Showing papers in "International Journal of Non-linear Mechanics in 1994"
TL;DR: In this paper, Tien et al. studied the dynamics of a shallow arch subjected to harmonic excitation in the presence of both external and 1:1 internal resonance, and the method of averaging was used to yield a set of autonomous equations of the second-order approximations to the response of the system.
Abstract: In this paper the work presented in Tien et al. [Int. J. Non-Linear Mech.29, 349–366 (1994)] is extended to study the dynamics of a shallow arch subjected to harmonic excitation in the presence of both external and 1:1 internal resonance. The method of averaging is used to yield a set of autonomous equations of the second-order approximations to the response of the system. The averaged equations are numerically examined to study the bifurcation behavior of the shallow arch system. In order to study the system with resonant fixed points, a new global perturbation technique developed by Kovacic and Wiggins [Physica D57, 185–225 (1992)] is used. This technique provides analytical results for the critical parameter values at which the dynamical system, through the Silnikov's type of homoclinic orbits, possesses a Smale horseshoe type of chaos.
111 citations
TL;DR: In this paper, the flow and heat transfer of an incompressible second-order fluid past a stretching sheet using Dandapat and Gupta's boundary layer solution is studied.
Abstract: This paper presents a study of the flow and heat transfer of an incompressible second-order fluid past a stretching sheet using Dandapat and Gupta's boundary layer solution. Two cases are studied, namely, (i) the sheet with constant surface temperature (CST case) and (ii) the sheet with prescribed surface temperature (PST case). The solutions for the temperature and the heat transfer characteristics for large Prandtl number (σ) are obtained using a Runge-Kutta method of fourth order with step size Δz = 0.01.
103 citations
TL;DR: In this paper, a criterion based on the concept of "modal buckling load" is proposed to determine which modes should be included in the analysis when the weighted residuals method is utilized to calculate the limit load.
Abstract: Buckling of initial imperfection sensitive structure — column on a non-linear elastic foundation — is investigated. A criterion based on the concept of “modal buckling load” is proposed to determine which modes should be included in the analysis when the weighted residuals method is utilized to calculate the limit load — maximum load the structure can support — for a given initial deflection. For stochastic analysis, a random field model is suggested for the uncertain initial imperfection, and Monte Carlo simulations are performed to obtain the probability density of the buckling load and the reliability of the column. Finally, a non-stochastic convex model of uncertainty is employed to describe a situation when only limited information is available on uncertain initial deflection, and the minimum buckling load is obtained for this model. The results from both the stochastic and the non-stochastic approaches are derived and critically contrasted.
83 citations
TL;DR: In this paper, Liapunov-Floquet (L-F) transformation matrices have been used for the analysis of general quasilinear systems with periodically varying parameters, where the state vector and the periodic matrix of the linear system equations are expanded in terms of shifted Chebyshev polynomials over the principal period.
Abstract: In this paper, a new analysis technique in the study of general quasilinear systems with periodically varying parameters is presented. The method is based on the fact that all quasilinear periodic systems can be replaced by similar systems whose linear parts are time-invariant, via the well-known Liapunov-Floquet (L-F) transformation. A general technique for the computation of the L-F transformation matrices is outlined. In this technique, the state vector and the periodic matrix of the linear system equations are expanded in terms of the shifted Chebyshev polynomials over the principal period. Such an expansion reduces the original problem to a set of linear algebraic equations from which the state transition matrix can be constructed over the period as an explicit function of time. Application of Floquet theory and use of symbolic software yields the L-F transformation matrix in a form suitable for algebraic manipulations. Once the transformation has been applied, the solution of the resulting system is obtained through an application of the time-dependent normal form theory . The method is suitable for both numerical and symbolic computations and in some cases approximate closed form solutions can be obtained. Two simple examples of quasilinear periodic systems—namely, a commutative system with quadratic nonlinearity and a Mathieu equation with cubic non-linearity—are used to demonstrate the effectiveness of the method. For verification, results obtained from the proposed technique are compared with the numerical solutions computed using a standard Runge-Kutta type algorithm. It is shown that the present technique is applicable to systems where the periodic matrix does not contain a small parameter, which is not the case with averaging and perturbation procedures. It can also be used even for those systems for which the generating solutions do not exist in the classical sense.
