direct parameter estimation for linear and non linear structures abstract a method is presented for directly estimating the physical parameter matrices i e mass stiffness and damping of linear and non linear structures by using measured time data the structure is modelled by an equivalent lumped parameter system with symmetric parameter matrices a least squares strategy which makes use of these symmetry relations is adopted to allow the estimation of all the parameters and requires excitation at only one location the validity of the method is demonstrated by using both computer simulation and experimental procedures

experimental modal analysis of non linear systems a feasibility study experimental modal analysis techniques are based on the theory of linear systems in this paper the feasibility of their application to non linear systems is examined a three degree of freedom non linear mechanical system was modeled on an analog computer and system responses due to band limited random excitation were examined by using a two channel fft analyzer frequency response functions natural frequencies and model responses were studied experimentally for a number of example cases this study shows that the measured transfer function plots can alert one to the fact that the structure is non linear measured modal response data base can still give a rough idea about the system behavior in addition measured coherence function estimates can be used as the warning singls for inherent non linearities

forced harmonic response analysis of nonlinear structures using describing functions a semianalytical quasilinear method based on the describing function formulation is proposed for the harmonic response analysis of structures with symmetrical nonlinearities the equations of motion are converted to a set of nonlinear algebraic equations the linear and nonlinear parts of the structure are dealt with separately the former being represented by the constant linear receptance matrix and the latter by the generalized quasilinear matrix which is updated at each iteration several examples dealing with cubic stiffness piecewise linear stiffness and coulomb friction type of nonlinearities are presented

on a micro macro model for polymeric fluids near equilibrium in this paper we study a micro macro model for polymeric fluid the system involves coupling between the macroscopic momentum equation and a microscopic evolution equation describing the combined effects of the microscopic potential and thermofluctuation we employ an energetic variation procedure to explore the relation between the macroscopic transport of the particles and the induced elastic stress due to the microscopic structure for the initial data not far from the equilibrium we prove the global existence and uniqueness of classical solutions to the system 2006 wiley periodicals inc

formulae for the derivatives of heat semigroups abstract formulae for the derivatives of solutions of diffusion equations are derived which clearly exhibit and allow estimation of the equations smoothing properties these also give formulae for the logarithmic gradient of the corresponding heat kernels extending and giving a very elementary proof of bismut s well known formula corresponding formulae are derived for solutions of heat equations for differential forms and their exterior derivatives

global regularity for a birkhoff rott α approximation of the dynamics of vortex sheets of the 2d euler equations abstract we present an regularization of the birkhoff rott equation induced by the two dimensional euler equations for the vortex sheet dynamics we show that an initially smooth self avoiding vortex sheet remains smooth for all times under the regularized dynamics provided the initial density of vorticity is an integrable function over the curve with respect to the arc length measure

on the orthogonalised reverse path method for nonlinear system identification abstract the problem of obtaining the underlying linear dynamic compliance matrix in the presence of nonlinearities in a general multi degree of freedom mdof system can be solved using the conditioned reverse path crp method introduced by richards and singh 1998 journal of sound and vibration 213 4 pp 673 708 the crp method also provides a means of identifying the coefficients of any nonlinear terms which can be specified a priori in the candidate equations of motion although the crp has proved extremely useful in the context of nonlinear system identification it has a number of small issues associated with it one of these issues is the fact that the nonlinear coefficients are actually returned in the form of spectra which need to be averaged over frequency in order to generate parameter estimates the parameter spectra are typically polluted by artefacts from the identification of the underlying linear system which manifest themselves at the resonance and anti resonance frequencies a further problem is associated with the fact that the parameter estimates are extracted in a recursive fashion which leads to an accumulation of errors the first minor objective of this paper is to suggest ways to alleviate these problems without major modification to the algorithm the results are demonstrated on numerically simulated responses from mdof systems in the second part of the paper a more radical suggestion is made to replace the conditioned spectral analysis which is the basis of the crp method with an alternative time domain decorrelation method the suggested approach the orthogonalised reverse path orp method is illustrated here using data from simulated single degree of freedom sdof and mdof systems

nonexistence of self similar singularities in the ideal magnetohydrodynamics in this paper we exclude the scenario of the apparition of finite time singularity in the form of self similar singularities in the ideal magnetohydrodynamic equations assuming suitable integrability conditions on the vorticity and the magnetic field we also consider the more refined possibility of asymptotically self similar singularities where the local classical solution converges to the self similar profile as we approach the possible time of singularity the scenario of asymptotically self similar singularity is also excluded under suitable conditions on the profile in the two dimensional magnetohydrodynamics the magnetic field evolution equations reduce to a divergence free transport equation for a scalar stream function this helps us to improve the above nonexistence theorems on the self similar singularities in the sense that we require merely weaker integrability conditions on the profile in order to prove the results

an eulerian lagrangian approach to the navier stokes equations we present a formulation of the incompressible viscous navier stokes equation based on a generalization of the inviscid weber formula in terms of a diffusive back to labels map and a virtual velocity we derive a generalization of the inviscid cauchy formula and obtain certain bounds for the objects introduced

