probabilistic solution of vibro impact stochastic duffing systems with a unilateral non zero offset barrier abstract this paper proposes a solution procedure for obtaining an approximate probability density function pdf of the response of vibro impact duffing systems with a unilateral non zero offset barrier the impact model is used with classical impacts with instantaneous velocity jumps and the excitation is modelled by a zero mean stationary gaussian white noise first the original vibro impact system is converted into a system without barriers by adopting the zhuravlev non smooth coordinate transformation second the pdf of the converted system is obtained by solving the fokker planck fp equation with the exponential polynomial closure epc method last the pdf of the original system is achieved using the well established methodology on seeking the pdf distribution of a function of a random variable a further parametric study is also conducted to evaluate the effectiveness of the proposed solution procedure comparison with the simulated result shows that when lightly inelastic impacts occur i e the restitution coefficient is very close to 1 0 the technique of the zhuravlev non smooth coordinate transformation is feasible the transformed system can adequately represent the original system consequently the epc method can provide a good approximate pdf solution the tail region of the pdf is also approximated well the analysis further shows that the non zero offset significantly affects the shape of the pdf distribution of displacement

the dynamics of repeated impacts with a sinusoidally vibrating table a deceptively simple difference equation is derived which approximately describes the motion of a small ball bouncing vertically on a massive sinusoidally vibrating plate in the case of perfect elastic impacts the equation reduces to the standard mapping which has been extensively studied by physicists in connection with the motions of particles constrained in potential wells it is shown that for sufficiently large excitation velocities and a coefficient of restitution close to one this deterministic dynamical system exhibits large families of irregular non periodic solutions in addition to the expected harmonic and subharmonic motions the physical significance of these and other chaotic motions which appear to occur frequently in non linear oscillations is discussed

impulsive control of sticking motion in van der pol one sided constraint system in this paper the impulsive control method is used to dominate sticking motion in van der pol one sided constraint system we find that the method can restrain sticking motion effectively and the method can also induce sticking motion for special parameters in addition numerical results show that the method is stable even for high levels of multiplicative noise or additive noise

a discrete model of a stochastic friction system abstract to investigate a random friction system must cost a large of time on computer so appropriate modeling of this system is of significance in practice in this paper an average model on simple random system with friction is firstly developed which is a discrete model by a two dimensional mean poincare map it is applied to describe random stick slip motion the numerical example shows that external noise can change the system behavior that model is extended to a mdof system only with a friction interface the example of 3 dof system exhibits the interesting behavior due to influence of external noise

algebraic design techniques for reliable stabilization in this paper we study two problems in feedback stabilization the first is the simultaneous stabilization problem which can be stated as follows given plants g 0 g 1 g l does there exist a single compensator c that stabilizes all of them the second is that of stabilization by a stable compensator or more generally a least unstable compensator given a plant g we would like to know whether or not there exists a stable compensator c that stabilizes g if not what is the smallest number of right half place poles counted according to their mcmillan degree that any stabilizing compensator must have we show that the two problems are equivalent in the following sense the problem of simultaneously stabilizing l 1 plants can be reduced to the problem of simultaneously stabilizing l plants using a stable compensator which in turn can be stated as the following purely algebraic problem given 2l matrices a 1 a l b 1 b l where a i b i are right coprime for all i does there exist a matrix m such that a i mb i is unimodular for all i conversely the problem of simultaneously stabilizing l plants using a stable compensator can be formulated as one of simultaneously stabilizing l 1 plants the problem of determining whether or not there exists an m such that a bm is unimodular given a right coprime pair a b turns out to be a special case of a question concerning a matrix division algorithm in a proper euclidean domain we give an answer to this question and we believe this result might be of some independent interest we show that given two n times m plants g 0 and g 1 we can generically stabilize them simultaneously provided either n or m is greater than one in contrast simultaneous stabilizability of two single input single output plants g 0 and g 1 is not generic