51 citations
TL;DR: In this paper, a continuousuum approximation is used to reduce the infinite set of ordinary differential equations of motion to a single approximate, non-linear partial differential equation, and the structure of the propagation and attenuation zones of the linearized system is found to affect the nonlinear localization.
Abstract: Forced localization in a periodic system consisting of an infinite number of coupled non-linear oscillators is examined. A “continuum approximation” is used to reduce the infinite set of ordinary differential equations of motion to a single approximate, non-linear partial differential equation. The structure of the propagation and attenuation zones of the linearized system is found to affect the non-linear localization. Harmonic excitations with general spatial distributions are considered and the localized responses of the chain are studied using exact and asymptotic techniques. Only certain classes of forcing distributions lead to spatial confinement of the forced responses, whereas other types of excitation give rise to spatially periodic or even chaotic harmonic motions of the chain. Systems with weak coupling between particles and/or strong non-linear effects have more profound localization characteristics. The theoretical predictions of the analysis are verified by direct numerical simulations of the equations of motion.
49 citations
TL;DR: In this paper, a non-linear diffusion-convection equation arising from the theory of transport in porous media is analyzed using the Lie group technique, and a complete classification of the functional forms of the transport coefficients is presented for which different symmetry groups are admitted.
Abstract: A non-linear diffusion-convection equation arising from the theory of transport in porous media is analyzed using the Lie group technique. A complete classification of the functional forms of the transport coefficients is presented for which different symmetry groups are admitted. For a number of interesting cases, symmetry reductions are performed leading to exact group-invariant solutions. In particular, the special case of the (integrable) Fokas-Yortsos equation is examined in the light of the “Painleve conjecture” concerning the symmetry reductions of integrable PDEs.
47 citations
TL;DR: In this article, the authors studied the well-posedness of the steady motions problem for a second-grade fluid in a bounded domain, with adherence conditions at the boundary, and proved the existence and uniqueness of steady classical solutions for any value of the normal stress moduli α1 and α2.
Abstract: In this paper we study the well-posedness of the steady motions problem for a second-grade fluid in a bounded domain, with adherence conditions at the boundary. We prove the existence and uniqueness of steady classical solutions for any value of the normal stress moduli α1 and α2, thus showing that the thermodynamical restrictions are not needed for the mathematical problem being well-set. Moreover, we find that such steady motions are exponentially non-linearly stable, provided α1 > 0.
46 citations
TL;DR: The generalized Emden-Fowler equation has a single point symmetry under a certain constraint on ƒ( x ). Although the order of the equation can be reduced by one, integration of the resulting Abel's equation of the second kind in closed form is not generally possible.
Abstract: The generalized Emden-Fowler equation y″ + p(x)y′ + r(x)y = ƒ(x)y″ has a single point symmetry under a certain constraint on ƒ( x ). Although the order of the equation can be reduced by one, integration of the resulting Abel's equation of the second kind in closed form is not generally possible. Under a stronger constraint there exist two symmetries G 1 and G 2 , such that [ G 1 , G 2 ] = ( cst ) G 2 and reduction to quadratures becomes trivial. The special cases n = 2 and n = — 3 are treated in detail.
44 citations
TL;DR: In this article, a similarity analysis of three-dimensional boundary layer equations of a class of non-Newtonian fluids in which the stress is an arbitrary function of rates of strain is made, and conditions for invariance and the form of the stress function for a two-dimensional case are also presented.