conjugate flow action functionals we present a new general framework to construct an action functional for a non potential field theory the key idea relies on representing the governing equations relative to a diffeomorphic flow of curvilinear coordinates which is assumed to be functionally dependent on the solution field such flow which will be called the conjugate flow evolves in space and time similarly to a physical fluid flow of classical mechanics and it can be selected in order to symmetrize the g teaux derivative of the field equations with respect to suitable local bilinear forms this is equivalent to requiring that the governing equations of the field theory can be derived from a principle of stationary action on a lie group manifold by using a general operator framework we obtain the determining equations of such manifold and the corresponding conjugate flow action functional in particular we study scalar and vector field theories governed by second order nonlinear partial differential equations the identification of transformation groups leaving the conjugate flow action functional invariant could lead to the discovery of new conservation laws in fluid dynamics and other disciplines

a stochastic representation for backward incompressible navier stokes equations by reversing the time variable we derive a stochastic representation for backward incompressible navier stokes equations in terms of stochastic lagrangian paths which is similar to constantin and iyer s forward formulations in constantin and iyer comm pure appl math lxi 330 345 2008 using this representation a self contained proof of local existence of solutions in sobolev spaces are provided for incompressible navier stokes equations in the whole space in two dimensions or large viscosity an alternative proof to the global existence is also given moreover a large deviation estimate for stochastic particle trajectories is presented when the viscosity tends to zero

a geometrical study of 3d incompressible euler flows with clebsch potentials a long lived euler flow and its power law energy spectrum abstract a simple initial condition for vorticity sin y z sin z x sin x y which has clebsch potentials has been identified to lead to a flow evolution with a very weak energy transfer this allows us to integrate the euler equations in time longer than commonly expected to reach a stage at which the total enstrophy attains its peak for the corresponding navier stokes flow it thereby enables us to study the relationship between the inviscid limit and totally inviscid behaviours numerically in spite of small energy dissipation rate the navier stokes flow shows a power law spectrum whose exponent is around 5 3 and 2 a similar behaviour is also observed for the euler flow in physical space this flow has groups of vorticity layers which hesitate to roll up

probabilistic approach to the lame equations of linear elastostatics abstract a probabilistic approach to systems of partial differential equations is developed on the basis of the well known feynman kac and bismut formulas providing explicit probabilistic representations of the solutions and of their derivatives of scalar differential equations some numerical examples are also included in particular the lame equations of elastostatics are solved and the results are compared with some known exact analytic solutions to demonstrate the efficiency of the approach

dimension free quasiconformal distortion in n space most distortion theorems for k quasiconformal mappings in rn n 2 depend on both n and k in an essential way with bounds that become infinite as n tends to oo the present authors obtain dimension free versions of four well known distortion theorems for quasiconformal mappings namely bounds for the linear dilatation the schwarz lemma the e distortion theorem and the r1 quasisymmetry property of these mappings they show that the upper estimates they have obtained in each of these four main results remain bounded as n tends to oo with k fixed the proofs are based on a dimensioncancellation property of the function t h r 1 r t k t 0 k 0 where r t is the capacity of a teichmiiller extremal ring in rn the authors also prove a dimension free distortion theorem for the absolute cross ratio under k quasiconformal mappings of n from which several other distortion theorems follow as special cases

singularities of euler flow not out of the blue does three dimensional incompressible euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time this blowup problem is still open after briefly reviewing what is known and pointing out some of the difficulties we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations we present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup

a large deviations principle for stochastic flows of viscous fluids abstract we study the well posedness of a stochastic differential equation on the two dimensional torus t 2 driven by an infinite dimensional wiener process with drift in the sobolev space l 2 0 t h 1 t 2 the solution corresponds to a stochastic lagrangian flow in the sense of diperna lions by taking into account that the motion of a viscous incompressible fluid on the torus can be described through a suitable stochastic differential equation of the previous type we study the inviscid limit by establishing a large deviations principle we show that as the viscosity goes to zero the lagrangian stochastic navier stokes flow approaches the euler deterministic lagrangian flow with an exponential rate function

an eulerian lagrangian approach for incompressible fluids local theory we study a formulation of the incompressible euler equations in terms of the inverse lagrangian map in this formulation the equations become a first order advective nonlinear system of partial differential equations

hilbert transform and its engineering applications the hilbert transform is useful in calculating instantaneous attributes of a time series especially the envelope amplitude and instantaneous frequency the instantaneous envelope is the amplitude of the complex hilbert transform the instantaneous frequency is the time rate of change of the instantaneous phase angle these properties can be applied to identify dynamic characteristics of a linear as well as a nonlinear system however the conventional discrete hilbert transform which is based on fast fourier transform and inverse fourier transform has shown the lack of accuracy for time derivative calculations in this paper we first introduce the hilbert transform and its applications to the nonlinear system parameter identification then we address the practical issues in applying the hilbert transform to engineering applications to increase the accuracy of the envelope detection we propose a hilbert transform technique based on local maxima interpolation analyses and simulations are carried out to demonstrate the advantages of the proposed technique finally we employ the proposed local maxima interpolation technique in identifying the nonlinear dynamic characteristics of industrial examples