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

dynamical behavior of vibro impact machinery near a point of codimension two bifurcation a dual component system with vibro impact is considered local codimension two bifurcation of maps involving a real eigenvalue and a complex conjugate pair escaping the unit circle simultaneously is analyzed by using the center manifold and normal form method for maps the period n single impact motion and poincare map of the vibro impact system are derived analytically stability and local bifurcation of single impact periodic motion are analyzed by using the poincare map a center manifold theorem technique is applied to reduce the poincare map to a three dimensional one and the normal form map associated with the codimension two bifurcation is obtained local behavior of the dual component system with vibro impact near the point of codimension two bifurcation is analyzed it is found that near the point of codimension two bifurcation there exists not only hopf bifurcation of period one single impact motion but also hopf bifurcation of period two double impact motion period doubling bifurcation of period one single impact motion commonly exists near the point of codimension two bifurcation however no period doubling cascade emerges due to change of the type of period two fixed points and occurrence of hopf bifurcation associated with period two fixed points the period two fixed points are symmetrical about the corresponding period one point the results from simulation show that the attracting invariant circles associated with period two points are symmetrical about the corresponding period one point near the value of hopf bifurcation of period two points two actual examples the impact forming machinery and inertial shaker are chosen to analyze further the phenomena of codimension two bifurcation of maps and local bifurcation analysis and numerical simulation are carried out to unfold dynamical behavior of these vibro impact systems near the points of codimension two bifurcations

stationary response of multi degree of freedom vibro impact systems under white noise excitations a multi degree of freedom vibro impact system under white noise excitations is formulated as a stochastically excited and dissipated hamiltonian system the constraints are modelled as non linear springs according to the hertz contact law the exact stationary solution of the system is derived under certain conditions the approximate stationary solutions of the system are also obtained by using the stochastic averaging methods for quasi hamiltonian systems it is shown that the stochastic averaging method for quasi non integrable hamiltonian systems is applicable if the non linear forces according to the hertz contact law take an important role in the response of the system while the stochastic averaging method for quasi integrable hamiltonian systems is applicable if the non linear forces can be neglected an example for stochastically excited two degree of freedom vibro impact system is given to illustrate the application of the procedures in detail

extreme point results for robust stabilization of interval plants with first order compensators it has been shown previously that a first order compensator robustly stabilizes an internal plant family if and only if it stabilizes all of the extreme plants these extreme plants are obtained by considering all possible combinations for the extreme values of the numerator and denominator coefficients in this work the authors prove a stronger result namely that it is necessary and sufficient to stabilize only sixteen of the extreme plants these sixteen plants are generated using the kharitonov polynomials associated with the numerator and denominator furthermore when additional information about the compensator is specified sign of the gain and signs and relative magnitudes of the pole and zero then in some cases it is necessary and sufficient to stabilize eight critical plants while in other cases it is necessary and sufficient to stabilize twelve critical plants

periodic solution finder for an impact oscillator with a drift in this paper an efficient semi analytical method is developed to compute periodic solutions for a new model of an impact oscillator with a drift which explains the progression mechanism in vibro impact systems and can be used to optimize their performance the method constructs a periodic response assuming that each period is comprised of a sequence of distinct phases for which analytical solutions are known for example a period may consist of the following sequential phases i contact with progression ii contact without progression iii no contact and iv contact without progression using this information a system of four piecewise linear first order differential equations is transformed to a system of non linear algebraic equations the method allows one to accurately predict a range of control parameters for which the best progression rates are obtained

stochastic response of vibro impact duffing oscillators under external and parametric gaussian white noises abstract this study presents a solution procedure for the stationary probability density function pdf of the response of vibro impact duffing oscillators under external and parametric gaussian white noises first the zhuravlev non smooth coordinate transformation is adopted to convert a vibro impact oscillator into an oscillator without barriers the stationary pdf of the converted oscillator is governed by the fokker planck fp equation the fp equation is solved by the exponential polynomial closure epc method illustrative examples are presented with vibro impact duffing oscillators under external and parametric gaussian white noises to show the effectiveness of the solution procedure the parametric excitation is acting in displacement and the constraint is a unilateral zero offset barrier the restitution coefficient of impacts is taken as 0 90 comparison with the simulated results shows that the proposed solution procedure can provide good approximate pdfs for displacement and velocity although a little difference exists in the tail of these pdfs this difference may be due to the weak approximation on the response of the vibro impact oscillators using a continuous markov process when the restitution coefficient is not very close to unity

melnikov s method for a general nonlinear vibro impact oscillator abstract melnikov s method is presented for a general nonlinear vibro impact oscillator and melnikov function for homoclinic orbits is obtained analytically a duffing vibro impact oscillator is given to illustrate the application of the procedures the obtained results are verified by the phase portrait poincare surface of section and bifurcation diagrams

simultaneous stabilization of linear single input single output plants the problem of designing a single compensator that simultaneously stabilizes several distinct plants is considered here using the fractional representation approach owing to saeks and murray 1982 1983 necessary and sufficient conditions for the existence of such a compensator are given and a design algorithm is proposed