Abstract: A similarity analysis of three-dimensional boundary layer equations of a class of non-Newtonian fluids in which the stress is an arbitrary function of rates of strain is made. It is shown that under scaling transformation, for an arbitrary stress function, only 90° of wedge flow leads to similarity solutions, whereas for a specific more restricted form, similarity solutions exist for arbitrary wedge angles. In the case of spiral group transformation, no similarity solutions exist if we force the stress function to remain arbitrary after the transformation, whereas for a specific more restricted form, similarity solutions exist for arbitrary wedge angles. For both transformations, similarity equations for power-law and Newtonian fluids are presented as special cases of the analysis. Finally the conditions for invariance and the form of the stress function for a two-dimensional case are also presented.
43 citations
TL;DR: In this article, the subhannonic motions of thin, axisymmetric, geometrically non-linear circular plates are analyzed and two types of forced sub-annonic motion are detected, namely subharmonic standing waves (SSW) and sub-harmonic travelling waves (STW).
Abstract: The subhannonic motions of thin, axisymmetric, geometrically non-linear circular plates are analyzed. It is well known that such cyclic systems possess pairs of degenerate modes in “1-1” internal resonance, i.e. modes having equal linearized natural frequencies. The non-linear interaction of such a pair of modes is examined by discretizing the non-linear partial differential equations of motion and then investigating the resulting set of non-linear ordinary differential equations analytically and numerically. Two types of forced subhannonic motions are detected, namely subharmonic standing waves (SSW) and subharmonic travelling waves (STW). Moreover, it is found that for sufficiently large values of frequency detuning of the forcing function the STW lose stability in a Hopf bifurcation, leading to quasi-periodic motions of the plate, i.e. to oscillations on a two-torus. The analytical results are confirmed by numerical integrations of the equations of motion and by numerical Poincare maps. The results reported in this work are expected to have applicability on the dynamic analysis and design ofthin, flexible disks, of the type used for data storage in the computer industry.
42 citations
TL;DR: In this paper, an analysis is made of the boundary layer flow of Reiner-Philippoff fluids and a general formulation is given which makes it possible to solve boundary layer equations for any body shape by a finite-difference technique.
Abstract: An analysis is made of the boundary layer flow of Reiner-Philippoff fluids. This work is an extension of a previous analysis by Hansen and Na [A.G. Hansen and T.Y. Na, Similarity solutions of laminar, incompressible boundary layer equations of non-Newtonian fluids. ASME 67-WA/FE-2, presented at the ASME Winter Annual Meeting, November (1967)], where the existence of similar solutions of the boundary layer equations of a class of general non-Newtonian fluids were investigated. It was found that similarity solutions exist only for the case of flow over a 90° wedge and, being similar, the solution of the non-linear boundary layer equations can be reduced to the solution of non-linear ordinary differential equations. In this paper, the more general case of the boundary layer flow of Reiner-Philippoff fluids over other body shapes will be considered. A general formulation is given which makes it possible to solve the boundary layer equations for any body shape by a finite-difference technique. As an example, the classical solution of the boundary layer flow over a flat plate, known as the Blasius solution, will be considered. Numerical results are generated for a series of values of the parameters in the Reiner-Philippoff model.
TL;DR: In this paper, an averaging method is developed to predict periodic solutions of strongly nonlinear and harmonically forced oscillators, restricted to the case of period-1 orbits, where the original governing equation is transformed into an autonomous set of differential equations governing the energy and resonant phase variables.
Abstract: An averaging method is developed to predict periodic solutions of strongly non-linear and harmonically forced oscillators. The analysis is restricted to the case of period-1 orbits. The original governing equation is transformed into an autonomous set of differential equations governing the energy and resonant phase variables. The form of the transformation is given by the unperturbed conservative orbits of the system. The scheme is applied to three examples, the non-linear pendulum, the single-well Duffing oscillator, and the canonical escape oscillator. For these examples, the analysis is performed by using Jacobian elliptic functions. These examples demonstrate the ability of the averaging method to predict both transient and steady-state behavior of the system. The method has been developed in view of studying the large excursions of the response of non-linear systems induced by random perturbations.
TL;DR: In this article, the non-linear dynamics of planar motions of cantilevered pipes conveying fluid is studied via a two-mode discretization of the governing partial differential equation, after first replacing nonlinear inertial terms by equivalent ones in the equations of motion through a perturbation procedure.