novel solutions of the helmholtz equation and their application to diffraction this paper presents a series of novel representations for the solutions of the helmholtz equation in a broad class of wedge like domains including those with curvilinear non flat faces these repr

a stochastic lagrangian representation of the three dimensional incompressible navier stokes equations in this paper we derive a probabilistic representation of the deterministic three dimensional navier stokes equations based on stochastic lagrangian paths the particle trajectories obey sdes driven by a uniform wiener process the inviscid weber formula for the euler equations of ideal fluids is used to recover the velocity field this method admits a self contained proof of local existence for the nonlinear stochastic system and can be extended to formulate stochastic representations of related hydrodynamic type equations including viscous burgers equations and lagrangian averaged navier stokes alpha models 2007 wiley periodicals inc

finite difference schemes for incompressible flows in the velocity impulse density formulation we consider finite difference schemes based on the impulse density variable we show that the original velocity impulse density formulation of oseledets is marginally ill posed for the inviscid flow and this has the consequence that some ordinarily stable numerical methods in other formulations become unstable in the velocity impulse density formulation we present numerical evidence of this instability we then discuss the construction of stable finite difference schemes by requiring that at the numerical level the nonlinear terms be convertible to similar terms in the primitive variable formulation finally we give a simplified velocity impulse density formulation which is free of these complications and yet retains the nice features of the original velocity impulse density formulation with regard to the treatment of boundary we present numerical results on this simplified formulation for the driven cavity flow on both the staggered and non staggered grids

experimental validation of the constant level method for identification of non linear multi degree of freedom systems system identification for non linear dynamical systems could find use in many applications such as condition monitoring finite element model validation and determination of stability the effectiveness of existing non linear system identification techniques is limited by various factors such as the complexity of the system under investigation and the type of non linearities present in this work the constant level identification approach which can identify multi degree of freedom systems featuring any type of non linear function including discontinuous functions is validated experimentally the method is shown to identify accurately an experimental dynamical system featuring two types of stiffness non linearity the full equations of motion are also extracted accurately even in the presence of a discontinuous non linearity

a probabilistic interpretation and stochastic particle approximations of the 3 dimensional navier stokes equations we develop a probabilistic interpretation of local mild solutions of the three dimensional navier stokes equation in the l p spaces when the initial vorticity field is integrable this is done by associating a generalized nonlinear diffusion of the mckean vlasov type with the solution of the corresponding vortex equation we then construct trajectorial chaotic stochastic particle approximations of this nonlinear process these results provide the first complete proof of convergence of a stochastic vortex method for the navier stokes equation in three dimensions and rectify the algorithm conjectured by esposito and pulvirenti in 1989 our techniques rely on a fine regularity study of the vortex equation in the supercritical l p spaces and on an extension of the classic mckean vlasov model which incorporates the derivative of the stochastic flow of the nonlinear process to explain the vortex stretching phenomenon proper to dimension three

an unconstrained hamiltonian formulation for incompressible fluid flow the equations governing the time evolution of an ideal fluid in material coordinates are expressed as an unconstrained canonical hamiltonian system the incompressibility of the flow is consequent upon certain first integrals of the motion the variable conjugate to the configuration field is not the usual linear momentum but is instead a quantity that is related to linear momentum through an auxiliary scalar field whose time derivative is the pressure the definition of the hamiltonian involves a minimization with respect to this auxiliary field the method of derivation may be generally applied to obtain unconstrained hamiltonian descriptions of lagrangian field equations subject to pointwise constraints

filtered viscous fluid equations abstract we analyze an eulerian lagrangian description of filtered incompressible fluid equations a viscous cauchy formula is derived for such equations the models discussed are related to vortex methods and to filtered kuzmin oseledets equations

on a leray α model of turbulence in this paper we introduce and study a new model for three dimensional turbulence the leray model this model is inspired by the lagrangian averaged navier stokes model of turbulence also known navier stokes model or the viscous camassa holm equations as in the case of the lagrangian averaged navier stokes model the leray model compares successfully with empirical data from turbulent channel and pipe flows for a wide range of reynolds numbers we establish here an upper bound for the dimension of the global attractor the number of degrees of freedom of the leray model of the order of l l d 12 7 where l is the size of the domain and l d is the dissipation length scale this upper bound is much smaller than what one would expect for three dimensional models i e l l d 3 this remarkable result suggests that the leray model has a great potential to become a good sub grid scale large eddy simulation model of turbulence we support this observation by studying analytically and computationally the energy spectrum and show that in addition to the usual k 5 3 kolmogorov power law the inertial range has a steeper power law spectrum for wavenumbers larger than 1 finally we propose a prandtl like boundary layer model induced by the leray model and show a very good agreement of this model with empirical data for turbulent boundary layers

