periodic response of piecewise non linear oscillators under harmonic excitation the method of equivalent linearization has been extended to obtain periodic responses of harmonically excited piecewise non linear oscillators a dual representation of the solution is used to greatly enhance the algebraic simplicity the stability analysis of the solutions so obtained is carried out by the method of error propagation these different systems having piecewise non linearity are considered numerical results are compared with those obtained from the simple harmonic balance method and direct numerical integration the proposed method not only gives better results than harmonic balance but is also capable of including super and subharmonics

a periodically forced piecewise linear oscillator a single degree of freedom non linear oscillator is considered the non linearity is in the restoring force and is piecewise linear with a single change in slope such oscillators provide models for mechanical systems in which components make intermittent contact a limiting case in which one slope approaches infinity an impact oscillator is also considered harmonic subharmonic and chaotic motions are found to exist and the bifurcations leading to them are analyzed

stability regions of nonlinear autonomous dynamical systems a topological and dynamical characterization of the stability boundaries for a fairly large class of nonlinear autonomous dynamic systems is presented the stability boundary of a stable equilibrium point is shown to consist of the stable manifolds of all the equilibrium points and or closed orbits on the stability boundary several necessary and sufficient conditions are derived to determine whether a given equilibrium point or closed orbit is on the stability boundary a method for finding the stability region on the basis of these results is proposed the method when feasible will find the exact stability region rather than a subset of it as in the lyapunov theory approach several examples are given to illustrate the theoretical prediction

analytical approximation of the primary resonance response of a periodically excited piecewise non linear linear oscillator an analytical approximate solution is constructed for the primary resonance response of a periodically excited non linear oscillator which is characterized by a combination of a weakly non linear and a linear differential equation without eliminating the secular terms a valid asymptotic expansion solution for the weakly non linear equation is analytically determined for the case of primary resonances then a symmetric periodic solution for the overall system is obtained by imposing continuity and matching conditions the stability characteristic of the symmetric periodic solution is investigated by examining the asymptotic behaviour of perturbations to the steady state solution the validity of the developed analysis is highlighted by comparing the first order approximate solutions with the results of numerical integration of the original equations

asymptotic motions and stability of the elastoplastic oscillator studied via maps abstract the response of the hysteretic oscillator subjected to periodic actions presents aspects that are not yet completely clarified in this paper the elastoplastic oscillator is studied with tools that are typical of nonlinear dynamics as developed in recent years using iterated maps including the poincare map it is shown how the dynamics concerned can be reduced to a one dimensional problem an extended numerical survey shows how the response of the elastoplastic oscillator is always asymptotically stable with the exception of a zero measure set in the space of control variables where it is only stable moreover solutions having a period other than that of the external force do not exist though for some parameter values higher harmonics may assume a very relevant role

an investigation of stability of a control surface with structural nonlinearities in supersonic flow using zubov s method it is well known that the presence of nonlinearities may significantly affect the aeroelastic response of an aerospace vehicle structure in this paper the aeroelastic behaviour at high mach numbers of an all moving control surface with a nonlinearity in the root support is investigated very often under certain flight conditions a stable equilibrium point corresponding to zero displacement of the structure together with an unstable limit cycle arising from a sub critical hopf bifurcation results from the presence of the nonlinearity the dynamic aeroelastic response to external excitation is also of interest and when sinusoidal forcing is applied the stable equilibrium point may then be replaced by a periodic attractor and the limit cycle by an unstable multi periodic solution with or without this forcing there is an attractor which will possess a domain of attraction in this paper the problem of estimating these domains of attraction is tackled using zubov s method in the absence of forcing the method is applied directly to the aeroelastic equations while for the forced system the method of averaging is applied to approximate the aeroelastic equations by an autonomous system the behaviour of the system with forcing is also investigated for flight speeds below a threshold which may occur where the unstable limit cycle of the unforced system disappears in this regime the nonlinear system may nevertheless still possess multiple attractors and their domains of attraction are investigated again using an averaged form of the aeroelastic equations in this study the nonlinearity in the root support was assumed to be due to a cubic hardening restoring moment the zubov approach which always yields conservative estimates was shown to be capable of rapidly giving a good indication of stability domain boundaries under many conditions although this investigation focuses on an aeroelastic system the general form of equations considered arises in many other settings so that the approach would be relevant to a whole range of engineering applications