hopf bifurcation of a two degree of freedom vibro impact system abstract the bifurcation problem of a two degree of freedom system vibrating against a rigid surface is studied in this paper it is shown that there exist hopf bifurcations in the vibro impact systems with two or more degrees of freedom under suitable system parameters in the paper a centre manifold theorem technique is applied to reduce the poincare map of the vibro impact system to a two dimensional one and then the theory of hopf bifurcation of maps in r 2 is applied to conclude the existence of hopf bifurcation of the vibro impact system the theoretical solutions are verified by numerical computations the quasi periodic response of the system represented by invariant circles in the projected poincare sections is obtained by numerical simulations and routes of quasi periodic impacts to chaos are stated briefly

analysis of a random repeated impact process a simple random repeated impact process similar to the classical random walk process is analyzed the process which consists of a ball bouncing on a randomly vibrating surface is analogous to loss of contact situations which can occur in linkages and vibrating tools it also has relevance to rolling contact where the rolling element may separate from the surface and sustain repeated impacts a coefficient of restitution is used to describe impact and some important assumptions limit the analysis to values of this coefficient greater than 0 8 the markov nature of the process is demonstrated by its one step memory it may be regarded as a discrete markov process imbedded in continuous time a difference equation governing the process is developed and analyzed for the case where the vibrating surface has a gaussian distribution of velocities with the one step transition probability density function a linear weighting function is used to account for the weighting effect of the ball s velocity on the distribution of table velocities at impact the resulting integral equation is solved iteratively to yield the probability density function of ball velocities after impact this information may then be used to predict the magnitudes of the impacts and the time between them

response probability density functions of duffing van der pol vibro impact system under correlated gaussian white noise excitations abstract this paper aims to investigate the stationary probability density functions pdfs of a duffing van der pol vibro impact system excited by correlated gaussian white noise with the help of non smooth transformation the stationary pdfs are formulated analytically by the stochastic averaging of energy envelope the analytical results are verified by numerical simulation results stochastic bifurcations for different parameters are considered and several special pdf forms are observed in this paper the first form is the shape of the pdf of total energy can be similar to a crater which has a minimum and a maximum the second one is the shape of the joint pdf with three peaks that is to say the section of joint pdf has three maximum and two minimum in addition the influence of the degree of the correlation of gaussian white noises is explored

bouncing ball problem stability of the periodic modes exploring all its ramifications we give an overview of the simple yet fundamental bouncing ball problem which consists of a ball bouncing vertically on a sinusoidally vibrating table under the action of gravity the dynamics is modeled on the basis of a discrete map of difference equations which numerically solved fully reveals a rich variety of nonlinear behaviors encompassing irregular nonperiodic orbits subharmonic and chaotic motions chattering mechanisms and also unbounded nonperiodic orbits for periodic motions the corresponding conditions for stability and bifurcation are determined from analytical considerations of a reduced map through numerical examples it is shown that a slight change in the initial conditions makes the ball motion switch from periodic to chaotic orbits bounded by a velocity strip v gamma 1 epsilon where gamma is the nondimensionalized shaking acceleration and epsilon the coefficient of restitution which quantifies the amount of energy lost in the ball table collision

an overview of extremal properties for robust control of interval plants abstract this paper provides an overview of recent results on the problem of robust stability and performance of feedback control systems in the presence of plant perturbations the widely studied class of interval plants is considered and an effort is made to cover different types of plant perturbations by incorporating in addition to the interval plant description unstructured norm bounded perturbations extremality results for these classes of uncertain systems are provided in a unifying framework robust performance of control systems is addressed in the paper and a number of extremal results are given for the computation of the structured singular value function for feedback systems with interval plants also classical frequency response analysis for interval plant controller families of open loop transfer functions is surveyed with the aim of providing extremal results for the computation of specification parameters such as phase or gain margins and sensitivity and complementary sensitivity peaks

chaos in a noisy mechanical system with stress relaxation the influence of periodic and random external excitation on the chaotic behaviour of a self excited system is considered the lyapunov exponents of the noisy system have been defined as random variables the properties of their distribution which allow one to quantify chaos have been determined