identification of dynamic characteristics of nonlinear joint based on the optimum equivalent linear frequency response function abstract identification of dynamic characteristics of local nonlinearities has been aimed in this paper the spirit of the identification method is established on optimum equivalent linear frequency response function oelf dynamic behavior of nonlinear elements in system is extracted from oelf using two different techniques the first technique is direct identification method dim in which no pre assumed model is considered for the nonlinearity s behavior and the second technique is model based identification method mim the second technique is introduced with two different formulations in order to take into account the practical limits due to the inaccessibility of nonlinearity location and or indeterminability of degree of freedom dynamic characteristics of common nonlinearity mechanisms like cubic stiffness pure slip and stick slip have been identified using the proposed technique and it has been shown that although the proposed identification technique is simple it does not require any sophisticated measurement hardwares and techniques as required by most of the identification methods proposed so far also the relation of this technique to harmonic balance method is discussed

a very smooth ride in a rough sea it has been known for some time that a 3d incompressible euler flow that has initially a barely smooth velocity field nonetheless has lagrangian fluid particle trajectories that are analytic in time for at least a finite time serfati c r acad sci paris serie i 320 175 180 1995 shnirelman glob stoch anal http arxiv org abs 1205 5837v1 2012 here an elementary derivation is given based on cauchy s form of the euler equations in lagrangian coordinates this form implies simple recurrence relations among the time taylor coefficients of the lagrangian map used here to derive bounds for the c1 holder norms of the coefficients and infer temporal analyticity of lagrangian trajectories when the initial velocity is c1

a probabilistic approach to the two dimensional navier stokes equations we turn the navier stokes equations for a 2 dimensional viscous incompressible fluid into a system of functional integrals in the trajectory space of a suitable diffusion process using probabilistic techniques as girsanov s transformation and bismut elworthy formula we prove the existence of a unique global solution of this system in a constructive way

identification of a continuous structure with a geometrical non linearity part i conditioned reverse path method particular effort has been spent in the field of identification of multi degree of freedom non linear systems the newly developed methods permit the structural analyst to consider increasingly complex systems the aim of this paper and a companion paper is to study by means of two methods a continuous non linear system consisting of an experimental cantilever beam with a geometrical non linearity in this paper part i the ability of the conditioned reverse path method which is a frequency domain technique to identify the behaviour of this structure is assessed the companion paper part ii is devoted to the application of proper orthogonal decomposition which is an updating technique to the test example

use of the hilbert transform in modal analysis of linear and non linear structures abstract an initial study into the application of the hilbert transform in modal analysis procedures is presented it is shown that typical structural non linearities such as non linear damping and stiffness can be detected and identified directly without the need to generate explicit models no assumptions regarding the degree of non linearity are made which is a restriction in the classical methods for dealing with non linearities the properties of the hilbert transform are discussed with respect to linear and non linear dynamical systems and a discrete transform developed from the continuous functions is derived in the frequency domain and adapted to modal analysis data in the form of mobility transfer functions truncation effects arising from limited frequency ranges of the mobility transfer functions are accounted for by employing correction terms in the frequency domain several examples are studied of single and multi mode systems with non linearities such as friction clearance and non linear stiffness these examples indicate that the hilbert transform offers a new method for extending modal analysis to the domains of non linear systems

global regularity and convergence of a birkhoff rott α approximation of the dynamics of vortex sheets of the two dimensional euler equations we present an regularization of the birkhoff rott equation br equation induced by the two dimensional euler equations for the vortex sheet dynamics we show the convergence of the solutions of euler equations to a weak solution of the euler equations for initial vorticity being a finite radon measure of fixed sign which includes the vortex sheets case we also show that provided the initial density of vorticity is an integrable function over the curve with respect to the arc length measure i an initially lipschitz chord arc vortex sheet curve evolving under the br equation remains lipschitz for all times ii an initially holder c1 0 1 chord arc curve remains in c1 for all times and finally iii an initially holder cn n 1 0 1 closed chord arc curve remains so for all times in all these cases the weak euler and the br descriptions of the vortex sheet motion are equivalent 2009 wiley periodicals inc

stochastic least action principle for the incompressible navier stokes equation abstract we formulate a stochastic least action principle for solutions of the incompressible navier stokes equation which formally reduces to hamilton s principle for the incompressible euler solutions in the case of zero viscosity we use this principle to give a new derivation of a stochastic kelvin theorem for the navier stokes equation recently established by constantin and iyer which shows that this stochastic conservation law arises from particle relabelling symmetry of the action we discuss issues of irreversibility energy dissipation and the inviscid limit of navier stokes solutions in the framework of the stochastic variational principle in particular we discuss the connection of the stochastic kelvin theorem with our previous martingale hypothesis for fluid circulations in turbulent solutions of the incompressible euler equations

on the euler equations of incompressible fluids euler equations of incompressible fluids use and enrich many branches of mathematics from integrable systems to geometric analysis they present important open physical and mathematical problems examples include the stable statistical behavior of ill posed free surface problems such as the rayleightaylor and kelvin helmholtz instabilities the paper describes some of the open problems related to the incompressible euler equations with emphasis on the blowup problem the inviscid limit and anomalous dissipation some of the recent results on the quasigeostrophic model are also mentioned

