exact nonlinear model reduction for a von karman beam slow fast decomposition and spectral submanifolds abstract we apply two recently formulated mathematical techniques slow fast decomposition sfd and spectral submanifold ssm reduction to a von karman beam with geometric nonlinearities and viscoelastic damping sfd identifies a global slow manifold in the full system which attracts solutions at rates faster than typical rates within the manifold an ssm the smoothest nonlinear continuation of a linear modal subspace is then used to further reduce the beam equations within the slow manifold this two stage mathematically exact procedure results in a drastic reduction of the finite element beam model to a one degree of freedom nonlinear oscillator we also introduce the technique of spectral quotient analysis which gives the number of modes relevant for reduction as output rather than input to the reduction process

model reduction of nonlinear systems with external periodic excitations via construction of invariant manifolds abstract a methodology for determining reduced order models of periodically excited nonlinear systems with constant as well as periodic coefficients is presented the approach is based on the construction of an invariant manifold such that the projected dynamics is governed by a fewer number of ordinary differential equations due to the existence of external and parametric periodic excitations however the geometry of the manifold varies with time as a result the manifold is constructed in terms of temporal and dominant state variables the governing partial differential equation pde for the manifold is nonlinear and contains time varying coefficients an approximate technique to find solution of this pde using a multivariable taylor fourier series is suggested it is shown that in certain cases it is possible to obtain various reducibility conditions in a closed form the case of time periodic systems is handled through the use of lyapunov floquet l f transformation application of the l f transformation produces a dynamically equivalent system in which the linear part of the system is time invariant however the nonlinear terms get multiplied by a truncated fourier series containing multiple parametric excitation frequencies this warrants some structural changes in the proposed manifold but the solution procedure remains the same two examples namely a 2 dof mass spring damper system and an inverted pendulum with periodic loads are used to illustrate applications of the technique for systems with constant and periodic coefficients respectively results show that the dynamics of these 2 dof systems can be accurately approximated by equivalent 1 dof systems using the proposed methodology

nonlinear model identification and spectral submanifolds for multi degree of freedom mechanical vibrations in a nonlinear oscillatory system spectral submanifolds ssms are the smoothest invariant manifolds tangent to linear modal subspaces of an equilibrium amplitude frequency plots of the dynamics on ssms provide the classic backbone curves sought in experimental nonlinear model identification we develop here a methodology to compute analytically both the shape of ssms and their corresponding backbone curves from a data assimilating model fitted to experimental vibration signals this model identification utilizes taken s delay embedding theorem as well as a least square fit to the taylor expansion of the sampling map associated with that embedding the ssms are then constructed for the sampling map using the parametrization method for invariant manifolds which assumes that the manifold is an embedding of rather than a graph over a spectral subspace using examples of both synthetic and real experimental data we demonstrate that this approach reproduces backbone curves with high accuracy

bounds for twisted symmetric square l functions iii abstract let f be a hecke modular form and let be a primitive character of conductor q l assume that q is an odd prime in this paper we prove the subconvex bound l 1 2 sym 2 f f q e q 3 l 1 4 1 36 e for any e 0 this can be compared with the recently established t aspect subconvexity of the symmetric square l functions

on the central value of symmetric square l functions let s k n be the space of cusp forms of weight k level n and character for f in s k n chi let l s sym2 f be the symmetric square l function and l s f otimes f be the rankin selberg square attached to f for fixed k 2 n prime and real primitive asymptotic formulas for the first and second moment of the central value of l s sym2 f and l s f otimes f over a basis of s k n are given as n as an application it is shown that a positive proportion of the central values l 1 2 sym2 f does not vanish

the circle method and bounds for l functions iii t aspect subconvexity for gl 3 l functions letbe a hecke maass cusp form for sl 3 z in this paper we will prove the following subconvex bound