a generalization of kharitonov s theorem robust stability of interval plants the robust stability problem is considered for interval plants in the case of single input multioutput or single output multi input systems a necessary and sufficient condition for the robust stabilization of such plants is developed using a generalization of v l kharitonov s theorem 1978 the generalization given provides necessary and sufficient conditions for the stability of a family of polynomials delta s q sub 1 s p sub 1 s q sub m s p sub m s where the q sub i are fixed and the p sub i are interval polynomials the coefficients of which are regarded as a point in parameter space which varies within a prescribed box this generalization called the box theorem reduces the question of the stability of the box in parameter space to the equivalent problem of the stability of a prescribed set of line segments it is shown that for special classes of polynomials q sub i s the set of line segments collapses to a set of points and this version of the box theorem in turn reduces to kharitonov s original theorem

fdtd analysis of periodic structures with arbitrary skewed grid an efficient finite difference time domain fdtd algorithm with a simple periodic boundary condition pbc is developed to analyze general periodic structures with arbitrary skewed grids the algorithm is easy to implement and efficient in both memory usage and computation time the stability criterion of the algorithm is angle independent and therefore it is suitable for implementing incidence with angle close to grazing as well as normal incidence the validity of this algorithm is verified through several numerical examples such as dipole and jerusalem cross frequency selective surfaces fss with various skew angles

on nonlinear boundary value problems ghosts parasites and discretizations ghosts are a new class of approximate solutions to nonlinear boundary value problems bvps they are bounded away from true solutions but solve an initial value problem ivp for the differential equations exactly and approximate boundary and internal matching conditions with exponentially decaying error as a control parameter approaches infinity ghosts are characteristic of singularly perturbed differential equations in which solutions are localized and the ivp depends sensitively on initial data they correspond to shifted localized solutions and they can be observed in imperfect physical experiments and be approximated by convergent discretization schemes the latter property suggests that the concept of the numerically irrelevant solution introduced for discretized second order bvps should be re examined

stochastic responses of vibro impact duffing oscillator excited by additive gaussian noise the stationary responses of vibro impact duffing oscillator excited by additive gaussian white noise are studied by using the quasi conservative averaging method with the help of a non smooth variable transformation and the dirac delta function the response probability density functions pdfs are formulated analytically meanwhile the results are validated numerically using monte carlo simulations

brief paper simultaneous lq control of a set of lti systems using constrained generalized sampled data hold functions in this paper sampled data control of a set of continuous time lti systems is considered it is assumed that a predefined guaranteed continuous time quadratic cost function which is in fact the sum of the performance indices for all systems is given the main objective here is to design a decentralized periodic output feedback controller with a prespecified form e g polynomial piecewise constant exponential etc which minimizes the above mentioned guaranteed cost function this problem is first formulated as a set of matrix inequalities and then by using a well known technique it is reformulated as a lmi problem the set of linear matrix inequalities obtained provides necessary and sufficient conditions for the existence of a decentralized optimal simultaneous stabilizing controller with the prespecified form rather than a general form moreover an algorithm is presented to solve the resultant lmi problem finally the efficiency of the proposed method is demonstrated in two numerical examples

implementation of the periodic boundary condition in the finite difference time domain algorithm for fss structures a method for implementing the general floquet boundary condition in the finite difference time domain algorithm fdtd is presented the floquet type of phase shift boundary condition is incorporated into the time domain analysis by illuminating the structure with a combination of sine and cosine excitations to generate a phasor representation of the solution at each time step with this approach the characteristics of periodic structures comprised of arbitrarily shaped inhomogeneous geometries can be computed for an arbitrary angle of incidence theoretical results are compared for various planar frequency selective surfaces fss and for one with a three dimensional element e g a thick double concentric square loop

maximum entropy principle and nonlinear stochastic oscillators abstract the principle of maximum entropy introduced first in physics and successfully applied in many other fields e g statistics reliability estimation simulation has recently been developed to analyze the systems governed by stochastic differential equations in this paper we extend the maximum entropy method to the general class of stochastic nonlinear systems and apply it to stochastic nonlinear oscillators to indicate the effectiveness of the method a comparison of the results with predictions of the known procedures is shown graphically the analysis presented in this paper indicates that the maximum entropy principle can serve as a general method for determining the probability distributions of a wide class of stochastic systems when we have only a partial information in the form of moments or equations for moments

solution to the champagne problem on the simultaneous stabilisation of three plants abstract in this paper the problem proposed in blondel et al siam j optim 32 2 572 590 as an illustration of the difficulty of the simultaneous stabilization problem has been solved the same problem was also mentioned in blondel and gevers math control signals systems 6 1994 135 145 where a bottle of good french champagne was offered for its solution a new open problem is proposed based on the solution of the problem in this paper