on the euler flow in 2 we give a global existence and uniqueness theorem for the euler flow in 2 for suitable initial velocity fields possibly diverging at infinity

variational principles for stochastic fluid dynamics this paper derives stochastic partial differential equations spdes for fluid dynamics from a stochastic variational principle svp the paper proceeds by taking variations in the svp to derive stochastic stratonovich fluid equations writing their ito representation and then investigating the properties of these stochastic fluid models in comparison with each other and with the corresponding deterministic fluid models the circulation properties of the stochastic stratonovich fluid equations are found to closely mimic those of the deterministic ideal fluid models as with deterministic ideal flows motion along the stochastic stratonovich paths also preserves the helicity of the vortex field lines in incompressible stochastic flows however these stratonovich properties are not apparent in the equivalent ito representation because they are disguised by the quadratic covariation drift term arising in the stratonovich to ito transformation this term is a geometric generalization of the quadratic covariation drift term already found for scalar densities in stratonovich s famous 1966 paper the paper also derives motion equations for two examples of stochastic geophysical fluid dynamics namely the euler boussinesq and quasi geostropic approximations

liouville type theorems for the forced euler equations and the navier stokes equations in this paper we study the liouville type properties for solutions to the steady incompressible euler equations with forces in mathbb r n if we assume single signedness condition on the force then we can show that a c 1 mathbb r n solution v p with v 2 p in l frac q 2 mathbb r n q in frac 3n n 1 infty is trivial v 0 for the solution of the steady navier stokes equations satisfying v x to 0 as x to infty the condition int mathbb r 3 delta v frac 6 5 dx infty which is stronger than the important d condition int mathbb r 3 nabla v 2 dx infty but both having the same scaling property implies that v 0 in the appendix we reprove theorem 1 1 chae commun math phys 273 203 215 2007 using the self similar euler equations directly

nonlinear structural identification by the reverse path spectral method abstract when dealing with nonlinear dynamical systems it is important to have efficient accurate and reliable tools for estimating both the linear and nonlinear system parameters from measured data an approach for nonlinear system identification widely studied in recent years is reverse path this method is based on broad band excitation and treats the nonlinear terms as feedback forces acting on an underlying linear system parameter estimation is performed in the frequency domain using conventional multiple input multiple output or multiple input single output techniques this paper presents a generalized approach to apply the method of reverse path on continuous mechanical systems with multiple nonlinearities the method requires few spectral calculations and is therefore suitable for use in iterative processes to locate and estimate structural nonlinearities the proposed method is demonstrated in both simulations and experiments on continuous nonlinear mechanical structures the results show that the method is effective on both simulated as well as experimental data

confinement of vorticity for the 2d euler α equations abstract in this article we consider weak solutions of the euler equations in the full plane we take as initial unfiltered vorticity an arbitrary nonnegative compactly supported bounded radon measure global well posedness for the corresponding initial value problem is due m oliver and s shkoller we show that for all time the support of the unfiltered vorticity is contained in a disk whose radius grows no faster than o t log t 1 4 this result is an adaptation of the corresponding result for the incompressible 2d euler equations with initial vorticity compactly supported nonnegative and p th power integrable p 2 due to d iftimie t sideris and p gamblin and independently to ph serfati

the anisotropic lagrangian averaged euler and navier stokes equations the purpose of this paper is twofold first we give a derivation of the lagrangian averaged euler lae and navier stokes lans equations this theory involves a spatial scale and the equations are designed to accurately capture the dynamics of the euler and navier stokes equations at length scales larger than while averaging the motion at scales smaller than the derivation involves an averaging procedure that combines ideas from both the material lagrangian and spatial eulerian viewpoints this framework allows the use of a variant of g i taylor s frozen turbulence hypothesis as the foundation for the model equations more precisely the derivation is based on the strong physical assumption that fluctutations are frozen into the mean flow in this article we use this hypothesis to derive the averaged lagrangian for the theory and all the terms up to and including order 2 are accounted for

identification of multi degree of freedom non linear systems under random excitations by the reverse path spectral method abstract conventional frequency response estimation methods such as the h 1 and h 2 methods often yield measured frequency response functions which are contaminated by the presence of non linearities and hence make it difficult to extract underlying linear system properties to overcome this deficiency a new spectral approach for identifying multi degree of freedom non linear systems is introduced which is based on a reverse path formulation as available in the literature for single degree of freedom non linear systems certain modifications are made in this article for a multi degree of freedom reverse path formulation that utilizes multiple input multiple output data from non linear systems when excited by gaussian random excitations conditioned h c 1 and h c 2 frequency response estimates now yield the underlying linear properties without contaminating effects from the non linearities once the conditioned frequency response functions have been estimated the non linearities which are described by analytical functions are also identified by estimating the coefficients of these functions identification of the local or distributed non linearities which exist at or away from the excitation locations is possible the new spectral approach is successfully tested on several example systems which include a three degree of freedom system with an asymmetric non linearity a three degree of freedom system with distributed non linearities and a five degree of freedom system with multiple non linearities and multiple excitations