non linear dynamics of an elasto plastic oscillator with kinematic and isotropic hardening abstract this contribution reports on a dynamic analysis of an elasto plastic oscillator kinematic and isotropic hardening are considered the equations of motion have five state variable associated with complementary conditions system dynamics is treated by performing a split in phase space in two parts this split is suggested by an analysis of the equations of motion near equilibrium points and permits conclusions about high dimensional dynamical system by analyzing subspaces with lower dimension this physical consideration is in close agreement with the operator split technique used for the numerical solution some numerical results are shown for free and forced vibrations of the oscillator with kinematic isotropic and kinematic isotropic hardening

asymptotic analysis of a computational method for time and frequency dependent radiative transfer we consider a time dependent energy dependent nonlinear radiative transfer problem in which opacities are large o 1 and interior sources are small o an asymptotic analysis of this problem as 0 leads to the equilibrium diffusion equation in the interior of the system along with boundary conditions and initial conditions for this equation we apply the same asymptotic analysis to a discrete version of the problem in which the frequency variable is discretized by the multigroup method the direction variable by the discrete ordinates method the time variable by the fully implicit method and the spatial variable by a subcell balance method we find that as 0 the discrete solution satisfies a robust discretized version of the correct equilibrium diffusion equation with boundary conditions and initial conditions that are remarkably accurate the analysis thus predicts that if a spatial grid is chosen that resolves interior temperature gradients then the numerical method obtains an accurate solution in the interior of the system even though the optical thickness of the spatial cells tends to and boundary layers in the transport solution are not resolved we go a step further to analyze problems that are optically thin at some photon frequencies but thick at others and show that once again the discrete solution is remarkably accurate we present numerical results that verify these and other predictions of the analyses

asymptotic diffusion limit of cell temperature discretisation schemes for thermal radiation transport abstract this paper attempts to unify the asymptotic diffusion limit analysis of thermal radiation transport schemes for a linear discontinuous representation of the material temperature reconstructed from cell centred temperature unknowns in a process known as source tilting the asymptotic limits of both monte carlo continuous in space and deterministic approaches based on linear discontinuous finite elements for solving the transport equation are investigated in slab geometry the resulting discrete diffusion equations are found to have nonphysical terms that are proportional to any cell edge discontinuity in the temperature representation based on this analysis it is possible to design accurate schemes for representing the material temperature for coupling thermal radiation transport codes to a cell centred representation of internal energy favoured by ale arbitrary lagrange eulerian hydrodynamics schemes

finite size effects and capillary waves in solid on solid models abstract for a 2d solid on solid model of interfaces with arbitrary nearest neighbour interactions finite size corrections to the surface tension and the interface profile can be traced to the stiffness alone these effects may be ascribed to capillary wave fluctuations governed by the standard capillary wave hamiltonian

periodic forced vibration of unsymmetrical piecewise linear systems by incremental harmonic balance method abstract a single degree of freedom non linear system subjected to periodic excitation is considered the non linearity is in the restoring force and is represented by unsymmetrical piecewise linear stiffness the steady state response are analysed by an extended incremental harmonic balance ihb method all possible subharmonic harmonic and superharmonic resonances and bifurcations are sought the results reveal some interesting details of responses of unsymmetrical piecewise linear systems comparisons with available published findings are also made

asymptotic analysis of the spatial discretization of radiation absorption and re emission in implicit monte carlo we perform an asymptotic analysis of the spatial discretization of radiation absorption and re emission in implicit monte carlo imc a monte carlo technique for simulating nonlinear radiative transfer specifically we examine the approximation of absorption and re emission by a spatially continuous artificial scattering process and either a piecewise constant or piecewise linear emission source within each spatial cell we consider three asymptotic scalings representing i a time step that resolves the mean free time ii a courant limit on the time step size and iii a fixed time step that does not depend on any asymptotic scaling for the piecewise constant approximation we show that only the third scaling results in a valid discretization of the proper diffusion equation which implies that imc may generate inaccurate solutions with optically large spatial cells if time steps are refined however we also demonstrate that for a certain class of problems the piecewise linear approximation yields an appropriate discretized diffusion equation under all three scalings we therefore expect imc to produce accurate solutions for a wider range of time step sizes when the piecewise linear instead of piecewise constant discretization is employed we demonstrate the validity of our analysis with a set of numerical examples