on the origin and bifurcations of stick slip oscillations a recently proposed model of macroscopic friction is investigated using methods of dynamical systems analysis particular emphasis is put on the bifurcations associated with the appearance of stick

chaotic dynamics of a bouncing ball abstract a detailed study of a mapping on a two dimensional manifold is made the mapping describes a system subject to periodic forcing in particular an imperfectly elastic ball bouncing on a vibrating platform quasiperiodic motion on a one dimensional manifold is proven and observed numerically at low forcing while at higher forcing smale horseshoes are present we examine the evolution of the attracting set with changing parameter spatial structure is oganised by fixed points of the mapping and sudden changes occur by crises a new type of chaos in which a trajectory alternates between two distinct chaotic regions is described and explained in terms of manifold collisions throughout we are concerned to examine the behaviour of lyapunov exponents typical behaviour of lyapunov exponents in the quasiperiodic regime under the influence of external noise is discussed at higher forcing a certain symmetry of the attractor allows an analytic expression for the exponents to be given

a feynman kac formula approach for computing expectations and threshold crossing probabilities of non smooth stochastic dynamical systems abstract we present a computational alternative to probabilistic simulations for non smooth stochastic dynamical systems that are prevalent in engineering mechanics as examples we target 1 stochastic elasto plastic problems which involve transitions between elastic and plastic states and 2 obstacle problems with noise which involve discrete impulses due to collisions with an obstacle we formally introduce a class of partial differential equations related to the feynman kac formula where the underlying stochastic processes satisfy variational inequalities modeling elasto plastic and obstacle oscillators we then focus on solving them numerically the main challenge in solving these equations is the non standard boundary conditions which describe the behavior of the underlying process on the boundary we illustrate how to use our approach to compute expectations and other statistical quantities such as the asymptotic growth rate of variance in asymptotic formulae for threshold crossing probabilities

brief condition and algorithm for simultaneous stabilization of linear plants this paper is devoted to the problem of simultaneous stabilization of linear plants it is shown that four given plants p i s i 1 2 3 4 are simultaneously stabilizable if and only if there exist four stable polynomials h i s i 1 2 3 4 which simultaneously satisfy two polynomial equations if we can only find four stable h i s i 1 2 3 4 to satisfy one of the two polynomial equations then two controllers c 1 s and c 2 s having a common denominator or numerator can be obtained such that c 1 s simultaneously stabilizes p 1 s and p 3 s and c 2 s simultaneously stabilizes p 2 s and p 4 s which is called the simultaneous stabilization in groups based on such two controllers a design algorithm for the simultaneous stabilization of three plants is induced

simultaneous stabilization of three or more plants conditions on the positive real axis do not suffice the problem of the simultaneous stabilizability of a finite family of single input single output time invariant systems by a time invariant controller is studied the link between stabilization and avoidance is shown and is used to derive necessary conditions for the simultaneous stabilization of k plants these necessary conditions are proved to be in general not sufficient this result also disproves a long standing conjecture on the stabilizability condition of a single plant with a stable minimum phase controller the main result is to show that unlike the case of two plants the existence of a simultaneous stabilizing controller for more than two plants is not guaranteed by the existence of a controller such that the closed loops have no real unstable poles

a survey of extreme point results for robustness of control systems abstract this paper surveys a subset of the body of research which was sparked by kharitonov s theorem the focal point is extreme point results for robust stability and robust performance that is we give conditions under which satisfaction of a performance specification is ascertained for a family of systems by only checking a finite subset of the extreme members of this family the results which are surveyed apply mainly to systems with structured real parametric uncertainty in addition a number of counterexamples are given to illustrate cases for which an extreme point result does not hold for such cases a solution via the so called edge theorem is often possible

extreme point results for robust stability of interval plants beyond first order compensators motivated by robust stabilization of interval plants this paper concentrates on a fundamental problem involving a family of polynomials obtained via convex combination of extreme members the objective is to provide conditions under which stability of the extremes implies stability of the entire family although the existing literature contains rather strong extreme point results involving frequency domain analysis there is a paucity of more direct coefficient space criteria this paper deals exclusively with the coefficient space problem to this end we introduce the so called alternating hurwitz minor condition ahmc and show how it can be used to enlarge the class of polynomial families for which extreme point results can be obtained subsequently the ramifications on robust stability of interval plants are fully explained in contrast to existing literature for first order compensators the ahmc makes it possible to deal with classes of higher order compensators in an extreme point context