Abstract: The non-linear dynamics of planar motions of cantilevered pipes conveying fluid is studied via a two-mode discretization of the governing partial differential equation, after first replacing non-linear inertial terms by equivalent ones in the equations of motion through a perturbation procedure. For hanging cantilevers, as the flow velocity U is incremented to a critical value, the undeformed vertical configuration of the pipe becomes unstable and bifurcates into stable periodic orbits through a Hopf bifurcation. For a “standing” cantilever, in which the flow discharges from the free, upper end, the pipe is statically unstable for small U if the pipe is sufficiently long; it regains stability through a subcritical pitchfork bifurcation at higher U , and this is followed by a Hopf bifurcation and periodic motions at still higher U . For some parameter values these two bifurcations occur simultaneously (double degeneracy). By using centre manifold theory and normal forms, it is shown that heteroclinic cycles exist in the reduced subsystem, suggesting the possible existence of chaotic behaviour. Melnikov computations give guidance as to the likely location of chaotic regimes in the parameter space. The numerical results, in which the existence of chaos is demonstrated, are presented in the form of phase portraits, bifurcation diagrams and Lyapunov exponents.
TL;DR: In this article, a single-degree-of-freedom parametrically excited system coupled with a Lanchester damper, a mass-dashpot device, is studied and the two equations governing the total system are solved using the method of multiple scales for the case of principal parametric resonance.
Abstract: A single-degree-of-freedom parametrically excited system coupled with a Lanchester damper, a mass-dashpot device, is studied. The two equations governing the total system are solved using the method of multiple scales for the case of principal parametric resonance. Steady-state solutions are obtained and the effect of the various system parameters examined. The stability analysis for the steady-state solution is also carried out. Results show that this damper can limit the maximum response of the main system and delay the onset of the force threshold necessary to trigger a non-trivial stable response.
TL;DR: Goodman and Cowin this paper studied the gravity flow of granular materials down a vertical pipe using a continuum model and developed equations for both the case of slip and no-slip at the boundary.
Abstract: The gravity flow of granular materials down a vertical pipe is studied using a continuum model [Goodman and Cowin, J. Fluid Mech. 45 , 321–339 (1971), Goodman and Cowin, Arch. Rational Mech. Anal. 44 , 249–266 (1972), Rajagopal and Massoudi, DOE/PETC/TR-90/3 (1990)]. The equations are developed for both the case of slip and no-slip at the boundary. The resulting boundary value problem is solved numerically. The effect of material parameters (compressibility, volume distribution, and viscous effects) on the volume fraction and velocity in the pipe are studied.
TL;DR: In this article, an analysis of the non-linear equilibrium behavior of thin rectangular plates subjected to in-plane normal loads and/or pressure loads is presented, where transverse displacements are represented as a series of buckling modes, and the difference in snap initiation loads is attributed to fundamental differences in the secondary instability mode shapes for the two cases.
Abstract: An investigation is made into various aspects of the snap phenomenon in buckled plates, wherein a plate loaded in postbuckling encounters a secondary instability which initiates a dynamic snap to a different waveform. First, a method is presented for the analysis of the geometrically non-linear equilibrium behavior of thin rectangular plates subjected to in-plane normal loads and/or pressure loads. Transverse displacements are represented as a series of buckling modes. The approach is similar to that used in numerous other investigations, but is significant in that it provides the ability to model a variety of boundary conditions and combined-load scenarios using a single, unified method. This method is then used to analyze the snap behavior of a uniaxially loaded aluminum plate of length-to-width ratio 5.4 which was previously tested and observed to snap in postbuckling. It is found that the snap-initiation load is significantly less when the loaded ends are clamped than when the loaded ends are simply supported, and the difference in snap initiation loads is attributed to fundamental differences in the secondary-instability mode shapes for the two cases. Geometric imperfections of small amplitude are found to affect greatly the snapinitiation load, and can even change the fundamental character of buckling and postbuckling response. Two approximate methods which have been used in the literature for predicting sudden changes in the postbuckling waveform are also discussed in the context of the current work.
