiré interferometry are derived from the basic laws of diffraction and interference. A vectorial representation of three-dimensional diffraction employs incidence and emergence vectors in the plane of the grating; the representation is especially well suited for this type of analysis. The corresponding equations for geometrical moiré are derived by a remarkably direct vectorial method. The analyses prove that the patterns of moiré interferometry and geometrical moiré are governed by identical relationships. Paper was presented at 1985 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 9–14, 1985.     

Peridynamic beams: A non-ordinary,state-based model

This paper develops a new peridynamic state based model to represent the bending of an Euler–Bernoulli beam. This model is non-ordinary and derived from the concept of a rotational spring between bonds. While multiple peridynamic material models capture the behavior of solid materials, this is the first 1D state based peridynamic model to resist bending. For sufficiently homogeneous and differentiable displacements, the model is shown to be equivalent to Eringen’s nonlocal elasticity. As the peridynamic horizon approaches 0, it reduces to the classical Euler–Bernoulli beam equations. Simple test cases demonstrate the model’s performance.