stochastic stationary responses of a viscoelastic system with impacts under additive gaussian white noise excitation abstract this paper is devoted to investigate the stochastic stationary responses of a viscoelastic system with impacts under additive gaussian white noise excitation first the viscoelastic force is approximated as equivalent stiffness and damping terms and the original system is replaced by a system without viscoelasticity then the stationary responses of the system are obtained analytically by using non smooth transformation and stochastic averaging method the theoretical analyses are verified by numerical results from monte carlo simulations effects of the noise intensity and the nonlinear damping coefficient on the system responses are investigated at last it is worthy noting that the restitution coefficient the linear damping coefficient and the viscoelastic parameters can induce the occurrence of stochastic p bifurcation

stochastic response of a vibro impact system with variable mass abstract vibro impact machines are found increasingly broad application in various sectors of the world and the mass disturbance exists in the process of impact which refers to the variable mass system that is a common phenomenon in life with the development of micro electronic and nanotechnology it is essential to investigate the effect of mass disturbance on this random vibro impact system response the stationary system response is the main topic of the manuscript in the light of the discontinuity of vibro impact system the nonsmooth transformation is introduced to overcome the discontinuity then the stochastic averaging technique is applied to the transformed stochastic variable mass system the approximate analytical response can be obtained by solving associated fokker planck kolmogorov equation a van der pol variable mass vibro impact system is given to illustrate the application and availability of this method meanwhile stochastic p bifurcation is studied and a reasonable interpretation is given to explain the procedure of stochastic p bifurcation

some dynamical properties of a classical dissipative bouncing ball model with two nonlinearities abstract some dynamical properties for a bouncing ball model are studied we show that when dissipation is introduced the structure of the phase space is changed and attractors appear increasing the amount of dissipation the edges of the basins of attraction of an attracting fixed point touch the chaotic attractor consequently the chaotic attractor and its basin of attraction are destroyed given place to a transient described by a power law with exponent 2 the parameter space is also studied and we show that it presents a rich structure with infinite self similar structures of shrimp shape

a partial differential equation with infinitely many periodic orbits chaotic oscillations of a forced beam this paper delineates a class of time periodically perturved evolution equations in a banach space whose associated poincar e map contains a smale horseshoe this implies that such systems possess periodic orbits with arbitrarily high period the method uses techniques originally due to melnikov and applies to systems of the form x f0 x f1 x t where x f0 x is hamiltonian and has a homoclinic orbit we give an example from structural mechanics sinusoidally forced vibrations of a buckled beam

on normal form calculations in impact oscillators normal form calculations are useful for analysing the dynamics close to bifurcations however the application to non smooth systems is a topic for current research here we consider a class of imp

dynamical analysis of a two parameter family for a vibro impact system in resonance cases abstract dynamics of a two degree of freedom vibro impact system in resonance is considered the dynamical model and poincare maps are established when a pair of complex conjugate eigenvalues of the jacobian matrix cross the unit circle and satisfy the resonant condition 0 4 1 or 0 3 1 the four dimensional map is reduced to a two dimensional one by the center manifold theorem and the reduced map is put into its normal form by the method of normal form the two parameter unfoldings of local dynamical behavior studied in this paper develop the results of one parameter family analysis the hopf and subharmonic bifurcation conditions of period n 1 motion are given the numerical simulation method confirms the theoretical analysis it is shown that there exists an invariant torus via hopf bifurcation and period 4 4 motions via subharmonic bifurcation as two controlling parameters varying near the critical point for the resonance 0 4 1 and that there exists an invariant circle and unstable fixed points of order 3 bifurcating from the fixed point and the system leads eventually to chaos in the resonance 0 3 1

a kharitonov like theorem for interval polynomial matrices abstract as an extension of kharitonov s theorem robust stability of interval polynomial matrices is studied here a polynomial matrix is said to be stable if its determinant has all roots with negative real parts the present paper shows that the robust stability of interval polynomial matrices is equivalent to that of the subclasses where each row column has only one element that involves kharitonov edge polynomials and all the other elements take on one of the four kharitonov vertex polynomials

simultaneous and strong simultaneous stabilisation of some classes of mimo systems sufficient conditions are derived for simultaneous stabilisability of any finite number of linear time invariant multi input multi output mimo systems four general mimo plant classes with arbitrary finite number of plants are shown to be simultaneously stabilisable using simple proportional derivative controllers for some of these plant classes it is also possible to achieve simultaneous stabilisation and asymptotic tracking of step input references with zero steady state error by using integral action controllers three of these general mimo plant classes are shown to be strongly simultaneously stabilisable