TL;DR: In this paper, solutions to the fourth-order non-linear system arising in combined free and forced convection flow of a second-order fluid, over a stretching sheet, are obtained.
Abstract: Solutions to the fourth-order non-linear systems arising in combined free and forced convection flow of a second-order fluid, over a stretching sheet, are obtained. Existence (or non-existence) and uniqueness (or non-uniqueness) results of the problem are obtained and discussed. Moreover, ranges of parametric values are obtained for which the system has a unique pair of solutions, a double pair of solutions, and infinitely many monotonically decaying solutions at infinity.
TL;DR: In this paper, the non-linear free vibrations of simply supported curved orthotropic panels are modeled using the Donnell-Mushtari-Vlasov shell relationships and a combination of the Galerkin procedure and the Lindstedt-Poincare perturbation technique is used to construct an approximate solution to the resulting nonlinear equations of motion.
Abstract: This paper studies the non-linear free vibrations of simply supported curved orthotropic panels. The panels are modeled using the Donnell-Mushtari-Vlasov shell relationships. A combination of the Galerkin procedure and the Lindstedt-Poincare perturbation technique is used to construct an approximate solution to the resulting non-linear equations of motion. Algebraic manipulations show that the panel exhibits a non-linear response only when both the involved axial and circumferential modes are axisymmetric. Numerical studies of a Graphite/Epoxy panel show that its response is softening, i.e. the non-linear natural frequency decreases as the amplitude of motion increases. They also show that the lower modes are more non-linear than the higher, mainly flexural modes. The presented results also show that for the studied panels, the non-linear effects are the strongest for shallow, thin, and short panels.
TL;DR: Closed-form solutions from elliptic integrals with the usual assumption of inextensional elastica are derived for large deflections of specially orthotropic and mid-surface symmetric laminated circular springs in the form of ring under in-plane and uniaxial tension as mentioned in this paper.
Abstract: Closed-form solutions from elliptic integrals with the usual assumption of inextensional elastica are derived for large deflections of specially orthotropic and mid-surface symmetric laminated circular springs in the form of ring under in-plane and uniaxial tension. A numerical procedure including the effects of flexural bending, mid-surface stretching and through-the-thickness parabolic distribution of transverse shear deformations has also been applied to study the non-linear problem. Comparison studies of the results obtained from the two methods are made with experimental data and the results are found to be in good agreement as far as the present investigation is concerned. Non-linear spring behavior are obtained and verified by experimental testing. Three E-glass woven cloth/epoxy composite springs with different aspect ratios are tested. The influence of elastic to shear modulus ratios and radius to thickness ratios to deformation are considered and discussed.
TL;DR: In this article, a perturbation theoretic approach of the Fokker-Planck-Kolmogorov equation is used to determine the eigenfunctions and eigenvalues of a general class of linear multidimensional systems.
Abstract: The method of stochastic averaging has been developed and applied in the past mainly based on Stratonovich-Khasminskii theorem. We examine in this paper the application of this method in the case of arbitrary colored Gaussian excitations, which can be considered as the output of multidimensional linear filters to white Gaussian noise. The method used is based on a perturbation theoretic approach of the Fokker-Planck-Kolmogorov equation, which governs the response probability density function. First, for oscillators with linear elastic forces and non-parametric excitation, it is shown that, to leading order of perturbation, the results obtained match those derived by application of Stratonovich-Khasminskii theorem in the case of broad-band excitation. Then, more general results are derived for nearly Hamiltonian systems perturbed by parametric excitations of uncorrelated colored noises. It is shown that the state probability density function is governed by a reduced equation in the “slow” Hamiltonian variable only, which depends on a number of parameters characterizing the colored noise excitations. Several examples are given for illustration. As a preliminary to these theoretical developments, the problem of determining the eigenfunctions and eigenvalues of the Fokker-Planck operator is addressed for a general class of linear multidimensional systems.