a new derivation of the frequency response function matrix for vibrating non linear systems abstract frequency response function matrices relate the inputs and the outputs of structural dynamic systems if a system is linear the frequency response function matrix is the same for any combination or types of inputs over the entire operating range furthermore the frequency response matrix of a linear vibrating system is a simple combination of temporal and spatial characteristics the modal frequencies modal vectors and modal scale factors when a system is non linear the inputs interact through an exchange of energy between the linear and non linear elements in the system no general combination of the temporal and spatial non linear characteristics has to date been proposed to describe these linear non linear interactions this article introduces a unifying perspective of non linearities as internal feedback forces that act together with the external forces to generate the response of the non linear system this perspective of the non linearities is spatial in nature and leads to two simple but conceptually powerful relationships between the frequency response function matrix of a non linear multiple degree of freedom system and its linear counterpart several single and multiple degree of freedom systems are used to demonstrate the use and interpretation of these relationships the broad implication of the new input output frequency response representation for both linear and non linear systems are also addressed in particular the merits of the spatial perspective of non linear systems and the new frequency response relationships are stated in the context of linear and non linear system characterization and identification one implication is that these relationships suggest there is an input output dependent temporal spatial modal decomposition of the frequency response function matrix for non linear systems

convective derivatives and reynolds transport in curvilinear time dependent coordinates a fully covariant formulation of kinematics and dynamics of fluid flows and heat transfer is developed in time dependent curvilinear coordinate systems these moving and deformable reference frames have the same properties of a fluid motion and they can be successfully applied to a variety of problems ranging from numerical analysis to theoretical physics the classical reynolds transport theorem the euler formula and the acceleration addition theorem are extended to these general types of coordinates through a generalization of the convected differentiation concept originally introduced by oldroyd 1950 proc r soc a 200 523 41 a rigorous formulation of dynamical equations and conservation laws in curvilinear time dependent coordinates could be the key for the construction of variational principles based on the method of constrained variations

navier stokes equation and diffusions on the group of homeomorphisms of the torus a stochastic variational principle for the two dimensional navier stokes equation is established the velocity field can be considered as a generalized velocity of a diffusion process with values on the volume preserving diffeomorphism group of the underlying manifold navier stokes equation is reinterpreted as a perturbed equation of geodesics for the l 2 norm the method described here should hold as well in higher dimensions

two examples on the property of the noise in the systems of equations of fluid mechanics abstract in this manuscript we discuss two examples of the results on the systems of partial differential equations in fluid mechanics obtained via stochastic analysis for which deterministic case has no counterpart firstly it is well known that for a deterministic system of equations in fluid mechanics a lack of diffusion creates a significant obstacle in proving its global well posedness for a sufficiently small initial data nevertheless it may be proven that for the hall magnetohydrodynamics system forced by l e vy noise such a result may be obtained even with zero viscous diffusion secondly we may extend the classical kelvin s conservation of circulation flow as well as the cauchy s vorticity transport formula on the euler equations to show that the solution to the damped euler equations enjoys the exact same identities except with an exponential decay in time such identities are known to completely fail once we add diffusion and consider the damped navier stokes equations nevertheless discovering a stochastic lagrangian formulation we are able to show that such identities indeed do continue to hold informally on average i e via a mathematical expectation over the random ensemble at earlier time the stochastic lagrangian formulation and these identities have also been obtained for the solution to the boussinesq system these results are rigorously proven in mohan and yamazaki 0000 and yamazaki 0000 in this manuscript we will not pursue complete proofs which were in particular mostly inspired by kim 2011 and iyer 2006 but rather elaborate on motivations difficulty and remaining open problems for future works

nonexistence of self similar singularities for the 3d incompressible euler equations we prove that there exists no self similar finite time blowing up solution to the 3d incompressible euler equations if the vorticity decays sufficiently fast near infinity in mathbb r 3 by a similar method we also show nonexistence of self similar blowing up solutions to the divergence free transport equation in mathbb r n this result has direct applications to the density dependent euler equations the boussinesq system and the quasi geostrophic equations for which we also show nonexistence of self similar blowing up solutions

nonexistence of self similar singularities in the viscous magnetohydrodynamics with zero resistivity abstract we are concerned on the possibility of finite time singularity in a partially viscous magnetohydrodynamic equations in r n n 2 3 namely the mhd with positive viscosity and zero resistivity in the special case of zero magnetic field the system reduces to the navier stokes equations in r n in this paper we exclude the scenario of finite time singularity in the form of self similarity under suitable integrability conditions on the velocity and the magnetic field we also prove the nonexistence of asymptotically self similar singularity this provides us information on the behavior of solutions near possible singularity of general type as described in corollary 1 1

global regularity for vortex patches we present a proof of chemin s 4 result which states that the boundary of a vortex patch remains smooth for all time if it is initially smooth