grazing and border collision in piecewise smooth systems a unified analytical framework a comprehensive derivation is presented of normal form maps for grazing bifurcations in piecewise smooth models of physical processes this links grazings with border collisions in nonsmooth maps contrary to previous literature piecewise linear maps correspond only to nonsmooth discontinuity boundaries all other maps have either square root or 3 2 type singularities

non periodic motion caused by grazing incidence in an impact oscillator the motion of a single degree of freedom periodically forced oscillator subjected to a rigid amplitude constraint is considered using analytical methods the singularities caused by grazing impact are studied it is shown that as a stable periodic orbit comes to grazing impact under the control of a single parameter a special type of bifurcation occurs the motion after the bifurcation may be non periodic and a criterion for this based on orientation and eigenvalues is given

bifurcations and chaos in a system with impacts abstract a vibro impact system is considered a body bounces on a flat horizontal surface of the vibro bench hopf bifurcations of the vibro impact system in two kinds of strong resonance cases 03 1 and 04 1 are investigated a poincare map of the system is established the period 1 single impact motion of the system and its stability are studied by analytical methods and hopf bifurcation values and intersecting conditions of the system in strong resonance cases are determined a center manifold theorem technique is applied to reduce the poincare map to a two dimensional one which is put into normal form by theory of normal forms by the theory of hopf and subharmonic bifurcations of fixed points in r 2 strong resonance dynamic behavior of the vibro impact system near points of resonance is analyzed in the resonance case of 03 1 the system generally exhibits unstable period 3 three impact motion in the resonance case of 04 1 the system can exhibit stable period 4 four impact motion and quasi periodic motion the theoretical analyses are verified by numerical solutions routes to chaos in two kinds of strong resonance cases considered are obtained by numerical simulations quasi periodic impacts in non resonance and weak resonance cases doubling periodic bifurcations and chaotic motions are also stated briefly by numerical analyses

the closed form stationary probability distribution of the stochastically excited vibro impact oscillators abstract the response of vibro impact oscillators under random excitations has been studied using various techniques in the last three decades however the results available were smooth approximations of the intrinsically non smooth response of vibroimpact oscillators this paper proposes a new procedure for the closed form stationary probability density function pdf of the response of single degree of freedom sdof vibro impact system under gaussian white noise excitation first the zhuravlev ivanov transformation is adopted to convert a vibro impact oscillator with one side barrier into an oscillator without barrier the probabilistic description of the system is subsequently defined through the corresponding fokker planck kolmogorov fpk equation the closed form stationary pdf of the response is obtained by solving the reduced fpk equation using the iterative method of weighted residue together with the concepts of the circulatory probability flow and the potential probability flow two examples are examined to illustrate the effectiveness of the proposed procedure good agreements are found between the analytical solutions and the simulated results of the pdfs and logarithmic pdfs of the examples the reported studies also show that the proposed procedure can deal with case of small restitution factor in particular the closed form solution for the case of small restitution factor is firstly obtained in literature and can be used as the benchmark problem to examine the other method in random vibration furthermore the approximate solutions obtained with the proposed procedure are piece wise form reflecting the true nature of the intrinsically discontinuous vibro impact oscillators

non stationary response of variable mass duffing oscillator with mass disturbance modeled as gaussian white noise abstract this paper investigates the probability density evolution process of a variable mass duffing oscillator with mass disturbance modeled as gaussian white noise for small mass disturbance the transition probability density function tpdf solution of the variable mass system is calculated by using path integration method the corresponding fokker planck kolmogorov fpk equation of the approximate mass disturbance system are derived the solution procedure based on gauss legendre quadrature integration rule is formulated to obtain and study the probabilistic solutions of the strongly and weekly nonlinear variable mass system different values of excitation intensity are used to examine the effectiveness of the proposed method compared with the monte carlo simulation results good agreement is achieved with the path integration method for large mass disturbance the probability density evolution process are numerically calculated via the fourth order runge kutta algorithm

the solution of a transcendental problem and its applications in simultaneous stabilization problems a complete solution for a special transcendental problem in system theory is given one of the interpretations of this transcendental problem is the following simultaneous stabilization problem under what conditions can two linear plants be simultaneously stabilized by a single compensator having no real poles in the closed right half of the complex plane these conditions are completely general and therefore encompass unstable non minimum phase plants as well it is shown that the simultaneous stabilizability of two given plants via such a restricted compensator depends solely on the locations of the real zeros of the two plants and their difference plant in the closed right half of the complex plane for two plants satisfying the stabilizability conditions a computational design procedure for constructing a desired compensator is provided and illustrated a computationally checkable necessary condition for simultaneous stabilization of three plants is also given