finite size corrections for inclined interfaces in two dimensions exact results for ising and solid on solid models analysis of finite size corrections for the surface tension and surface stiffness coefficients in two dimensional models with inclined interfaces is presented we obtain a universal leading contribution proportional to lnl l for the 2d system of sizel by explicit calculations for restricted and unrestricted solid on solid models and the square lattice ising model we demonstrate the gaussian nature of rough interfaces with fixed ends and derive the leading 1 l type corrections for appropriate surface quantities

asymptotic degeneracy of the transfer matrix spectrum for systems with interfaces relation to surface stiffness and step free energy two and three dimensional ising type systems are considered in the finite cross section cylindrical geometry an interface is forced along the cylinder strip in 2d by the antiperiodic or boundary conditions detailed predictions are presented for the largest asymptotically degenerate set of the transfer matrix eigenvalues for rough interfaces i e for 0 t tc in 2d tr t tc in 3d the eigenvalues are split algebraically and the spectral gaps are governed by thesurface stiffness coefficient for rigid interfaces i e 0 t tr in 3d the eigenvalues are split exponentially with the gaps determined by thestep free energy

application of the centre manifold theory in non linear aeroelasticity abstract in this study a frequency relation for limit cycle oscillations of a two degree of freedom aeroelastic system with structural non linearities represented by cubic restoring spring forces is derived the centre manifold theory is applied to reduce the original system of nine dimensional first order ordinary differential equations to a governing system in two dimensions near the bifurcation point the principle of normal form is used to simplify the non linear terms of the lower dimensional system using the frequency relation and the amplitude frequency relationships derived from a previous study limit cycle oscillations lcos for self excited systems can be predicted analytically the mathematical technique proposed here has been applied to investigate lco near a hopf bifurcation for an aeroelastic system with cubic restoring forces not only that an excellent agreement is obtained compared to the numerical results from solving the original system of eight non linear differential equations by runge kutta time integration scheme but we also demonstrate that the use of a mathematical approach leads to a better understanding of non linear aeroelasticity

exact solution of the discrete 1 1 dimensional sos model with field and surface interactions we present the solution of a linear solid on solid sos model configurations are partially directed walks on a two dimensional square lattice and we include a linear surface tension a magnetic field and surface interaction terms in the hamiltonian there is a wetting transition at zero field and as expected the behavior is similar to a continuous model solved previously the solution is in terms ofq series most closely related to theq hypergeometric functions1 1

exact solution of stable periodic one contact per n cycles motion of a damped linear oscillator contacting a unilateral elastic stop abstract an exact closed form solution is developed for a damped single degree of freedom oscillator contacting a unilateral stop once per n cycles of external sinusoidal force it also includes the effect of a bias force this solution was obtained by using two different displacement and velocity expressions for a two region piecewise linear oscillator continuity of displacement and velocity at the boundary of the two regions and periodicity conditions contact duration is assumed as known and four simultaneous equations in four unknowns are solved exactly the four unknowns are the amplitude of the sinusoidal force phase angle of the assumed first contact entry velocity and exit velocity of mass the stability of motion is checked using the closed form expressions of elements of a 2 2 stability governing matrix these elements are obtained by considering only first order perturbations in periodic motion theoretical predictions based on exact solution agree with previous results and with results obtained using a numerical simulation approach the relationships are studied in detail between input parameters such as amplitude of external force frequency of external force bias force gap spring stiffness and damping constants and output parameters such as contact duration phase angle at contact entry velocity exit velocity maximum displacement and minimum displacement

the steady state responses of s d o f viscous elasto plastic oscillator under sinusoidal loadings abstract depending on the values of the parameters identified the long term behaviour of s d o f viscous elasto plastic oscillator under sinusoidal loadings may be periodic with or without intermediate unloading or elastic shakedown including purely elastic a new phase plane estimation method for the steady state elasto plastic solutions is presented we identify three dimensionless ratios namely damping ratio force ratio rf and frequency ratio rw as well as an elastic phase duration variable y the new estimate offers closed form formulae for the force ratio rf and for the ductility ratio in terms of rw and y applying numerical method to the function rf rf rw y we can obtain the inverse function y y rw rf such that the variation of in terms of rw and rf can be evaluated alternatively for a given ductility ratio we can solve rw y numerically for y and then obtain the curves of rf versus rw to meet specified ductility ratio the proposed method can estimate the steady state responses for any applied load and forcing frequency the results calculated are in very good agreement with the exact time marching solutions a simple criterion of parameters values for elastic shakedown is derived by which we can calculate the maximum driving force amplitude to avoid structures oscillating in the plastic range it is found that there exists a best driving force amplitude for maximizing dissipation efficiency the distribution of periodic points with and without intermediate unloading in the parametric plane rw 1 rf for given damping ratio is clarified the intermediate unloading motion region locates within the range of r w 1 2 3 and its size decreases when damping ratio increases