feasibility of identifying non linear vibratory systems consisting of unknown polynomial forms abstract system identification techniques for non linear systems may require a priori knowledge of the nature and mathematical form of the non linearities however for practical systems this is not always possible as a result non linearities are often approximated and questions remain as to whether a reasonably accurate model can be determined concurrently under experimental conditions some means of quantifying the amount of measurement noise present in the identification process must also be obtained to resolve such issues a discrete non linear system problem is formulated in the presence of uncorrelated noise and critically examined from the standpoint of identification coherence functions are introduced which are based on a reverse path spectral approach recently developed by the authors for multi degree of freedom systems these coherence functions as calculated from conditioned spectra indicate the extent of uncorrelated noise present and the accuracy of assumed mathematical models employed for describing non linear systems using several example simulation systems including a system with a continuous non linearity described by a non integer exponent both temporal and spectral identification techniques are employed to study the issues described above

on a model for the navier stokes equations using magnetization variables abstract it is known that in a classical setting the navier stokes equations can be reformulated in terms of so called magnetization variables w that satisfy 1 t w p w w p w w w 0 and relate to the velocity u via a leray projection u p w we will prove the equivalence of these formulations in the setting of weak solutions that are also in l 0 t h 1 2 l 2 0 t h 3 2 on the 3 dimensional torus our main focus is the proof of global well posedness in h 1 2 for a new variant of 1 where p w is replaced by w in the second nonlinear term 2 t w p w w 1 2 w 2 w 0 this is based on a maximum principle analogous to a similar property of the burgers equations

stochastic lagrangian particle approach to fractal navier stokes equations in this article we study the fractal navier stokes equations by using the stochastic lagrangian particle path approach in constantin and iyer comm pure appl math lxi 330 345 2008 more precisely a stochastic representation for the fractal navier stokes equations is given in terms of stochastic differential equations driven by levy processes based on this representation a self contained proof for the existence of a local unique solution for the fractal navier stokes equation with initial data in mathbb w 1 p is provided and in the case of two dimensions or large viscosity the existence of global solutions is also obtained in order to obtain the global existence in any dimensions for large viscosity the gradient estimates for levy processes with time dependent and discontinuous drifts are proved

a random walk interpretation of the incompressible navier stokes equations we introduce a stochastic process that roughly corresponds to a random walk of a single molecule in an incompressible fluid and we consider the force per unit volume exerted by such a molecule on the rest of the fluid averaging this force over an ensemble of such molecules and taking an appropriate limit we obtain the navier stokes equations the origin of time irreversibility in these equations is discussed in terms of the model

analyticity of lagrangian trajectories for well posed inviscid incompressible fluid models abstract we discuss general incompressible inviscid models including the euler equations the surface quasi geostrophic equation incompressible porous medium equation and boussinesq equations all these models have classical unique solutions at least for short time we show that they have real analytic lagrangian paths more precisely we show that as long as a solution of any of these equations is in a class of regularity that assures holder continuous gradients of velocity the corresponding lagrangian paths are real analytic functions of time the method of proof is conceptually straightforward and general and we address the combinatorial issues head on

quaternions and ideal flows after a review of some of the recent works by holm and gibbon on quaternions and their application to lagrangian flows particularly the incompressible euler equations and the equations of ideal mhd this paper investigates the compressible and relativistic euler equations using these methods

on the convergence rate of the euler α an inviscid second grade complex fluid model to the euler equations we study the convergence rate of the solutions of the incompressible euler an inviscid second grade complex fluid equations to the corresponding solutions of the euler equations as the regularization parameter approaches zero first we show the convergence in hs s n 2 1 in the whole space and that the smooth euler solutions exist at least as long as the corresponding solution of the euler equations next we estimate the convergence rate for two dimensional vortex patch with smooth boundaries

a stochastic perturbation of inviscid flows we prove existence and regularity of the stochastic flows used in the stochastic lagrangian formulation of the incompressible navier stokes equations with periodic boundary conditions and consequently obtain a ck local existence result for the navier stokes equations our estimates are independent of viscosity allowing us to consider the inviscid limit we show that as 0 solutions of the stochastic lagrangian formulation with periodic boundary conditions converge to solutions of the euler equations at the rate of o sqrt nu t

global small solutions of 2 d incompressible mhd system abstract in this paper we consider the global wellposedness of 2 d incompressible magneto hydrodynamical system with smooth initial data which is close to some non trivial steady state it is a coupled system between the navier stokes equations and a free transport equation with a universal nonlinear coupling structure the main difficulty of the proof lies in exploring the dissipative mechanism of the system to achieve this and to avoid the difficulty of propagating anisotropic regularity for the free transport equation we first reformulate our system 1 1 in the lagrangian coordinates 2 19 then we employ anisotropic littlewood paley analysis to establish the key a priori l 1 r lip r 2 estimate for the lagrangian velocity field y t with this estimate we can prove the global wellposedness of 2 19 with smooth and small initial data by using the energy method we emphasize that the algebraic structure of 2 19 is crucial for the proofs to work the global wellposedness of the original system 1 1 then follows by a suitable change of variables

a variational formulation for the navier stokes equation in this paper we prove a new variational principle for the navier stokes equation which asserts that its solutions are critical points of a stochastic control problem in the group of area preserving diffeomorphisms this principle is a natural extension of the results by arnold ebin and marsden concerning the euler equation