TL;DR: In this paper, a pendulum is attached to one mass of a chain of a masses, connected by n linear springs, and one of the masses is harmonically excited; the stability of the semi-trivial solutions, representing vibration of n masses without pendulum oscillation, is investigated in general.
Abstract: A pendulum is attached to one mass of a chain of a masses, connected by n linear springs. One of the masses is harmonically excited. The stability of the semi-trivial solutions, representing vibration of n masses without pendulum oscillation, is investigated in general. Using this approach, the occurrence of all autoparametric resonances can be determined. As an illustration, a two-mass subsystem with two degrees of freedom, where the pendulum is attached to the upper mass, is analysed.
TL;DR: In this article, the global dynamics of free non-linear vibration of thin rotating rings are investigated and a complete picture of the dynamics is provided, including stability considerations and effects from internal and external damping.
Abstract: Global dynamics of free non-linear vibration of thin rotating rings are investigated. The rings are circular and exhibit in-plane motions, involving two bending modes with the same circumferential wave number. First, a set of weakly non-linear equations of motion is derived by employing an energy principle. Then the emphasis is placed on obtaining averaged equations for the amplitudes and phases of the modes examined. For a rapidly rotating ring the resulting equations do not involve internal resonance. However, as the spin speed is decreased to small values, the case of the slowly rotating ring is approached, involving 1:1 internal resonance. Finally, the vibration of the stationary ring is obtained as a limiting case of the slowly rotating ring. Based on these sets of averaged equations, the possible types of solutions are first determined and their geometrical and physical interpretation is provided. Then the focus is shifted towards investigating the interrelation and transition between the motions of the stationary, the slowly rotating and the rapidly rotating ring. A complete picture of the dynamics is provided, including stability considerations and effects from internal and external damping.
TL;DR: For axial deformation of rotating rods, it was shown in this article that unless this linear model is valid for large strain, it is impossible to determine whether this instability really exists; because, as the angular speed is increased, the strain becomes large well short of the critical speed.
Abstract: For strain sufficiently small such that Hooke's Law is valid, it is shown that only a linear model for axial deformation of rotating rods can be derived. As discussed in the literature, this linear model exhibits an instability when the angular speed reaches a certain critical value. However, unless this linear model is valid for large strain, it is impossible to determine whether this instability really exists; because, as the angular speed is increased, the strain becomes large well short of the critical speed. Next, axial deformation of rotating rods is analyzed using two strain energy functions to model non-linear elastic behavior. The first of these functions is the usual quadratic strain energy function augmented with a cubic term. With this model it is shown that no instability exists if the non-linearity is stiffening (i.e. if the coefficient of the cubic term is positive), although the strain can become large. If the non-linearity is of the softening variety, then the critical angular speed drops as the degree of softening increases. Still, the strains are large enough that, except for rubber-like materials, a non-linear elastic model is not likely to be appropriate. The second strain energy function is based on the square of the logarithmic strain and yields a softening model. It quite accurately models the behavior of certain rubber rods which exhibit the instability within the validated range of elongation.
TL;DR: In this article, acceleration waves were investigated in a thermo-microstretch fluid and it was shown that only longitudinal waves may occur in the case of a single-dimensional model.
Abstract: Acceleration waves are investigated in a thermo-microstretch fluid. It turns out that only longitudinal waves may occur.
TL;DR: In this article, a constitutive relationship for the stress tensor of granular materials during the rapid flow regime that includes the effects of density gradient and fluctuating velocity is proposed.
Abstract: A constitutive relationship for the stress tensor of granular materials during the rapid flow regime that includes the effects of density gradient and fluctuating velocity is proposed. Explicit expressions for various stress components are obtained and simple examples are presented. It is shown that this continuum-based model is capable of predicting normal stress differences. The model also contains many earlier models as its special cases. The results also show that the model may be suitable for analyzing rapid, as well as slow, motions of granular materials.