the analysis of periodic structures with 2 d staggered grid by fdtd algorithm according to the characteristics of the periodic structures with 2 d staggered grids two new effective construction methods dealing with periodic boundaries based on the finite difference time domain fdtd algorithm are developed in this work the two proposed staggered periodic boundary conditions pbc arrangement methods can be implemented combining the existing boundary data mapping techniques applying these arrangements coordinate change and additional ladder boundary grids can be avoided during the modeling process so that the simulation accuracy of the fdtd algorithm can be maintained the performance and the applicable conditions of the two staggered pbc arrangement methods are discussed and compared with the common nonstaggered pbc arrangement method the validity of the two staggered pbc arrangement methods is verified through the numerical examples of an infinite dielectric slab and jerusalem cross frequency selective surface with 2 d staggered girds

spectral fdtd a novel technique for the analysis of oblique incident plane wave on periodic structures this paper introduces a new technique which calculates the reflection coefficient for the plane wave incident on planar periodic structures the method referred to as spectral finite difference time domain sfdtd replaces the conventional single angle incident wave with a constant transverse wavenumber ctw wave because the transverse wavenumbers are constant the fields have no delay in the transverse plane x y plane and pbc periodic boundary condition can be directly implemented in the time domain for both oblique and normal incident waves the stability criterion for this new fdtd technique is angle independent and therefore this method works well for incident angles close to grazing spl theta 90 spl deg as well as normal incident spl theta 0 spl deg this shows the efficiency of the method compared to other available fdtd techniques for the same purpose that force a more restricted stability criterion as angles turns to grazing the validity of this method is verified by comparing the reflection coefficient calculated by this method with the analytical results of a grounded slab the results of this technique are also compared with method of moments for a periodic array of metallic patches and a good agreement is observed a periodic array of metallic patches above a pec plate is analyzed and the reflection coefficient is calculated over a wide frequency band for angles varying from 0 spl deg to close to 90 spl deg

fractional representation algebraic geometry and the simultaneous stabilization problem an explicit relationship between the fractional representation approach feedback system design and the algebro geometric approach to system theory is formulated and used to derive a global solution to the feedback system problem these techniques are then applied to the simultaneous stabilization problem yielding a natural geometric criterion for a set of plants to be simultaneously stabilized by a single compensator

a consistent method for the solution to reduced fpk equation in statistical mechanics abstract the solution of reduced fokker planck kolmogorov fpk equation resulted from the problems in statistical mechanics is formulated as an exponential function of polynomials in state variables special measure is taken to satisfy the fpk equation in the weak sense of integration with the assumed function examples are given to show the application of the method to the non linear systems with additive random excitations and those with both additive and multiplicative random excitations the pdf solutions obtained with the proposed method and conventional gaussian closure method are compared with the exact solutions in special cases numerical results showed that the solutions obtained with the proposed method are much close to the exact solutions even for highly non linear system and the system with both additive and multiplicative excitations the convergence of the approximate pdf solutions obtained with the method is also shown with numerical results

repeated impacts on a sinusoidally vibrating table reappraised the traditional simplification for one commonly used example of repeated collisions is shown to be frequently erroneous for a wide range of coefficients of restitution bounded errors arise from an artificial speed constraint whose circumvention is largely but not exclusively unpredictable therefore exact equations whose periodic solutions are detailed should be generally used

the periodicity of chaotic impact oscillators in hausdorff phase spaces it is well known that non periodic behavior is one of the most puzzling characteristics of chaotic oscillators so far chaotic dynamical systems have been investigated in euclidean spaces in this paper the concept of non autonomous dynamical systems and that of hausdorff phase spaces are proposed the behavior of chaotic impact oscillators is investigated in hausdorff phase spaces it is discovered that although the non autonomous dynamical systems described by chaotic impact oscillators are non periodic in euclidean phase spaces they are periodic in hausdorff phase spaces this shows that euclidean spaces in which we stayed for hundreds of years may no longer be suitable for the investigation into chaotic phenomena in addition the periodicity of chaotic dynamical systems in hausdorff metric spaces induces a new class of strange invariant sets in euclidean spaces such strange invariant sets may be an ideal symbol of chaotic dynamical systems