normal modes of vibration for non linear continuous systems a definition and a constructive methodology for normal nodes of motion are developed for a class of vibratory systems the dynamics of which are governed by non linear partial differential equations the definition for normal modes is given in terms of the dynamics on two dimensional invariant manifolds in the system phase space these manifolds are a continuation of the planes which represent the well known normal modes of the linearized system a local asymptotic approximation of the geometric structure of these manifolds can be obtained by using an approach which follows that used for generating center manifolds the procedure also provides the non linear ordinary differential equations which govern individual modal dynamics and a physical description of the system configuration when it is undergoing a modal motion in this paper the general theory is described for the application of vibrations of continuous media but it can easily be extended to other situations in order to demonstrate the power of the approach and its unique procedural aspects three examples involving beam vibrations are worked out in detail the examples are conservative simply supported beams the first demonstrates the methodology for a linear beam model the second is a beam on a non linear elastic foundation and the third example is a beam with non linear torsional springs attached at each end in these examples the simplicity of the mode shapes of the linearized model yields relatively simple calculations that do not obscure the important features of the procedure

non linear free vibration identification via the hilbert transform a time domain non parametric method for non linear vibration system identification based on the hilbert transform is introduced the method is demonstrated using computer simulations of different types of non linear elastic and damping dynamic systems

dynamic testing of nonlinear vibrating structures using nonlinear normal modes modal testing and analysis is well established for linear systems the objective of this paper is to progress toward a practical experimental modal analysis ema methodology of nonlinear mechanical structures in this context nonlinear normal modes nnms offer a solid theoretical and mathematical tool for interpreting a wide class of nonlinear dynamical phenomena yet they have a clear and simple conceptual relation to the classical linear normal modes lnms a nonlinear extension of force appropriation techniques is developed in this study in order to isolate one single nnm during the experiments with the help of time frequency analysis the energy dependence of nnm modal curves and their frequencies of oscillation are then extracted from the time series the proposed methodology is demonstrated using two numerical benchmarks a two degree of freedom system and a planar cantilever beam with a cubic spring at its free end

a method of non linear modal identification from frequency response tests abstract a theoretical and experimental study of non linear systems governed by duffing s equation is presented in this study the non linear mode concept is applied to analyze the steady state responses of non linear systems driven by a harmonic force in the experimental context the present modal identification method permits one to obtain the non linear modal parameters from frequency response tests by comparison with other methods the proposed methods appear very interesting in regard to computational time formulation and they necessitate fewer computer resources an experimental work on the steady state response of a beam with local non linear stiffness is reported to justify the methods many aspects of theoretical results such as the dependence of resonance frequency on the amplitude of vibration and the downward and upward jump phenomena are observed experimentally the experimental results are in good agreement with theoretical predictions based upon the non linear modal superposition approach

model reduction to spectral submanifolds and forced response calculation in high dimensional mechanical systems abstract we show how spectral submanifold ssm theory can be used to extract forced response curves without any numerical simulation in high degree of freedom periodically forced mechanical systems we use multivariate recurrence relations to construct the ssms achieving a major speed up relative to earlier autonomous ssm algorithms the increase in computational efficiency promises to close the current gap between studying lower dimensional academic examples and analyzing larger systems obtained from finite element modeling as we illustrate on two different discretized damped forced beam models using the exact reduction procedure via ssms for obtaining forced response curves we further demonstrate speed gains of several orders in magnitude relative to the available state of the art continuation packages while retaining accuracy

continuous wavelet transform for modal identification using free decay response abstract this paper deals with the use of the continuous wavelet transform for system identification purposes the wavelet analysis of the free responses of a linear mechanical system allows the estimation of its natural frequencies viscous damping ratios and mode shapes using either the modulus or the phase of the wavelet transform a complete procedure for modal identification from free responses based on the continuous wavelet transform is presented two difficulties during the implementation of this technique are highlighted the edge effect and the choice of the time frequency localization of the wavelet transform some upper and lower bounds for the mother wavelet s parameters are given in order to improve the numerical computation three complex valued mother wavelets are studied and the full procedure is applied to a damped discrete system the correct choice of the mother wavelet s parameters leads to an accurate identification of the modal parameters