stability and dynamics of a harmonically excited elastic perfectly plastic oscillator this paper deals with the stability and the dynamics of a harmonically excited elastic perfectly plastic oscillator the hysteretical system is written as a non smooth forced autonomous system it is shown that the dimension of the phase space can be reduced using adapted variables free vibrations of such a system are then considered for the damped system the extended direct method of liapounov is applied to this non smooth mechanical system and the asymptotic stability of the origin is proven in the new phase space the forced vibration of such an oscillator is treated by numerical approach by using the time locating techniques the stability of the limit cycles is analytically investigated with a perturbation approach the boundary between elastoplastic shakedown and alternating plasticity is given in closed form it is shown that this boundary corresponds to a bifurcation boundary for the undamped system period doubling bifurcation finally the equivalent damping of this hysteretic system is characterized from dynamical properties

finite size effects in fluid interfaces abstract finite size effects in interfacial tensions of fluid interfaces are studied at the level of the gaussian model of capillary waves it had been suggested that such effects might play a significant role in surface tension measurements based on capillary rise between plates spaced a small distance d apart the finite size correction varying as 1 d a reconsideration of the thermodynamics of finite size effects followed by careful calculations for the gaussian model leads to a finite size correction varying as 1 d 2 with a universal coefficient the effect is too small to be noticeable in current experiments the calculations have been carried out for rectangular interfaces of arbitrary dimensionality and general shapes subject to many types of boundary conditions they also have implications for finite size effects in interfacial free energies in two and three dimensional ising models which are discussed in connection with recent exact and monte carlo calculations

dirac delta representation by exact parametric equations application to impulsive vibration systems a pair of closed parametric equations are proposed to represent the heaviside unit step function differentiating the step equations results in two additional parametric equations that are also hereby proposed to represent the dirac delta function these equations are expressed in algebraic terms and are handled by means of elementary algebra and elementary calculus the proposed delta representation complies exactly with the values of the definition it complies also with the sifting property and the requisite unit area and its laplace transform coincides with the most general form given in the tables furthermore it leads to a very simple method of solution of impulsive vibrating systems either linear or belonging to a large class of nonlinear problems two example solutions are presented

a linear discontinuous spatial differencing scheme forsnradiative transfer calculations various types of linear discontinuous spatial differencing schemes have been developed for thesn discrete ordinates equations approximating the linear boltzmann transport equation it has been shown through an asymptotic analysis that the 1d slab geometry lumped linear discontinuous scheme not only goes over to a convergent and robust differencing of the diffusion equation in the monoenergetic thick diffusion limit but it also yields the correct interior solution even when boundary layers are left unresolved by the spatial mesh in the present work we generalize this scheme to obtain a 1d slab geometry lumped linear discontinuous scheme for the nonlinear radiative transfer equation and the associated material temperature equation we present a full nonlinear energy dependent asymptotic analysis of the behavior of this scheme in the thick equilibrium diffusion limit we find that this scheme goes over to a convergent and robust differencing of the equilibrium diffusion equation on the interior of the mesh but it does not yield the exact interior solution when boundary layers are left unresolved by the spatial mesh nevertheless the interior solution obtained with spatially unresolved boundary layers is always well behaved and fairly accurate computational results are presented which test the predictions of our asymptotic analysis and demonstrate the efficiency of our solution technique

an iterative procedure for nonlinear flutter analysis an iterative procedure in the frequency domain is presented for flutter analysis of large dynamic systems with multiple structural nonlinearities the major components of the procedure are the describing function approach for system linearization a structural dynamics modification method for shifting system mode shapes and frequencies and a complex eigenvalue algorithm for solution of the flutter equation the purpose of the procedure is to achieve alignment of the oscillatory amplitude in each nonlinear spring with the describing function prediction of stiffness before computing the final stability characteristics the result is a system tuned to the flutter frequency at the time of instability to support the development and validation of the procedure several describing functions are formulated and a quantitative measure of the errors in each is presented validation of the iterative method is accomplished through examples involving dynamic systems of increasing complexity coupled with various representations of the unsteady aerodynamic forces both numerical simulations and experimental data are used to compare with the iterative predictions in the cases studied the agreement is good to excellent with the method accurately predicting the amplitude of a limit cycle flutter as well as the initial disturbance required to produce flutter