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

the geometric approaches to the possible singularities in the inviscid fluid flows we consider the possible generation of singularities of a vector field transported by diffeomorphisms with derivatives of uniformly bounded determinants we find relations between the directions of the vector field and the eigenvectors of the derivative of the back to label map near the singularity we also find an invariant when we follow the motion of the integral curves of the vector field for the 3d incompressible euler equations these results have immediate implications about the directions of the vortex stretching and the material stretching near the possible singularities we also have similar applications to other inviscid fluid equations such as the 2d quasi geostrophic equation and the 3d magnetohydrodynamics equations

spatial truncation in models of non linear vibrating systems abstract vibrating systems usually have an infinite number of degrees of freedom d o f since a finite number of measurement d o f can only capture certain deformation patterns the spatial characteristics of vibrating systems are only partially observed experimentally this research examines the effects of spatial truncation on non linear system identification it is demonstrated that truncation produces frequency correlated noise which cannot be extracted completely with traditional techniques the goal of this article is to provide insight and practical recommendations for diagnosing and or compensating for errors due to spatial truncation in multiple d o f systems the article demonstrates that non linear dynamic systems should be instrumented with sufficient sensors model properties should be used in addition to the input output data and additional temporal data should be collected to help diagnose errors due to spatial truncation

frequency domain arx model and multi harmonic frf estimators for non linear dynamic systems abstract non linear dynamic systems respond at frequencies other than the excitation frequency however standard frequency response function estimators for linear systems do not accommodate this harmonic distortion a new multi harmonic frequency response function estimator that utilizes discrete frequency models for non linear systems is introduced here the multi harmonic estimator relates the frequency response at each frequency to the input and output spectra within a given frequency band in the same way that autoregressive exogenous input models relate inputs and outputs at particular samples in the time domain overdetermined least mean squares calculations are used to minimize model error throughout a frequency band rather than at a single frequency as in the corresponding linear estimators the resulting multi harmonic frequency response function models are non parametric e g vary with amplitude when linear functions are used and parametric when non linear functions are used a new sensitive indicator for experimentally characterizing non linearity is introduced

a random space time birth particle method for 2d vortex equations with external field we consider incompressible 2d navier stokes equations in the whole plane with external nonconservative forces fields the initial data and external field are functions assumed to satisfy only slight integrability properties we develop a probabilistic interpretation of these equations based on the associated vortex equation in order to construct a numerical particle method to approximate the solutions more precisely we relate the vortex equation with additional term to a nonlinear process with random space time birth which provides a probabilistic description of the creation of vorticity we then introduce interacting particle systems defined for a regularized interaction kernel whose births are chosen randomly in time and space by a coupling method we show that these systems are approximations of the nonlinear process and obtain precise convergence estimates from this result we deduce a stochastic numerical particle method to obtain the vorticity and also to recover the velocity field the results are either pathwise or of weak convergence depending on the integrability of the data we illustrate our results with simulations

aeroelastic response of nonlinear wing sections using a functional series technique this paper addresses the problem of the determination of the subcritical aeroelastic response and flutter instability of nonlinear two dimensional lifting surfaces in an incompressible flow field via indicial functions and volterra series approach the related aeroelastic governing equations are based upon the inclusion of structural and damping nonlinearities in plunging and pitching of the linear unsteady aerodynamics and consideration of an arbitrary time dependent external pressure pulse unsteady aeroelastic nonlinear kernels are determined and based on these frequency and time histories of the subcritical aeroelastic response are obtained and in this context the influence of the considered nonlinearities is emphasized conclusions and results displaying the implications of the considered effects are supplied

a unified matrix polynomial approach to modal identification abstract one important current focus of modal identification is a reformulation of modal parameter estimation algorithms into a single consistent mathematical formulation with a corresponding set of definitions and unifying concepts particularly a matrix polynomial approach is used to unify the presentation with respect to current algorithms such as the least squares complex exponential lsce the polyreference time domain ptd ibrahim time domain itd eigensystem realization algorithm era rational fraction polynomial rfp polyreference frequency domain pfd and the complex mode indication function cmif methods using this unified matrix polynomial approach umpa allows a discussion of the similarities and differences of the commonly used methods the use of least squares ls total least squares tls double least squares dls and singular value decomposition svd methods is discussed in order to take advantage of redundant measurement data eigenvalue and svd transformation methods are utilized to reduce the effective size of the resulting eigenvalue eigenvector problem as well

representations of non linear systems the narmax model abstract input output representations of non linear discrete time systems are discussed it is shown that the narmax non linear autoregressive moving average with exogenous inputs model is a general and natural representation of non linear systems and contains as special cases several existing non linear models the problem of approximating non linear input output systems is also addressed and several properties of non linear models are highlighted using simple examples