some non existence results for a class of stochastic partial differential equations abstract consider the following stochastic partial differential equation t u t x l u t x u t x f t x t 0 and x r d the operator l is the generator of a strictly stable process and f is the random forcing term which is assumed to be gaussian under some additional conditions most notably on and the initial condition we show non existence of global random field solutions our results complement those of p chow in 5 and 6 among others

subconvexity for twisted l functions on gl 3 let q be a large prime and chi the quadratic character modulo q let phi be a self dual cuspidal hecke eigenform for rm sl 3 bbb z and f a hecke maa ss cusp form for gamma 0 q subseteq rm sl 2 bbb z we consider the twisted l functions l s phi times f times chi and l s phi times chi on rm gl 3 times rm gl 2 and rm gl 3 with conductors q 6 and q 3 respectively we prove the subconvexity bounds l 1 2 phi times f times chi ll phi f varepsilon q 5 4 varepsilon quad l 1 2 it phi times chi ll phi t varepsilon q 5 8 varepsilon for any varepsilon 0

bounds for gl 3 gl 2 l functions and gl 3 l functions in this paper we will give the subconvexity bounds for self dual gl 3 l functions in the t aspect as well as subconvexity bounds for self dual gl 3 gl 2 l functions in the gl 2 spectral aspect

identification of a continuous structure with a geometrical non linearity part ii proper orthogonal decomposition 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 the companion paper part i 1 the ability of the conditioned reverse path method which is a frequency domain technique to identify the behaviour of this structure is assessed this paper part ii is devoted to the application of proper orthogonal decomposition which is an updating technique to the test example

the circle method and bounds for l functions ii subconvexity for twists of gl 3 l functions let pi be a rm sl 3 bbb z hecke maass cusp form let chi chi 1 chi 2 be a dirichlet character with chi i primitive modulo m i suppose m 1 m 2 are primes such that sqrt m 2 m 4 delta m 1 m 2m 3 delta where m m 1m 2 and 0 delta 1 28 in this paper we will prove the following subconvex bound l left 1 over 2 pi otimes chi right ll pi varepsilon m 3 over 4 delta varepsilon

singularity and existence for a wave system of nematic liquid crystals abstract in this paper we prove the global existence and singularity formation for a wave system modeling nematic liquid crystals in one space dimension in our model although the viscous damping term is included the solution with smooth initial data still has gradient blowup in general even when the initial energy is arbitrarily small

nonlinear normal modes for vibratory systems under harmonic excitation this paper considers the use of numerically constructed invariant manifolds to determine the response of nonlinear vibratory systems that are subjected to harmonic excitation the approach is an extension of the nonlinear normal mode nnm formulation previously developed by the authors for free oscillations wherein an auxiliary system that models the excitation is used to augment the equations of motion in this manner the excitation is simply treated as an additional system state yielding a system with an extra degree of freedom dof whose response is known a reduced order model for the forced system is then determined by the usual nnm procedure and an efficient galerkin based solution method is used to numerically construct the attendant invariant manifolds the technique is illustrated by determining the frequency response for a simple 2 dof mass spring system with cubic nonlinearities and for a discretized beam model with 12 dof the results show that this method provides very accurate responses over a range of frequencies near resonances

a spectral characterization of nonlinear normal modes belgian network dysco dynamical systems control and optimization funded by the interuniversity attraction poles programme initiated by the belgian science policy office

one sided maximal inequalities for a stock process abstract let x x t t 0 be a stock process assumed a geometric brownian motion with drift 0 and volatility 0 starting at x 0 for any stopping time for x we establish an upper moment bound for e x max 0 t x t under certain restrictions our method of proof employs the gronwall inequality and a comparison principle for a system of first order nonlinear differential equations the bound obtained in this paper extends an existing result and the method of proof employed is quite new

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