asymptotic equilibrium diffusion analysis of time dependent monte carlo methods for grey radiative transfer the equations of nonlinear time dependent radiative transfer are known to yield the equilibrium diffusion equation as the leading order solution of an asymptotic analysis when the mean free path and mean free time of a photon become small we apply this same analysis to the fleck cummings carter forest and n kaoua monte carlo approximations for grey frequency independent radiative transfer although monte carlo simulation usually does not require the discretizations found in deterministic transport techniques monte carlo methods for radiative transfer require a time discretization due to the nonlinearities of the problem if an asymptotic analysis of the equations used by a particular monte carlo method yields an accurate time discretized version of the equilibrium diffusion equation the method should generate accurate solutions if a time discretization is chosen that resolves temperature changes even if the time steps are much larger than the mean free time of a photon this analysis is of interest because in many radiative transfer problems it is a practical necessity to use time steps that are large compared to a mean free time our asymptotic analysis shows that i the n kaoua method has the equilibrium diffusion limit ii the carter forest method has the equilibrium diffusion limit if the material temperature change during a time step is small and iii the fleck cummings method does not have the equilibrium diffusion limit we include numerical results that verify our theoretical predictions

asymptotic diffusion limit of the symbolic monte carlo method for the transport equation we use asymptotic analysis to study the diffusion limit of the symbolic implicit monte carlo simc method for the transport equation for standard simc with piecewise constant basis functions we demonstrate mathematically that the solution converges to the solution of a wrong diffusion equation nevertheless a simple extension to piecewise linear basis functions enables to obtain the correct solution we present numerical examples which illustrate the analysis

on the approximate solution of a piecewise nonlinear oscillator under super harmonic resonance abstract an approximate solution for the super harmonic resonance response of a periodically excited nonlinear oscillator with a piecewise nonlinear linear characteristic is constructed using both a matching method and a modified averaging method the validity of the developed analysis is confirmed by comparing the approximate solutions with the results of direct numerical integration of the original equation

exact single equation closed form solution of vibrating systems with piecewise linear springs abstract a method is proposed to obtain in a single equation the exact closed form displacement response of both free undamped vibrations and the steady state of forced damped vibrations of systems with piecewise linear springs with continuous or discontinuous force deflection relations since the displacement equations are exact the velocity and acceleration responses are obtained by differentiation the solutions are complete they apply for any value of time from zero to infinity the method is designed to be handled in a computer either in conjunction with a symbolic mathematics package like mathematica or maple or without it however for certain relatively simple problems the phase plane solution may be obtained even in a hand held calculator two examples are presented

an investigation of stability in the large behaviour of a control surface with structural non linearities in supersonic flow it is well known that the presence of non linearities may significantly affect the aeroelastic response of an aerospace vehicle structure in this paper the aeroelastic behaviour at high mach numbers of an all moving control surface with a non linearity in the root support is investigated very often a stable equilibrium point corresponding to zero displacement of the structure together with an unstable limit cycle arising from a sub critical hopf bifurcation results from the presence of the non linearity the stable equilibrium point will then possess a domain of attraction in this paper this situation is investigated by first applying the averaging method to obtain a new set of aeroelastic equations in which the limit cycle is replaced by an unstable equilibrium point a fourth order power series approximation to the stable manifold in the neighbourhood of this equilibrium point is then determined from the stable manifold predictions of the domain of attraction of the stable equilibrium point may then be made the method is applied to two examples in which the non linearity in the root support was due to either a cubic hardening restoring moment or the presence of freeplay the approximation to the stable manifold was sufficient to enable significant information about the domain of attraction of the stable equilibrium point of the control surface to be obtained agreement with predictions from numerical integration of the aeroelastic equations in the time domain was shown to be generally good in the cases considered though outside the region of validity of the stable manifold expansion discrepancies will occur the averaging method was shown to be sufficiently accurate for this analysis even when the non linearities could not be considered as weak

lyapunov stability theory of nonsmooth systems this paper develops nonsmooth lyapunov stability theory and lasalle s invariance principle for a class of nonsmooth lipschitz continuous lyapunov functions and absolutely continuous state trajectories computable tests based on filipov s differential inclusion and clarke s generalized gradient are derived the primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right hand side such as in nonsmooth dynamic systems or variable structure control