TL;DR: In this paper, the authors proposed the differential quadrature method (DQM) to solve the Navier-Stokes equation without time-dependence, and showed that the DQM can be used for the inhomogeneous biharmonic stream function.
Abstract: Many numerical solutions have been obtained for the problem of a steady state flow of viscous incompressible fluid in a driven cavity. Unlike most finite difference methods (FDMs), the differential quadrature method (DQM) uses new iterative techniques for numerically solving the Navier-Stokes equation without time-dependence. The DQM solves directly for the inhomogeneous biharmonic stream function, and the applicability of the method is demonstrated. The method of differential quadrature allows for simpler formulations and less computational effort compared with finite difference methods and leads to quite accurate results for a coarse mesh model at low Reynolds number.
TL;DR: In this paper, a comprehensive investigation of the bifurcational behavior of a pre-buckled beam is presented, where the Galerkin method is applied to convert the partial differential equation into a set of ordinary differential equations.
Abstract: A comprehensive investigation of the bifurcational behaviour of a pre-buckled beam is presented. The Galerkin method is applied to convert the partial differential equation into a set of ordinary differential equations. A two-mode solution is sought that includes both symmetric and antisymmetric modes of the structural system. Both the pre-buckled amplitude and external loading are used as control parameters in the bifurcational plane. Qualitative as well as quantitative measures of critical points are obtained. Catastrophes associated with the global bifurcation are established. In the light of various established theories, global bifurcational analysis is performed. Further, the existence of a chaotic attractor is shown and routes to the chaos are interpreted using catastrophe classifications.
TL;DR: In this paper, the accuracy of a new linearization technique in contrast to the classical linearization scheme is examined for a Duffing oscillator under colored noise excitation, and the results obtained by the two linearization schemes are compared in terms of percentage error with reference to the numerical results obtained through Monte Carlo simulation.
Abstract: The accuracy of a new linearization technique in contrast to the classical linearization scheme is examined for a Duffing oscillator under colored noise excitation. The results obtained by the two linearization schemes are compared in terms of percentage error with reference to the numerical results obtained through Monte Carlo simulation. This application confirms a superior performance of new linearization technique compared to the classical one.
TL;DR: In this article, the main cause of concrete pavement blowups are axial compression forces induced into the pavement by a rise in temperature and moisture, and the analysis is similar to the one used by Kerr and Dallis and Kerr and Shade.
Abstract: The main cause of concrete pavement blowups are axial compression forces induced into the pavement by a rise in temperature and moisture. Recently, Kerr and Dallis and Kerr and Shade presented analyses based on the notion that blowups are caused by lift-off buckling of the pavement. The cases analyzed were: (1) continuously reinforced concrete pavements and (2) concrete pavements weakened by a transverse joint or crack. The present paper contains an analysis of another case, when a long continuously reinforced concrete pavement adjoins a rigid structure, like a bridge abutment. The analysis is similar to the one used by Kerr and Shade. The resulting formulation is non-linear and is solved exactly, in closed form. The obtained results are evaluated numerically and are compared with those of a long continuously reinforced pavement, in order to show the effect of the rigid structure on the pavement response.
TL;DR: In this paper, it is shown that the stochastic differential equation is generally associated with singular boundaries, in the sense that either the diffusion coefficient vanishes, or the drift coefficient becomes unbounded.
Abstract: The investigation of motion stability of a non-linear system can be simplified by focusing on a crucial variable, such as the amplitude or the total energy of the dominant mode. For a randomly excited system, this can be accomplished using the stochastic averaging or quasi-conservative averaging procedure. It is shown that the stochastic differential equation so obtained is generally associated with singular boundaries, in the sense that either the diffusion coefficient vanishes, or the drift coefficient becomes unbounded. The condition for a dynamical system to be asymptotically stable in probability can be determined from the behaviors of sample functions at or near the two boundaries. Criteria are established to characterize various sample behaviors near each type of singular boundary. A procedure is also developed to obtain the conditions for the asymptotic stability of the statistical moments. Examples are given for illustration.